71
BSDE Representation and Discretization for Hamilton-Jacobi-Bellman PDE Idris KHARROUBI CEREMADE, CNRS, UMR 7534 Universit´ e Paris Dauphine and CREST March 31, 2014
BSDE Representation and Discretization
for Hamilton-Jacobi-Bellman PDE
Idris KHARROUBI
CEREMADE, CNRS, UMR 7534
Universite Paris Dauphine
and CREST
March 31, 2014
Contents
1 Introduction 2
I BSDE representation for HJB equations 8
2 BSDE with nonpositive jumps 9
2.1 Formulation and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Existence and approximation by penalization . . . . . . . . . . . . . . . . . 11
2.3 Dual representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Nonlinear IPDE and Feynman-Kac formula 19
3.1 The Markovian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Viscosity property of the penalized BSDE . . . . . . . . . . . . . . . . . . . 25
3.3 The non dependence of the function v in the variable a . . . . . . . . . . . . 29
3.4 Viscosity properties of the minimal solution to the constrained BSDE . . . 34
II Discretization of fully nonlinear HJB equations via BSDEs withnonpositive jumps 39
4 Discretization of the nonpositive jump constraint 40
4.1 Discretely jump-constrained BSDE . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Convergence of discretely jump-constrained BSDE . . . . . . . . . . . . . . 46
4.2.1 Convergence result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2 Rate of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Approximation scheme for jump-constrained BSDE and stochastic con-
trol problem 59
5.1 The forward regime switching process . . . . . . . . . . . . . . . . . . . . . 59
5.2 BSDE scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Error estimate for the discrete time scheme . . . . . . . . . . . . . . . . . . 63
5.4 Approximate optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1
Chapter 1
Introduction
Let us consider the Hamilton-Jacobi-Bellman (HJB) equation in the form:
∂v
∂t+ supa∈A
[b(x, a).Dxv +
1
2tr(σσᵀ(x, a)D2
xv) + f(x, a)]
= 0, on [0, T )× Rd, (1.0.1)
v(T, x) = g(x), x ∈ Rd,
where A is a subset of Rq. It is well-known (see e.g. [38]) that such nonlinear PDE is the
dynamic programming equation associated to the stochastic control problem with value
function defined by:
v(t, x) := supα
E[ ∫ T
tf(Xt,x,α
s , αs)ds+ g(Xt,x,αT )
], (1.0.2)
where Xt,x,α is the solution to the controlled diffusion:
dXαs = b(Xα
s , αs)ds+ σ(Xαs , αs)dWs,
starting from x at t, and given a predictable control process α valued in A.
Our main goal is to provide a numerical scheme for the approximation of the nonlinear
HJB equation using Backward Stochastic Differential Equation (BSDEs). To this end one
has first to built a probabilistic representation via BSDEs, namely the so-called nonlinear
Feynman-Kac formula, which involves a simulatable forward process. One can then hope
to use such representation for deriving a probabilistic numerical scheme for the solution to
HJB equation, whence the stochastic control problem. Such issues have attracted a lot of
interest and generated an important literature over the recent years. Actually, there is a
crucial distinction between the case where the diffusion coefficient is controlled or not.
Consider first the case where σ(x) does not depend on a ∈ A, and assume that σσᵀ(x)
is of full rank. Denoting by θ(x, a) = σᵀ(x)(σσᵀ(x))−1b(x, a) a solution to σ(x)θ(x, a) =
b(x, a), we notice that the HJB equation reduces into a semi-linear PDE:
∂v
∂t+
1
2tr(σσᵀ(x)D2
xv) + F (x, σᵀDxv) = 0, (1.0.3)
where F (x, z) = supa∈A[f(x, a)+θ(x, a).z] is the θ-Fenchel-Legendre transform of f . In this
case, we know from the seminal works by Pardoux and Peng [33], [34], that the (viscosity)
2
solution v to the semi-linear PDE (1.0.3) is connected to the BSDE:
Yt = g(X0T ) +
∫ T
tF (X0
s , Zs)ds−∫ T
tZsdWs, t ≤ T, (1.0.4)
through the relation Yt = v(t,X0t ), with a forward diffusion process
dX0s = σ(X0
s )dWs.
This probabilistic representation leads to a probabilistic numerical scheme for the resolution
of (1.0.3) by discretization and simulation of the BSDE (1.0.4), see [9]. Alternatively,
when the function F (x, z) is of polynomial type on z, the semi-linear PDE (1.0.3) can be
numerically solved by a forward Monte-Carlo scheme relying on marked branching diffusion,
as recently pointed out in [22]. Moreover, as showed in [15], the solution to the BSDE (1.0.4)
admits a dual representation in terms of equivalent change of probability measures as:
Yt = ess supα
EPα[ ∫ T
tf(X0
s , αs)ds+ g(X0T )∣∣Ft], (1.0.5)
where for a control α, Pα is the equivalent probability measure to P under which
dX0s = b(X0
s , αs)ds+ σ(X0s )dWα
s ,
with Wα a Pα-Brownian motion by Girsanov’s theorem. In other words, the process X0 has
the same dynamics under Pα than the controlled processXα under P, and the representation
(1.0.5) can be viewed as a weak formulation (see [14]) of the stochastic control problem
(1.0.2) in the case of uncontrolled diffusion coefficient.
The general case with controlled diffusion coefficient σ(x, a) associated to fully nonlinear
PDE is challenging and led to recent theoretical advances. Consider the motivating example
from uncertain volatility model in finance formulated here in dimension 1 for simplicity of
notations:
dXαs = αsdWs,
where the control process α is valued in A = [a, a] with 0 ≤ a ≤ a < ∞, and define the
value function of the stochastic control problem:
v(t, x) := supα
E[g(Xt,x,αT )], (t, x) ∈ [0, T ]× R.
The associated HJB equation takes the form:
∂v
∂t+G(D2
xv) = 0, (t, x) ∈ [0, T )× R, v(T, x) = g(x), x ∈ R, (1.0.6)
where G(M) = 12 supa∈A[a2M ] = a2M+−a2M−. The unique (viscosity) solution to (1.0.6)
is represented in terms of the so-called G-Brownian motion B, and G-expectation EG,
concepts introduced in [36]:
v(t, x) = EG[g(x+BT−t)
].
3
Moreover, G-expectation is closely related to second order BSDE studied in [41], namely
the process Yt = v(t, Bt) satisfies a 2BSDE, which is formulated under a nondominated
family of singular probability measures given by the law of Xα under P. This gives a nice
theory and representation for nonlinear PDE, but it requires a non degeneracy assumption
on the diffusion coefficient, and does not cover general HJB equation (i.e. control both on
drift and diffusion arising for instance in portfolio optimization). On the other hand, it is
not clear how to simulate G-Brownian motion.
We provide here an alternative BSDE representation including general HJB equation,
formulated under a single probability measure (thus avoiding nondominated singular mea-
sures), and under which the forward process can be simulated. The idea, used in [25] for
quasi variational inequalities arising in impulse control problems, is the following. We intro-
duce a Poisson random measure µ(dt, da) on R+ ×A with finite intensity measure λ(da)dt
associated to the marked point process (τi, ζi)i, independent of W , and consider the pure
jump process (It)t equal to the mark ζi valued in A between two jump times τi and τi+1.
We next consider the forward regime switching diffusion process
dXs = b(Xs, Is)ds+ σ(Xs, Is)dWs,
and observe that the (uncontrolled) pair process (X, I) is Markov. Let us then consider the
BSDE with jumps w.r.t the Brownian-Poisson filtration F = FW,µ:
Yt = g(XT ) +
∫ T
tf(Xs, Is)ds−
∫ T
tZsdWs −
∫ T
t
∫AUs(a)µ(ds, da), (1.0.7)
where µ is the compensated measure of µ. This linear BSDE is the Feynman-Kac formula
for the linear integro-partial differential equation (IPDE):
∂v
∂t+ b(x, a).Dxv +
1
2tr(σσᵀ(x, a)D2
xv) (1.0.8)
+
∫A
(v(t, x, a′)− v(t, x, a))λ(da′) + f(x, a) = 0, (t, x, a) ∈ [0, T )× Rd ×A,
v(T, x, a) = g(x), (x, a) ∈ Rd ×A, (1.0.9)
through the relation: Yt = v(t,Xt, It). Now, in order to pass from the above linear IPDE
with the additional auxiliary variable a ∈ A to the nonlinear HJB PDE (4.2.18), we con-
strain the jump component to the BSDE (1.0.7) to be nonpositive, i.e.
Ut(a) ≤ 0, ∀(t, a). (1.0.10)
Then, since Ut(a) represents the jump of Yt = v(t,Xt, It) induced by a jump of the random
measure µ, i.e of I, and assuming that v is continuous, the constraint (1.0.10) means that
Ut(a) = v(t,Xt, a)− v(t,Xt, It−) ≤ 0 for all (t, a). This formally implies that v(t, x) should
not depend on a ∈ A. Once we get the non dependence of v in a, the equation (1.0.8)
becomes a PDE on [0, T )×Rd with a parameter a ∈ A. By taking the supremum over a ∈A in (1.0.8), we then obtain the nonlinear HJB equation (4.2.18).
4
Inspired by the above discussion, we now introduce the following general class of BSDE
with nonpositive jumps, which is a non Markovian extension of (1.0.7)-(1.0.10):
Yt = ξ +
∫ T
tF (s, Ys, Zs, Us)ds+KT −Kt (1.0.11)
−∫ T
tZsdWs −
∫ T
t
∫AUs(a)µ(ds, da), 0 ≤ t ≤ T, a.s.
with
Ut(a) ≤ 0 , dP⊗ dt⊗ λ(da) a.e. on Ω× [0, T ]×A. (1.0.12)
The solution to this BSDE is a quadruple (Y,Z, U,K) where, besides the usual component
(Y,Z, U), the fourth component K is a predictable nondecreasing process, which makes
the A-constraint (1.0.12) feasible. We thus look at the minimal solution (Y, Z, U,K) in the
sense that for any other solution (Y , Z, U , K) to (1.0.11)-(1.0.12), we must have Y ≤ Y .
Through a penalization method, we construct the unique minimal solution as the limit
of a sequence (Y n, Zn, Un,Kn)n of Lipschitz BSDEs with jumps. In a markovian framework
the general constrained BSDE takes the following form:
Yt = g(XT ) +
∫ T
tf(Xs, Is, Ys, Zs)ds+KT −Kt (1.0.13)
−∫ T
tZsdWs −
∫ T
t
∫AUs(a)µ(ds, da), 0 ≤ t ≤ T,
with
Ut(a) ≤ 0 , dP⊗ dt⊗ λ(da) a.e. on Ω× [0, T ]×A. (1.0.14)
Its minimal solution is then proved to be the Feynman-Kac representation of the PDE
∂v
∂t+ supa∈A
[b(., a).Dxv +
1
2tr(σσᵀ(., a)D2
xv) + f(x, a, v, σᵀ(x, a)Dxv)
]= 0
on [0, T ) × Rd, through the relation: Yt = v(t,Xt). This equation clearly extends HJB
equation (4.2.18) by incorporating the terms v and Dxv in the function f .
In the second part of the lectures, we use this representation to set an approximation of
solutions to HJB equations. Namely, we provide and analyze a discrete-time approximation
scheme for the minimal solution to (1.0.13)-(1.0.14), and thus an approximation scheme for
the HJB equation. In the non-constrained jump case, approximations schemes for BSDE
have been studied in the papers [21], [8], which extended works in [9], [44] for BSDEs in
a Brownian framework. The issue is now to deal with the nonpositive jump constraint in
(1.0.13)-(1.0.14), and we propose a discrete time approximation scheme of the form:
Y πT = YπT = g(Xπ
T )
Zπtk = E[Y πtk+1
Wtk+1−Wtk
tk+1−tk
∣∣Ftk]Yπtk = E
[Y πtk+1
∣∣Ftk]+ (tk+1 − tk) f(Xπtk, Itk , Yπtk , Z
πtk
)
Y πtk
= ess supa∈A
E[Yπtk∣∣Ftk , Itk = a
], k = 0, . . . , n− 1,
(1.0.15)
5
where π = t0 = 0 < . . . < tk < . . . < tn = T is a partition of the time interval [0, T ],
with modulus |π|, and Xπ is the Euler scheme of X (notice that I is perfectly simulatable
once we know how to simulate the distribution λ(da)/∫A λ(da) of the jump marks). The
interpretation of this scheme is the following. The first three lines in (1.0.15) correspond to
the standard scheme (Yπ, Zπ) for a discretization of a BSDE with jumps (see [8]), where we
omit here the computation of the jump component. The last line in (1.0.15) for computing
the approximation Y π of the minimal solution Y corresponds precisely to the minimality
condition for the nonpositive jump constraint and should be understood as follows. By the
Markov property of the forward process (X, I), the solution (Y,Z,U) to the BSDE with
jumps (without constraint) is in the form Yt = ϑ(t,Xt, It) for some deterministic function
ϑ. Assuming that ϑ is a continuous function, the jump component of the BSDE, which is
induced by a jump of the forward component I, is equal to Ut(a) = ϑ(t,Xt, a)−ϑ(t,Xt, It−).
Therefore, the nonpositive jump constraint means that: ϑ(t,Xt, It−) ≥ ess supa∈A
ϑ(t,Xt, a).
The minimality condition is thus written as:
Yt = v(t,Xt) = ess supa∈A
ϑ(t,Xt, a) = ess supa∈A
E[Yt|Xt, It = a],
whose discrete time version is the last line in scheme (1.0.15). We mainly consider the
case where f(x, a, y) does not depend on z, and our aim is to analyze the discrete time
approximation error on Y , where we split the error between the positive and negative
parts:
Errπ+(Y ) :=(
maxk≤n−1
E[(Ytk − Y
πtk
)2+
]) 12, Errπ−(Y ) :=
(maxk≤n−1
E[(Ytk − Y
πtk
)2−
]) 12.
We do not study directly the error on Z, and instead focus on the approximation of
an optimal control for the HJB equation, which is more relevant in practice. It appears
that the maximization step in the scheme (1.0.15) provides a control in feedback form
a(tk, Xπtk
), k ≤ n − 1, which approximates the optimal control with an estimated error
bound. The analysis of the error on Y proceeds as follows. We first introduce the solu-
tion (Y π,Yπ,Zπ,Uπ) of a discretely jump-constrained BSDE. This corresponds formally to
BSDEs for which the nonpositive jump constraint operates only a finite set of times, and
should be viewed as the analog of discretely reflected BSDEs defined in [1] and [7] in the
context of the approximation for reflected BSDEs. By combining BSDE methods and PDE
approach with comparison principles, and further with the shaking coefficients method of
Krylov [29] and Barles, Jacobsen [5], we prove the monotone convergence of this discretely
jump-constrained BSDE towards the minimal solution to the BSDE with nonpositive jump
constraint. We also obtained a convergence rate without any ellipticity condition on the
diffusion coefficient σ. We next focus on the approximation error between the discrete time
scheme in (1.0.15) and the discretely jump-constrained BSDE. The standard argument for
studying rate of convergence of such error consists in getting an estimate of the error at
time tk: E[|Y πtk− Y π
tk|2] in function of the same estimate at time tk+1, and then conclude by
induction together with classical estimates for the forward Euler scheme.
However, due to the supremum in the conditional expectation in the scheme (1.0.15) for
passing from Yπ to Y π, such argument does not work anymore. Indeed, taking the supre-
mum is a nonlinear operation, which violates the law of iterated conditional expectations.
6
Therefore, we cannot obtain directly the error at time tk as a function of that at time tk+1.
Instead, we consider the auxiliary error control at time tk:
Eπk (Y) := E[ess supa∈A
Et1,a[. . . ess sup
a∈AEtk,a
[|Yπtk − Y
πtk|2]. . .]],
where Etk,a[.] denotes the conditional expectation E[.|Ftk , Itk = a], and we are able to
express Eπk (Y) in function of Eπk+1(Y). We define similarly an error control Eπk (X) for the
forward Euler scheme, and prove that it converges to zero with a rate |π|. Proceeding
by induction, we then obtain a rate of convergence |π| for Eπk (Y), and consequently for
E[|Y πtk− Y π
tk|2]. This leads finally to a rate |π|
12 for Errπ−(Y ), |π|
110 for Errπ+(Y ), and so |π|
110
for the global error Errπ(Y ) = Errπ+(Y ) + Errπ−(Y ). In fact, as noticed in Remark 5.3.4,
we believe that one can obtain a better rate at least of the order |π|16 . Anyway, our result
improves the convergence rate of the mixed Monte-Carlo finite difference scheme proposed
in [17], where the authors obtained a rate |π|14 on one side and |π|
110 on the other side under
a nondegeneracy condition.
We conclude this introduction by pointing out that the above discrete time scheme is
not yet directly implemented in practice, and requires the estimation and computation of
the conditional expectations together with the supremum. Actually, simulation-regression
methods on basis functions defined on Rd × A appear to be very efficient, and provide
approximate optimal controls in feedback forms via the maximization operation in the last
step of the scheme (1.0.15). We refer to [24] for analysis and illustrations with several
numerical tests arising in superreplication of options under uncertain volatility and correla-
tion. Notice that since it relies on the simulation of the forward process (X, I), our scheme
does not suffer the curse of dimensionality encountered in finite difference scheme or con-
trolled Markov chains methods (see [30], [6]), and takes advantage of the high-dimensional
properties of Monte-Carlo methods.
7
Part I
BSDE representation for HJB
equations
8
Chapter 2
BSDE with nonpositive jumps
We introduce in this chapter a new class of BSDEs driven by a Brownian motion and a
Poisson, where the jump component of the solution is subject to a constraint.
2.1 Formulation and assumptions
Let (Ω,F ,P) be a complete probability space on which are defined a d-dimensional Brownian
motion W = (Wt)t≥0, and an independent integer valued Poisson random measure µ on
R+ × A, where A is a Borelian subset of Rq, endowed with its Borel σ-field B(A). We
assume that the random measure µ has the intensity measure λ(da)dt for some σ-finite
measure λ on (A,B(A)) satisfying
λ(A) :=
∫Aλ(da) < ∞ .
We set µ(dt, da) = µ(dt, da)− λ(da)dt, the compensated martingale measure associated to
µ, and denote by F = (Ft)t≥0 the completion of the natural filtration generated by W and
µ.
We fix a finite time duration T <∞ and we denote by P the σ-algebra of F-predictable
subsets of Ω× [0, T ]. Let us introduce some additional notations. We denote by
• S2 the set of real-valued cadlag adapted processes Y = (Yt)0≤t≤T such that ‖Y ‖S2 :=(
E[
sup0≤t≤T |Yt|2]) 1
2< ∞.
• Lp(0,T), p≥ 1, the set of real-valued adapted processes (φt)0≤t≤T such that E[ ∫ T
0 |φt|pdt]
< ∞.
• Lp(W), p ≥ 1, the set of Rd-valued P-measurable processes Z = (Zt)0≤t≤T such that
‖Z‖Lp(W)
:=(E[ ∫ T
0 |Zt|pdt]) 1
p<∞.
• Lp(µ), p ≥ 1, the set of P ⊗B(A)-measurable maps U : Ω× [0, T ]×E → R such that
‖U‖Lp(µ)
:=(E[ ∫ T
0
(∫A |Ut(a)|2λ(da)
) p2 dt]) 1
p<∞.
9
• L2(λ) is the set of B(A)-measurable maps u : E → R such that |u|L2(λ)
:=( ∫
A |u(a)|2λ(da)) 1
2
< ∞.
• K2 the closed subset of S2 consisting of nondecreasing processes K = (Kt)0≤t≤T with
K0 = 0.
We are then given two objects:
1. A terminal condition ξ, which is an FT -measurable random variable.
2. A generator function F : Ω × [0, T ] × R × Rd × L2(λ) → R, which is a P ⊗ B(R) ⊗B(Rd)⊗ B(L2(λ))-measurable map.
We shall impose the following assumption on these objects:
(H0)
(i) The random variable ξ and the generator function F satisfy the square integrability
condition:
E[|ξ|2]
+ E[ ∫ T
0|F (t, 0, 0, 0)|2dt
]< ∞ .
(ii) The generator function F satisfies the uniform Lipschitz condition: there exists a
constant CF such that
|F (t, y, z, u)− F (t, y′, z′, u′)| ≤ CF(|y − y′|+ |z − z′|+ |u− u′|
L2(λ)
),
for all t ∈ [0, T ], y, y′ ∈ R, z, z′ ∈ Rd and u, u′ ∈ L2(λ).
(iii) The generator function F satisfies the monotonicity condition:
F (t, y, z, u)− F (t, y, z, u′) ≤∫Aγ(t, e, y, z, u, u′)(u(a)− u′(a))λ(da) ,
for all t ∈ [0, T ], z ∈ Rd, y ∈ R and u, u′ ∈ L2(λ), where γ : [0, T ] × Ω × E × R ×Rd × L2(λ) × L2(λ) → R is a P ⊗ B(A) ⊗ B(R) ⊗ B(Rd) ⊗ B(L2(λ)) ⊗ B(L2(λ))-
measurable map satisfying: C1 ≤ γ(t, a, y, z, u, u′) ≤ C2, for all a ∈ A, with two
constants −1 < C1 ≤ 0 ≤ C2.
Let us now introduce our class of Backward Stochastic Differential Equations (BSDE)
with partially nonpositive jumps written in the form:
Yt = ξ +
∫ T
tF (s, Ys, Zs, Us)ds+KT −Kt (2.1.1)
−∫ T
tZsdWs −
∫ T
t
∫AUs(a)µ(ds, da), 0 ≤ t ≤ T, a.s.
with
Ut(a) ≤ 0 , dP⊗ dt⊗ λ(da) a.e. on Ω× [0, T ]× E. (2.1.2)
10
Definition 2.1.1 A minimal solution to the BSDE with terminal data/generator (ξ, F ) and
A-nonpositive jumps is a quadruple of processes (Y,Z, U,K) ∈ S2 × L2(W)× L2(µ)×K2
satisfying (2.1.1)-(2.1.2) such that for any other quadruple (Y , Z, U , K) ∈ S2 × L2(W) ×L2(µ)×K2 satisfying (2.1.1)-(2.1.2), we have
Yt ≤ Yt, 0 ≤ t ≤ T, a.s.
Remark 2.1.1 Notice that when it exists, there is a unique minimal solution. Indeed, by
definition, we clearly have uniqueness of the component Y . The uniqueness of Z follows by
identifying the Brownian parts and the finite variation parts, and then the uniqueness of
(U,K) is obtained by identifying the predictable parts and by recalling that the jumps of µ
are inaccessible. By misuse of language, we say sometimes that Y (instead of the quadruple
(Y,Z, U,K)) is the minimal solution to (2.1.1)-(2.1.2). 2
In order to ensure that the problem of getting a minimal solution is well-posed, we shall
need to assume:
(H1) There exists a quadruple (Y , Z, K, U) ∈ S2 × L2(W) × L2(µ) × K2 satisfying
(2.1.1)-(2.1.2).
We shall see later in Lemma 3.1.6 how such condition is satisfied in a Markovian frame-
work.
2.2 Existence and approximation by penalization
In this section, we prove the existence of a minimal solution to (2.1.1)-(2.1.2), based on
approximation via penalization. For each n ∈ N, we introduce the penalized BSDE with
jumps
Y nt = ξ +
∫ T
tF (s, Y n
s , Zns , U
ns )ds+Kn
T −Knt (2.2.3)
−∫ T
tZns dWs −
∫ T
t
∫AUns (a)µ(ds, da), 0 ≤ t ≤ T,
where Kn is the nondecreasing process in K2 defined by
Knt = n
∫ t
0
∫A
[Uns (a)]+λ(da)ds, 0 ≤ t ≤ T.
Here [u]+ = max(u, 0) denotes the positive part of u. Notice that this penalized BSDE can
be rewritten as
Y nt = ξ +
∫ T
tFn(s, Y n
s , Zns , U
ns )ds−
∫ T
tZns dWs −
∫ T
t
∫AUns (a)µ(ds, da), 0 ≤ t ≤ T,
where the generator Fn is defined by
Fn(t, y, z, u) = F (t, y, z, u) + n
∫A
[u(a)]+λ(da),
11
for all (t, y, z, u) ∈ [0, T ] × R × Rd × L2(λ). Under (H0)(ii)-(iii) and since λ(E) < ∞, we
see that Fn is Lipschitz continuous w.r.t. (y, z, u) for all n ∈ N. Therefore, we obtain from
Lemma 2.4 in [43], that under (H0), BSDE (2.2.3) admits a unique solution (Y n, Zn, Un) ∈S2 × L2(W)× L2(µ) for any n ∈ N.
Lemma 2.2.1 Let Assumption (H0) holds. The sequence (Y n)n is nondecreasing, i.e. Y nt
≤ Y n+1t for all t ∈ [0, T ] and all n ∈ N.
Proof. Fix n ∈ N, and observe that
Fn(t, e, y, z, u) ≤ Fn+1(t, e, y, z, u),
for all (t, e, y, z, u) ∈ [0, T ]×E × R× Rd × L2(λ). Under Assumption (H0), we can apply
the comparison Theorem 2.5 in [40], which shows that Y nt ≤ Y n+1
t , 0 ≤ t ≤ T , a.s. 2
The next result shows that the sequence (Y n)n is upper-bounded by any solution to the
constrained BSDE.
Lemma 2.2.2 Let Assumption (H0) holds. For any quadruple (Y , Z, U , K) ∈ S2×L2(W)×L2(µ)×K2 satisfying (2.1.1)-(2.1.2), we have
Y nt ≤ Yt, 0 ≤ t ≤ T, n ∈ N. (2.2.4)
Proof. Fix n ∈ N, and consider a quadruple (Y , Z, U , K) ∈ S2 × L2(W) × L2(µ) ×K2
solution to (2.1.1)-(2.1.2). Then, U clearly satisfies∫ t
0
∫A[Us(a)]+λ(da)ds = 0 for all t ∈
[0, T ], and so (Y , Z, U , K) is a supersolution to the penalized BSDE (2.2.3), i.e:
Yt = ξ +
∫ T
tFn(s, Ys, Zs, Us)ds+ KT − Kt
−∫ T
tZsdWs −
∫ T
t
∫AUs(a)µ(ds, da), 0 ≤ t ≤ T.
By a slight adaptation of the comparison Theorem 2.5 in [40] under (H0), we obtain the
required inequality: Y nt ≤ Yt, 0 ≤ t ≤ T . 2
We now establish a priori uniform estimates on the sequence (Y n, Zn, Un,Kn)n.
Lemma 2.2.3 Under (H0) and (H1), there exists some constant C depending only on T
and the monotonicity condition of F in (H0)(iii) such that
‖Y n‖2S2
+ ‖Zn‖2L2(W)
+ ‖Un‖2L2(µ)
+ ‖Kn‖2S2
≤ C(E|ξ|2 + E
[ ∫ T
0|F (t, 0, 0, 0)|2dt
]+ E
[sup
0≤t≤T|Yt|2
]), ∀n ∈ N. (2.2.5)
Proof. In what follows we shall denote by C > 0 a generic positive constant depending
only on T , and the linear growth condition of F in (H0)(ii), which may vary from line to
12
line. By applying Ito’s formula to |Y nt |2, and observing that Kn is continuous and ∆Y n
t =∫A U
nt (a)µ(t, da), we have
E|ξ|2 = E|Y nt |2 − 2E
∫ T
tY ns F (s, Y n
s , Zns , U
ns )ds− 2E
∫ T
tY ns dK
ns + E
∫ T
t|Zns |2ds
+ E∫ T
t
∫A
|Y ns− + Uns (a)|2 − |Y n
s−|2 − 2Y ns−U
ns (a)
µ(da, ds)
= E|Y nt |2 + E
∫ T
t|Zns |2ds+ E
∫ T
t
∫A|Uns (a)|2λ(da)ds
−2E∫ T
tY ns F (s, Y n
s , Zns , U
ns )ds− 2E
∫ T
tY ns dK
ns , 0 ≤ t ≤ T.
From (H0)(iii), the inequality Y nt ≤ Yt by Lemma 2.2.2 under (H1), and the inequality
2ab ≤ 1αa
2 + αb2 for any constant α > 0, we have:
E|Y nt |2 + E
∫ T
t|Zns |2ds+ E
∫ T
t
∫A|Uns (a)|2λ(da)ds
≤ E|ξ|2 + CE∫ T
t|Y ns |(|F (s, 0, 0, 0)|+ |Y n
s |+ |Zns |+ |Uns |L2(λ)
)ds
+1
αE[
sups∈[0,T ]
|Ys|2]
+ αE|KnT −Kn
t |2.
Using again the inequality ab ≤ a2
2 + b2
2 , and (H0)(i), we get
E|Y nt |2 +
1
2E∫ T
t|Zns |2ds+
1
2E∫ T
t
∫A|Uns (a)|2λ(da)ds (2.2.6)
≤ CE∫ T
t|Y ns |2ds+ E|ξ|2 +
1
2E∫ T
0|F (s, 0, 0, 0)|2ds+
1
αE[
supt∈[0,T ]
|Yt|2]
+ αE|KnT −Kn
t |2 .
Now, from the relation (2.2.3), we have:
KnT −Kn
t = Y nt − ξ −
∫ T
tF (s, Y n
s , Zns , U
ns )ds
+
∫ T
tZns dWs +
∫ T
t
∫AUns (a)µ(ds, da).
Thus, there exists some positive constant C1 depending only on the linear growth condition
of F in (H0)(ii) such that
E|KnT −Kn
t |2 ≤ C1
(E|ξ|2 + E
∫ T
0|F (s, 0, 0, 0)|2ds+ E|Y n
t |2
+ E∫ T
t
(|Y ns |2 + |Zns |2 + |Uns |2L2(λ)
)ds), 0 ≤ t ≤ T. (2.2.7)
Hence, by choosing α > 0 s.t. C1α ≤ 14 , and plugging into (2.2.6), we get
3
4E|Y n
t |2 +1
4E∫ T
t|Zns |2ds+
1
4E∫ T
t
∫A|Uns (a)|2λ(da)ds
≤ CE∫ T
t|Y ns |2ds+
5
4E|ξ|2 +
1
4E∫ T
0|F (s, 0, 0, 0)|2ds+
1
αE[
sups∈[0,T ]
|Ys|2], 0 ≤ t ≤ T.
13
Thus application of Gronwall’s lemma to t 7→ E|Y nt |2 yields:
sup0≤t≤T
E|Y nt |2 + E
∫ T
0|Znt |2dt+ E
∫ T
0
∫A|Unt (a)|2λ(da)dt
≤ C(E|ξ|2 + E
∫ T
0|F (t, 0, 0, 0)|2dt+ E
[supt∈[0,T ]
|Yt|2]), (2.2.8)
which gives the required uniform estimates (2.2.5) for (Zn, Un)n and also (Kn)n by (2.2.7).
Finally, by writing from (2.2.3) that
sup0≤t≤T
|Y nt | ≤ |ξ|+
∫ T
0|F (t, Y n
t , Znt , U
nt )|dt+Kn
T
+ sup0≤t≤T
∣∣∣ ∫ t
0Zns dWs
∣∣∣+ sup0≤t≤T
∣∣∣ ∫ t
0
∫AUns (a)µ(ds, da)
∣∣∣,we obtain the required uniform estimate (2.2.5) for (Y n)n by Burkholder-Davis-Gundy in-
equality, linear growth condition in (H0)(ii), and the uniform estimates for (Zn, Un,Kn)n.
2
We can now state the main result of this paragraph.
Theorem 2.2.1 Under (H0) and (H1), there exists a unique minimal solution (Y,Z, U,K)
∈ S2 × L2(W) × L2(µ) × K2 with K predictable, to (2.1.1)-(2.1.2). Y is the increasing
limit of (Y n)n and also in L2(0,T), Kt is the weak limit of (Knt )n in L2(Ω,Ft,P) for all
t ∈ [0, T ], and for any p ∈ [1, 2),
‖Zn − Z‖Lp(W)
+ ‖Un − U‖Lp(µ)
−→ 0,
as n goes to infinity.
Proof. By the Lemmata 2.2.1 and 2.2.2, (Y n)n converges increasingly to some adapted
process Y , satisfying: ‖Y ‖S2 < ∞ by the uniform estimate for (Y n)n in Lemma 2.2.3 and
Fatou’s lemma. Moreover by dominated convergence theorem, the convergence of (Y n)nto Y also holds in L2(0,T). Next, by the uniform estimates for (Zn, Un,Kn)n in Lemma
2.2.3, we can apply the monotonic convergence Theorem 3.1 in [16], which extends to the
jump case the monotonic convergence theorem of Peng [35] for BSDE. This provides the
existence of (Z,U) ∈ L2(W)×L2(µ), and K predictable, nondecreasing with E[K2T ] < ∞,
such that the sequence (Zn, Un,Kn)n converges in the sense of Theorem 2.2.1 to (Z,U,K)
satisfying:
Yt = ξ +
∫ T
tF (s, Ys, Zs, Us)ds+KT −Kt
−∫ T
tZsdWs −
∫ T
t
∫AUs(a)µ(ds, da), 0 ≤ t ≤ T.
Thus, the process Y is the difference of a cad-lag process and the nondecreasing process K,
and by Lemma 2.2 in [35], this implies that Y and K are also cad-lag, hence respectively
14
in S2 and K2. Moreover, from the strong convergence in L1(µ) of (Un)n to U and since
λ(E) <∞, we have
E∫ T
0
∫A
[Uns (a)]+λ(da)ds −→ E∫ T
0
∫A
[Us(a)]+λ(da)ds,
as n goes to infinity. Since KnT = n
∫ T0
∫A[Uns (a)]+λ(da)ds is bounded in L2(Ω,FT,P), this
implies
E∫ T
0
∫A
[Us(a)]+λ(da)ds = 0,
which means that the A-nonpositive constraint (2.1.2) is satisfied. Hence, (Y, Z,K,U) is a
solution to the constrained BSDE (2.1.1)-(2.1.2), and by Lemma 2.2.2, Y = limY n is the
minimal solution. Finally, the uniqueness of the solution (Y,Z, U,K) is given by Remark
2.1.1. 2
2.3 Dual representation
In this section, we consider the case where the generator function F (t, ω) does not depend
on y, z, u. Our main goal is to provide a dual representation of the minimal solution to the
BSDE with nonpositive jumps in terms of a family of equivalent probability measures.
Let V be the set of essentially bounded P⊗B(A)-measurable processes valued in (0,∞),
and consider for any ν ∈ V, the Doleans-Dade exponential local martingale
Lνt := E(∫ .
0
∫A
(νs(a)− 1)µ(ds, da))t
= exp(∫ t
0
∫A
ln νs(a)µ(ds, da)−∫ t
0
∫A
(νs(a)− 1)λ(da)ds), 0 ≤ t ≤ T.(2.3.9)
When Lν is a true martingale, i.e. E[LνT ] = 1, it defines a probability measure Pν equivalent
to P on (Ω,FT ) with Radon-Nikodym density:
dPν
dP
∣∣∣Ft
= Lνt , 0 ≤ t ≤ T, (2.3.10)
and we denote by Eν the expectation operator under Pν . Notice that W remains a Brownian
motion under Pν , and the effect of the probability measure Pν , by Girsanov’s Theorem, is
to change the compensator λ(da)dt of µ under P to νt(a)λ(da)dt under Pν . We denote by
µν(dt, da) = µ(dt, da)− νt(a)λ(da)dt the compensated martingale measure of µ under Pν .
We then introduce the subset Vn of V as the elements ν ∈ V essentially bounded by
n+ 1, for n ∈ N.
Lemma 2.3.4 For any ν ∈ V, Lν is a uniformly integrable martingale, and LνT is square
integrable.
15
Proof. Several sufficient criteria for Lν to be a uniformly integrable martingale are known.
We refer for example to the recent paper [39], which shows that if
SνT := exp(∫ T
0
∫A|νt(a)− 1|2λ(da)dt
)is integrable, then Lν is uniformly integrable. By definition of V, we see that for ν ∈ V, SνTis essentially bounded since ν is essentially bounded and λ(E) < ∞. Moreover, from the
explicit form (2.3.9) of Lν , we have |LνT |2 = Lν2
T SνT , and so E|LνT |2 ≤ ‖SνT ‖∞. 2
We can then associate to each ν ∈ V the probability measure Pν through (2.3.10). We
first provide a dual representation of the penalized BSDEs in terms of such Pν . To this
end, we need the following Lemma.
Lemma 2.3.5 Let φ ∈ L2(W) and ψ ∈ L2(µ). Then for every ν ∈ V, the processes∫ .0 φtdWt and
∫ .0
∫A ψt(a)µν(dt, da) are Pν-martingales.
Proof. Fix φ ∈ L2(W) and ν ∈ V and denote by Mφ the process∫ .
0 φtdWt. Since
W remains a Pν-Brownian motion, we know that Mφ is a Pν-local martingale. From
Burkholder-Davis-Gundy and Cauchy Schwarz inequalites, we have
Eν[
supt∈[0,T ]
|Mφt |]≤ CEν
[√〈Mφ〉T
]= CE
[LνT
√∫ T
0|φt|2dt
]≤ C
√E[|LνT |2
]√E[ ∫ T
0|φt|2dt
]< ∞,
since LνT is square integrable by Lemma 2.3.4, and φ ∈ L2(W). This implies that Mφ is Pν-
uniformly integrable, and hence a true Pν-martingale. The proof for∫ .
0
∫A φt(a)µν(dt, da)
follows exactly the same lines and is therefore omitted. 2
Proposition 2.3.1 For all n ∈ N, the solution to the penalized BSDE (2.2.3) is explicitly
represented as
Y nt = ess sup
ν∈VnEν[ξ +
∫ T
tF (s)ds
∣∣∣Ft], 0 ≤ t ≤ T. (2.3.11)
Proof. Fix n ∈ N. For any ν ∈ Vn, and by introducing the compensated martingale
measure µν(dt, da) = µ(dt, da) − (νt(a) − 1)λ(da)dt under Pν , we see that the solution
(Y n, Zn, Un) to the BSDE (2.2.3) satisfies:
Y nt = ξ +
∫ T
t
[F (s) +
∫A
(n[Uns (a)]+ − (νs(a)− 1)Uns (a)
)λ(da)
]ds (2.3.12)
−∫ T
tZns dWs −
∫ T
t
∫AUns (a)µν(ds, da).
By taking expectation in (2.3.12) under Pν (∼ P), we then get from Lemma 2.3.5:
Y nt = Eν
[ξ +
∫ T
t
(F (s) +
∫A
(n[Uns (a)]+ − (νs(a)− 1)Uns (a)
)λ(da)
)ds∣∣∣Ft]. (2.3.13)
16
Now, observe that for any ν ∈ Vn, hence valued in [1, n+ 1], we have
n[Unt (a)]+ − (νt(a)− 1)Unt (a) ≥ 0, dP⊗ dt⊗ λ(da) a.e.
which yields by (2.3.13):
Y nt ≥ ess sup
ν∈VnEν[ξ +
∫ T
tF (s)ds
∣∣∣Ft]. (2.3.14)
On the other hand, let us consider the process ν∗ ∈ Vn defined by
ν∗t (a) = 1Ut(a)≤0 + (n+ 1)1Ut(a)>0, 0 ≤ t ≤ T, e ∈ E.
By construction, we clearly have
n[Unt (a)]+ − (ν∗t (a)− 1)Unt (a) = 0, for all 0 ≤ t ≤ T, e ∈ E,
and thus for this choice of ν = ν∗ in (2.3.13):
Y nt = Eν
∗[ξ +
∫ T
tF (s)ds
∣∣∣Ft].Together with (2.3.14), this proves the required representation of Y n. 2
Remark 2.3.2 Arguments in the proof of Proposition 2.3.1 shows that the relation (2.3.11)
holds for general generator function F depending on (y, z, u), i.e.
Y nt = ess sup
ν∈VnEν[ξ +
∫ T
tF (s, Y n
s , Zns , U
ns )ds
∣∣∣Ft] ,which is in this case an implicit relation for Y n. Moreover, the essential supremum in this
dual representation is attained for some ν∗, which takes extreme values 1 or n+1 depending
on the sign of Un, i.e. of bang-bang form. 2
Let us then focus on the limiting behavior of the above dual representation for Y n when
n goes to infinity.
Theorem 2.3.2 Under (H1), the minimal solution to (2.1.1)-(2.1.2) is explicitly repre-
sented as
Yt = ess supν∈V
Eν[ξ +
∫ T
tF (s)ds
∣∣∣Ft], 0 ≤ t ≤ T. (2.3.15)
Proof. Let (Y,Z, U,K) ∈ be the minimal solution to (2.1.1)-(2.1.2). Let us denote by Y
the process defined in the r.h.s of (2.3.15). Since Vn ⊂ V, it is clear from the representation
(2.3.11) that Y nt ≤ Yt, for all n. Recalling from Theorem 2.2.1 that Y is the pointwise limit
of Y n, we deduce that Yt = limn→∞ Ynt ≤ Yt, 0 ≤ t ≤ T .
17
Conversely, for any ν ∈ V, let us consider the compensated martingale measure µν(dt, da)
= µ(dt, da)− (νt(a)− 1)λ(da)dt under Pν , and observe that (Y,Z, U,K) satisfies:
Yt = ξ +
∫ T
t
[F (s)−
∫A
(νs(a)− 1)Us(a)λ(da)]ds+KT −Kt (2.3.16)
−∫ T
tZsdWs −
∫ T
t
∫AUs(a)µν(ds, da).
Thus, by taking expectation in (2.3.16) under Pν from Lemma 2.3.5, and recalling that K
is nondecreasing, we have:
Yt ≥ Eν[ξ +
∫ T
t
(F (s)−
∫A
(νs(a)− 1)Us(a)λ(da))ds∣∣∣Ft]
≥ Eν[ξ +
∫ T
tF (s)ds
∣∣∣Ft],since ν is valued in [1,∞), and U satisfies the nonpositive constraint (2.1.2). Since ν is
arbitrary in V, this proves the inequality Yt ≥ Yt, and finally the required relation Y = Y .
2
18
Chapter 3
Nonlinear IPDE and Feynman-Kac
formula
In this chapter, we shall show how minimal solutions to our BSDE class with partially
nonpositive jumps provides actually a new probabilistic representation (or Feynman-Kac
formula) to fully nonlinear integro-partial differential equation (IPDE) of Hamilton-Jacobi-
Bellman (HJB) type, when dealing with a suitable Markovian framework.
3.1 The Markovian framework
We first assume that
(HA) A is compact, its interior A is connex, and A = Adh(A), the closure of its interior.
∫Aλ(da) < ∞ .
We also assume that
(Hλ)
(i) The measure λ supports the whole set A: for any a ∈ A and any open neighborhood
O of a in Rq we have λ(O ∩ A) > 0.
(ii) The boundary of A: ∂A = A \ A, is negligible w.r.t. λ, i.e. λ(∂A) = 0.
Given some measurable functions b : Rd × Rq → Rd, σ : Rd × Rq → Rd×d and β :
Rd × Rq × L → Rd, we introduce the forward Markov regime-switching jump-diffusion
process (X, I) governed by:
dXs = b(Xs, Is)ds+ σ(Xs, Is)dWs, (3.1.1)
dIs =
∫A
(a− Is−
)µ(ds, da). (3.1.2)
19
In other words, I is the pure jump process valued in A associated to the Poisson random
measure π, which changes the coefficients of jump-diffusion process X. We make the usual
assumptions on the forward jump-diffusion coefficients:
(HFC) There exists a constant C such that
|b(x, a)− b(x′, a′)|+ |σ(x, a)− σ(x′, a′)| ≤ C(|x− x′|+ |a− a′|
),
for all x, x′ ∈ Rd and a, a′ ∈ Rq.
Remark 3.1.3 We do not make any ellipticity assumption on σ. In particular, some lines
and columns of σ may be equal to zero, and so there is no loss of generality by considering
that the dimension of X and W are equal. We can also make the coefficients b, σ and β
depend on time with the following standard procedure: we introduce the time variable as
a state component Θt = t, and consider the forward Markov system:
dXs = b(Xs,Θs, Is)ds+ σ(Xs,Θs, Is)dWs,
dΘs = ds
dIs =
∫A
(a− Is−
)π(ds, da).
which is of the form given above, but with an enlarged state (X,Θ, I) (with degenerate
noise), and with the resulting assumptions on b(x, θ, a) and σ(x, θ, a). 2
Under these conditions, existence and uniqueness of a solution (Xt,x,as , It,as )t≤s≤T to
(3.1.1)-(3.1.2) starting from (x, a) ∈ Rd × Rq at time s = t ∈ [0, T ], is well-known, and we
have the standard estimate: for all p ≥ 2, there exists some positive constant Cp s.t.
E[
supt≤s≤T
|Xt,x,as |p + |It,as |p
]≤ Cp(1 + |x|p + |a|p) , (3.1.3)
for all (t, x, a) ∈ [0, T ]× Rd × Rq.
In this Markovian framework, the terminal data and generator of our class of BSDE are
given by two continuous functions g: Rd × Rq → R and f : Rd × Rq × R × Rd → R. We
make the following assumptions on the BSDE coefficients:
(HBC) There exists some constant C s.t.
|f(x, a, y, z)− f(x′, a′, y′, z′)|≤ C
(|x− x′|+ |a− a′|+ |y − y′|+ |z − z′|
),
for all x, x′ ∈ Rd, y, y′ ∈ R, z, z′ ∈ Rd and a, a′ ∈ Rq.
In this framework, the BSDE problem (2.1.1)-(2.1.2) takes the following form: find the
minimal solution (Y,Z, U,K) ∈ S2 × L2(W)× L2(µ)×K2 to
Yt = g(XT , IT ) +
∫ T
tf(Xs, Is, Ys, Zs
)ds+KT −Kt
−∫ T
tZs.dWs −
∫ T
t
∫LUs(a)µ(ds, da), (3.1.4)
20
with
Ut(a) ≤ 0 , dP⊗ dt⊗ λ(da) a.e. (3.1.5)
The main goal of this chapter is to relate the BSDE (3.1.4) with nonpositive jumps
(3.1.5) to the following nonlinear IPDE of HJB type:
− ∂w
∂t− supa∈A
[Law + f
(., a, w, σᵀ(., a)Dxw)
]= 0, on [0, T )× Rd, (3.1.6)
w(T, x) = supa∈A
g(x, a), x ∈ Rd, (3.1.7)
where
Law(t, x) = b(x, a).Dxw(t, x) +1
2tr(σσᵀ(x, a)D2
xw(t, x))
for (t, x, a) ∈ [0, T ]× Rd × Rq.
Notice that under (HBC) and (3.1.3) (which follows from (HFC)), the generator
F (t, ω, y, z, u) = f(Xt(ω), It(ω), y, z) and the terminal condition ξ = g(XT , IT ) satisfy
clearly Assumption (H0). Let us now show that Assumption (H1) is satisfied. More
precisely, we have the following result.
Lemma 3.1.6 Let Assumptions (HFC), (HBC1) hold. Then, for any initial condition
(t, x, a) ∈ [0, T ]×Rd×Rq, there exists a solution (Y t,x,as , Zt,x,as , U t,x,as , Kt,x,a
s ), t ≤ s ≤ T to
the BSDE (3.1.4)-(3.1.5) when (X, I) = (Xt,x,as , It,as ), t ≤ s ≤ T, with Y t,x,a
s = v(s,Xt,x,as )
for some deterministic function v on [0, T ]× Rd satisfying a polynomial growth condition:
sup(t,x)∈[0,T ]×Rd
|v(t, x)|1 + |x|2
< ∞ . (3.1.8)
Proof. Under (HBC1) and since A is compact, we observe that
Cf,g := supx∈Rd,a∈A
|g(x, a)|+ |f(x, a, y, z)|1 + |x|+ |y|+ |z|
< ∞. (3.1.9)
Let us then consider the smooth function v(t, x) = Ceρ(T−t)(1 + |x|2) for some positive
constants C and ρ to be determined later. We claim that for C and ρ large enough, the
function v is a classical supersolution to (3.1.6)-(3.1.7). Indeed, observe first that from the
growth condition on g in (3.1.9), there exists C > 0 s.t. g(x) := supa∈A g(x, a) ≤ C(1+ |x|2)
for all x ∈ Rd. For such C, we then have: v(T, .) ≥ g. On the other hand, we see after
straightforward calculation that there exists a positive constant C depending only on C,
Cf,g, and the linear growth condition in x on b and σ by (HFC) (recall that A is compact),
such that
−∂v∂t− supa∈A
[Lav + f
(., a, v, σᵀ(., a)Dxv
)]≥ (ρ− C)v
≥ 0,
21
by choosing ρ ≥ C. Let us now define the quadruple (Y , Z, U , K) by:
Yt = v(t,Xt) for t < T, YT = g(XT , IT ),
Zt = σᵀ(Xt− , It−)Dxv(t,Xt−), t ≤ T,Ut = 0, t ≤ T
Kt =
∫ t
0
[− ∂v
∂t(s,Xs)− LIs v(s,Xs)− f(Xs, Is, Zs)
]ds, t < T
KT = KT− + v(T,XT )− g(XT , IT ).
From the supersolution property of v to (3.1.6)-(3.1.7), the process K is nondecreasing.
Moreover, from the polynomial growth condition on v, linear growth condition on b, σ,
growth condition (3.1.9) on f , g and the estimate (3.1.3), we see that (Y , Z, U , K) lies in
S2×L2(W)×L2(µ)×K2. Finally, by applying Ito’s formula to v(t,Xt), we conclude that
(Y , Z, U , R, K) is solution a to (3.1.4), and the constraint (3.1.5) is trivially satisfied. 2
Under (HFC) and (HBC), we then get from Theorem 2.2.1 the existence of a unique
minimal solution (Y t,x,as , Zt,x,as , U t,x,as ,Kt,x,a
s ), t ≤ s ≤ T to (3.1.4)-(3.1.5) when (X, I) =
(Xt,x,as , It,as ), t ≤ s ≤ T. Moreover, as we shall see in the next paragraph, this minimal
solution is written in this Markovian context as: Y t,x,as = v(s,Xt,x,a
s , It,x,as ) where v is the
deterministic function defined on [0, T ]× Rd × Rq → R by:
v(t, x, a) := Y t,x,at , (t, x, a) ∈ [0, T ]× Rd × Rq. (3.1.10)
We aim at proving that the function v defined by (3.1.10) does not depend actually on its
argument a, and is a solution in a sense to be precised to the parabolic IPDE (3.1.6)-(3.1.7).
Notice that we do not have a priori any smoothness or even continuity properties on v.
To this end, we first recall the definition of (discontinuous) viscosity solutions to (3.1.6)-
(3.1.7). For a locally bounded function w on [0, T )×Rd, we define its lower semicontinuous
(lsc for short) envelope w∗, and upper semicontinuous (usc for short) envelope w∗ by
w∗(t, x) = lim inf(t′, x′)→ (t, x)
t′ < T
w(t′, x′) and w∗(t, x) = lim sup(t′, x′)→ (t, x)
t′ < T
w(t′, x′),
for all (t, x) ∈ [0, T ]× Rd.
Definition 3.1.2 (Viscosity solutions to (3.1.6)-(3.1.7))
(i) A function w, lsc (resp. usc) on [0, T ] × Rd, is called a viscosity supersolution (resp.
subsolution) to (3.1.6)-(3.1.7) if
w(T, x) ≥ (resp. ≤) supa∈A
g(x, a) ,
for any x ∈ Rd, and(− ∂ϕ
∂t− supa∈A
[Laϕ+ f(., a, w, σᵀ(., a)Dxϕ)
])(t, x) ≥ (resp. ≤) 0,
22
for any (t, x) ∈ [0, T )× Rd and any ϕ ∈ C1,2([0, T ]× Rd) such that
(w − ϕ)(t, x) = min[0,T ]×Rd
(w − ϕ) (resp. max[0,T ]×Rd
(w − ϕ)) .
(ii) A locally bounded function w on [0, T )×Rd is called a viscosity solution to (3.1.6)-(3.1.7)
if w∗ is a viscosity supersolution and w∗ is a viscosity subsolution to (3.1.6)-(3.1.7).
We can now state the main result of this chapter.
Theorem 3.1.3 Assume that conditions (HA), (Hλ), (HFC) and (HBC) hold. The
function v in (3.1.10) does not depend on the variable a on [0, T )× R× A i.e.
v(t, x, a) = v(t, x, a′), ∀ a, a′ ∈ A,
for all (t, x) ∈ [0, T ) × Rd. Let us then define by misuse of notation the function v on
[0, T )× Rd by:
v(t, x) = v(t, x, a), (t, x) ∈ [0, T )× Rd, (3.1.11)
for any a ∈ A. Then v is a viscosity solution to (3.1.6)-(3.1.7).
Remark 3.1.4 1. Once we have a uniqueness result for the fully nonlinear IPDE (3.1.6)-
(3.1.7), Theorem 3.1.3 provides a Feynman-Kac representation of this unique solution by
means of the minimal solution to the BSDE (3.1.4)-(3.1.5). This suggests consequently an
original probabilistic numerical approximation of the nonlinear IPDE (3.1.6)-(3.1.7) by dis-
cretization and simulation of the minimal solution to the BSDE (3.1.4)-(3.1.5). This issue,
especially the treatment of the nonpositive jump constraint, has been recently investigated
in [23] and [24], where the authors analyze the convergence rate of the approximation
scheme, and illustrate their results with several numerical tests arising for instance in the
super-replication of options in uncertain volatilities and correlations models. We mention
here that a nice feature of our scheme is the fact that the forward process (X, I) can be
easily simulated: indeed, notice that the jump times of I follow a Poisson distribution of
parameter λ :=∫A λ(da), and so the pure jump process I is perfectly simulatable once
we know how to simulate the distribution λ(da)/λ of the jump marks. Then, we can use
a standard Euler scheme for simulating the component X. Our scheme does not suffer
the curse of dimensionality encountered in finite difference methods or controlled Markov
chains, and takes advantage of the high dimensional properties of Monte-Carlo methods.
2. We do not address here comparison principles (and so uniqueness results) for the general
parabolic nonlinear IPDE (3.1.6)-(3.1.7). In the case where the generator function f(x, a)
does not depend on (y, z, u) (see Remark 3.1.5 below), comparison principle is proved in
[37], and the result can be extended by same arguments when f(x, a, y, z) depends also on
y, z under the Lipschitz condition of (HBC). 2
Remark 3.1.5 Stochastic control problem
23
1. Let us now consider the particular and important case where the generator f(x, a) does
not depend on (y, z). We then observe that the nonlinear IPDE (3.1.6) is the Hamilton-
Jacobi-Bellman (HJB) equation associated to the following stochastic control problem: let
us introduce the controlled jump-diffusion process:
dXαs = b(Xα
s , αs)ds+ σ(Xαs , αs)dWs, (3.1.12)
where W is a Brownian motion independent of a random measure ϑ on a filtered probability
space (Ω,F ,F0,P), the control α lies in AF0 , the set of F0-predictable process valued in A,
and define the value function for the control problem:
w(t, x) := supα∈AF0
E[ ∫ T
tf(Xt,x,α
s , αs)ds+ g(Xt,x,αT , αT )
], (t, x) ∈ [0, T ]× Rd,
where Xt,x,αs , t ≤ s ≤ T denotes the solution to (3.1.12) starting from x at s = t, given
a control α ∈ AF0 . It is well-known (see e.g. [37] or [32]) that the value function w is
characterized as the unique viscosity solution to the dynamic programming HJB equation
(3.1.6)-(3.1.7), and therefore by Theorem 3.1.3, w = v. In other words, we have provided a
representation of fully nonlinear stochastic control problem, including especially control in
the diffusion term, possibly degenerate, in terms of minimal solution to the BSDE (3.1.4)-
(3.1.5).
2. Combining the BSDE representation of Theorem 3.1.3 together with the dual repre-
sentation in Theorem 2.3.2, we obtain an original representation for the value function of
stochastic control problem:
supα∈AF0
E[ ∫ T
0f(Xα
t , αt)dt+ g(XαT , αT )
]= sup
ν∈VAEν[ ∫ T
0f(Xt, It)dt+ g(XT , IT )
]The r.h.s. in the above relation may be viewed as a weak formulation of the stochastic
control problem. Indeed, it is well-known that when there is only control on the drift, the
value function may be represented in terms of control on change of equivalent probability
measures via Girsanov’s theorem for Brownian motion. Such representation is called weak
formulation for stochastic control problem, see [14]. In the general case, when there is
control on the diffusion coefficient, such “Brownian” Girsanov’s transformation can not be
applied, and the idea here is to introduce an exogenous process I valued in the control set
A, independent of W and ϑ governing the controlled state process Xα, and then to control
the change of equivalent probability measures through a Girsanov’s transformation on this
auxiliary process.
3. Non Markovian extension. An interesting issue is to extend our BSDE representation
of stochastic control problem to a non Markovian context, that is when the coefficients
b, σ and β of the controlled process are path-dependent. In this case, we know from the
recent works by Ekren, Touzi, and Zhang [13] that the value function to the path-dependent
stochastic control is a viscosity solution to a path-dependent fully nonlinear HJB equation.
One possible approach for getting a BSDE representation to path-dependent stochastic
control, would be to prove that our minimal solution to the BSDE with nonpositive jumps
is a viscosity solution to the path-dependent fully nonlinear HJB equation, and then to
24
conclude with a uniqueness result for path-dependent nonlinear PDE. However, to the best
of our knowledge, there is not yet such comparison result for viscosity supersolution and
subsolution of path-dependent nonlinear PDEs. Instead, we recently proved in [20] by
purely probabilistic arguments that the minimal solution to the BSDE with nonpositive
jumps is equal to the value function of a path-dependent stochastic control problem, and
our approach circumvents the delicate issue of dynamic programming principle and viscosity
solution in the non Markovian context. Our result is also obtained without assuming that
σ is non degenerate, in contrast with [13] (see their Assumption 4.7). 2
The rest of this paper is devoted to the proof of Theorem 3.1.3.
3.2 Viscosity property of the penalized BSDE
Let us consider the Markov penalized BSDE associated to (3.1.4)-(3.1.5):
Y nt = g(XT , IT ) +
∫ T
tf(Xs, Is, Y
ns , Z
ns )ds+ n
∫ T
t
∫A
[Uns (a)]+λ(da)ds
−∫ T
tZns dWs −
∫ T
t
∫AUns (a)µ(ds, da) , (3.2.13)
and denote by (Y n,t,x,as , Zn,t,x,as , Un,t,x,as ), t ≤ s ≤ T the unique solution to (3.2.13) when
(X, I) = (Xt,x,as , It,as ), t ≤ s ≤ T for any initial condition (t, x, a) ∈ [0, T ] × Rd × Rq.
From the Markov property of the jump-diffusion process (X, I), we recall from [3] that
Y n,t,x,as = vn(s,Xt,x,a
s , It,as ), t ≤ s ≤ T , where vn is the deterministic function defined on
[0, T ]× Rd × Rq by:
vn(t, x, a) := Y n,t,x,at , (t, x, a) ∈ [0, T ]× Rd × Rq. (3.2.14)
From the convergence result (Theorem 2.2.1) of the penalized solution, we deduce that the
minimal solution of the constrained BSDE is actually in the form Y t,x,as = v(s,Xt,x,a
s , It,as ),
t ≤ s ≤ T , with a deterministic function v defined in (3.1.10).
Moreover, from the uniform estimate (2.2.5) and Lemma 3.1.6, there exists some positive
constant C s.t. for all n,
|vn(t, x, a)|2 ≤ C(E|g(Xt,x,a
T , It,aT )|2 + E[ ∫ T
t|f(Xt,x,a
s , It,as , 0, 0)|2ds]
+ E[
supt≤s≤T
|v(s,Xt,x,as )|2
]),
for all (t, x, a) ∈ [0, T ]×Rd×Rq. From (HBC) we get that g and f satisfy a linear growth
condition. Using (3.1.8) for v, and the estimate (3.1.3) for (X, I), we obtain that vn, and
thus also v by passing to the limit, satisfy a polynomial growth condition: there exists some
positive constant Cv such that for all n:
|vn(t, x, a)|+ |v(t, x, a)| ≤ Cv(1 + |x|2 + |a|2
), ∀(t, x, a) ∈ [0, T ]× Rd × Rq. (3.2.15)
25
We now consider the parabolic semi-linear penalized IPDE for any n:
− ∂vn∂t
(t, x, a)− Lavn(t, x, a)− f(x, a, vn, σ
ᵀ(x, a)Dxvn) (3.2.16)
−∫A
[vn(t, x, a′)− vn(t, x, a)]λ(da′)
−n∫A
[vn(t, x, a′)− vn(t, x, a)]+λ(da′) = 0, on [0, T )× Rd × Rq,
vn(T, ., .) = g, on Rd × Rq. (3.2.17)
From Theorem 3.4 in Barles et al. [3], we have the well-known property that the
penalized BSDE with jumps (2.2.3) provides a viscosity solution to the penalized IPDE
(3.2.16)-(3.2.17). Moreprecisely, we have the following result.
Proposition 3.2.2 Let Assumptions (HFC) and (HBC) hold. The function vn in (3.2.14)
is a continuous viscosity solution to (3.2.16)-(3.2.17), i.e. it is continuous on [0, T ]×Rd×Rq,a viscosity supersolution (resp. subsolution) to (3.2.17):
vn(T, x, a) ≥ (resp. ≤) g(x, a) ,
for any (x, a) ∈ Rd × Rq, and a viscosity supersolution (resp. subsolution) to (3.2.16):
− ∂ϕ
∂t(t, x, a)− Laϕ(t, x, a) (3.2.18)
−f(x, a, vn(t, x, a), σᵀ(x, a)Dxϕ(t, x, a))
−∫A
[ϕ(t, x, a′)− ϕ(t, x, a)]λ(da′)− n∫A
[ϕ(t, x, a′)− ϕ(t, x, a)]+λ(da′) ≥ (resp. ≤) 0,
for any (t, x, a) ∈ [0, T )× Rd × Rq and any ϕ ∈ C1,2([0, T ]× (Rd × Rq)) such that
(vn − ϕ)(t, x, a) = min[0,T ]×Rd×Rq
(vn − ϕ) (resp. max[0,T ]×Rd×Rq
(vn − ϕ)) . (3.2.19)
In contrast to local PDEs with no integro-differential terms, we cannot restrict in general
the global minimum (resp. maximum) condition on the test functions for the definition of
viscosity supersolution (resp. subsolution) to local minimum (resp. maximum) condition.
In our IPDE case, the nonlocal terms appearing in (3.2.16) involve the values w.r.t. the
variable a only on the set A. Therefore, we are able to restrict the global extremum
condition on the test functions to extremum on [0, T ] × Rd × A. More precisely, we have
the following equivalent definition of viscosity solutions, which will be used later.
Lemma 3.2.7 Assume that (Hλ), (HFC), and (HBC) hold. In the definition of vn being
a viscosity supersolution (resp. subsolution) to (3.2.16) at a point (t, x, a) ∈ [0, T )×Rd× A,
we can replace condition (3.2.19) by:
0 = (vn − ϕ)(t, x, a) = min[0,T ]×Rd×A
(vn − ϕ) (resp. max[0,T ]×Rd×A
(vn − ϕ)) ,
and suppose that the test function ϕ is in C1,2,0([0, T ]× Rd × Rq).
26
Proof. We treat only the supersolution case as the subsolution case is proved by same
arguments, and proceed in two steps.
Step 1. Fix (t, x, a) ∈ [0, T ) × Rd × Rq, and let us show that the viscosity supersolution
inequality (3.2.18) also holds for any test function ϕ in C1,2,0([0, T ]× Rd × Rq) s.t.
(vn − ϕ)(t, x, a) = min[0,T ]×Rd×Rq
(vn − ϕ) . (3.2.20)
We may assume w.l.o.g. that the minimum for such test function ϕ is zero, and let us
define for r > 0 the function ϕr by
ϕr(t′, x′, a′) = ϕ(t′, x′, a′)(
1− Φ( |x′|2 + |a′|2
r2
))− CvΦ
( |x′|2 + |a′|2
r2
)(1 + |x′|p + |a′|p
),
where Cv > 0 and p ≥ 2 are the constant and degree appearing in the polynomial growth
condition (3.2.15) for vn, Φ : R+ → [0, 1] is a function in C∞(R+) such that Φ|[0,1] ≡ 0 and
Φ|[2,+∞) ≡ 1. Notice that ϕr ∈ C1,2,0([0, T ]× Rd × Rq),
(ϕr, Dxϕr, D2
xϕr) −→ (ϕ,Dxϕ,D
2xϕ) as r →∞ (3.2.21)
locally uniformly on [0, T ]× Rd × Rq, and that there exists a constant Cr > 0 such that
|ϕr(t′, x′, a′)| ≤ Cr(1 + |x′|p + |a′|p
)(3.2.22)
for all (t′, x′, a′) ∈ [0, T ]×Rq×Rd. Since Φ is valued in [0, 1], we deduce from the polynomial
growth condition (3.2.15) satisfied by vn and (3.2.20) that ϕr ≤ vn on [0, T ]× Rd × Rq for
all r > 0. Moreover, we have ϕr(t, x, a) = ϕ(t, x, a) (= vn(t, x, a)) for r large enough.
Therefore we get
(vn − ϕr)(t, x, a) = min[0,T ]×Rd×Rq
(vn − ϕr) , (3.2.23)
for r large enough, and we may assume w.l.o.g. that this minimum is strict. Let (ϕrk)k be
a sequence of function in C1,2([0, T ]× (Rd × Rq)) satisfying (3.2.22) and such that
(ϕrk, Dxϕrk, D
2xϕ
rk) −→ (ϕr, Dxϕ
r, D2xϕ
r) as k →∞, (3.2.24)
locally uniformly on [0, T ] × Rd × Rq. From the growth conditions (3.2.15) and (3.2.22)
on the continuous functions vn and ϕrk, we can assume w.l.o.g. (up to an usual negative
perturbation of the function ϕkr for large (x′, a′)), that there exists a bounded sequence
(tk, xk, ak)k in [0, T ]× Rd × Rq such that
(vn − ϕrk)(tk, xk, ak) = min[0,T ]×Rd×Rq
(vn − ϕrk) . (3.2.25)
The sequence (tk, xk, ak)k converges up to a subsequence, and thus, by (3.2.23), (3.2.24)
and (3.2.25), we have
(tk, xk, ak) → (t, x, a), as k →∞ . (3.2.26)
27
Now, from the viscosity supersolution property of vn at (tk, xk, ak) with the test function
ϕrk, we have
−∂ϕrk∂t
(tk, xk, ak)− Lakϕrk(tk, xk, ak)
−f(xk, ak, vn(tk, xk, ak), σᵀ(xk, ak)Dxϕ
rk(tk, xk, ak))
−∫A
[ϕrk(tk, xk, a′)− ϕrk(tk, xk, ak)]λ(da′)
−n∫A
[ϕrk(tk, xk, a′)− ϕrk(tk, xk, ak)]+λ(da′) ≥ 0,
Sending k and r to infinity, and using (3.2.21), (3.2.24) and (3.2.26), we obtain the viscosity
supersolution inequality at (t, x, a) with the test function ϕ.
Step 2. Fix (t, x, a) ∈ [0, T )×Rd× A, and let ϕ be a test function in C1,2([0, T ]×(Rd×Rq))such that
0 = (vn − ϕ)(t, x, a) = min[0,T ]×Rd×A
(vn − ϕ). (3.2.27)
By same arguments as in (3.2.22), we can assume w.l.o.g. that ϕ satisfies the polynomial
growth condition:
|ϕ(t′, x′, a′)| ≤ C(1 + |x′|p + |a′|p), (t′, x′, a′) ∈ [0, T ]× Rd × Rq,
for some positive constant C. Together with (3.2.15), and since A is compact, we have
(vn − ϕ)(t′, x′, a′) ≥ −C(1 + |x′|p + |dA(a′)|p), (3.2.28)
for all (t′, x′, a′) ∈ [0, T ] × Rd × Rq, where dA(a′) is the distance from a′ to A. Fix ε > 0
and define the function ϕε ∈ C1,2,0([0, T ]× Rd × Rq) by
ϕε(t′, x′, a′) = ϕ(t′, x′, a′)− Φ
(dAε(a′)ε
)C(1 + |x′|p + |dA(a′)|p)
for all (t′, x′, a′) ∈ [0, T ]× Rd × Rq, where
Aε =a′ ∈ A : d∂A(a′) ≥ ε
, (3.2.29)
and Φ : R+ → [0, 1] is a function in C∞(R+) such that Φ|[0, 12
] ≡ 0 and Φ|[1,+∞) ≡ 1. Notice
that
(ϕε, Dxϕε, D2xϕε) −→ (ϕ,Dxϕ,D
2xϕ) as ε→ 0, (3.2.30)
locally uniformly on [0, T ]×Rd × A. We notice from (3.2.28) and the definition of ϕε that
ϕε ≤ vn on [0, T ] × Rd × Acε. Moreover, since ϕε ≤ ϕ on [0, T ] × Rd × Rq, ϕε = ϕ on
[0, T ]× Rd × Aε and a ∈ A, we get by (3.2.27) for ε small enough
0 = (vn − ϕε)(t, x, a) = min[0,T ]×Rd×Rq
(vn − ϕε) .
28
From Step 1, we then have
−∂ϕε∂t
(t, x, a)− Laϕε(t, x, a)
−f(x, a, vn(t, x, a), σᵀ(x, a)Dxϕε(t, x, a))
−∫A
[ϕε(t, x, a′)− ϕε(t, x, a)]λ(da′)− n
∫A
[ϕε(t, x, a′)− ϕε(t, x, a)]+λ(da′) ≥ 0 .
By sending ε to zero with (3.2.30), and using a ∈ A with (Hλ)(ii), we get the required
viscosity subsolution inequality at (t, x, a) for the test function ϕ. 2
3.3 The non dependence of the function v in the variable a
In this subsection, we aim to prove that the function v(t, x, a) does not depend on a.
From the relation defining the Markov BSDE (3.1.4), and since for the minimal solution
(Y t,x,a, Zt,x,a, U t,x,a,Kt,x,a) to (3.1.4)-(3.1.5), the process Kt,x,a is predictable, we observe
that the A-jump component U t,x,a is expressed in terms of Y t,x,a = v(., Xt,x,a, It,x,a) as:
U t,x,as (a′) = v(s,Xt,x,as− , a′)− v(s,Xt,x,a
s− , It,x,as− ), t ≤ s ≤ T, a′ ∈ A,
for all (t, x, a) ∈ [0, T ]× Rd × Rq. From the A-nonpositive constraint (3.1.5), this yields
E[ ∫ t+h
t
∫A
[v(s,Xt,x,a
s , a′)− v(s,Xt,x,as , It,x,as )
]+λ(da′)ds
]= 0,
for any h > 0. If we knew a priori that the function v was continuous on [0, T )× Rd × A,
we could obtain by sending h to zero in the above equality divided by h (and by dominated
convergence theorem), and from the mean-value theorem:∫A
[v(t, x, a′)− v(t, x, a)
]+λ(da′) = 0.
Under condition (Hλ)(i), this would prove that v(t, x, a) ≥ v(t, x, a′) for any a, a′ ∈ A, and
thus the function v would not depend on a in A.
Unfortunately, we are not able to prove directly the continuity of v from its very defi-
nition (3.1.10), and instead, we shall rely on viscosity solutions approach to derive the non
dependence of v(t, x, a) in a ∈ A. To this end, let us introduce the following first-order
PDE:
−∣∣Dav(t, x, a)
∣∣ = 0 , (t, x, a) ∈ [0, T )× Rd × A. (3.3.31)
Lemma 3.3.8 Let assumptions (Hλ), (HFC) and (HBC) hold. The function v is a
viscosity supersolution to (3.3.31): for any (t, x, a) ∈ [0, T ) × Rd × A and any function
ϕ ∈ C1,2([0, T ]× (Rd × Rq)) such that (v − ϕ)(t, x, a) = min[0,T ]×Rd×Rq(v − ϕ), we have
−∣∣Daϕ(t, x, a)
∣∣ ≥ 0, i.e. Daϕ(t, x, a) = 0.
29
Proof. We know that v is the pointwise limit of the nondecreasing sequence of functions
(vn). By continuity of vn, the function v is lsc and we have (see e.g. [2] p. 91):
v = v∗ = lim infn→∞ ∗vn, (3.3.32)
where
lim infn→∞ ∗vn(t, x, a) := lim inf
n→∞(t′, x′, a′)→ (t, x, a)
t′ < T
vn(t′, x′, a′), (t, x, a) ∈ [0, T ]× Rd × Rq .
Let (t, x, a) ∈ [0, T ) × Rd × A, and ϕ ∈ C1,2([0, T ] × (Rd × Rq)), such that (v − ϕ)(t, x, a)
= min[0,T ]×Rd×Rq(v − ϕ). We may assume w.l.o.g. that this minimum is strict:
(v − ϕ)(t, x, a) = strict min[0,T ]×Rd×Rq
(v − ϕ) . (3.3.33)
Up to a suitable negative perturbation of ϕ for large (x, a), we can assume w.l.o.g. that
there exists a bounded sequence (tn, xn, an)n in [0, T ]× Rd × Rq such that
(vn − ϕ)(tn, xn, an) = min[0,T ]×Rd×Rq
(vn − ϕ) . (3.3.34)
From (4.2.22), (3.3.33), and (3.3.34), we then have, up to a subsequence:
(tn, xn, an, vn(tn, xn, an)) −→ (t, x, a, v(t, x, a)) as n→∞ . (3.3.35)
Now, from the viscosity supersolution property of vn at (tn, xn, an) with the test function
ϕ, we have by (3.3.34):
−∂ϕ∂t
(tn, xn, an)− Lanϕ(tn, xn, an)
−f(xn, an, vn(tn, xn, an), σᵀ(xn, an)Dxϕ(tn, xn, an))
−∫A
[ϕ(tn, xn, a′)− ϕ(tn, xn, an)]λ(da′)
−n∫A
[ϕ(tn, xn, a′)− ϕ(tn, xn, an)]+λ(da′) ≥ 0,
which implies ∫A
[ϕ(tn, xn, a′)− ϕ(tn, xn, an)]+λ(da′)
≤ 1
n
[− ∂ϕ
∂t(tn, xn, an)− Lanϕ(tn, xn, an)
− f(xn, an, vn(tn, xn, an), σᵀ(xn, an)Dxϕ(tn, xn, an))
−∫A
[ϕ(tn, xn, a′)− ϕ(tn, xn, an)]λ(da′)
].
Sending n to infinity, we get from (3.3.35), the continuity of coefficients b, σ, β and f , and
the dominated convergence theorem:∫A
[ϕ(t, x, a′)− ϕ(t, x, a)]+λ(da′) = 0 .
30
Under (Hλ), this means that ϕ(t, x, a) = maxa′∈A ϕ(t, x, a′). Since a ∈ A, we deduce that
Daϕ(t, x, a) = 0. 2
We notice that the PDE (3.3.31) involves only differential terms in the variable a.
Therefore, we can freeze the terms (t, x) ∈ [0, T ) × Rd in the PDE (3.3.31), i.e. we can
take test functions not depending on the variables (t, x) in the definition of the viscosity
solution, as shown in the following Lemma.
Lemma 3.3.9 Let assumptions (Hλ), (HFC) and (HBC) hold. For any (t, x) ∈ [0, T )×Rd, the function v(t, x, .) is a viscosity supersolution to
−∣∣Dav(t, x, a)
∣∣ = 0 , a ∈ A,
i.e. for any a ∈ A and any function ϕ ∈ C2(Rq) such that (v(t, x, .)−ϕ)(a) = minRq(v(t, x, .)−ϕ), we have: −
∣∣Daϕ(a)∣∣ ≥ 0 (and so = 0).
Proof. Fix (t, x) ∈ [0, T )× Rd, a ∈ A and ϕ ∈ C2(Rq) such that
(v(t, x, .)− ϕ)(a) = minRq
(v(t, x, .)− ϕ) . (3.3.36)
As usual, we may assume w.l.o.g. that this minimum is strict and that ϕ satisfies the
growth condition supa′∈Rq|ϕ(a′)|1+|a′|2 < ∞. Let us then define for n ≥ 1, the function ϕn ∈
C1,2([0, T ]× (Rd × Rq)) by
ϕn(t′, x′, a′) = ϕ(a′)− n(|t′ − t|2 + |x′ − x|4
)− |a′ − a|4
for all (t′, x′, a′) ∈ [0, T ]×Rd ×Rq. From the growth condition (3.2.15) on the lsc function
v, and the growth condition on the continuous function ϕ, one can find for any n ≥ 1 an
element (tn, xn, an) of [0, T ]× Rd × Rq such that
(v − ϕn)(tn, xn, an) = min[0,T ]×Rd×Rq
(v − ϕn).
In particular, we have
v(t, x, a)− ϕ(a) = (v − ϕn)(t, x, a) ≥ (v − ϕn)(tn, xn, an) (3.3.37)
= v(tn, xn, an)− ϕ(an) + n(|tn − t|2 + |xn − x|4) + |an − a|4
≥ v(tn, xn, an)− v(t, x, an) + v(t, x, a)− ϕ(a)
+ n(|tn − t|2 + |xn − x|4) + |an − a|4
by (3.3.36), which implies from the growth condition (3.2.15) on v:
n(|tn − t|2 + |xn − x|4) + |an − a|4 ≤ C(1 + |xn − x|2 + |an − a|2).
Therefore, the sequences n(|tn− t|2 + |xn−x|4)n and (|a−an|4)n are bounded and (up to
a subsequence) we have: (tn, xn, an) −→ (t, x, a∞) as n goes to infinity, for some a∞ ∈ Rq.Actually, since v(t, x, a)−ϕ(a) ≥ v(tn, xn, an)−ϕ(an) by (3.3.37), we obtain by sending n
to infinity and since the minimum in (3.3.36) is strict, that a∞ = a, and so:
(tn, xn, an) −→ (t, x, a) as n→∞ .
31
On the other hand, from Lemma 3.3.8 applied to (tn, xn, an) with the test function ϕn, we
have
0 = Daϕn(tn, xn, an) = Daϕ(an)− 4(an − a)|an − a|3,
for all n ≥ 1. Sending n to infinity we get the required result: Daϕ(a) = 0. 2
We are now able to state the main result of this subsection.
Proposition 3.3.3 Let assumptions (HA), (Hλ), (HFC) and (HBC) hold. The func-
tion v does not depend on the variable a on [0, T )× Rd × A:
v(t, x, a) = v(t, x, a′) , a, a′ ∈ A,
for any (t, x) ∈ [0, T )× Rd.
Proof. We proceed in four steps.
Step 1. Approximation by inf-convolution.
We introduce the family of functions (un)n defined by
un(t, x, a) = infa′∈A
[v(t, x, a′) + n|a− a′|4
], (t, x, a) ∈ [0, T ]× Rd ×A.
It is clear that the sequence (un)n is nondecreasing and upper-bounded by v. Moreover,
since v is lsc, we have the pointwise convergence of un to v on [0, T ]× Rd ×A. Indeed, fix
some (t, x, a) ∈ [0, T ] × Rd × A. Since v is lsc, there exists a sequence (an)n valued in A
such that
un(t, x, a) = v(t, x, an) + n|a− an|4 ,
for all n ≥ 1. Since A is compact, the sequence (an) converges, up to a subsequence, to
some a∞ ∈ A. Moreover, since un is upper-bounded by v and v is lsc, we see that a∞ = a
and
un(t, x, a) −→ v(t, x, a) as n→∞ . (3.3.38)
Step 2. A test function for un seen as a test function for v.
For r, δ > 0 let us define the integer N(r, δ) by
N(r, δ) = minn ∈ N : n ≥ 2Cv(1 + 2−1 + rp + 2 maxa∈A |a|2)(
δ2
)4 + Cv
where Cv is the constant in the growth condition (3.2.15), and define the set Aδ by
Aδ =a ∈ A : d(a, ∂A) := min
a′∈∂A|a− a′| > δ
.
Fix (t, x) ∈ [0, T ) × Rd. We now prove that for any δ > 0, n ≥ N(|x|, δ), a ∈ Aδ and
ϕ ∈ C2(Rq) such that
0 = (un(t, x, .)− ϕ)(a) = minRq
(un(t, x, .)− ϕ) , (3.3.39)
32
there exists an ∈ A and ψ ∈ C2(Rq) such that
0 = (v(t, x, .)− ψ)(an) = minRq
(v(t, x, .)− ψ) , (3.3.40)
and
Daψ(an) = Daϕ(a). (3.3.41)
To this end we proceed in two substeps.
Substep 2.1. We prove that for any δ > 0, (t, x, a) ∈ [0, T ) × Rd × Aδ, and any n ≥N(|x|, δ):
argmina′∈A
v(t, x, a′) + n|a′ − a|4
⊂ A .
Fix (t, x, a) ∈ [0, T )× Rd × Aδ and let an ∈ A such that
v(t, x, an) + n|an − a|4 = mina′∈A
[v(t, x, a′) + n|a′ − a|4
].
Then we have
v(t, x, an) + n|an − a|4 ≤ v(t, x, a),
and by (3.2.15), this gives
−Cv(1 + |x|p + 2 maxa∈A|a|2 + 2|an − a|2) + n|an − a|4 ≤ Cv(1 + |x|2 + |a|2) .
Then using the inequality 2αβ ≤ α2 + β2 to the product 2αβ = 2p−1|an − a|p, we get:
(n− Cv)|an − a|4 ≤ 2Cv(1 + 2−1 + |x|2 + 2 maxa∈A|a|2) .
For n ≥ N(|x|, δ), we get from the definition of N(r, δ):
|an − a| ≤δ
2,
which shows that an ∈ A since a ∈ Aδ.Substep 2.2. Fix δ > 0, (t, x, a) ∈ [0, T )× Rd × Aδ, and ϕ ∈ C2(Rq) satisfying (3.3.39).
Let us then choose an ∈ argmin v(t, x, a′) +n|a′−a|4 : a′ ∈ A, and define ψ ∈ C2(Rq) by:
ψ(a′) = ϕ(a+ a′ − an)− n|an − a|4, a′ ∈ Rq.
It is clear that ψ satisfies (3.3.41). Moreover, we have by (3.3.39) and the inf-convolution
definition of un:
ψ(a′) ≤ un(t, x, a+ a′ − an)− n|an − a|4 ≤ v(t, x, a′) , a′ ∈ Rq.
Moreover, since an ∈ A attains the infimum in the inf-convolution definition of un(t, x, a),
we have
ψ(an) = ϕ(a)− n|an − a|4 = un(t, x, a)− n|an − a|4 = v(t, x, an) ,
33
which shows (3.3.40).
Step 3. The function un does not depend locally on the variable a. From Step 2 and Lemma
3.3.9, we obtain that for any fixed (t, x) ∈ [0, T )×Rd, the function un(t, x, .) inherits from
v(t, x, .) the viscosity supersolution to
−∣∣Daun(t, x, a)
∣∣ = 0, a ∈ Aδ, (3.3.42)
for any δ > 0, n ≥ N(|x|, δ). Let us then show that un(t, x, .) is locally constant in the
sense that for all a ∈ Aδ:
un(t, x, a) = un(t, x, a′), ∀a′ ∈ B(a, η), (3.3.43)
for all η > 0 such that B(a, η) ⊂ Aδ. We first notice from the inf-convolution definition
that un(t, x, .) is semi concave on Aδ. From Theorem 2.1.7 in [10], we deduce that un(t, x, .)
is locally Lipschitz continuous on Aδ. By Rademacher theorem, this implies that un(t, x, .)
is differentiable almost everywhere on Aδ. Therefore, by Corollary 2.1 (ii) in [2], and the
viscosity supersolution property (3.3.42), we get that this relation (3.3.42) holds actually
in the classical sense for almost all a′ ∈ Aδ. In other words, un(t, x, .) is a locally Lipschitz
continuous function with derivatives equal to zero almost everywhere on Aδ. This means
that it is locally constant (easy exercise in analysis left to the reader).
Step 4. From the convergence (3.3.38) of un to v, and the relation (3.3.43), we get by sending
n to infinity that for any δ > 0 the function v satisfies: for any (t, x, a) ∈ [0, T )× Rd × Aδ
v(t, x, a) = v(t, x, a′)
for all η > 0 such that B(a, η) ⊂ Aδ and all a′ ∈ B(a, η). Then by sending δ to zero
we obtain that v does not depend on the variable a locally on [0, T )× Rd × A. Since A is
assumed to be connex, we obtain that v does not depend on the variable a on [0, T )×Rd×A.
2
3.4 Viscosity properties of the minimal solution to the con-
strained BSDE
From Proposition 3.3.3, we can define by misuse of notation the function v on [0, T )× Rd
by
v(t, x) = v(t, x, a), (t, x) ∈ [0, T )× Rd, (3.4.44)
for any a ∈ A. Moreover, by the growth condition (3.2.15), we have
sup(t,x)∈[0,T ]×Rd
|v(t, x)|1 + |x|2
< ∞. (3.4.45)
The aim of this section is to prove that the function v is a viscosity solution to (3.1.6)-
(3.1.7).
34
Proof of the viscosity supersolution property to (3.1.6). We first notice from (4.2.22)
and (3.4.44) that v is lsc and
v(t, x) = v∗(t, x) = lim infn→∞ ∗vn(t, x, a) (3.4.46)
for all (t, x, a) ∈ [0, T ]×Rd×A. Let (t, x) be a point in [0, T )×Rd, and ϕ ∈ C1,2([0, T ]×Rd),such that
(v − ϕ)(t, x) = min[0,T ]×Rd
(v − ϕ) .
We may assume w.l.o.g. that ϕ satisfies sup(t,x)∈[0,T ]×Rd|ϕ(t,x)|1+|x|p <∞. Fix some a ∈ A, and
define for ε > 0, the test function
ϕε(t′, x′, a′) = ϕ(t′, x′)− ε(|t′ − t|2 + |x′ − x|4 + |a′ − a|4),
for all (t′, x′, a′) ∈ [0, T ]×Rd×Rq. Since ϕε(t, x, a) = ϕ(t, x), and ϕε ≤ ϕ with equality iff
(t′, x′, a′) = (t, x, a), we then have
(v − ϕε)(t, x, a) = strict min[0,T ]×Rd×Rq
(v − ϕε). (3.4.47)
From the growth conditions on the continuous functions vn and ϕ, there exists a bounded
sequence (tn, xn, an)n (we omit the dependence in ε) in [0, T ]× Rd × Rq such that
(vn − ϕε)(tn, xn, an) = min[0,T ]×Rd×Rq
(vn − ϕε) . (3.4.48)
From (3.4.46) and (3.4.47), we obtain by standard arguments that up to a subsequence:
(tn, xn, an, vn(tn, xn, an)) −→ (t, x, a, v(t, x)), as n goes to infinity.
Now from the viscosity supersolution property of vn at (tn, xn, an) with the test function
ϕε, we have
−∂ϕε
∂t(tn, xn, an)− Lanϕε(tn, xn, an)
−f(xn, an, vn(tn, xn, an), σᵀ(xn, an)Dxϕ
ε(tn, xn, an))
−∫A
[ϕε(tn, xn, a′)− ϕε(tn, xn, an)]λ(da′)
− n∫A
[ϕε(tn, xn, a′)− ϕε(tn, xn, an)]+λ(da′) ≥ 0 .
Sending n to infinity in the above inequality, we get from the definition of ϕε and the
dominated convergence Theorem:
− ∂ϕε
∂t(t, x, a)− Laϕε(t, x, a)
−f(x, a, v(t, x), σᵀ(x, a)Dxϕ
ε(t, x, a))
(3.4.49)
+ε
∫A|a′ − a|4λ(da′) ≥ 0 .
35
Sending ε to zero, and since ϕε(t, x, a) = ϕ(t, x), we get
−∂ϕ∂t
(t, x)− Laϕ(t, x)− f(x, a, v(t, x), σᵀ(x, a)Dxϕ(t, x)
)≥ 0 .
Since a is arbitrarily chosen in A, we get from (HA) and the continuity of the coefficients
b, σ, γ and f in the variable a
−∂ϕ∂t
(t, x)− supa∈A
[Laϕ(t, x) + f
(x, a, v(t, x), σᵀ(x, a)Dxϕ(t, x)
)]≥ 0 ,
which is the viscosity supersolution property. 2
Proof of the viscosity subsolution property to (3.1.6). Since v is the pointwise limit
of the nondecreasing sequence of continuous functions (vn), and recalling (3.4.44), we have
by [2] p. 91:
v∗(t, x) = lim supn→∞
∗vn(t, x, a) (3.4.50)
for all (t, x, a) ∈ [0, T ]× Rd × A, where
lim supn→∞
∗vn(t, x, a) := lim supn→∞
(t′, x′, a′)→ (t, x, a)
t′ < T, a′ ∈ A
vn(t′, x′, a′).
Fix (t, x) ∈ [0, T )× Rd and ϕ ∈ C1,2([0, T ]× Rd) such that
(v∗ − ϕ)(t, x) = max[0,T ]×Rd
(v∗ − ϕ) . (3.4.51)
We may assume w.l.o.g. that this maximum is strict and that ϕ satisfies
sup(t,x)∈[0,T ]×Rd
|ϕ(t, x)|1 + |x|p
< ∞ . (3.4.52)
Fix a ∈ A and consider a sequence (tn, xn, an)n in [0, T )× Rd × A such that
(tn, xn, an, vn(tn, xn, an)) −→ (t, x, a, v∗(t, x)) as n→∞. (3.4.53)
Let us define for n ≥ 1 the function ϕn ∈ C1,2,0([0, T ]× Rd × Rq) by
ϕn(t′, x′, a′) = ϕ(t′, x′) + n(dAηn (a′)
ηn∧ 1 + |t′ − tn|2 + |x′ − xn|4
)where Aηn is defined by (3.2.29) for ε = ηn and (ηn)n is a positive sequence converging to
0 s.t. (such sequence exists by (Hλ)(ii)):
n2λ(A \Aηn) −→ 0 as n→∞ . (3.4.54)
From the growth conditions (3.4.45) and (3.4.52) on v and ϕ, we can find a sequence
(tn, xn, an) in [0, T ]× Rd ×A such that
(vn − ϕn)(tn, xn, an) = max[0,T ]×Rd×A
(vn − ϕn) , n ≥ 1 . (3.4.55)
36
Using (3.4.50) and (3.4.51), we obtain by standard arguments that up to a subsequence
n( 1
ηndAηn (an) + |tn − tn|2 + |xn − xn|4
)−→ 0 as n→∞ , (3.4.56)
and
vn(tn, xn, an) −→ v∗(t, x) as n→∞ .
We deduce from (3.4.56) and (3.4.53) that, up to a subsequence:
(tn, xn, an) −→ (t, x, a), as n→∞ . (3.4.57)
for some a ∈ A. Moreover, for n large enough, we have an ∈ A. Indeed, suppose that, up
to a subsequence, an ∈ ∂A for n ≥ 1. Then we have 1ηndAηn (an) ≥ 1, which contradicts
(3.4.56). Now, from the viscosity subsolution property of vn at (tn, xn, an) with the test
function ϕn satisfying (3.4.55), Lemma 3.2.7, and since an ∈ A, we have:
− ∂ϕn∂t
(tn, xn, an)− Lanϕn(tn, xn, an)
−f(xn, an, vn(tn, xn, an), σᵀ(xn, an)Dxϕ(tn, xn)) (3.4.58)
−(n+ 1)n
∫A
(dAηn (a′)
ηn∧ 1)λ(da′) ≤ 0 ,
for all n ≥ 1. From (3.4.54) we get
(n+ 1)n
∫A
(dAηn (a′)
ηn∧ 1)λ(da′) −→ 0 as n→∞ (3.4.59)
Sending n to infinity into (3.4.58), and using (3.4.50), (3.4.57) and (3.4.59), we get
−∂ϕ∂t
(t, x)− Laϕ(t, x)− f(x, a, v∗(t, x), σᵀ(x, a)Dxϕ(t, x)) ≤ 0 .
Since a ∈ A, this gives
−∂ϕ∂t
(t, x)− supa∈A
[Laϕ(t, x) + f
(x, a, v∗(t, x), σᵀ(x, a)Dxϕ(t, x)
)]≤ 0 ,
which is the viscosity subsolution property. 2
Proof of the viscosity supersolution property to (3.1.7). Let (x, a) ∈ Rd × A. From
(3.4.46), we can find a sequence (tn, xn, an)n valued in [0, T )× Rd × Rq such that
(tn, xn, an, vn(tn, xn, an)) −→ (T, x, a, v∗(T, x)) as n→∞ .
The sequence of continuous functions (vn)n being nondecreasing and vn(T, .) = g we have
v∗(T, x) ≥ limn→∞
v1(tn, xn, an) = g(x, a) .
Since a is arbitrarily chosen in A, we deduce that v∗(T, x) ≥ supa∈A g(x, a) = supa∈A g(x, a)
by (HA) and continuity of g in a. 2
37
Proof of the viscosity subsolution property to (3.1.7). Let x ∈ Rd. Then we can find
by (3.4.50) a sequence (tn, xn, an)n in [0, T )× Rd × A such that
(tn, xn, vn(tn, xn, an)) → (T, x, v∗(T, x)) , as n→∞ . (3.4.60)
Define the function h : [0, T ]× Rd → R by
h(t, x) =√T − t+ sup
a∈Ag(x, a)
for all (t, x) ∈ [0, T )×Rd. From (HFC) and (HBC), we see that h is a continuous viscosity
supersolution to (3.2.16)-(3.2.17), on [T − η, T ]× B(x, η) for η small enough. We can then
apply Theorem 3.5 in [3] which gives that
vn ≤ h on [T − η, T ]× B(x, η)×A
for all n ≥ 0. By applying the above inequality at (tn, xn, an), and sending n to infinity,
together with (3.4.60), we get the required result. 2
38
Part II
Discretization of fully nonlinear
HJB equations via BSDEs with
nonpositive jumps
39
Chapter 4
Discretization of the nonpositive
jump constraint
In this chapter, we present a approximation of the constraint imposed to the jump com-
ponent of the solution to the BSDE. We first express this approximated constraint as a
constraint on the the component Y operating only at the times on a fixed grid. We then
show that the solution satisfying the approximating constraint converges as soon as the
time mesh of the grid goes to zero.
4.1 Discretely jump-constrained BSDE
We introduce in this section discretely jump-constrained BSDE. The nonpositive jump
constraint operates only at the times of the grid π = t0 = 0 < t1 < . . . < tn = T of [0, T ],
and we look for a quadruple (Y π,Yπ,Zπ,Uπ) ∈ S2 × S2 × L2(W)× L2(µ) satisfying:
Y πT = YπT = g(XT ) (4.1.1)
and
Yπt = Y πtk+1
+
∫ tk+1
tf(Xs, Is,Yπs ,Zπs )ds (4.1.2)
−∫ tk+1
tZπs dWs −
∫ tk+1
t
∫AUπs (a)µ(ds, da) ,
Y πt = Yπt 1(tk,tk+1)(t) + ess sup
a∈AE[Yπt∣∣Xt, It = a
]1tk(t) , (4.1.3)
for all t ∈ [tk, tk+1) and all 0 ≤ k ≤ n− 1.
Notice that at each time tk of the grid, the condition is not known a priori to be
square integrable since it involves a supremum over A, and the well-posedness of the BSDE
(4.1.1)-(4.1.2)-(4.1.3) is not a direct and standard issue. We shall use a PDE approach for
proving the existence and uniqueness of a solution. Let us consider the system of integro-
partial differential equations (IPDEs) for the functions vπ and ϑπ defined recursively on
[0, T ]× Rd ×A by:
40
• A terminal condition for vπ and ϑπ:
vπ(T, x, a) = ϑπ(T, x, a) = g(x) , (x, a) ∈ Rd ×A , (4.1.4)
• A sequence of IPDEs for ϑπ−Laϑπ − f
(x, a, ϑπ, σᵀ(x, a)Dxϑ
π)
−∫A
(ϑπ(t, x, a′)− ϑπ(t, x, a)
)λ(da′) = 0, (t, x, a) ∈ [tk, tk+1)× Rd ×A,
ϑπ(t−k+1, x, a) = supa′∈A ϑπ(tk+1, x, a
′) (x, a) ∈ Rd ×A(4.1.5)
for k = 0 . . . , n− 1,
• the relation between vπ and ϑπ:
vπ(t, x, a) = ϑπ(t, x, a)1(tk,tk+1)(t) + supa′∈A
ϑπ(t, x, a′)1tk(t) , (4.1.6)
for all t ∈ [tk, tk+1) and k = 0 . . . , n− 1. The rest of this section is devoted to the proof of
existence and uniqueness of a solution to (4.1.4)-(4.1.5)-(4.1.6), together with some uniform
Lipschitz properties, and its connection to the discretely jump-constrained BSDE (4.1.1)-
(4.1.2)-(4.1.3).
For any L-Lipschitz continuous function ϕ on Rd ×A, and k ≤ n− 1, we denote:
Tkπ[ϕ](t, x, a) := w(t, x, a), (t, x, a) ∈ [tk, tk+1)× Rd ×A, (4.1.7)
where w is the unique continuous viscosity solution on [tk, tk+1]×Rd×A with linear growth
condition in x to the integro partial differential equation (IPDE):−Law − f(x, a, w, σᵀDxw)
−∫A
(w(t, x, a′)− w(t, x, a)
)λ(da′) = 0, (t, x, a) ∈ [tk, tk+1)× Rd ×A,
w(t−k+1, x, a) = ϕ(x, a), (x, a) ∈ Rd ×A ,
(4.1.8)
and we extend by continuity Tkπ[ϕ](tk+1, x, a) = ϕ(x, a). The existence and uniqueness
of such a solution w to the semi linear IPDE (4.1.8), and its nonlinear Feynman-Kac
representation in terms of BSDE with jumps, is obtained e.g. from Theorems 3.4 and 3.5
in [3].
Lemma 4.1.1 There exists a constant C such that for any L-Lipschitz continuous function
ϕ on Rd ×A, and k ≤ n− 1, we have
|Tkπ[ϕ](t, x, a)− Tkπ[ϕ](t, x′, a′)| ≤ max(L, 1)√
1 + |π|eC|π|(|x− x′|+ |a− a′|) ,
for all t ∈ [tk, tk+1), and (x, a), (x′, a′) ∈ Rd ×A.
Proof. Fix t ∈ [tk, tk+1), k ≤ n−1, (x, a), (x′, a′) ∈ Rd×A, and ϕ an L-Lipschitz continuous
function on Rd × A. Let (Y ϕ, Zϕ, Uϕ) and (Y ϕ,′ , Zϕ,′, Uϕ,
′) be the solutions on [t, tk+1] to
41
the BSDEs
Y ϕs = ϕ(Xt,x,a
tk+1, It,atk+1
) +
∫ tk+1
sf(Xt,x,a
r , It,ar , Y ϕr , Z
ϕr )dr
−∫ tk+1
sZϕr dWr −
∫ tk+1
s
∫AUϕr (a)µ(dr, da), t ≤ s ≤ tk+1,
Y ϕ,′s = ϕ(Xt,x′,a′
tk+1, It,a
′
tk+1) +
∫ tk+1
sf(Xt,x′,a′
r , It,a′
r , Y ϕ,′r , Zϕ,
′r )dr
−∫ tk+1
sZϕ,
′r dWr −
∫ tk+1
s
∫AUϕ,
′r (a)µ(dr, da), t ≤ s ≤ tk+1
From Theorems 3.4 and 3.5 in [3], we have the identification:
Y ϕt = Tkπ[ϕ](t, x, a) and Y ϕ,′
t = Tkπ[ϕ](t, x′, a′) . (4.1.9)
We now estimate the difference between the processes Y ϕand Y ϕ,′ , and set δY ϕ = Y ϕ−Y ϕ,′ ,
δZϕ = Zϕ − Zϕ,′ , δX = Xt,x,a −Xt,x′,a′ , δI = It,a − It,a′ . By Ito’s formula, the Lipschitz
condition of f and ϕ, and Young inequality, we have
E[|δY ϕ
s |2]
+ E[ ∫ tk+1
s|δZϕs |2ds
]≤ L2E
[|δXT |2 + |δIT |2
]+ C
∫ tk+1
sE[|δY ϕ
r |2]dr
+1
2E[ ∫ tk+1
s
(|δXr|2 + |δIr|2 + |δZϕr |2
)dr],
for any s ∈ [t, tk+1]. Now, from classical estimates on jump-diffusion processes we have
supr∈[t,tk+1]
E[|δXr|2 + |δIr|2
]≤ eC|π|
(|x− x′|2 + |a− a′|2
),
and thus:
E[|δY ϕ
s |2]≤ (L2 + |π|)eC|π|
(|x− x′|2 + |a− a′|2
)+ C
∫ tk+1
sE[|δY ϕ
r |2]dr ,
for all s ∈ [t, tk+1]. By Gronwall’s Lemma, this yields
sups∈[t,tk+1]
E[|δY ϕ
s |2]≤ (L2 + |π|)e2C|π|(|x− x′|2 + |a− a′|2
),
which proves the required result from the identification (4.1.9):
|Tkπ[ϕ](t, x, a)− Tkπ[ϕ](t, x′, a′)| ≤√L2 + |π|eC|π|(|x− x′|+ |a− a′|)
≤ max(L, 1)√
1 + |π|eC|π|(|x− x′|+ |a− a′|).
2
Proposition 4.1.1 There exists a unique viscosity solution ϑπ with linear growth condition
to the IPDE (4.1.4)-(4.1.5), and this solution satisfies:
|ϑπ(t, x, a)− ϑπ(t, x′, a′)|
≤ max(L2, 1)
√(e2C|π|(1 + |π|)
)n−k(|x− x′|+ |a− a′|
), (4.1.10)
for all k = 0, . . . , n− 1, t ∈ [tk, tk+1), (x, a), (x′, a′) ∈ Rd ×A.
42
Proof. We prove by a backward induction on k that the IPDE (4.1.4)-(4.1.5) admits a
unique solution on [tk, T ]× Rd ×A, which satisfies (4.1.10).
• For k = n−1, we directly get the existence and uniqueness of ϑπ on [tn−1, T ]×Rd×A from
Theorems 3.4 and 3.5 in [3], and we have ϑπ = Tn−1π [g] on [tn−1, T ) × Rd × A. Moreover,
we also get by Lemma 4.1.1:
|ϑπ(t, x, a)− ϑπ(t, x′, a′)| ≤ max(L2, 1)√e2C|π|(1 + |π|)
(|x− x′|+ |a− a′|
)for all t ∈ [tn−1, tn), (x, a), (x′, a′) ∈ Rd ×A.
• Suppose that the result holds true at step k + 1 i.e. there exists a unique function ϑπ on
[tk+1, T ]×Rd×A with linear growth and satisfying (4.1.4)-(4.1.5) and (4.1.10). It remains
to prove that ϑπ is uniquely determined by (4.1.5) on [tk, tk+1)×Rd×A and that it satisfies
(4.1.10) on [tk, tk+1)×Rd ×A. Since ϑπ satisfies (4.1.10) at time tk+1, we deduce that the
function
ψk+1(x) := supa∈A
ϑπ(tk+1, x, a), x ∈ Rd,
is also Lipschitz continuous, and satisfies by the induction hypothesis:
|ψk+1(x)− ψk+1(x′)| ≤ max(L2, 1)
√(e2C|π|(1 + |π|)
)n−k−1|x− x′|, (4.1.11)
for all x, x′ ∈ Rd. Under (HFC) and (HBC), we can apply Theorems 3.4 and 3.5 in
[3], and we get that ϑπ is the unique viscosity solution with linear growth to (4.1.5) on
[tk, tk+1)× Rd × A, with ϑπ = Tkπ[ψk+1]. Thus it exists and is unique on [tk, T ]× Rd × A.
From Lemma 4.1.1 and (4.1.11), we then get
|ϑπ(t, x, a)− ϑπ(t, x′, a′)| = |Tkπ[ψk+1](t, x, a)− Tkπ[ψk+1](t, x′, a′)|
≤ max(L2, 1)
√(e2C|π|(1 + |π|)
)n−k−1
√(1 + |π|)e2C|π|
(|x− x′|+ |a− a′|
)≤ max(L2, 1)
√(e2C|π|(1 + |π|)
)n−k(|x− x′|+ |a− a′|
)for any t ∈ [tk, tk+1) and (x, a), (x′, a′) ∈ Rd × A, which proves the required induction
inequality at step k. 2
Remark 4.1.1 The function ϑπ(t, x, .) is continuous on A, for each (t, x), and so the
function vπ is well-defined by (4.1.6). Moreover, the function ϑπ may be written recursively
as: ϑπ(T, ., .) = g on Rd ×A,
ϑπ = Tkπ[vπ(tk+1, .)], on [tk, tk+1)× Rd ×A,(4.1.12)
for k = 0, . . . , n− 1. In particular, ϑπ is continuous on (tk, tk+1)× Rd ×A, k ≤ n− 1. 2
As a consequence of the above proposition, we obtain the uniform Lipschitz property
of ϑπ and vπ, with a Lipschitz constant independent of π.
43
Corollary 4.1.1 There exists a constant C (independent of |π|) such that
|ϑπ(t, x, a)− ϑπ(t, x′, a′)|+ |vπ(t, x, a)− vπ(t, x′, a′)| ≤ C(|x− x′|+ |a− a′|
),
for all t ∈ [0, T ], x, x′ ∈ Rd, a, a′ ∈ Rd.
Proof. Recalling that n|π| is bounded, we see that the sequence appearing in (4.1.10):((e2C|π|(1+ |π|)
)n−k)0≤k≤n−1
is bounded uniformly in |π| (or n), which shows the required
Lipschitz property of ϑπ. Since A is assumed to be compact, this shows in particular that
the function vπ defined by the relation (4.1.6) is well-defined and finite. Moreover, by
noting that
| supa∈A
ϑπ(t, x, a)− supa∈A
ϑπ(t, x′, a)| ≤ supa∈A|ϑπ(t, x, a)− ϑπ(t, x′, a)|
for all (t, x) ∈ [0, T ]× Rd, we also obtain the required Lipschitz property for vπ. 2
We now turn to the existence of a solution to the discretely jump-constrained BSDE.
Proposition 4.1.2 The BSDE (4.1.1)-(4.1.2)-(4.1.3) admits a unique solution (Y π,Yπ,Zπ,Uπ)
in S2 × S2 × L2(W)× L2(µ). Moreover we have
Yπt = ϑπ(t,Xt, It), and Y πt = vπ(t,Xt, It) (4.1.13)
for all t ∈ [0, T ].
Proof. We prove by backward induction on k that (Y π,Yπ,Zπ,Uπ) is well defined and
satisfies (4.1.13) on [tk, T ].
• Suppose that k = n − 1. From Corollary 2.3 in [3], we know that (Yπ,Zπ,Uπ), exists
and is unique on [tn−1, T ]. Moreover, from Theorems 3.4 and 3.5 in [3], we get Yπt =
Tkπ[g](t,Xt, It) = ϑπ(t,Xt, It) on [tn−1, T ]. By (4.1.3), we then have for all t ∈ [tn−1, T ):
Y πt = 1(tn−1,T )(t) ϑ
π(t,Xt, It) + 1tn−1(t) ess supa∈A
ϑπ(t,Xt, a)
= 1(tn−1,T )(t) ϑπ(t,Xt, It) + 1tn−1(t) sup
a∈Aϑπ(t,Xt, a) = vπ(t,Xt, It),
since the essential supremum and supremum coincide by continuity of a 7→ ϑπ(t,Xt, a) on
the compact set A.
• Suppose that the result holds true for some k ≤ n− 1. Then, we see that (Yπ,Zπ,Uπ) is
defined on [tk−1, tk) as the solution to a BSDE driven by W and µ with a terminal condition
vπ(tk, Xtk). Since vπ satisfies a linear growth condition, we know again by Corollary 2.3
in [3] that (Yπ,Zπ,Uπ), thus also Y π, exists and is unique on [tk−1, tk). Moreover, using
again Theorems 3.4 and 3.5 in [3], we get (4.1.13) on [tk−1, tk). 2
We end this section with a conditional regularity result for the discretely jump-constrained
BSDE.
44
Proposition 4.1.3 There exists some constant C such that
supt∈[tk,tk+1)
Etk[|Yπt − Yπtk |
2]
+ supt∈(tk,tk+1]
Etk[|Y πt − Y π
tk+1|2]≤ C(1 + |Xtk |
2)|π|,
for all k = 0, . . . , n− 1.
Proof. Fix k ≤ n− 1. By Ito’s formula, we have for all t ∈ [tk, tk+1):
Etk[|Yπt − Yπtk |
2]
= 2Etk[ ∫ t
tk
f(Xs, Is,Yπs ,Zπs )(Yπtk − Yπs )ds
]+ Etk
[ ∫ t
tk
|Zπs |2]
+ Etk[ ∫ t
tk
∫A|Uπs (a)|2λ(da)ds
]≤ Etk
[ ∫ t
tk
|Yπs − Yπtk |2]
+ C|π|(
1 + Etk[
sups∈[tk,tk+1]
|Xs|2])
+ C|π|Etk[
sups∈[tk,tk+1]
(|Yπs |2 + |Zπs |2 +
∫A|Uπs (a)|2λ(da)
)],
by the linear growth condition on f (recall also that A is compact), and Young inequality.
Now, by standard estimate for X under growth linear condition on b and σ, we have:
Etk[
sups∈[tk,tk+1]
|Xs|2]≤ C(1 + |Xtk |
2). (4.1.14)
We also know from Proposition 4.2 in [8], under (H1) and (H2), that there exists a
constant C depending only on the Lipschitz constants of b, σ f and vπ(tk+1, .) (which does
not depend on π by Corollary 4.1.1), such that
Etk[
sups∈[tk,tk+1]
(|Yπs |2 + |Zπs |2 +
∫A|Uπs (a)|2λ(da)
)]≤ C(1 + |Xtk |
2). (4.1.15)
We deduce that
Etk[|Yπt − Yπtk |
2]≤ Etk
[ ∫ t
tk
|Yπs − Yπtk |2]
+ C|π|(1 + |Xtk |2),
and we conclude for the regularity of Yπ by Gronwall’s lemma. Finally, from the defini-
tion (4.1.2)-(4.1.3) of Y π and Yπ, Ito isometry for stochastic integrals, and growth linear
condition on f , we have for all t ∈ (tk, tk+1):
Etk[|Y πt − Y π
tk+1|2]
= Etk[|Yπt − Y π
tk+1|2]
≤ 3Etk[ ∫ tk+1
tk
(|f(Xs, Is,Yπs ,Zπs )|2 + |Zπs |2 +
∫A|Uπs (a)|2λ(da)
)ds]
≤ C|π|Etk[1 + sup
s∈[tk,tk+1]
(|Xs|2 + |Yπs |2 + |Zπs |2 +
∫A|Uπs (a)|2λ(da)
)]≤ C|π|(1 + |Xtk |
2),
where we used again (4.1.14) and (4.1.15). This ends the proof. 2
45
4.2 Convergence of discretely jump-constrained BSDE
This section is devoted to the convergence of the discretely jump-constrained BSDE towards
the minimal solution to the BSDE with nonpositive jump.
Under (HFC) and (HBC), we have seen in Chapter 3 the existence and uniqueness of
a minimal solution (Y,Z, U,K) ∈ S2 × L2(W)× L2(µ)× L2(µ) toYt = g(XT ) +
∫ Tt f(Xs, Is, Ys, Zs)ds+KT −Kt
−∫ Tt ZsdWs −
∫ Tt
∫A Us(a)µ(ds, da), 0 ≤ t ≤ T.
Ut(a) ≤ 0 dP⊗ dt⊗ λ(da) a.e.
(4.2.16)
Moreover, the minimal solution Y is in the form
Yt = v(t,Xt), 0 ≤ t ≤ T, (4.2.17)
where v : [0, T ] × Rd → R is a viscosity solution with linear growth to the fully nonlinear
HJB type equation:− supa∈A
[Lav + f(x, a, v, σᵀ(x, a)Dxv)
]= 0, on [0, T )× Rd,
v(T, x) = g, on Rd,(4.2.18)
where
Lav =∂v
∂t+ b(x, a).Dxv +
1
2tr(σσᵀ(x, a)D2
xv).
We shall make the standing assumption that comparison principle holds for (4.2.18).
(HC) Let w (resp. w) be a lower-semicontinuous (resp. upper-semicontinuous) viscosity
supersolution (resp. subsolution) with linear growth condition to (4.2.18). Then, w ≥ w.
When f does not depend on y, z, i.e. (4.2.18) is the usual HJB equation for a stochastic
control problem, Assumption (HC) holds true, see [18] or [38]. In the general case, we
refer to [12] for sufficient conditions to comparison principles. Under (HC), the function
v in (4.2.17) is the unique viscosity solution to (4.2.18), and is in particular continuous.
Actually, we have the standard Holder and Lipschitz property (see Appendix in [29] or [5]):
|v(t, x)− v(t′, x′)| ≤ C(|t− t′|
12 + |x− x′|
), (t, t′) ∈ [0, T ], x, x′ ∈ Rd. (4.2.19)
This implies that the process Y is continuous, and thus the jump component U = 0. In
the sequel, we shall focus on the approximation of the remaining components Y and Z of
the minimal solution to (4.2.16).
4.2.1 Convergence result
Lemma 4.2.2 We have the following assertions:
1) The familly (ϑπ)π is nondecreasing and upper bounded by v: for any grids π and π′ such
that π ⊂ π′, we have
ϑπ(t, x, a) ≤ ϑπ′(t, x, a) ≤ v(t, x) , (t, x, a) ∈ [0, T ]× Rd ×A .
46
2) The familly (ϑπ)π satisfies a uniform linear growth condition: there exists a constant C
such that
|ϑπ(t, x, a)| ≤ C(1 + |x|),
for any (t, x, a) ∈ [0, T ]× Rd ×A and any grid π.
Proof. 1) Let us first prove that ϑπ ≤ v. Since v is a (continuous) viscosity solution
to the HJB equation (4.2.18), and v does not depend on a, we see that v is a viscosity
supersolution to the IPDE in (4.1.5) satisfied by ϑπ on each interval [tk, tk+1). Now, since
v(T, x) = ϑπ(T, x, a), we deduce by comparison principle for this IPDE (see e.g. Theorem
3.4 in [3]) on [tn−1, T )×Rd×A that v(t, x) ≥ ϑπ(t, x, a) for all t ∈ [tn−1, T ], (x, a) ∈ Rd×A.
In particular, v(t−n−1, x) = v(tn−1, x) ≥ supa∈A ϑπ(tn−1, x, a) = ϑπ(t−n−1, x, a). Again, by
comparison principle for the IPDE (4.1.5) on [tn−2, tn−1) × Rd × A, it follows that v(t, x)
≥ ϑπ(t, x, a) for all t ∈ [tn−2, tn−1], (x, a) ∈ Rd × A. By backward induction on time, we
conclude that v ≥ ϑπ on [0, T ]× Rd ×A.
Let us next consider two partitions π = (tk)0≤k≤n and π′ = (t′k)0≤k≤n′ of [0, T ] with π
⊂ π′, and denote by m = maxk ≤ n′ : t′m /∈ π. Thus, all the points of the grid π and
π′ coincide after time t′m, and since ϑπ and ϑπ′
are viscosity solution to the same IPDE
(4.1.5) starting from the same terminal data g, we deduce by uniqueness that ϑπ = ϑπ′
on
[t′m, T ] × Rd × A. Then, we have ϑπ′(t′−m , x, a) = supa∈A ϑ
π(t′m, x, a) = supa∈A ϑπ(t′m, x, a)
≥ ϑπ(t−m, x, a) since ϑπ is continuous outside of the points of the grid π (recall Remark
4.1.1). Now, since ϑπ and ϑπ′
are viscosity solution to the same IPDE (4.1.5) on [t′m−1, tm),
we deduce by comparison principle that ϑπ′ ≥ ϑπ on [t′m−1, t
′m] × Rd × A. Proceeding by
backward induction, we conclude that ϑπ′ ≥ ϑπ on [0, T ]× Rd ×A.
2) Denote by π0 = t0 = 0, t1 = T the trivial grid of [0, T ]. Since ϑπ0 ≤ ϑπ ≤ v and ϑπ0
and v satisfy a linear growth condition, we get (recall that A is compact):
|ϑπ(t, x, a)| ≤ |ϑπ0(t, x, a)|+ |v(t, x)| ≤ C(1 + |x|),
for any (t, x, a) ∈ [0, T ]× Rd ×A and any grid π. 2
In the sequel, we denote by ϑ the increasing limit of the sequence (ϑπ)π when the grid
increases by becoming finer, i.e. its modulus |π| goes to zero. The next result shows that
ϑ does not depend on the variable a in A.
Proposition 4.2.4 The function ϑ is l.s.c. and does not depend on the variable a ∈ A:
ϑ(t, x, a) = ϑ(t, x, a′) , (t, x) ∈ [0, T ]× Rd, a, a′ ∈ A .
To prove this result we use the following lemma. Observe by definition (4.1.6) of vπ
that the function vπ does not depend on a on the grid times π, and we shall denote by
misuse of notation: vπ(tk, x), for k ≤ n, x ∈ Rd.
Lemma 4.2.3 There exists a constant C (not depending on π) such that
|ϑπ(t, x, a)− vπ(tk+1, x)| ≤ C(1 + |x|)|π|12
for all k = 0, . . . , n− 1, t ∈ [tk, tk+1), (x, a) ∈ Rd ×A.
47
Proof. Fix k = 0, . . . , n−1, t ∈ [tk, tk+1) and (x, a) ∈ Rd×A. Let (Y, Z, U) be the solution
to the BSDE
Ys = vπ(tk+1, Xt,x,atk+1
) +
∫ tk+1
sf(Xt,x,a
s , It,as , Ys, Zs)ds
−∫ tk+1
sZsdWs −
∫ tk+1
s
∫AUs(a′)µ(ds, da′) , s ∈ [t, tk+1] .
From Proposition 4.1.2, Markov property and uniqueness of a solution to the BSDE (4.1.1)-
(4.1.2)-(4.1.3) we have: Ys = ϑπ(s,Xt,x,as , It,as ), for s ∈ [t, tk+1], and so:
|ϑπ(t, x, a)− vπ(tk+1, x)| =∣∣Yt − vπ(tk+1, x)
∣∣≤ E|vπ(tk+1, X
t,x,atk+1
)− vπ(tk+1, x)|
+ E[ ∫ tk+1
t
∣∣f(Xt,x,as , It,as , Ys, Zs)
∣∣ds]. (4.2.20)
From Corollary 4.1.1, we have
E|vπ(tk+1, Xt,x,atk+1
)− vπ(tk+1, x)| ≤ C√
E[|Xt,x,atk+1− x|2] ≤ C
√|π| . (4.2.21)
Moreover, by the growth linear condition on f in (H2), and on ϑπ in Lemma 4.2.2, we
have
E[ ∫ tk+1
t
∣∣f(Xs, Is, Ys, Zs)∣∣ds] ≤ CE
[ ∫ tk+1
t
(1 + |Xt,x,a
s |+ |Zs|)ds].
By classical estimates, we have
sups∈[t,T ]
E[|Xt,x,a
s |2]≤ C(1 + |x|2).
Moreover, under (H1) and (H2), we know from Proposition 4.2 in [8] that there exists a
constant C depending only on the Lipschitz constants of b, σ f and vπ(tk+1, .) such that
E[
sups∈[tk,tk+1]
|Zs|2]≤ C(1 + |x|2).
This proves that
E[ ∫ tk+1
t
∣∣f(Xs, Is, Ys, Zs)∣∣ds] ≤ C(1 + |x|)|π| .
Combining this last estimate with (4.2.20) and (4.2.21), we get the result 2
Proof of Proposition 4.2.4. The function ϑ is l.s.c. as the supremum of the l.s.c.
functions ϑπ. Fix (t, x) ∈ [0, T )×Rd and a, a′ ∈ A. Let (πp)p be a sequence of subdivisions
of [0, T ] such that |πp| ↓ 0 as p ↑ ∞. We define the sequence (tp)p of [0, T ] by
tp = mins ∈ πp : s > t
, p ≥ 0 .
48
Since |πp| → 0 as p → ∞ we get tp → t as p → +∞. We then have from the previous
lemma:
|ϑπp(t, x, a)− ϑπp(t, x, a′)| ≤ |ϑπp(t, x, a)− vπp(tp, x)|+ |vπp(tp, x)− ϑπp(t, x, a′)|≤ 2C|πp|
12 .
Sending p to ∞ we obtain that ϑ(t, x, a) = ϑ(t, x, a′). 2
Corollary 4.2.2 We have the identification: ϑ = v, and the sequence (vπ)π also converges
to v.
Proof. We proceed in two steps.
Step 1. The function ϑ is a supersolution to (4.2.18). Since ϑπk(T, .) = g for all k ≥ 1, we
first notice that ϑ(T, .) = g. Next, since ϑ does not depend on the variable a, we have
ϑπ(t, x, a) ↑ ϑ(t, x) as |π| ↓ 0
for any (t, x, a) ∈ [0, T ]× Rd ×A. Moreover, since the function ϑ is l.s.c, we have
ϑ = ϑ∗ = lim inf|π|→0
∗ϑπ, (4.2.22)
where
lim inf|π|→0
∗ϑπ(t, x, a) := lim inf
|π| → 0
(t′, x′, a′)→ (t, x, a)
t′ < T
ϑπ(t′, x′, a′), (t, x, a) ∈ [0, T ]× Rd × Rq .
Fix now some (t, x) ∈ [0, T ] × Rd and a ∈ A and (p, q,M) ∈ J2,−ϑ(t, x), the limiting
parabolic subjet of ϑ at (t, x) (see definition in [12]). From standard stability results, there
exists a sequence (πk, tk, xk, ak, pk, qk,Mk)k such that
(pk, qk,Mk) ∈ J2,−ϑπk(tk, xk, ak)
for all k ≥ 1 and
(tk, xk, ak, ϑπk(tk, xk, ak)) −→ (t, x, a, ϑ(t, x, a)) as k →∞, |πk| → 0.
From the viscosity supersolution property of ϑπk to (4.1.5) in terms of subjets, we have
−pk − b(xk, ak).qk −1
2tr(σσᵀ(xk, ak)Mk)− f
(xk, ak, ϑ
πk(tk, xk, ak), σᵀ(xk, ak)qk)
−∫A
(ϑπk(tk, xk, a
′)− ϑπk(tk, xk, ak))λ(da′) ≥ 0
for all k ≥ 1. Sending k to infinity and using (4.2.22), we get
−p− b(x, a).q − 1
2tr(σσᵀ(x, a)M)− f
(x, a, ϑ(t, x), σᵀ(x, a)q) ≥ 0.
49
Since a is arbitrary in A, this shows
−p− supa∈A
[b(x, a).q +
1
2tr(σσᵀ(x, a)M) + f
(x, a, ϑ(t, x), σᵀ(x, a)q)
]≥ 0,
i.e. the viscosity supersolution property of ϑ to (4.2.18).
Step 2. Comparison. Since the PDE (4.2.18) satisfies a comparison principle, we have
from the previous step ϑ ≥ v, and we conclude with Lemma 4.2.2 that ϑ = v. Finally, by
definition (4.1.6) of vπ and from Lemma 4.2.2, we clearly have ϑπ ≤ vπ ≤ v, which also
proves that (vπ)π converges to v. 2
In terms of the discretely jump-constrained BSDE, the convergence result is formulated
as follows:
Proposition 4.2.5 We have Yπt ≤ Y πt ≤ Yt, 0 ≤ t ≤ T , and
E[
supt∈[0,T ]
|Yt − Yπt |2]
+ E[
supt∈[0,T ]
|Yt − Y πt |2]
+ E[ ∫ T
0|Zt −Zπt |2dt
]→ 0,
as |π| goes to zero.
Proof. Recall from (4.2.17) and (4.1.13) that we have the representation:
Yt = v(t,Xt), Y πt = vπ(t,Xt, It), Yπt = ϑ(t,Xt, It), (4.2.23)
and the first assertion of Lemma (4.2.2), we clearly have: Yπt ≤ Y πt ≤ Yt, 0 ≤ t ≤ T .
The convergence in S2 for Yπ to Y and Y π to Y comes from the above representation
(4.2.23), the pointwise convergence of ϑπ and vπ to v in Corollary 4.2.2, and the dominated
convergence theorem by recalling that 0 ≤ (v − vπ)(t, x, a) ≤ (v − ϑπ)(t, x, a) ≤ v(t, x) ≤C(1 + |x|). Let us now turn to the component Z. By definition (4.1.1)-(4.1.2)-(4.1.3) of the
discretely jump-constrained BSDE we notice that Yπ can be written on [0, T ] as:
Yπt = g(XT ) +
∫ T
tf(Xs, Is,Yπs ,Zπs )−
∫ T
tZπs dWs −
∫ T
t
∫AUπs (a)µ(ds, da) +KπT −Kπt ,
where Kπ is the nondecreasing process defined by: Kπt =∑
tk≤t(Yπtk− Yπtk), for t ∈ [0, T ].
Denote by δY = Y − Yπ, δZ = Z − Zπ, δU = U − Uπ and δK = K − Kπ. From Ito’s
formula, Young Inequality and (H2), there exists a constant C such that
E[|δYt|2
]+
1
2E[ ∫ T
t|δZs|2ds
]+
1
2E[ ∫ T
t|δUs(a)|2λ(da)ds
]≤ C
∫ T
tE[|δYs|2
]ds+
1
εE[
sups∈[0,T ]
|δYs|2]
+ εE[∣∣δKT − δKt
∣∣2] (4.2.24)
for all t ∈ [0, T ], with ε a constant to be chosen later. From the definition of δK we have
δKT − δKt = δYt −∫ T
t
(f(Xs, Is, Ys, Zs)− f(Xs, Is,Yπs ,Zπs )
)ds
+
∫ T
0δZsdWs +
∫ T
t
∫AδUs(a)µ(ds, da) .
50
Therefore, by (H2), we get the existence of a constant C ′ such that
E[∣∣δKT − δKt
∣∣2] ≤ C ′(E[
sups∈[0,T ]
|δYs|2]
+ E[ ∫ T
t|δZs|2ds
]+ E
[ ∫ T
t|δUs(a)|2λ(da)ds
])Taking ε = C′
4 and plugging this last inequality in (4.2.24), we get the existence of a
constant C ′′ such that
E[ ∫ T
t|δZs|2ds
]+ E
[ ∫ T
t|δUs(a)|2λ(da)ds
]≤ C ′′
(E[
sups∈[0,T ]
|δYs|2]), (4.2.25)
which shows the L2(W ) convergence of Zπ to Z from the S2 convergence of Yπ to Y . 2
4.2.2 Rate of convergence
We next provide an error estimate for the convergence of the discretely jump-constrained
BSDE. We shall combine BSDE methods and PDE arguments adapted from the shaking co-
efficients approach of Krylov [29] and switching systems approximation of Barles, Jacobsen
[5]. We make further assumptions:
(HFC’) The functions b and σ are uniformly bounded:
supx∈Rd,a∈A
|b(x, a)|+ |σ(x, a)| < ∞.
(HBC’) The function f does not depend on z: f(x, a, y, z) = f(x, a, y) for all (x, a, y, z) ∈Rd ×A× R× Rd and
(i) the functions f(., ., 0) and g are uniformly bounded:
supx∈Rd,a∈A
|f(x, a, 0)|+ |g(x)| < ∞,
(ii) for all (x, a) ∈ Rd ×A, y 7→ f(x, a, y) is convex.
Under these assumptions, we obtain the rate of convergence for vπ and ϑπ towards v.
Theorem 4.2.1 Under (HFC’) and (HBC’), there exists a constant C such that
0 ≤ v(t, x)− vπ(t, x, a) ≤ v(t, x)− ϑπ(t, x, a) ≤ C|π|110
for all (t, x, a) ∈ [0, T ]× Rd × A and all grid π with |π| ≤ 1. Moreover, when f(x, a) does
not depend on y, the rate of convergence is improved to |π|16 .
Before proving this result, we give as corollary the rate of convergence for the discretely
jump-constrained BSDE.
Corollary 4.2.3 Under (HFC’) and (HBC’), there exists a constant C such that
E[
supt∈[0,T ]
|Yt − Yπt |2]
+ E[
supt∈[0,T ]
|Yt − Y πt |2]
+ E[ ∫ T
0|Zt −Zπt |2dt
]≤ C|π|
15 .
for all grid π with |π| ≤ 1, and the above rate is improved to |π|13 when f(x, a) does not
depend on y.
51
Proof. From the representation (4.2.23), and Theorem 4.2.1, we immediately have
E[
supt∈[0,T ]
|Yt − Yπt |2]
+ E[
supt∈[0,T ]
|Yt − Y πt |2]≤ C|π|
15 , (4.2.26)
(resp. |π|13 when f(x, a) does not depend on y). Finally, the convergence rate for Z follows
from the inequality (4.2.25). 2
Remark 4.2.2 The above convergence rate |π|110 is the optimal rate that one can prove
in our generalized stochastic control context with fully nonlinear HJB equation by PDE
approach and shaking coefficients technique, see [29], [5], [17] or [42]. However, this rate
may not be the sharpest one. In the case of continuously reflected BSDEs, i.e. BSDEs
with upper or lower constraint on Y , it is known that Y can be approximated by discretely
reflected BSDEs, i.e. BSDEs where reflection on Y operates a finite set of times on the grid
π, with a rate |π|12 (see [1]). The standard arguments for proving this rate is based on the
representation of the continuously (resp. discretely) reflected BSDE as optimal stopping
problems where stopping is possible over the whole interval time (resp. only on the grid
times). In our jump-constrained case, we know from Chapter 3 that, when f(x, a) does
not depend on y and z, the minimal solution to the BSDE with nonpositive jumps has the
stochastic control representation
v(t, x) = supα
E[ ∫ T
tf(Xα
s , αs)ds+ g(XαT )∣∣∣Xα
t = x], (4.2.27)
with controlled diffusion in Rd:
dXαt = b(Xα
t , αt)dt+ σ(Xαt , αt)dWt,
and where α is an adapted control process valued in A. We shall prove an analog rep-
resentation for discretely jump-constrained BSDEs, and this helps to improve the rate of
convergence from |π|110 to |π|
16 . 2
The rest of this section is devoted to the proof of Theorem 4.2.1. We first consider the
special case where f(x, a) does not depend on y, and then address the case f(x, a, y).
Proof of Theorem 4.2.1 in the case f(x, a).
In the case where f(x, a) does not depend on y, z, by (linear) Feynman-Kac formula for ϑπ
solution to (4.1.5), and by definition of vπ in (4.1.6), we have:
vπ(tk, x) = supa∈A
E[ ∫ tk+1
tk
f(Xtk,x,at , Itk,at )dt+ vπ(tk+1, X
tk,x,atk+1
)], k ≤ n− 1, x ∈ Rd.
By induction, this dynamic programming relation leads to the following stochastic control
problem with discrete time policies:
vπ(tk, x) = supα∈AπF
E[ ∫ T
tk
f(Xtk,x,αt , Iαt )dt+ g(Xtk,x,α
T )],
52
where AπF is the set of discrete time processes α = (αtj )j≤n−1, with αtj Ftj -measurable,
valued in A, and
Xtk,x,αt = x+
∫ t
tk
b(Xt,x,αs , Iαs )ds+
∫ t
tk
σ(Xtk,x,αs , Iαs )dWs, tk ≤ t ≤ T,
Iαt = αtj +
∫(tj ,t]
∫A
(a− Iαs−)µ(ds, da), tj ≤ t < tj+1, j ≤ n− 1.
In other words, vπ(tk, x) corresponds to the value function for a stochastic control problem
where the controller can act only at the dates tj of the grid π, and then let the regime of
the coefficients of the diffusion evolve according to the Poisson random measure µ. Let us
introduce the following stochastic control problem with piece-wise constant control policies:
vπ(tk, x) = supα∈AπF
E[ ∫ T
tk
f(Xtk,x,αt , Iαt )dt+ g(Xtk,x,α
T )],
where for α = (αtj )j≤n−1 ∈ AπF:
Xtk,x,αt = x+
∫ t
tk
b(Xt,x,αs , Iαs )ds+
∫ t
tk
σ(Xtk,x,αs , Iαs )dWs, tk ≤ t ≤ T,
Iαt = αtj , tj ≤ t < tj+1, j ≤ n− 1.
It is shown in [28] that vπ approximates the value function v for the controlled diffusion
problem (4.2.27), solution to the HJB equation (4.2.18), with a rate |π|16 :
0 ≤ v(tk, x)− vπ(tk, x) ≤ C|π|16 , (4.2.28)
for all tk ∈ π, x ∈ Rd. Now, recalling that A is compact and λ(A) < ∞, it is clear that
there exists some positive constant C such that for all α ∈ AπF, j ≤ n− 1:
E[
supt∈[tj ,tj+1)
|Iαt − Iαt |2]≤ C|π|,
and then by standard arguments under Lipschitz condition on b, σ:
E[
supt∈[tj ,tj+1]
|Xtk,x,αt − Xtk,x,α
t |2]≤ C|π|, k ≤ j ≤ n− 1, x ∈ Rd.
By the Lipschitz conditions on f and g, it follows that
|vπ(tk, x)− vπ(tk, x)| ≤ C|π|12 ,
and thus with (4.2.28):
0 ≤ supx∈Rd
(v − vπ)(tk, x) ≤ C|π|16 .
Finally, by combining with the estimate in Lemma 4.2.3, which gives actually under (HBC’)(i):
|ϑπ(t, x, a)− vπ(tk+1, x)| ≤ C|π|12 , t ∈ [tk, tk+1), (x, a) ∈ Rd ×A,
53
together with the 1/2-Holder property of v in time (see (4.2.19)), we obtain:
sup(t,x,a)∈[0,T ]×Rd×A
(v − ϑπ)(t, x, a) ≤ C(|π|16 + |π|
12 ) ≤ C|π|
16 ,
for |π| ≤ 1. This ends the proof. 2
Let us now turn to the case where f(x, a, y) may also depend on y. We cannot rely
anymore on the convergence rate result in [28]. Instead, recalling that A is compact and
since σ, b and f are Lipschitz in (x, a), we shall apply the switching system method of
Barles and Jacobsen [5], which is a variation of the shaken coefficients method and smooth-
ing technique used in Krylov [29], in order to obtain approximate smooth subsolution to
(4.2.18). By Lemmas 3.3 and 3.4 in [5], one can find a family of smooth functions (wε)0<ε≤1
on [0, T ]× Rd such that:
sup[0,T ]×Rd
|wε| ≤ C, (4.2.29)
sup[0,T ]×Rd
|wε − w| ≤ Cε13 , (4.2.30)
sup[0,T ]×Rd
|∂β0t D
βwε| ≤ Cε1−2β0−∑di=1 β
i, β0 ∈ N, β = (β1, . . . , βd) ∈ Nd, (4.2.31)
for some positive constant C independent of ε, and by convexity of f in (HBC’)(ii), for
any ε ∈ (0, 1], (t, x) ∈ [0, T ]× Rd, there exists at,x,ε ∈ A such that:
− Lat,x,εwε(t, x)− f(x, at,x,ε, wε(t, x)) ≥ 0. (4.2.32)
Recalling the definition of the operator Tkπ in (4.1.7), we define for any function ϕ on
[0, T ]× Rd ×A, Lipschitz in (x, a):
Tπ[ϕ](t, x, a) := Tkπ[ϕ(tk+1, ., .)](t, x, a), t ∈ [tk, tk+1), (x, a) ∈ Rd ×A,
for k = 0, . . . , n− 1, and
Sπ[ϕ](t, x, a) :=1
|π|
[ϕ(t, x)− Tπ[ϕ](t, x, a)
+(tk+1 − t)(Laϕ(t, x) + f(x, a, ϕ(t, x)
)],
for (t, x, a) ∈ [tk, tk+1)× Rd ×A, k ≤ n− 1.
We have the following key error bound on §π.
Lemma 4.2.4 Let (HFC’) and (HBC’)(i) hold. There exists a constant C such that
|Sπ[ϕε](t, x, a)| ≤ C(|π|
12 (1 + ε−1) + |π|ε−3
), (t, x, a) ∈ [0, T ]× Rd ×A,
for any family (ϕε)ε of smooth functions on [0, T ]× Rd satisfying (4.2.29) and (4.2.31).
54
Proof. Fix (t, x, a) ∈ [0, T ]×Rd ×A. If t = T , we have |Sπ[ϕε](t, x, a)| = 0. Suppose that
t < T and fix k ≤ n such that t ∈ [tk, tk+1). Given a smooth function ϕε satisfying (4.2.29)
and (4.2.31), we split:
|Sπ[ϕε](t, x, a)| ≤ Aε(t, x, a) +Bε(t, x, a),
where
Aε(t, x, a) :=1
|π|
∣∣∣Tπ[ϕε](t, x, a)− E[ϕε(tk+1, X
t,x,atk+1
)]− (tk+1 − t)f(x, a, ϕε(t, x)
)∣∣∣ ,and
Bε(t, x, a) :=1
|π|
∣∣∣E[ϕε(tk+1, Xt,x,atk+1
)]− ϕε(t, x)− (tk+1 − t)Laϕε(t, x)
∣∣∣,and we study each term Aε and Bε separately.
1. Estimate on Aε(t, x, a).
Define (Y ϕε , Zϕε , Uϕε) as the solution to the BSDE on [t, tk+1]:
Y ϕεs = ϕε(tk+1, X
t,x,atk+1
) +
∫ tk+1
sf(Xt,x,a
r , It,ar , Y ϕεr )dr
−∫ tk+1
sZϕεr dWr −
∫ tk+1
s
∫AUϕεr (a)µ(dr, da) , s ∈ [t, tk+1]. (4.2.33)
From Theorems 3.4 and 3.5 in [3], we have Y ϕεt = Tπ[ϕε](t, x, a), and by taking expectation
in (4.2.33), we thus get:
Y ϕεt = Tπ[ϕε](t, x, a) = E
[ϕε(tk+1, X
t,x,atk+1
)]
+ E[ ∫ tk+1
tf(Xt,x,a
s , It,as , Y ϕεs )ds
]and so:
Aε(t, x, a) ≤ 1
|π|E[ ∫ tk+1
t
∣∣f(Xt,x,as , It,as , Y ϕε
s )− f(x, a, ϕε(t, x))∣∣ds]
≤ C(E[
sups∈[t,tk+1]
|Xt,x,as − x|+ |It,as − a|
]+ E
[sup
s∈[t,tk+1]|Y ϕεs − ϕε(t, x)|
]),
by the Lipschitz continuity of f . From standard estimate for SDE, we have (recall that the
coefficients b and σ are bounded under (HFC’) and A is compact):
E[
sups∈[t,tk+1]
|Xt,x,as − x|+ |It,as − a|
]≤ C|π|
12 . (4.2.34)
Moreover, by (4.2.33), the boundedness condition in (HBC’)(i) together with the Lipschitz
condition of f , and Burkholder-Davis-Gundy inequality, we have:
E[
sups∈[t,tk+1]
|Y ϕεs − ϕε(t, x)|
]≤ E
[|ϕε(tk+1, X
t,x,atk+1
)− ϕε(t, x)|]
+ C|π|E[1 + sup
s∈[t,tk+1]|Y ϕεs |]
+ C|π|(E[
sups∈[t,tk+1]
|Zϕεs |2] + E[
sups∈[t,tk+1]
∫A|Uϕεs (a)|2λ(da)]
).
55
From standard estimate for the BSDE (4.2.33), we have:
E[
sups∈[t,tk+1]
|Y ϕεs |2
]≤ C,
for some positive constant C depending only on the Lipschitz constant of f , the upper bound
of |f(x, a, 0, 0)| in (HBC’)(i), and the upper bound of |ϕε| in (4.2.29). Moreover, from the
estimate in Proposition 4.2 in [8] about the coefficients Zϕε and Uϕε of the BSDE with
jumps (4.2.33), there exists some constant C depending only on the Lipschitz constant of
b, σ, f , and of the Lipschitz constant of ϕε(tk+1, .) (which does not depend on ε by (4.2.31)),
such that:
E[
sups∈[t,tk+1]
|Zϕεs |2] + E[
sups∈[t,tk+1]
∫A|Uϕεs (a)|2λ(da)] ≤ C.
From (4.2.31), we then have:
E[
sups∈[t,tk+1]
|Y ϕεs − ϕε(t, x)|
]≤ C
(|tk+1 − t|ε−1 + E
[|Xt,x,a
tk+1− x|
]+ |π|
)≤ C|π|
12(1 + ε−1
),
by (4.2.34). This leads to the error bound for Aε(t, x, a):
Aε(t, x, a) ≤ C|π|12 (1 + ε−1).
2. Estimate on Bε(t, x, a).
From Ito’s formula we have
Bε(t, x, a) =1
|π|
∣∣∣E[ ∫ tk+1
t
(LI
t,as ϕε(s,X
t,x,as )− Laϕε(t, x)
)ds]∣∣∣
≤ B1ε (t, x, a) +B2
ε (t, x, a)
where
B1ε (t, x, a) =
1
|π|E[ ∫ tk+1
t
∣∣∣(b(Xt,x,as , It,as
)− b(x, a)).Dxϕε(s,X
t,x,as )
+1
2tr[(σσᵀ(Xt,x,a
s , It,as )− σσᵀ(x, a))D2xϕε(t, x)
]∣∣∣ds]and
B2ε (t, x, a) =
1
|π|E[ ∫ tk+1
t
∣∣Lat,xϕε(s,Xt,x,as )− Lat,xϕε(t, x)
∣∣ds] ,with Lat,x defined by
Lat,xϕε(t′, x′) =∂ϕε∂t
(t′, x′) + b(x, a).Dxϕε(t′, x′) +
1
2tr(σσᵀ(x, a)D2
xϕε(t′, x′)
).
Under (HFC), (HFC’), and by (4.2.31), we have
B1ε (t, x, a) ≤ C(1 + ε−1)E
[sup
s∈[t,tk+1]|Xt,x,a
s − x|+ |It,as − a|]
≤ C(1 + ε−1)|π|12 ,
56
where we used again (4.2.34). On the other hand, since ϕε is smooth, we have from Ito’s
formula
B2ε (t, x, a) =
1
|π|E[ ∫ tk+1
t
∣∣∣ ∫ s
tLI
t,ar Lat,xφ(r,Xt,x,a
r )dr∣∣∣ds] .
Under (HFC’) and by (4.2.31), we then see that
B2ε (t, x, a) ≤ C|π|ε−3,
and so:
Bε(t, x, a) ≤ C(|π|
12 (1 + ε−1) + |π|ε−3
).
Together with the estimate for Aε(t, x, a), this proves the error bound for |Sπ[ϕε](t, x, a)|.2
We next state a maximum principle type result for the operator Tπ.
Lemma 4.2.5 Let ϕ and ψ be two functions on [0, T ]×Rd×A, Lipschitz in (x, a). Then,
there exists some positive constant C independent of π such that
sup(t,x,a)∈[tk,tk+1]×Rd×A
(Tπ[ϕ]− Tπ[ψ])(t, x, a) ≤ eC|π| sup(x,a)∈Rd×A
(ϕ− ψ)(tk+1, x, a) ,
for all k = 0, . . . , n− 1.
Proof. Fix k ≤ n− 1, and set
M := sup(x,a)∈Rd×A
(ϕ− ψ)(tk+1, x, a).
We can assume w.l.o.g. that M < ∞ since otherwise the required inequality is trivial. Let
us denote by ∆v the function
∆v(t, x, a) = Tπ[ϕ](t, x, a)− Tπ[ψ](t, x, a),
for all (t, x, a) ∈ [tk, tk+1]× Rd × A. By definition of Tπ, and from the Lipschitz condition
of f , we see that ∆v is a viscosity subsolution to−La∆v(t, x, a)− C
(|∆v(t, x, a)|+ |D∆v(t, x, a)|
)−∫A
(∆v(t, x, a′)−∆v(t, x, a)
)λ(da′) = 0, for (t, x, a) ∈ [tk, tk+1)× Rd ×A,
∆v(tk+1, x, a) ≤M , for (x, a) ∈ Rd ×A .
(4.2.35)
Then, we easily check that the function Φ defined by
Φ(t, x, a) = MeC(tk+1−t) , (t, x, a) ∈ [tk, tk+1]× Rd ×A ,
is a solution to−LaΦ(t, x, a)− C
(|Φ(t, x, a)|+ |DΦ(t, x, a)|
)−∫A
(Φ(t, x, a′)− Φ(t, x, a)
)λ(da′) = 0, for (t, x, a) ∈ [tk, tk+1)× Rd ×A,
Φ(tk+1, x, a) = M , for (x, a) ∈ Rd ×A .
(4.2.36)
57
From the comparison theorem in [3] for viscosity solutions of semi-linear IPDEs, we get
that ∆v ≤ Φ on [tk, tk+1]× Rd ×A, which proves the required inequality. 2
Proof of Theorem 4.2.1. By (4.1.6) and (4.1.12), we observe that vπ is a fixed point of
Tπ, i.e.
Tπ[vπ] = vπ.
On the other hand, by (4.2.32), and the estimate of Lemma 4.2.4 applied to wε, we have:
wε(t, x)− Tπ[wε](t, x, at,x,ε) ≤ |π|§π[wε](t, x, at,x,ε) ≤ C|π|S(π, ε)
where we set: S(π, ε) = (|π|32 (1 + ε−1) + |π|2ε−3). Fix k ≤ n−1. By Lemma 4.2.5, we then
have for all t ∈ [tk, tk+1], x ∈ Rd:
wε(t, x)− vπ(t, x, at,x,ε) = wε(t, x)− Tπ[wε](t, x, at,x,ε) + (Tπ[wε]− Tπ[vπ])(t, x, at,x,ε)
≤ C|π|S(π, ε) + eC|π| sup(x,a)∈Rd×A
(wε − vπ)(tk+1, x, a). (4.2.37)
Recalling by its very definition that vπ does not depend on a ∈ A on the grid times of π,
and denoting then Mk := supx∈Rd(wε − vπ)(tk, x), we have by (4.2.37) the relation:
Mk ≤ C|π|S(π, ε) + eC|π|Mk+1.
By induction, this yields:
supx∈Rd
(wε − vπ)(tk, x) ≤ CeCn|π| − 1
eC|π| − 1|π|S(π, ε) + eCn|π| sup
x∈Rd(wε − vπ)(T, x)
≤ CS(π, ε) + C supx∈Rd
(wε − v)(T, x),
since n|π| is bounded and v(T, x) = vπ(T, x) (= g(x)). From (4.2.30), we then get:
supx∈Rd
(v − vπ)(tk, x) ≤ C(ε
13 + |π|
12 (1 + ε−1) + |π|ε−3
).
By minimizing the r.h.s of this estimate with respect to ε, this leads to the error bound
when taking ε = |π|310 ≤ 1:
supx∈Rd
(v − vπ)(tk, x) ≤ C|π|110 .
Finally, by combining with the estimate in Lemma 4.2.3, which gives actually under (HBC’)(i):
|ϑπ(t, x, a)− vπ(tk+1, x)| ≤ C|π|12 , t ∈ [tk, tk+1), (x, a) ∈ Rd ×A,
together with the 1/2-Holder property of v in time (see (4.2.19)), we obtain:
sup(t,x,a)∈[0,T ]×Rd×A
(v − ϑπ)(t, x, a) ≤ C(|π|110 + |π|
12 ) ≤ C|π|
110 .
This ends the proof. 2
58
Chapter 5
Approximation scheme for
jump-constrained BSDE and
stochastic control problem
In this chapter, we study the discrete time approximation of the solution to the discretely
constrained BSDE. We fist deal with the approximation of the forward process. We then
provide a discrete-time scheme for the discretely constrained BSDE. We finally show that
the scheme converges to the constrained solution as soon as the discrete-time mesh goes to
zero.
5.1 The forward regime switching process
In this section, we consider the discrete time approximation of the forward process (X, I)
on [0, T ] . Recall that it is defined byXt = X0 +
∫ t0 b(Xs, Is)ds+
∫ t0 σ(Xs, Is)dWs
It = I0 +∫
(0,t]
∫A(a− Is−)µ(ds, da),
(5.1.1)
for all t ∈ [0, T ].
In the sequel, we shall denote by C1 a generic positive constant which depends only on
the Lipschitz constant of b and σ, T , (X0, I0) and λ(A) < ∞, and may vary from lines to
lines. Under (HFC), we have the existence and uniqueness of a solution to (5.1.1), and
in the sequel, we shall denote by (Xt,x,a, It,a) the solution to (5.1.1) starting from (x, a) at
time t.
Remark 5.1.3 We do not make any ellipticity assumption on σ. In particular, some lines
and columns of σ may be equal to zero, and so there is no loss of generality by considering
that the dimension d of X and W are equal. 2
Denoting by (Tn, ιn)n the jump times and marks associated to µ, we observe that I is
59
explicitly written as:
It = I01[0,T1)(t) +∑n≥1
ιn1[Tn,Tn+1)(t), 0 ≤ t ≤ T,
where the jump times (Tn)n evolve according to a Poisson distribution of parameter λ :=∫A λ(da) <∞, and the i.i.d. marks (ιn)n follow a probability distribution λ(da) := λ(da)/λ.
Assuming that one can simulate the probability distribution λ, we then see that the pure
jump process I is perfectly simulated. Given a partition π = t0 = 0 < . . . < tk < . . . tn =
T of [0, T ], we shall use the natural Euler scheme Xπ for X, defined by:
Xπ0 = X0
Xπtk+1
= Xπtk
+ b(Xπtk, Itk)(tk+1 − tk) + σ(Xπ
tk, Itk)(Wtk+1
−Wtk),
for k = 0, . . . , n−1. We denote as usual by |π| = maxk≤n−1(tk+1−tk) the modulus of π, and
assume that n|π| is bounded by a constant independent of n, which holds for instance when
the grid is regular, i.e. (tk+1−tk) = |π| for all k ≤ n−1. We also define the continuous-time
version of Xπ by setting:
Xπt = Xπ
tk+ b(Xπ
tk, Itk)(t− tk) + σ(Xπ
tk, Itk)(Wt −Wtk), t ∈ [tk, tk+1], k < n.
By standard arguments, see e.g. [27], one can obtain under (HFC) the L2-error estimate
for the above Euler scheme:
E[
supt∈[tk,tk+1]
∣∣Xt − Xπtk
∣∣2] ≤ C1|π|, k < n.
For our purpose, we shall need a stronger result, and introduce the following error control
for the Euler scheme:
Eπk (X) := E[ess supa∈A
Et1,a[. . . ess sup
a∈AEtk,a
[sup
t∈[tk,tk+1]|Xt − Xπ
tk|2]. . .]], (5.1.2)
where Etk,a[.] denotes the conditional expectation E[.|Ftk , Itk = a]. We also denote by Etk [.]
the conditional expectation E[.|Ftk ]. Since Itk is Ftk -measurable, and by the law of iterated
conditional expectations, we notice that
E[
supt∈[tk,tk+1]
∣∣Xt − Xπtk
∣∣2] ≤ Eπk (X), k < n.
Lemma 5.1.6 Under (HFC), we have
maxk<nEπk (X) ≤ C1|π|.
Proof. From the definition of the Euler scheme, and under the growth linear condition in
(HFC), we easily see that
Etk[∣∣Xπ
tk+1
∣∣2] ≤ C1
(1 +
∣∣Xπtk
∣∣2), k < n. (5.1.3)
60
From the definition of the continuous-time Euler scheme, and by Burkholder-Davis-Gundy
inequality, it is also clear that
Etk[
supt∈[tk,tk+1]
∣∣Xπt − Xπ
tk
∣∣2] ≤ C1(1 +∣∣Xπ
tk
∣∣2)|π|, k < n. (5.1.4)
We also have the standard estimate for the pure jump process I (recall that A is assumed
to be compact and λ(A) < ∞):
Etk[
supt∈[tk,tk+1]
∣∣Is − Itk ∣∣2] ≤ C1|π|. (5.1.5)
Let us denote by ∆Xt = Xt − Xπt , and apply Ito’s formula to |∆Xt|2 so that for all t ∈
[tk, tk+1]:
|∆Xt|2 = |∆Xtk |2 +
∫ t
tk
2(b(Xs, Is)− b(Xπ
tk, Itk)
).∆Xs +
∣∣σ(Xs, Is)− σ(Xπtk, Itk)
∣∣2ds+ 2
∫ t
tk
(∆Xs)′(σ(Xs, Is)− σ(Xπ
tk, Itk)
)dWs
≤ |∆Xtk |2 + C1
∫ t
tk
|∆Xs|2 + |Xπs − Xπ
tk|2 + |Is − Itk |
2ds
+ 2
∫ t
tk
(∆Xs)′(σ(Xs, Is)− σ(Xπ
tk, Itk)
)dWs,
from the Lipschitz condition on b, σ in (HFC). By taking conditional expectation in the
above inequality, we then get:
Etk[|∆Xt|2
]≤ |∆Xtk |
2 + C1
∫ t
tk
Etk[|∆Xs|2 + |Xπ
s − Xπtk|2 + |Is − Itk |
2]ds
≤ |∆Xtk |2 + C1(1 +
∣∣Xπtk
∣∣2)|π|2 + C1
∫ t
tk
Etk[|∆Xs|2
]ds, t ∈ [tk, tk+1],
by (5.1.4)-(5.1.5). From Gronwall’s lemma, we thus deduce that
Etk[|∆Xtk+1
|2]≤ eC1|π||∆Xtk |
2 + C1(1 +∣∣Xπ
tk
∣∣2)|π|2, k < n. (5.1.6)
Since the right hand side of (5.1.6) does not depend on Itk , this shows that
ess supa∈A
Etk,a[|∆Xtk+1
|2]≤ eC1|π||∆Xtk |
2 + C1(1 +∣∣Xπ
tk
∣∣2)|π|2.By taking conditional expectation w.r.t. Ftk−1
in the above inequality, using again estimate
(5.1.6) together with (5.1.3) at step k − 1, and iterating this backward procedure until the
initial time t0 = 0, we obtain:
E[ess supa∈A
Et1,a[. . . ess sup
a∈AEtk,a
[|∆Xtk+1
|2]. . .]]
≤ eC1n|π||∆X0|2 + C1(1 + |X0|2)|π|2 eC1n|π| − 1
eC1|π| − 1≤ C1|π|, (5.1.7)
61
since ∆X0 = 0 and n|π| is bounded.
Moreover, the process X satisfies the standard conditional estimate similarly as for the
Euler scheme:
Etk[∣∣Xtk+1
∣∣2] ≤ C1
(1 +
∣∣Xtk
∣∣2),Etk[
supt∈[tk,tk+1]
∣∣Xt −Xtk
∣∣2] ≤ C1(1 +∣∣Xtk
∣∣2)|π|, k < n,
from which we deduce by backward induction on the conditional expectations:
E[ess supa∈A
Et1,a[. . . ess sup
a∈AEtk,a
[sup
t∈[tk,tk+1]
∣∣Xt −Xtk
∣∣2] . . . ]] ≤ C1|π|. (5.1.8)
Finally, by writing that supt∈[tk,tk+1] |Xt − Xπtk|2 ≤ 2 supt∈[tk,tk+1] |Xt − Xtk |2 + 2∆Xtk ,
taking successive condition expectations w.r.t to Ft` and essential supremum over It` = a,
for ` going recursively from k to 0, we get:
Etk[
supt∈[tk,tk+1]
|Xt − Xπtk|2]≤ 2E
[ess supa∈A
Et1,a[. . . ess sup
a∈AEtk,a
[sup
t∈[tk,tk+1]
∣∣Xt −Xtk
∣∣2] . . . ]]+ 2E
[ess supa∈A
Et1,a[. . . ess sup
a∈AEtk−1,a
[|∆Xtk |
2]. . .]]
≤ C1|π|,
by (5.1.7)-(5.1.8), which ends the proof. 2
5.2 BSDE scheme
We consider the discrete time approximation of the discretely jump-constrained BSDE in
the case where f(x, a, y) does not depend on z, and define the scheme (Y π, Yπ, Zπ) by
induction on the grid π = t0 = 0 < . . . < tk < . . . < tn = T by:Y πT = YπT = g(Xπ
T )
Yπtk = Etk[Y πtk+1
]+ f(Xπ
tk, Itk , Yπtk)∆tk
Y πtk
= ess supa∈A
Etk,a[Yπtk], k = 0, . . . , n− 1,
(5.2.9)
where ∆tk = tk+1 − tk, Etk [.] stands for E[.|Ftk ], and Etk,a[.] for E[.|Ftk , Itk = a].
By induction argument, we easily see that Yπtk is a deterministic function of (Xπtk, Itk),
while Y πtk
is a deterministic function of Xπtk
, for k = 0, . . . , n, and by the Markov pro-
perty of the process (Xπ, I), the conditional expectations in (5.2.9) can be replaced by the
corresponding regressions:
Etk[Y πtk+1
]= E
[Y πtk+1
∣∣Xπtk, Itk
]and Etk,a
[Yπtk ] = E
[Yπtk∣∣Xπ
tk, Itk = a
].
We then have:
Yπtk = ϑπk(Xπtk, Itk), Y π
tk= vπk (Xπ
tk),
62
for some sequence of functions (ϑπk)k and (vπk )k defined respectively on Rd × A and Rd by
backward induction:vπn(x, a) = ϑπn(x) = g(x)
ϑπk(x, a) = E[vπk+1(Xπ
tk+1, Itk+1
)∣∣(Xπ
tk, Itk) = (x, a)
]+ f(x, a, ϑπk(x, a))∆tk
vπk (x) = supa∈A ϑπk(x, a), k = 0, . . . , n− 1.
(5.2.10)
There are well-known different methods (Longstaff-Schwartz least square regression, quan-
tization, Malliavin integration by parts, see e.g. [1], [21], [9]) for computing the above
conditional expectations, and so the functions ϑπk and vπk . It appears that in our context,
the simulation-regression method on basis functions defined on Rd × A, is quite suitable
since it allows us to derive at each time step k ≤ n − 1, a functional form ak(x), which
attains the supremum over A in ϑπk(x, a). We shall see later in this section that the feedback
control (ak(x))k provides an approximation of the optimal control for the HJB equation
associated to a stochastic control problem when f(x, a) does not depend on y. We refer to
our companion paper [?] for the details about the computation of functions ϑπk , vπk , ak by
simulation-regression methods, and the associated error analysis.
5.3 Error estimate for the discrete time scheme
The main result of this section is to state an error bound between the component Y π of
the discretely jump-constrained BSDE and the solution (Y π, Yπ) to the above discrete time
scheme.
Theorem 5.3.2 There exists some constant C such that:
E[∣∣Y π
tk− Y π
tk
∣∣2]+ supt∈(tk,tk+1]
E[∣∣Y π
t − Y πtk+1
∣∣2]+ supt∈[tk,tk+1)
E[∣∣Y π
t − Yπtk∣∣2] ≤ C|π|,
for all k = 0, . . . , n− 1.
The above convergence rate |π|12 in the L2−norm for the discretization of the discretely
jump-constrained BSDE is the same as for standard BSDE, see [9], [44]. By combining
with the convergence result in Section 4, we finally obtain an estimate on the error due to
the discrete time approximation of the minimal solution Y to the BSDE with nonpositive
jumps. We split the error between the positive and negative parts:
Errπ+(Y ) := maxk≤n−1
(E[(Ytk − Y
πtk
)2+
]+ supt∈(tk,tk+1]
E[(Yt − Y π
tk+1
)2+
]+ supt∈[tk,tk+1)
E[(Yt − Yπtk
)2+
]) 12
Errπ−(Y ) := maxk≤n−1
(E[(Ytk − Y
πtk
)2−
]+ supt∈(tk,tk+1]
E[(Yt − Y π
tk+1
)2−
]+ supt∈[tk,tk+1)
E[(Yt − Yπtk
)2−
]) 12.
Corollary 5.3.4 We have:
Errπ−(Y ) ≤ C|π|12 .
63
Moreover, under (HFC’) and (HBC’),
Errπ+(Y ) ≤ C|π|110 ,
and when f(x, a) does not depend on y:
Errπ+(Y ) ≤ C|π|16 .
Proof. Recall from Proposition 4.2.5 that Yπt ≤ Y πt ≤ Yt, 0 ≤ t ≤ T . Then, we have:
(Ytk − Y πtk
)− ≤ |Y πtk− Y π
tk|, (Yt − Y π
tk+1)− ≤ |Y π
t − Y πtk+1|, and (Ytk − Yπtk)− ≤ |Y π
tk− Yπtk |, for
all k ≤ n− 1, and t ∈ [0, T ]. The error bound on Errπ−(Y ) follows then from the estimation
in Theorem 5.3.2. The error bound on Errπ−(Y ) follows from Corollary 4.2.3 and Theorem
5.3.2. 2
Remark 5.3.4 In the particular case where f depends only on (x, a), our discrete time
approximation scheme is a probabilistic scheme for the fully nonlinear HJB equation as-
sociated to the stochastic control problem (4.2.27). As in [29], [5] or [17], we have non
symmetric bounds on the rate of convergence. For instance, in [17], the authors obtained
a convergence rate |π|14 on one side and |π|
110 on the other side, while we improve the rate
to |π|12 for one side, and |π|
16 on the other side. This induces a global error Errπ(Y ) =
Errπ+(Y ) + Errπ−(Y ) of order |π|16 , which is derived without any non degeneracy condition
on the controlled diffusion coefficient. 2
Proof of Theorem 5.3.2.
Let us introduce the continuous time version of (5.2.9). By the martingale representation
theorem, there exists Zπ ∈ L2(W ) and Uπ ∈ L2(µ) such that
Y πtk+1
= Etk[Y πtk+1
]+
∫ tk+1
tk
Zπt dWt +
∫ tk+1
tk
∫AUπt (a)µ(dt, da), k < n,
and we can then define the continuous-time processes Y π and Yπ by:
Yπt = Y πtk+1
+ (tk+1 − t)f(Xπtk, Itk , Y
πtk
) (5.3.11)
−∫ tk+1
tZπt dWt −
∫ tk+1
t
∫AUπt (a)µ(dt, da), t ∈ [tk, tk+1),
Y πt = Y π
tk+1+ (tk+1 − t)f(Xπ
tk, Itk , Y
πtk
) (5.3.12)
−∫ tk+1
tZπt dWt −
∫ tk+1
t
∫AUπt (a)µ(dt, da), t ∈ (tk, tk+1],
for k = 0, . . . , n− 1. Denote by δY πt = Y π
t − Y πt , δYπt = Yπt − Yπt , δZπt = Zπt − Zπt , δUπt =
Uπt − Uπt and δft = f(Xt, It,Yπt ) − f(Xπtk, Itk , Yπtk) for t ∈ [tk, tk+1). Recalling (4.1.2) and
(5.3.11), we have by Ito’s formula:
∆t := Etk[|δYπt |2 +
∫ tk+1
t|δZπs |2ds+
∫ tk+1
t
∫A|δUπs (a)|2λ(da)ds
]= Etk
[|δY π
tk+1|2∣∣]+ Etk
[ ∫ tk+1
t2δYπs δfs
]ds
64
for all t ∈ [tk, tk+1). By the Lipschitz continuity of f in (H2) and Young inequality, we
then have:
∆t ≤ Etk[|δY π
tk+1|2∣∣]+ Etk
[ ∫ tk+1
tη|δYπs |2ds+
C
ηπ|δYπtk |
2]
+C
ηEtk[ ∫ tk+1
t
(|Xs − Xπ
tk|2 + |Is − Itk |
2 + |Yπs − Yπtk |2)ds].
From Gronwall’s lemma, and by taking η large enough, this yields for all k ≤ n− 1:
Etk[|δYπtk |
2]≤ eC|π|Etk
[|δY π
tk+1|2∣∣]+ CBk (5.3.13)
where
Bk = Etk[ ∫ tk+1
tk
(|Xs − Xπ
tk|2 + |Is − Itk |
2 + |Yπs − Yπtk |2)ds]
≤ C|π|(Etk[
sups∈[tk,tk+1]
|Xs − Xπtk|2]
+ |π|(1 + |Xtk |)), (5.3.14)
by (5.1.5) and Proposition 4.1.3. Now, by definition of Y πtk+1
and Y πtk+1
, we have
|δY πtk+1|2 ≤ ess sup
a∈AEtk+1,a
[|δYπtk+1
|2]. (5.3.15)
By plugging (5.3.14), (5.3.15) into (5.3.13), taking conditional expectation with respect to
Itk = a, and taking essential supremum over a, we obtain:
ess supa∈A
Etk,a[|δYπtk |
2]≤ eC|π|ess sup
a∈AEtk,a
[ess supa∈A
Etk+1,a
[|δYπtk+1
|2]
+ C|π|(
ess supa∈A
Etk,a[
sups∈[tk,tk+1]
|Xs − Xπtk|2]
+ |π|(1 + |Xtk |)).
By taking conditional expectation with respect to Ftk−1, and Itk−1
= a, taking essential
supremum over a in the above inequality, and iterating this backward procedure until time
t0 = 0, we obtain:
Eπk (Y) ≤ eC|π|Eπk+1(Y) + C|π|(Eπk (X) + |π|(1 + E[|Xtk |])
)≤ eC|π|Eπk+1(Y) + C|π|2, k ≤ n− 1, (5.3.16)
where we recall the auxiliary error control Eπk (X) on X in (5.1.2) and its estimate in Lemma
5.1.6, and set:
Eπk (Y) := E[ess supa∈A
Et1,a[. . . ess sup
a∈AEtk,a
[|δYπtk |
2]. . .]].
By a direct induction on (5.3.16), and recalling that n|π| is bounded, we get
Eπk (Y) ≤ C(Eπn (Y) + |π|
)≤ C(Eπn (X) + |π|
)≤ C|π|,
65
since g is Lipschitz, and using again the estimate in Lemma 5.1.6. Observing that E[|δY πtk|2],
E[|δYπtk |2] ≤ Eπk (Y), we get the estimate:
maxk≤n
E[|Y πtk− Y π
tk|2]
+ E[|Yπtk − Y
πtk|2]≤ C|π|.
Moreover, by Proposition 4.1.3, we have
supt∈[tk,tk+1)
E[|Yπt − Yπtk |
2]
+ supt∈(tk,tk+1]
E[|Y πt − Y π
tk+1|2]≤ C(1 + E[|Xtk |])|π|
≤ C(1 + |X0|)|π|.
This implies finally that:
sups∈(tk,tk+1]
E[|Y πt − Y π
tk+1|2]≤ 2 sup
s∈(tk,tk+1]E[|Y πt − Y π
tk+1|2]
+ 2E[|Y πtk+1− Y π
tk+1|2]
≤ C|π|,
as well as
sups∈[tk,tk+1)
E[|Y πt − Yπtk |
2]≤ 2 sup
s∈[tk,tk+1)E[|Y πt − Yπtk |
2]
+ 2E[|Yπtk − Y
πtk|2]
≤ C|π|.
2
5.4 Approximate optimal control
We now consider the special case where f(x, a) does not depend on y, so that the discrete
time scheme (1.0.15) is an approximation for the value function of the stochastic control
problem:
V0 := supα∈A
J(α) = Y0, (5.4.17)
J(α) = E[ ∫ T
0f(Xα
t , αt)dt+ g(XαT )],
where A is the set of G-adapted control processes α valued in A, and Xα is the controlled
diffusion in Rd:
Xαt = X0 +
∫ t
0b(Xα
s , αs)ds+
∫ t
0σ(Xα
s , αs)dWs, 0 ≤ t ≤ T.
(Here G = (Gt)0≤t≤T denotes some filtration under which W is a standard Brownian mo-
tion). Let us now define the discrete time version of (5.4.17). We introduce the set Aπ of
discrete time processes α = (αtk)k with αtk Gtk -measurable, and valued in A. For each α
∈ Aπ, we consider the controlled discrete time process (Xπ,αtk
)k of Euler type defined by:
Xπ,αtk
= X0 +
k−1∑j=0
b(Xπ,αtj
, αtj )∆tj +
k−1∑j=0
σ(Xπ,αtj
, αtj )∆Wtj , k ≤ n,
66
where ∆Wtj = Wtj+1 −Wtj , and the gain functional:
Jπ(α) = E[ n−1∑k=0
f(Xπ,αtk
, αtk)∆tk + g(Xπ,αtn )
].
Given any α ∈ Aπ, we define its continuous time piecewise-constant interpolation α ∈ Aby setting: αt = αtk , for t ∈ [tk, tk+1) (by misuse of notation, we keep the same notation
α for the discrete time and continuous time interpolation). By standard arguments similar
to those for Euler scheme of SDE, there exists some positive constant C such that for all α
∈ Aπ, k ≤ n− 1:
E[
supt∈[tk,tk+1]
∣∣Xαt −X
π,αtk
∣∣2] ≤ C|π|,
from which we easily deduce by Lipschitz property of f and g:∣∣J(α)− Jπ(α)| ≤ C|π|12 , ∀α ∈ Aπ. (5.4.18)
Let us now consider at each time step k ≤ n− 1, the function ak(x) which attains the
supremum over a ∈ A of ϑπk(x, a) in the scheme (5.2.10), so that:
vπk (x) = ϑπk(x, ak(x)
), k = 0, . . . , n− 1.
Let us define the process (Xπtk
)k by: Xπ0 = X0,
Xπtk+1
= Xπtk
+ b(Xπtk, ak(X
πtk
))∆tk + σ(Xπtk, ak(X
πtk
))∆Wtk , k ≤ n− 1,
and notice that Xπ = Xπ,α, where α ∈ Aπ is a feedback control defined by:
αtk = ak(Xπtk
) = ak(Xπ,αtk
), k = 0, . . . , n.
Next, we observe that the conditional law of Xπtk+1
given (Xπtk
= x, Itk = ak(Xπtk
) = ak(x))
is the same than the conditional law of Xπ,αtk+1
given Xπ,αtk
= x, for k ≤ n− 1, and thus the
induction step in the scheme (5.2.9) or (5.2.10) reads as:
vπk (Xπ,αtk
) = E[vπk+1(Xπ,α
tk+1)∣∣Xπ,α
tk
]+ f(Xπ,α
tk, αtk)∆tk, k ≤ n− 1.
By induction, and law of iterated conditional expectations, we then get:
Y π0 = vπ0 (X0) = Jπ(α). (5.4.19)
Consider the continuous time piecewise-constant interpolation α ∈ A defined by: αt = αtk ,
for t ∈ [tk, tk+1). By (5.4.17), (5.4.18), (5.4.19), and Corollary 5.3.4, we finally obtain:
0 ≤ V0 − J(α) = Y0 − Y π0 + Jπ(α)− J(α)
≤ C|π|16 + C|π|
12 ≤ C|π|
16 ,
for |π| ≤ 1. In other words, for any small ε > 0, we obtain an ε-approximate optimal control
α for the stochastic control problem (5.4.17) by taking |π| of order ε6.
67
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