Approximation rate of BSDEs using random...

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Université Savoie Mont Blanc

Approximation rate of BSDEs using randomwalk

Céline Labart, Rennes, 2018-07-06

Joint work with Christel Geiss and Antti Luoto (University ofJÿvaskÿla, Finland)

Introduction Numerical scheme and convergence results Numerical Examples Generalization to a diffusion process

Outline

Introduction

Numerical scheme and convergence results

Numerical Examples

Generalization to a diffusion process

Introduction

Numerical Examples

Random walk approximation of the Brownian motion

Let tk := kh, k = 0, · · · ,n be a regular grid of [0,T ], where h = Tn and

define

Bnt :=√

h[t/h]∑k=1

εk , (εk )k=1,··· ,n i.i.d. Bernoulli r.v.:

P(εk = 1) = P(εk = −1) =12

• Donsker’s Theorem:

(Bnt )t∈[0,T ] → (Bt )t∈[0,T ] in distribution.

• Wanted:

Bnt → Bt in L2 for all t ∈ [0,T ] with a convergence rate.

Random walk constructed by Skorohod embedding

τ0 = 0, τk := inf{t > τk−1 : |Bt − Bτk−1 | =√

− 2 h

h ● ● ● ●

● ●

Properties of (Bτk )k and (τk )k

• (Bτk − Bτk−1 )k≥1d= (√

hεk )k≥1

• (τk − τk−1)k are i.i.d. ∼ τ := inf{t > 0 : |Bt | =√

h}• It holds E(τk ) = tk , E|τk − tk |2 = 2

3 tk h, k = 1, · · · ,n.

E|Bτk − Btk |2 = E|τk − tk | ≤√

23 tk h

• random walk constructed from B: Bnt :=√

h∑[t/h]

k=1 (Bτk − Bτk−1 )

Since Bntk = Bτk we have for t ∈ [tk , tk+1)

E|Bnt − Bt |2 ≤ 2E|Bτk − Btk |2 + 2E|Btk − Bt |2

≤ 2√

23 tk T

n + 2 Tn ≤ 4 T√

Forward Backward Stochastic Differential Equations

Xt = x +∫ t

0b(r ,Xr )dr +

0σ(r ,Xr )dBr , 0 ≤ t ≤ T

Yt = g(XT ) +∫ T

tf (s,Xs,Ys,Zs)ds −

tZsdBs

replace (Bt )t∈[0,T ] by a random walk (Bnt )t∈[0,T ] what kind of convergence

((X nt )t∈[0,T ], (Y n

t )t∈[0,T ], (Z nt )t∈[0,T ])→ ((Xt )t∈[0,T ], (Yt )t∈[0,T ], (Zt )t∈[0,T ])?

Briand, Delyon and Mémin (2001)If b, σ, f and g are Lipschitz and (Bn

t )t∈[0,T ] such that

sup0≤t≤T |Bnt − Bt | → 0, n→∞, in probability, then

sup0≤t≤T

|Y nt − Yt |2 +

s − Zs|2ds → 0 when n→∞ in probability.

Other results

• Toldo (2005) extends the previous result to BSDEs with randomterminal time

• Numerical schemes : Ma, Protter, San Martin and Torres (2002),Peng, Xu (2008) and Mémin, Peng, Xu (2008) (Implicit and explicitschemes for BSDEs and RBSDEs), Martinez, San Martin and Torres(2011) (RBSDEs), Janczak (2008, 2009) (generalized RBSDEs withrandom terminal time)

=⇒ convergence in probability or weak convergence: no rate

Introduction

Numerical Examples

DiscretizationWe consider the BSDE

Yt = g(BT ) +∫ T

tf (s,Ys,Zs)ds −

tZsdBs, 0 ≤ t ≤ T

and its approximation

Y ntk = g(Bn

T ) + hn−1∑m=k

f (tm+1,Y ntm ,Z

ntm )−

n−1∑m=k

Z ntm (Bn

tm+1− Bn

tm )︸︷︷︸law=√

hεm+1

For n large enough: ∃! (Fτk )k -adapted solution (Y ntk ,Z

ntk )n−1

k=0 .The corresponding scheme:

Y ntn := g

√hεk

Z ntk−1

:=Eτk−1 [Y n

tkεk ]

, k = 1, ...,n

Y ntk−1

:= Eτk−1 [Yntk ] + f (tk ,Y n

tk−1,Z n

tk−1)h

Discretization : approximation of Z

From Zhang’s paper (2005), we have

Zt = Et

(g(BT )

BT − Bt

T − t

(∫ T

tf (s,Ys,Zs)

Bs − Bt

s − tds

LemmaFor all k = 0,1, . . . ,n − 1, we have

Z ntk = Eτk

tn − Bntk

tn − tk

)+ hEτk

(n−1∑

f (tm+1,Y ntm ,Z

Bntm − Bn

tktm − tk

Convergence resultsAssume: ∀(t , y , z), (t ′, y ′, z ′) ∈ [0,T ]× R2,

|f (t , y , z)− f (t ′, y ′, z ′)| ≤ Lf (√

t − t ′ + |y − y ′| + |z − z ′|),

∀(x , x ′) ∈ R2, |g(x)− g(x ′)| ≤ Cg(1 + |x |p0 + |x ′|p0 )|x − x ′|α.

Theorem (C.G., Labart, Luoto)

sup0≤t<T

E|Yt − Y nt |2 ≤ C1h

E∫ T

0|Zt − Z n

t |2dt ≤ C2hβ for β ∈ (0, α2 )

where C1 depends on Lf ,T ,Cg ,p0 and α and C2 additionally on β.

The reason for β :E|Zt − Z n

t |2 ≤ C0hα2

T−tk+ C1

(T−t)1−α2

1t 6=tk for t ∈ [tk , tk+1), k = 0, ...,n− 1,

Introduction

Numerical Examples

Example 1

dYt = −(Yt + Zt )dt + ZtdBt ,

YT = B2T

The explicit solution is given by

Yt = eT−t ((Bt + T − t)2 + T − t), Zt = 2eT−t (Bt + T − t)

We compute by Monte Carlo E(|Yt − Y nt |2) and E(|Zt − Z n

t |2) for differentvalues of n.

g is a locally Lipschitz function : the convergence should go faster than1√n .

Figure: log(Error on Y ) for different values of log(n)

We get a slope of −0.46

Example 1 : error on Z

Figure: log(Error on Z ) for different values of log(n)

We get a slope of −0.48

Example 2

dYt = −(Yt + Zt )dt + ZtdBt ,

YT =√|BT |

The solution is given by

Yt = eT−t

2 E(√|BT−t + Bt |eBT−t )

We compute by Monte Carlo E(|Yt − Y nt |2) and E(|Zt − Z n

t |2) for differentvalues of n.

g is a 12 -Hölder function : the convergence should go faster 1

Example 2 : error on Y

Figure: log(Error on Y ) for different values of log(n)

We get a slope of −0.56!!!!

Introduction

Numerical Examples

The general case

Xt = x +∫ t

0b(r ,Xr )dr +

0σ(r ,Xr )dBr , 0 ≤ t ≤ T

Yt = g(XT ) +∫ T

tf (s,Xs,Ys,Zs)ds −

tZsdBs

• b, σ ∈ C0,2b and 1

2 -Hölder in time, unif. in space• the first and second derivatives w.r.t. the space variable are

assumed to be γ-Hölder continuous (for some γ ∈]0,1], w.r.t. theparabolic metric d((x , t), (x ′, t ′)) = (|x − x ′|2 + |t − t ′|) 1

2 ) on allcompact subsets of [0,T ]× R.

• σ(t , x) ≥ δ > 0

Numerical Scheme for X and Y

X n0 = x ,

X ntk = x +

k∑j=1

b(tj ,X ntj−1

)h +√

σ(tj ,X ntj−1

Y ntk = g(X n

T ) + hn−1∑m=k

f (tm+1,X ntm ,Y

ntm ,Z

ntm )−

n−1∑m=k

Z ntmεm+1.

Y ntk = Eτk

(g(X n

tn ) + hn−1∑m=k

f (tm+1,X ntm ,Y

ntm ,Z

Z ntk = Eτk g(X n

tn )εk+1√h

+ Eτk

hn−1∑

f (tm+1,X ntm ,Y

ntm ,Z

ntm )εk+1

Approximation of Z

Zt = Et

(g(XT )N t

T +∫ T

tf (r ,Xr ,Yr ,Zr )N t

)σ(t ,Xt ), Zhang (2005)

where N tv := 1

∫ vt

∇Xsσ(s,Xs)∇Xt

dBs is the Malliavin weight of first order

Z ntk :=Eτk

(g(X n

T )Nn,τkτn

+ hn−1∑

f (tm+1,X ntm ,Y

ntm ,Z

ntm )Nn,τk

Nn,τkτm

tm − tk

m∑l=k+1

∇X ntl−1

σ(tl−1,X ntl−1

)∇X ntk

(Bntl − Bn

tl−1)

But Z ntk 6= Z n

tk , and E|Z ntk − Z n

tk |2 → 0 only for g′′ α-Hölder, f sufficiently

smooth

Convergence Results

Theorem (C. Geiss, C.L., A. Luoto)Let b, σ and f satisfy the above assumptions. Let g′′ be a locallyα-Hölder continuous function and assume additionally that all first andsecond partial derivatives w.r.t. the variables x , y , z of b(t , x), σ(t , x) andf (t , x , y , z) exist and are bounded Lipschitz functions w.r.t. thesevariables, uniformly in time. Then for all t ∈ [0,T ) and large enough n, wehave

E0,x |Yt − Y nt |2 ≤ Cψ(x)h

E0,x |Zt − Z nt |2 ≤ Cψ(x)h

12∧α,

where ψ(x) := K (1 + |x |p0+1).

Key results : regularity for fractional smoothness

Theorem (C. Geiss, S. Geiss, E. Gobet (2012))If g is α-Hölder and f Lipschitz, it holds for 0 ≤ t < s < T and x ∈ R,

‖Ys − Yt‖Lp(Pt,x ) ≤ c4ψ(x)(∫ s

t(T − r )α−1dr

‖Zs − Zt‖Lp(Pt,x ) ≤ c5ψ(x)(∫ s

t(T − r )α−2dr

Properties of the associated PDEut (t , x) + σ2(t,x)

2 uxx (t , x) + b(t , x)ux (t , x) + f (t , x ,u(t , x), (σux )(t , x)) = 0,

t ∈ [0,T ), x ∈ R,u(T , x) = g(x), x ∈ R

Properties of u and ux (Zhang (2005)) uxx (C. Geiss, C.L, A.Luoto)

1. Yt = u(t ,Xt ) and Z t,xs = ux (s,X t,x

s )σ(s,X t,xs )

whereu(t , x) = Et,x

(g(XT ) +

∫ Tt f (r ,Xr ,Yr ,Zr )dr

2. u is continuous on [0,T ]× R, ux and uxx are continuous on[0,T )× R,

3. |u(t , x)| ≤ c1ψ(x), |ux (t , x)| ≤ c2ψ(x)

(T−t)1−α

2, |uxx (t , x)| ≤ c3ψ(x)

(T−t)1−α2,

∂ ixu(t , x) = Et,x

[g(XT )N t,i

T +∫ T

t f (r ,Xr ,Yr ,Zr )Nt,ir dr

], where N t,i

r denotes

the Malliavin weight of the i th order

Conclusion

By using the Skorohod embedding to approximate the Brownian motion,we manage to prove,

• if f is Lipschitz and g is locally α-Hölder and YT = g(BT ) that• the rate of convergence = α

4 for the L2-error on Y and < α4 for the

L2-error on Z ,• if YT = g(XT ), X a diffusion process with nice enough b and σ, and

• g to be locally C2,α

• f has w.r.t. x , y , z Lipschitz continuous second partial derivatives,then

• the rate of convergence = 14 for the L2-error on Y and = 1

4 ∧ α2 for the

L2-error on Z

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