On pathwise stochastic integration
Rafa l Marcin Lochowski
Afican Institute for Mathematical Sciences,Warsaw School of Economics
UWC seminar
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 1 / 29
The Riemann-Stieltjes integral
The Riemann-Stieltjes integral of a deterministic function f : [a, b]→ R(integrand) with respect to another deterministic function g : [a, b]→ R(integrator) is defined as the limit of sums
n∑i=1
f (si ) g (ti )− g (ti−1) ,
where si ∈ [ti−1; ti ] , π = a = t0 < t1 < ... < tn = b , as the mesh of thepartition π, mesh(π) := maxi=1,2,...,n |ti − ti−1| , goes to 0.
The Riemann-Stieltjes integral, denoted by
(RS)
∫[a;b]
f dg
may not exist for a given pair (f , g). The simplest situation when ithappens is when the total variation of g is infinite.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 2 / 29
The Riemann-Stieltjes integral
The Riemann-Stieltjes integral of a deterministic function f : [a, b]→ R(integrand) with respect to another deterministic function g : [a, b]→ R(integrator) is defined as the limit of sums
n∑i=1
f (si ) g (ti )− g (ti−1) ,
where si ∈ [ti−1; ti ] , π = a = t0 < t1 < ... < tn = b , as the mesh of thepartition π, mesh(π) := maxi=1,2,...,n |ti − ti−1| , goes to 0.The Riemann-Stieltjes integral, denoted by
(RS)
∫[a;b]
f dg
may not exist for a given pair (f , g). The simplest situation when ithappens is when the total variation of g is infinite.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 2 / 29
The total variation and the existence of the R-S integral
Recall that the total variation of a deterministic function g : [a, b]→ R isdefined as
TV (g , [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
|g (ti )− g (ti−1)| ,
The finiteness of TV (g , [a; b]) together with the continuity of f guaranteethe existence of (RS)
∫[a;b] f dg .
However, still, for a bounded, Borel-measurable integrand f and finite(total) variation integrator g , (RS)
∫[a;b] f dg may not exist.
This may happen e.g. when the jumps of the function f coincide with thejumps of g (the invention of an appropriate example might be a good,instructive exercise for students).
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 3 / 29
The total variation and the existence of the R-S integral
Recall that the total variation of a deterministic function g : [a, b]→ R isdefined as
TV (g , [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
|g (ti )− g (ti−1)| ,
The finiteness of TV (g , [a; b]) together with the continuity of f guaranteethe existence of (RS)
∫[a;b] f dg .
However, still, for a bounded, Borel-measurable integrand f and finite(total) variation integrator g , (RS)
∫[a;b] f dg may not exist.
This may happen e.g. when the jumps of the function f coincide with thejumps of g (the invention of an appropriate example might be a good,instructive exercise for students).
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 3 / 29
The refinement of the R-S integral - the Lebesgue-Stieltjesintegral
A fine refinement of the R-S integral is the Lebesgue-Stieltjes integral.The construction is made by the introduction of the measure space([a; b],B ([a; b]) , µg ) , where B ([a; b]) denotes the σ-field of Borel subsetsof [a; b] and µg is a signed, σ-finite measure on [a; b].To define µg we consier the cadlag version of g (right-continuous with leftlimits) which we will also denote by g and for a ≤ c ≤ d ≤ b define
µg (c ; d ] := g(d)− g(c).
Now we extend the measure µg to all Borel subsets of [a; b](Caratheodory’s extension) and define the Lebesgue-Stieltjes integral asthe usual Lebesgue integral of f with respect to the measure µg :
(LS)
∫[a;b]
f dg :=
∫[a;b]
f dµg .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 4 / 29
The refinement of the R-S integral - the Lebesgue-Stieltjesintegral
A fine refinement of the R-S integral is the Lebesgue-Stieltjes integral.The construction is made by the introduction of the measure space([a; b],B ([a; b]) , µg ) , where B ([a; b]) denotes the σ-field of Borel subsetsof [a; b] and µg is a signed, σ-finite measure on [a; b].To define µg we consier the cadlag version of g (right-continuous with leftlimits) which we will also denote by g and for a ≤ c ≤ d ≤ b define
µg (c ; d ] := g(d)− g(c).
Now we extend the measure µg to all Borel subsets of [a; b](Caratheodory’s extension) and define the Lebesgue-Stieltjes integral asthe usual Lebesgue integral of f with respect to the measure µg :
(LS)
∫[a;b]
f dg :=
∫[a;b]
f dµg .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 4 / 29
The Lebesgue-Stieltjes integral - properties
Now, finiteness of TV (g , [a; b]) together with boundedness andBorel-measurability of the integrand f guarantee the existence of(LS)
∫[a;b] f dg .
If both, f and g are cadlag and have finite total variation we haveimportant integration by parts formula
(LS)
∫(a;b]
f (t−)dg(t) =f (b)g(b)− f (a)g(a) (1)
−(LS)
∫(a;b]
g(t−)df (t)−∑
a<s≤b∆f (s)∆g(s).
where ∆f (s) = f (s)− f (s−), ∆g(s) = g(s)− g(s−).Notice, that using this formula one may propose a reasonable value for(LS)
∫(a;b] f (t−)dg(t) whenever f and g are cadlag, TV (f , [a; b]) < +∞
and g is bounded! (The same observation may be used to differentiatedistributions but it is another story...)
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 5 / 29
The Lebesgue-Stieltjes integral - properties
Now, finiteness of TV (g , [a; b]) together with boundedness andBorel-measurability of the integrand f guarantee the existence of(LS)
∫[a;b] f dg .
If both, f and g are cadlag and have finite total variation we haveimportant integration by parts formula
(LS)
∫(a;b]
f (t−)dg(t) =f (b)g(b)− f (a)g(a) (1)
−(LS)
∫(a;b]
g(t−)df (t)−∑
a<s≤b∆f (s)∆g(s).
where ∆f (s) = f (s)− f (s−), ∆g(s) = g(s)− g(s−).
Notice, that using this formula one may propose a reasonable value for(LS)
∫(a;b] f (t−)dg(t) whenever f and g are cadlag, TV (f , [a; b]) < +∞
and g is bounded! (The same observation may be used to differentiatedistributions but it is another story...)
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 5 / 29
The Lebesgue-Stieltjes integral - properties
Now, finiteness of TV (g , [a; b]) together with boundedness andBorel-measurability of the integrand f guarantee the existence of(LS)
∫[a;b] f dg .
If both, f and g are cadlag and have finite total variation we haveimportant integration by parts formula
(LS)
∫(a;b]
f (t−)dg(t) =f (b)g(b)− f (a)g(a) (1)
−(LS)
∫(a;b]
g(t−)df (t)−∑
a<s≤b∆f (s)∆g(s).
where ∆f (s) = f (s)− f (s−), ∆g(s) = g(s)− g(s−).Notice, that using this formula one may propose a reasonable value for(LS)
∫(a;b] f (t−)dg(t) whenever f and g are cadlag, TV (f , [a; b]) < +∞
and g is bounded! (The same observation may be used to differentiatedistributions but it is another story...)
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 5 / 29
The Lebesgue-Stieltjes integral still insufficient
(A standard) Brownian motion B is one of the simplest continuous-timeprocess with continuous trajectories, widely used in stochastic modellingand optimisation.Unfortunately, its paths have a.s. infinite total variation. When one wantsto integrate locally finite variation deterministic function or locally finitevariation stochastic process with respect to a Brownian trajectory Bt ,t ∈ [0;T ], one may use the relation (1) (this idea is due to Paley, Wienerand Zygmund).
Unfortunately, this is still not sufficient to calculate (or at least to give areasonable meaning if the calculations were too hard) e.g.
(LS)
∫(a;b]
BtdBt .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 6 / 29
The Lebesgue-Stieltjes integral still insufficient
(A standard) Brownian motion B is one of the simplest continuous-timeprocess with continuous trajectories, widely used in stochastic modellingand optimisation.Unfortunately, its paths have a.s. infinite total variation. When one wantsto integrate locally finite variation deterministic function or locally finitevariation stochastic process with respect to a Brownian trajectory Bt ,t ∈ [0;T ], one may use the relation (1) (this idea is due to Paley, Wienerand Zygmund).Unfortunately, this is still not sufficient to calculate (or at least to give areasonable meaning if the calculations were too hard) e.g.
(LS)
∫(a;b]
BtdBt .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 6 / 29
Functions with finite and infinite total variation - examples
0.0 0.2 0.4 0.6 0.8 1.0
−1.0
−0.5
0.0
0.5
t
Figure : A typical path of a Brownian motion (green) and a function with finitetotal variation (blue)
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 7 / 29
A standard Brownian motion - an ”axiomatic” definition
A standard Brownian motion is a stochastic process Bt , t ≥ 0 defined onsome (rich enough) probability space (Ω,F ,P) with the followingproperties
B0 ≡ 0;
B has continuous paths, i.e. the probability P that for ω ∈ Ω, thepath
[0; +∞) 3 t 7→ Bt(ω) ∈ R
is continuous equals 1;
for any 0 ≤ s < t < u increments Bu − Bt and Bt − Bs areindependent;
for any 0 ≤ s < t the increment Bt − Bs has normal distribution withmean 0 and variance t − s, N (0, t − s).
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 8 / 29
Finiteness of the quadratic variation of a Brownian motion
Cruicial observation which is sometimes utilised to define a stochasticintegral with respect to the Brownian motion or even with respect to muchmore general family of processes - semimartingales - is the finiteness oftheir quadratic variation.But even with the quadratic variation we must be careful. When we define
V2 (B, [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
(Bti − Bti−1
)2,
then V2 (B, [a; b]) is a.s. infinite.
However, one may relatively easily prove that for any sequence of
partitions πk =a = tk0 < tk1 < ... < tkn(k) = b
, with mesh(πk) ↓ 0 as
k ↑ +∞ we have
n(k)∑i=1
(Btki− Btki−1
)2→P [B]b − [B]a := b − a,
where →P denotes the convergence in probability.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 9 / 29
Finiteness of the quadratic variation of a Brownian motion
Cruicial observation which is sometimes utilised to define a stochasticintegral with respect to the Brownian motion or even with respect to muchmore general family of processes - semimartingales - is the finiteness oftheir quadratic variation.But even with the quadratic variation we must be careful. When we define
V2 (B, [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
(Bti − Bti−1
)2,
then V2 (B, [a; b]) is a.s. infinite.However, one may relatively easily prove that for any sequence of
partitions πk =a = tk0 < tk1 < ... < tkn(k) = b
, with mesh(πk) ↓ 0 as
k ↑ +∞ we have
n(k)∑i=1
(Btki− Btki−1
)2→P [B]b − [B]a := b − a,
where →P denotes the convergence in probability.Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 9 / 29
ψ- variation of a Brownian motion
For x > 0 define
ψ(x) :=x2
ln(ln(1/x) ∨ 2)
and ψ(0) := 0 then, by the result of S. J. Taylor (Exact asymptoticestimates of Brownian path variation, Duke Math. J. 39), almost surely:
Vψ (B, [a; b]) := supn
sup0≤t0<t1<...<tn≤T
n∑i=1
ψ(∣∣Bti − Bti−1
∣∣) < +∞, (2)
moreover, ψ is a function with the greatest possible order at 0, for which(2) holds.
Remark
Moreover, when mesh(π(n)
)→ 0, then
limn→∞
supn
sup0≤t0<t1<...<tn≤T
n∑i=1
ψ
(Bt
(n)i
− Bt
(n)i−1
)= T a.s.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 10 / 29
ψ- variation of a Brownian motion
For x > 0 define
ψ(x) :=x2
ln(ln(1/x) ∨ 2)
and ψ(0) := 0 then, by the result of S. J. Taylor (Exact asymptoticestimates of Brownian path variation, Duke Math. J. 39), almost surely:
Vψ (B, [a; b]) := supn
sup0≤t0<t1<...<tn≤T
n∑i=1
ψ(∣∣Bti − Bti−1
∣∣) < +∞, (2)
moreover, ψ is a function with the greatest possible order at 0, for which(2) holds.
Remark
Moreover, when mesh(π(n)
)→ 0, then
limn→∞
supn
sup0≤t0<t1<...<tn≤T
n∑i=1
ψ
(Bt
(n)i
− Bt
(n)i−1
)= T a.s.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 10 / 29
A small detour - why the quadratic variation [B] of theBrownian motion on the interval [a; b] equals b − a?
To give a hint why the quadratic variation of the Brownian motion on theinterval [a; b] equals b − a let us recall the simplest construction of astandard Brownian motion (due to Donsker we know that this constructionworks, though the convergence is relatively slow).
Pick (very) large integer n;
set t = 0, 1n ,
2n , ...,
nn ,
n+1n , ...; dt = 1
n ;
set B0 = 0 and for t = 0, 1n ,
2n , ...,
nn ,
n+1n , ...
Bt+dt = Bt+1/n = Bt +
1√n
with probability 1/2;
− 1√n
with probability 1/2.
Now notice that (dBt)2 = dt (the simplest version of Ito’s formula) and
thus ∑t=1/n,2/n,...,T
(dBt)2 =
∑t=1/n,2/n,...,T
dt = [B]T = T .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 11 / 29
A small detour - why the quadratic variation [B] of theBrownian motion on the interval [a; b] equals b − a?
To give a hint why the quadratic variation of the Brownian motion on theinterval [a; b] equals b − a let us recall the simplest construction of astandard Brownian motion (due to Donsker we know that this constructionworks, though the convergence is relatively slow).
Pick (very) large integer n;
set t = 0, 1n ,
2n , ...,
nn ,
n+1n , ...; dt = 1
n ;
set B0 = 0 and for t = 0, 1n ,
2n , ...,
nn ,
n+1n , ...
Bt+dt = Bt+1/n = Bt +
1√n
with probability 1/2;
− 1√n
with probability 1/2.
Now notice that (dBt)2 = dt (the simplest version of Ito’s formula) and
thus ∑t=1/n,2/n,...,T
(dBt)2 =
∑t=1/n,2/n,...,T
dt = [B]T = T .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 11 / 29
A small detour - why the quadratic variation [B] of theBrownian motion on the interval [a; b] equals b − a?
To give a hint why the quadratic variation of the Brownian motion on theinterval [a; b] equals b − a let us recall the simplest construction of astandard Brownian motion (due to Donsker we know that this constructionworks, though the convergence is relatively slow).
Pick (very) large integer n;
set t = 0, 1n ,
2n , ...,
nn ,
n+1n , ...; dt = 1
n ;
set B0 = 0 and for t = 0, 1n ,
2n , ...,
nn ,
n+1n , ...
Bt+dt = Bt+1/n = Bt +
1√n
with probability 1/2;
− 1√n
with probability 1/2.
Now notice that (dBt)2 = dt (the simplest version of Ito’s formula) and
thus ∑t=1/n,2/n,...,T
(dBt)2 =
∑t=1/n,2/n,...,T
dt = [B]T = T .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 11 / 29
Another small detour - Young’s integral
When the integrand has finite p-variation, Vp the integrator has finiteq-variation, Vq, p > 1, q > 1 and 1/p + 1/q > 1, then one still may define
(RYS)
∫[a;b]
f dg .
where (RYS)∫
denotes some refinement of the Riemann-Stieltjes integral(the refinement is made to avoid problems with discontinuities) and wheng is continuous, it coincides with the Riemann-Stieltjes integral.
This is proved with the following Love-Young inequality:∣∣∣∣∣n∑
i=1
f (si ) g(ti )− g(ti−1) − f (si0) g(b)− g(a)
∣∣∣∣∣ ≤ ζ (p−1 + q−1),
for any partition π = a = t0 < t1 < ... < tn = b , si ∈ [ti−1; ti ] ,i , i0 ∈ 1, 2, . . . , n . Here
ζ(r) =∞∑k=1
k−r .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 12 / 29
Another small detour - Young’s integral
When the integrand has finite p-variation, Vp the integrator has finiteq-variation, Vq, p > 1, q > 1 and 1/p + 1/q > 1, then one still may define
(RYS)
∫[a;b]
f dg .
where (RYS)∫
denotes some refinement of the Riemann-Stieltjes integral(the refinement is made to avoid problems with discontinuities) and wheng is continuous, it coincides with the Riemann-Stieltjes integral.This is proved with the following Love-Young inequality:∣∣∣∣∣
n∑i=1
f (si ) g(ti )− g(ti−1) − f (si0) g(b)− g(a)
∣∣∣∣∣ ≤ ζ (p−1 + q−1),
for any partition π = a = t0 < t1 < ... < tn = b , si ∈ [ti−1; ti ] ,i , i0 ∈ 1, 2, . . . , n . Here
ζ(r) =∞∑k=1
k−r .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 12 / 29
The Young integral is still insufficient
Since the ψ- variation of a Brownian motion is finite then any p- variationwith p > 2 of the Brownian motion is locally finite. (Just to make itprecise let us write a formula for p-variation:
Vp (B, [a; b]) := supn
sup0≤t0<t1<...<tn≤T
n∑i=1
∣∣Bti − Bti−1
∣∣p .)
Thus, the Young integral may be used for the pathwise integration
(RYS)
∫[a;b]
XsdBs
whenever X is a stochastic process with locally finite q-variation, where1 ≤ q < 2.Unfortunately, this is still unsufficient for calcuation of the integral∫
[a;b] BsdBs .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 13 / 29
The Young integral is still insufficient
Since the ψ- variation of a Brownian motion is finite then any p- variationwith p > 2 of the Brownian motion is locally finite. (Just to make itprecise let us write a formula for p-variation:
Vp (B, [a; b]) := supn
sup0≤t0<t1<...<tn≤T
n∑i=1
∣∣Bti − Bti−1
∣∣p .) Thus, the Young integral may be used for the pathwise integration
(RYS)
∫[a;b]
XsdBs
whenever X is a stochastic process with locally finite q-variation, where1 ≤ q < 2.Unfortunately, this is still unsufficient for calcuation of the integral∫
[a;b] BsdBs .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 13 / 29
Stochastic integral - the quadratic variation approach
For simplicity we will fix on integration with respect to the Brownianmotion B.We consider a family of stochastic processes X with caglad (leftcontinuous with right limits) paths [0; +∞) 3 t 7→ Xt ∈ R such that forevery process X ∈ X the following hold
1 for every u > t ≥ 0 the increment Bu − Bt is independent of thevalues of Xs , 0 ≤ s ≤ t;
2 ‖X‖H := E∫ +∞
0 X 2s d[B]s = E
∫ +∞0 X 2
s ds =∫ +∞
0 EX 2s ds < +∞.
It appears that H = (X , ‖ · ‖H) is a Hilbert space and the family of simpleprocesses of the form
K = K−1 · 10 ++∞∑i=0
Ki · 1(ti ,ti+1],
where 0 = t0 < t1 < t2 < . . . with ti ↑ +∞ as i ↑ +∞ and Bu − Bti ,u ≥ ti being independent from Ki is dense in H.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 14 / 29
Stochastic integral - the quadratic variation approach
For simplicity we will fix on integration with respect to the Brownianmotion B.We consider a family of stochastic processes X with caglad (leftcontinuous with right limits) paths [0; +∞) 3 t 7→ Xt ∈ R such that forevery process X ∈ X the following hold
1 for every u > t ≥ 0 the increment Bu − Bt is independent of thevalues of Xs , 0 ≤ s ≤ t;
2 ‖X‖H := E∫ +∞
0 X 2s d[B]s = E
∫ +∞0 X 2
s ds =∫ +∞
0 EX 2s ds < +∞.
It appears that H = (X , ‖ · ‖H) is a Hilbert space and the family of simpleprocesses of the form
K = K−1 · 10 ++∞∑i=0
Ki · 1(ti ,ti+1],
where 0 = t0 < t1 < t2 < . . . with ti ↑ +∞ as i ↑ +∞ and Bu − Bti ,u ≥ ti being independent from Ki is dense in H.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 14 / 29
Stochastic integral - the quadratic variation approach cont.
Now, for any t > 0 and every simple process K we define∫ t
0KdB := K−1 · B0 +
n−1∑i=0
Ki ·(Bti+1 − Bti
)+ Kn · (Bt − Btn) , (3)
whenever tn ≤ t < tn+1.
By this definition and the independence of the increments Bti+1 − Bti fromKi we easily calculate
E(∫ t
0KdB
)2
≤ E(∫ +∞
0KdB
)2
= ‖K‖2H. (4)
The equality (4) is called Ito’s isometry. Now, for any X ∈ X and ε > 0we find a simple process Kε such that ‖X − Kε‖H < ε and define∫ t
0XdB = lim
ε↓0
∫ t
0KεdB.
By (4) this limit exists and is unique.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 15 / 29
Stochastic integral - the quadratic variation approach cont.
Now, for any t > 0 and every simple process K we define∫ t
0KdB := K−1 · B0 +
n−1∑i=0
Ki ·(Bti+1 − Bti
)+ Kn · (Bt − Btn) , (3)
whenever tn ≤ t < tn+1.By this definition and the independence of the increments Bti+1 − Bti fromKi we easily calculate
E(∫ t
0KdB
)2
≤ E(∫ +∞
0KdB
)2
= ‖K‖2H. (4)
The equality (4) is called Ito’s isometry. Now, for any X ∈ X and ε > 0we find a simple process Kε such that ‖X − Kε‖H < ε and define∫ t
0XdB = lim
ε↓0
∫ t
0KεdB.
By (4) this limit exists and is unique.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 15 / 29
Stochastic integral - the quadratic variation approach cont.
Now, for any t > 0 and every simple process K we define∫ t
0KdB := K−1 · B0 +
n−1∑i=0
Ki ·(Bti+1 − Bti
)+ Kn · (Bt − Btn) , (3)
whenever tn ≤ t < tn+1.By this definition and the independence of the increments Bti+1 − Bti fromKi we easily calculate
E(∫ t
0KdB
)2
≤ E(∫ +∞
0KdB
)2
= ‖K‖2H. (4)
The equality (4) is called Ito’s isometry. Now, for any X ∈ X and ε > 0we find a simple process Kε such that ‖X − Kε‖H < ε and define∫ t
0XdB = lim
ε↓0
∫ t
0KεdB.
By (4) this limit exists and is unique.Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 15 / 29
Stochastic integral - the quadratic variation approach cont.
The construction presented on two previous slides may be extended tomartingales, local martingales and semimartingales.
A continuous-time martingale Mt , t ≥ 0, is a special (cadlag) process,conditional increments of which, Mt −Ms , 0 ≤ s < t, with respect to theavailable information - ”filtration” till moment s, Fs , are centered
E [Mt −Ms | Fs ] = 0.
Every such a process has finite quadratic variation [M]. Moreover, theprocess M2 − [M] is also a martingale.A local martingale is a (cadlag) process, which stopped at theappropriate stopping times is a martingale.A semimartingale is a (cadlag) process which is a sum of a localmartingale and a locally finite variation process.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 16 / 29
Stochastic integral - the quadratic variation approach cont.
The construction presented on two previous slides may be extended tomartingales, local martingales and semimartingales.A continuous-time martingale Mt , t ≥ 0, is a special (cadlag) process,conditional increments of which, Mt −Ms , 0 ≤ s < t, with respect to theavailable information - ”filtration” till moment s, Fs , are centered
E [Mt −Ms | Fs ] = 0.
Every such a process has finite quadratic variation [M]. Moreover, theprocess M2 − [M] is also a martingale.
A local martingale is a (cadlag) process, which stopped at theappropriate stopping times is a martingale.A semimartingale is a (cadlag) process which is a sum of a localmartingale and a locally finite variation process.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 16 / 29
Stochastic integral - the quadratic variation approach cont.
The construction presented on two previous slides may be extended tomartingales, local martingales and semimartingales.A continuous-time martingale Mt , t ≥ 0, is a special (cadlag) process,conditional increments of which, Mt −Ms , 0 ≤ s < t, with respect to theavailable information - ”filtration” till moment s, Fs , are centered
E [Mt −Ms | Fs ] = 0.
Every such a process has finite quadratic variation [M]. Moreover, theprocess M2 − [M] is also a martingale.A local martingale is a (cadlag) process, which stopped at theappropriate stopping times is a martingale.
A semimartingale is a (cadlag) process which is a sum of a localmartingale and a locally finite variation process.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 16 / 29
Stochastic integral - the quadratic variation approach cont.
The construction presented on two previous slides may be extended tomartingales, local martingales and semimartingales.A continuous-time martingale Mt , t ≥ 0, is a special (cadlag) process,conditional increments of which, Mt −Ms , 0 ≤ s < t, with respect to theavailable information - ”filtration” till moment s, Fs , are centered
E [Mt −Ms | Fs ] = 0.
Every such a process has finite quadratic variation [M]. Moreover, theprocess M2 − [M] is also a martingale.A local martingale is a (cadlag) process, which stopped at theappropriate stopping times is a martingale.A semimartingale is a (cadlag) process which is a sum of a localmartingale and a locally finite variation process.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 16 / 29
Semimartingales
The family of semimartingales is rich enough to encompass majority ofprocesses used in stochastic modelling (except maybe fractional Brownianmotions).Moreover, the deep result of Bichteler and Dellacherie states that thisfamily is in some sense the broadest possible family of good integrators.Namely, when we define for a given integrator M and every simple processthe integral
∫ t0 KdM with the formula analogous to the formula (3), then
the transformation
K 7→∫ t
0KdM
is continuous if and only if M is a semimartingale.
To deal with continuity we need to define topologies: the space of simpleprocesses is endowed with uniform convergence in (t, ω) topology and thespace of integrals is topologized by convergence in probability.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 17 / 29
Semimartingales
The family of semimartingales is rich enough to encompass majority ofprocesses used in stochastic modelling (except maybe fractional Brownianmotions).Moreover, the deep result of Bichteler and Dellacherie states that thisfamily is in some sense the broadest possible family of good integrators.Namely, when we define for a given integrator M and every simple processthe integral
∫ t0 KdM with the formula analogous to the formula (3), then
the transformation
K 7→∫ t
0KdM
is continuous if and only if M is a semimartingale.To deal with continuity we need to define topologies: the space of simpleprocesses is endowed with uniform convergence in (t, ω) topology and thespace of integrals is topologized by convergence in probability.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 17 / 29
Few properties of a stochastic integral
The stochastic integral is now defined for any semimartingale M and any(predictable) caglad process X . For example
∫ t0 BtdBt = 1
2B2t − 1
2 t.We have important
Theorem (counterpart of the Lebesgue dominated convergence)
For any caglad (predictable) processes X 1, X 2, . . . , such that X i ≤ |X | fori = 1, 2, . . . , where X is some caglad (predictable) process andlimi↑+∞ X i → 0 pointwise then we have
sup0≤s≤T
∣∣∣∣∫ s
0X idM
∣∣∣∣→P 0.
For any p > 0 we also have important Burkholder-Davis-Gundy’s inequality
E sup0≤s≤T
∣∣∣∣∫ s
0XdM
∣∣∣∣p ≤ Cp · E∣∣∣∣∫ T
0X 2d[M]
∣∣∣∣p/2
.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 18 / 29
Few properties of a stochastic integral
The stochastic integral is now defined for any semimartingale M and any(predictable) caglad process X . For example
∫ t0 BtdBt = 1
2B2t − 1
2 t.We have important
Theorem (counterpart of the Lebesgue dominated convergence)
For any caglad (predictable) processes X 1, X 2, . . . , such that X i ≤ |X | fori = 1, 2, . . . , where X is some caglad (predictable) process andlimi↑+∞ X i → 0 pointwise then we have
sup0≤s≤T
∣∣∣∣∫ s
0X idM
∣∣∣∣→P 0.
For any p > 0 we also have important Burkholder-Davis-Gundy’s inequality
E sup0≤s≤T
∣∣∣∣∫ s
0XdM
∣∣∣∣p ≤ Cp · E∣∣∣∣∫ T
0X 2d[M]
∣∣∣∣p/2
.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 18 / 29
Wong-Zakai’s pathwise approach to the stochastic integral
Since many years mathematicians tried to define the stochastic integral ina pathwise way.One of the earliest of such attemps is due to Wong and Zakai (1965). ForT > 0 they considered the following approximation of Brownian paths:
(A) for all s ∈ [0;T ], Bns → Bs pointwise as n ↑ +∞, where Bn,
n = 1, 2, . . . , are continuous and have locally bounded variation;
(B) (A) and there exists such a bounded process Z that for all s ∈ [0;T ],|Bn
s | ≤ Zs ;
and stated the following approximation theorem.
Theorem (Wong-Zakai (1965))
Let ψ(t, x) has continuous partial derivatives ∂ψ∂t and ∂ψ
∂x and let Bn satisfy
(B), then for the Lebesgue-Stieltjes integrals∫ T
0 ψ (t,Bnt )dBn
t , a.s.,
limn→∞
∫ T
0ψ (t,Bn
t )dBnt =
∫ T
0ψ (t,Bt)dBt +
1
2
∫ T
0
∂ψ
∂x(t,Bt)dt.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 19 / 29
Wong-Zakai’s pathwise approach to the stochastic integral
Since many years mathematicians tried to define the stochastic integral ina pathwise way.One of the earliest of such attemps is due to Wong and Zakai (1965). ForT > 0 they considered the following approximation of Brownian paths:
(A) for all s ∈ [0;T ], Bns → Bs pointwise as n ↑ +∞, where Bn,
n = 1, 2, . . . , are continuous and have locally bounded variation;
(B) (A) and there exists such a bounded process Z that for all s ∈ [0;T ],|Bn
s | ≤ Zs ;
and stated the following approximation theorem.
Theorem (Wong-Zakai (1965))
Let ψ(t, x) has continuous partial derivatives ∂ψ∂t and ∂ψ
∂x and let Bn satisfy
(B), then for the Lebesgue-Stieltjes integrals∫ T
0 ψ (t,Bnt )dBn
t , a.s.,
limn→∞
∫ T
0ψ (t,Bn
t )dBnt =
∫ T
0ψ (t,Bt)dBt +
1
2
∫ T
0
∂ψ
∂x(t,Bt) dt.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 19 / 29
Wong-Zakai’s pathwise approach to the stochasticintegral, cont.
The correction term in Wong-Zakai’s theorem,∫ T
0∂ψ∂x (t,Bt)dt, is simply
the quadratic covariation of the processes Bt and ψ (t,Bt) .For two semimartingales X and Y the quadratic covariation, [X ,Y ], isthe (unique) limit in the probability of the sums
n(k)∑i=1
(Xtki− Xtki−1
)·(Ytki− Ytki−1
),
where the sequence of partitions πk =a = tk0 < tk1 < ... < tkn(k) = b
is
such that mesh(πk) ↓ 0 as k ↑ +∞.
For two cadlag semimartingales X and Y the integral
(S)
∫ T
0Y−dX :=
∫ T
0Y−dX +
1
2[X ,Y ]
is called the Stratonovich integral.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 20 / 29
Wong-Zakai’s pathwise approach to the stochasticintegral, cont.
The correction term in Wong-Zakai’s theorem,∫ T
0∂ψ∂x (t,Bt)dt, is simply
the quadratic covariation of the processes Bt and ψ (t,Bt) .For two semimartingales X and Y the quadratic covariation, [X ,Y ], isthe (unique) limit in the probability of the sums
n(k)∑i=1
(Xtki− Xtki−1
)·(Ytki− Ytki−1
),
where the sequence of partitions πk =a = tk0 < tk1 < ... < tkn(k) = b
is
such that mesh(πk) ↓ 0 as k ↑ +∞.For two cadlag semimartingales X and Y the integral
(S)
∫ T
0Y−dX :=
∫ T
0Y−dX +
1
2[X ,Y ]
is called the Stratonovich integral.Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 20 / 29
Disadvantages of the Wong-Zakai construction
The Wong-Zakai construction works only for very limited family ofintegrators: Brownian motions and processes of the formXt =
∫ t0 µsds +
∫ t0 σsdBs and integrands must be functions of the
integrators.
The generalisation for any pair of semimartingale integrator andintegrand is impossible.
It is relatively easy to give an example of two sequences ofcontinuous, with locally finite variation, bounded (and adapted to thenatural Brownian filtration) processes Bn and Bn such that Bn ⇒ B,Bn ⇒ B uniformly but ∫ 1
0BndBn
diverges (Lochowski (2013)).
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 21 / 29
A modification of the Wong-Zakai construction
In Lochowski (2013) for any cadlag process X the following construction isconsidered. For any c > 0 we there exists a process X c such that
(i) X c has locally finite total variation;
(ii) X c has cadlag paths;
(iii) for every T ≥ 0 there exists such KT < +∞ that for every t ∈ [0;T ] ,
|Xt − X ct | ≤ KT c ;
(iv) for every T ≥ 0 there exists such LT < +∞ that for every t ∈ [0;T ] ,
|∆X ct | ≤ LT |∆Xt | ,
where ∆X ct = X c
t − X ct−, ∆Xt = Xt − Xt−;
(v) the process X c is adapted to the natural filtration of X .
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 22 / 29
A modification of the Wong-Zakai construction, cont.
Further, in Lochowski (2013) it was shown that if processes X and Y arecadlag semimartingales then the sequence of pathwise Lebesgue-Stieltjesintegrals ∫ T
0Y−dX
c →Pc↓0
∫ T
0Y−dX + [X ,Y ]contT .∫ T
0 Y−dX denotes here the (semimartingale) stochastic integral and[X ,Y ]cont denotes here the continuous part of [X ,Y ], i.e.
[X ,Y ]contT = [X ,Y ]T −∑
0<s≤T∆Xs∆Ys .
Moreover, when c(n) > 0 and∑+∞
n=1 c(n)2 < +∞ then the convergence of∫ T0 Y−dX
c(n) to∫ T
0 Y−dX + [X ,Y ]contT holds almost surely.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 23 / 29
A modification of the Wong-Zakai construction, cont.
Further, in Lochowski (2013) it was shown that if processes X and Y arecadlag semimartingales then the sequence of pathwise Lebesgue-Stieltjesintegrals ∫ T
0Y−dX
c →Pc↓0
∫ T
0Y−dX + [X ,Y ]contT .∫ T
0 Y−dX denotes here the (semimartingale) stochastic integral and[X ,Y ]cont denotes here the continuous part of [X ,Y ], i.e.
[X ,Y ]contT = [X ,Y ]T −∑
0<s≤T∆Xs∆Ys .
Moreover, when c(n) > 0 and∑+∞
n=1 c(n)2 < +∞ then the convergence of∫ T0 Y−dX
c(n) to∫ T
0 Y−dX + [X ,Y ]contT holds almost surely.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 23 / 29
Drawbacks of the construction presented
Unfortunately, the construction presented does not work for any cagladintegrand Y .It is possible to construct a continuous, bounded (and adapted to thenatural Brownian filtration) process Y and a sequence Bc(n), n = 1, 2, . . . ,satisfying all conditions (i)-(v) for X = B such that the integral∫ 1
0Y dBc(n)
diverges (cf. Lochowski (2013)).
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 24 / 29
Bichteler’s construction
The remarkable Bichteler’s approach provides pathwise construction forintegration of any adapted cadlag process Y with cadlag semimartingaleintegrator X and is based on the approximation
limn→∞
sup0≤s≤T
∣∣∣∣∣Y0X0 +∞∑i=1
Yτni−1∧s
(Xτni ∧s − Xτni−1∧s
)−∫ s
0Y−dX
∣∣∣∣∣ = 0 a.s.,
where τn = (τni ) , i = 0, 1, 2, . . . , is the following sequence of stoppingtimes: τn0 = 0 and for i = 1, 2, . . . ,
τn = inft > τni−1 :
∣∣∣Yt − Yτni−1
∣∣∣ ≥ 2−n.
Remark
In fact, given c(n) > 0,∑∞
n=1 c2 (n) < +∞, Bichteler’s construction works
for any sequence τn = (τni ) , i = 0, 1, 2, . . . , of stopping times, such that
τn0 = 0 and for i = 1, 2, . . . , τni = inft > τni−1 :
∣∣∣Yt − Yτni−1
∣∣∣ ≥ c (n).
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 25 / 29
Bichteler’s construction
The remarkable Bichteler’s approach provides pathwise construction forintegration of any adapted cadlag process Y with cadlag semimartingaleintegrator X and is based on the approximation
limn→∞
sup0≤s≤T
∣∣∣∣∣Y0X0 +∞∑i=1
Yτni−1∧s
(Xτni ∧s − Xτni−1∧s
)−∫ s
0Y−dX
∣∣∣∣∣ = 0 a.s.,
where τn = (τni ) , i = 0, 1, 2, . . . , is the following sequence of stoppingtimes: τn0 = 0 and for i = 1, 2, . . . ,
τn = inft > τni−1 :
∣∣∣Yt − Yτni−1
∣∣∣ ≥ 2−n.
Remark
In fact, given c(n) > 0,∑∞
n=1 c2 (n) < +∞, Bichteler’s construction works
for any sequence τn = (τni ) , i = 0, 1, 2, . . . , of stopping times, such that
τn0 = 0 and for i = 1, 2, . . . , τni = inft > τni−1 :
∣∣∣Yt − Yτni−1
∣∣∣ ≥ c (n).
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 25 / 29
Bichteler’s construction
The remarkable Bichteler’s approach provides pathwise construction forintegration of any adapted cadlag process Y with cadlag semimartingaleintegrator X and is based on the approximation
limn→∞
sup0≤s≤T
∣∣∣∣∣Y0X0 +∞∑i=1
Yτni−1∧s
(Xτni ∧s − Xτni−1∧s
)−∫ s
0Y−dX
∣∣∣∣∣ = 0 a.s.,
where τn = (τni ) , i = 0, 1, 2, . . . , is the following sequence of stoppingtimes: τn0 = 0 and for i = 1, 2, . . . ,
τn = inft > τni−1 :
∣∣∣Yt − Yτni−1
∣∣∣ ≥ 2−n.
Remark
In fact, given c(n) > 0,∑∞
n=1 c2 (n) < +∞, Bichteler’s construction works
for any sequence τn = (τni ) , i = 0, 1, 2, . . . , of stopping times, such that
τn0 = 0 and for i = 1, 2, . . . , τni = inft > τni−1 :
∣∣∣Yt − Yτni−1
∣∣∣ ≥ c (n).
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 25 / 29
Norvaisa’s integral for functions with finite quadraticvariation
In a long (171 pages!) paper, Norvaisa develops (among other results) atheory of an integral for deterministic functions with finite λ - quadraticvariation.
A function f : [a; b]→ R has finite λ - quadratic variation if the sums
n(k)∑i=1
(f (tki )− f (tki−1)
)2
converge for any nested partitions i.e. πk ⊂ πk+1 with tk0 = a, tkn(k) = b
and mesh(πk)→ 0 as k ↑ +∞.
Sample paths of a Brownian motion have this property with probability Pequal 1 - a result due to Paul Levy.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 26 / 29
Norvaisa’s integral for functions with finite quadraticvariation
In a long (171 pages!) paper, Norvaisa develops (among other results) atheory of an integral for deterministic functions with finite λ - quadraticvariation.
A function f : [a; b]→ R has finite λ - quadratic variation if the sums
n(k)∑i=1
(f (tki )− f (tki−1)
)2
converge for any nested partitions i.e. πk ⊂ πk+1 with tk0 = a, tkn(k) = b
and mesh(πk)→ 0 as k ↑ +∞.
Sample paths of a Brownian motion have this property with probability Pequal 1 - a result due to Paul Levy.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 26 / 29
Norvaisa’s integral for functions with finite quadraticvariation
In a long (171 pages!) paper, Norvaisa develops (among other results) atheory of an integral for deterministic functions with finite λ - quadraticvariation.
A function f : [a; b]→ R has finite λ - quadratic variation if the sums
n(k)∑i=1
(f (tki )− f (tki−1)
)2
converge for any nested partitions i.e. πk ⊂ πk+1 with tk0 = a, tkn(k) = b
and mesh(πk)→ 0 as k ↑ +∞.
Sample paths of a Brownian motion have this property with probability Pequal 1 - a result due to Paul Levy.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 26 / 29
Norvaisa’s integral for functions with finite quadraticvariation, cont.
Further, he defines Left-Cauchy λ integral of g with respect to f ,(LC )
∫ ba gdf , as a limit of the sums
n(k)∑i=1
g(tki−1)(f (tki )− f (tki−1)
)and proves its existence for the integrands of the form g(t) = ψ(f (t)),where ψ is a C 1 function.
He manages to prove a version of Ito’s formula for this integral. But stillthis is far from the theory of an integral for any (locally bounded)deterministic integrands and finite quadratic variation integrators.
The reason for this might be the fact that the analogous construction forsemimartingales does not require only finiteness of the quadratic variationof the integrator but also a centering property of the increments of a localmartingale part.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 27 / 29
Norvaisa’s integral for functions with finite quadraticvariation, cont.
Further, he defines Left-Cauchy λ integral of g with respect to f ,(LC )
∫ ba gdf , as a limit of the sums
n(k)∑i=1
g(tki−1)(f (tki )− f (tki−1)
)and proves its existence for the integrands of the form g(t) = ψ(f (t)),where ψ is a C 1 function.
He manages to prove a version of Ito’s formula for this integral. But stillthis is far from the theory of an integral for any (locally bounded)deterministic integrands and finite quadratic variation integrators.
The reason for this might be the fact that the analogous construction forsemimartingales does not require only finiteness of the quadratic variationof the integrator but also a centering property of the increments of a localmartingale part.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 27 / 29
Norvaisa’s integral for functions with finite quadraticvariation, cont.
Further, he defines Left-Cauchy λ integral of g with respect to f ,(LC )
∫ ba gdf , as a limit of the sums
n(k)∑i=1
g(tki−1)(f (tki )− f (tki−1)
)and proves its existence for the integrands of the form g(t) = ψ(f (t)),where ψ is a C 1 function.
He manages to prove a version of Ito’s formula for this integral. But stillthis is far from the theory of an integral for any (locally bounded)deterministic integrands and finite quadratic variation integrators.
The reason for this might be the fact that the analogous construction forsemimartingales does not require only finiteness of the quadratic variationof the integrator but also a centering property of the increments of a localmartingale part.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 27 / 29
Some references
Bichteler, K., (1981) Stochastic integration and Lp− theory ofsemimartingales. Ann. Probab., 9(1):49–89.
Karandikar, R. L., (1995) On pathwise stochastic integration. Stoch.Process. Appl., 57(1):11–18.
Lochowski, R., M., (2013) Pathwise stochatic integration with finitevariation processes uniformly approximating cadlag processes,submitted.
Norvaisa, R., (2008) Quadratic Variation, p-variation and Integrationwith Applications to Stock Price Modelling, arXiv.
Wong, E. and Zakai, M., (1965) On the convergence of ordinaryintegrals to stochastic integrals. Ann. Math. Statist., 36:1560–1564.
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 28 / 29
Thank you!
Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 29 / 29