Math 562 Fall 2020rsong/562f20/562-10-07.pdf · General Info 5.1 The Construction of Stochastic Integrals We often use the notation (H M) t = Z t 0 H sdM s and call H M the stochastic

General Info 5.1 The Construction of Stochastic Integrals

Math 562 Fall 2020

Renming Song

University of Illinois at Urbana-Champaign

October 07, 2020


Outline


Outline

1 General Info

2 5.1 The Construction of Stochastic Integrals


I posted HW4 in my homepage. HW4 is due 10/16 at noon. I also setup HW4 in the course Moodle page.

HW3 is graded now.


I posted HW4 in my homepage. HW4 is due 10/16 at noon. I also setup HW4 in the course Moodle page.

HW3 is graded now.


Outline

1 General Info

2 5.1 The Construction of Stochastic Integrals


Recall that P is the progressive σ-field on Ω× R+. If M ∈ H2, we useL2(M) to denote the collection of all progressive processes H suchthat

E[∫ ∞

0H2

s d〈M,M〉s]<∞

with the convention that two progressive processes H and H ′

satisfying this integrability condition are identified if Hs = H ′s,d〈M,M〉s a.e. a.s.

We can view L2(M) as an ordinary L2 space, namely

L2(M) = L2(Ω× R+,P,dPd〈M,M〉s),

where dPd〈M,M〉s refers to the finite measure on (Ω× R+,P) thatassigns the mass

E[∫ ∞

01A(ω, s)d〈M,M〉s

]to a set A ∈ P.


Recall that P is the progressive σ-field on Ω× R+. If M ∈ H2, we useL2(M) to denote the collection of all progressive processes H suchthat

E[∫ ∞

0H2

s d〈M,M〉s]<∞

with the convention that two progressive processes H and H ′

satisfying this integrability condition are identified if Hs = H ′s,d〈M,M〉s a.e. a.s.

We can view L2(M) as an ordinary L2 space, namely

L2(M) = L2(Ω× R+,P,dPd〈M,M〉s),

where dPd〈M,M〉s refers to the finite measure on (Ω× R+,P) thatassigns the mass

E[∫ ∞

01A(ω, s)d〈M,M〉s

]to a set A ∈ P.


Just like any L2 space, the space L2(M) is a Hilbert space for thescalar product

(H,K )L2(M) = E[∫ ∞

0HsKsd〈M,M〉s

]and the associated norm is

‖H‖L2(M) =

(E[∫ ∞

0H2

s d〈M,M〉s])

.

Definition 5.2An elementary process is a progressive process of the form

Hs(ω) =

p−1∑j=0

H(j)(ω)1(tj ,tj+1](s)

where 0 = t0 < t1 < · · · < tp = t and for every j = 0, . . . ,p − 1, H(j) isa bounded Ftj -measurable random variable.


Just like any L2 space, the space L2(M) is a Hilbert space for thescalar product

(H,K )L2(M) = E[∫ ∞

0HsKsd〈M,M〉s

]and the associated norm is

‖H‖L2(M) =

(E[∫ ∞

0H2

s d〈M,M〉s])

.

Definition 5.2An elementary process is a progressive process of the form

Hs(ω) =

p−1∑j=0

H(j)(ω)1(tj ,tj+1](s)

where 0 = t0 < t1 < · · · < tp = t and for every j = 0, . . . ,p − 1, H(j) isa bounded Ftj -measurable random variable.


The family E of all elementary processes forms a linear subspace ofL2(M).

Proposition 5.3

For every M ∈ H2, E is dense in L2(M).

Proof of Proposition 5.3

It suffices to show that, if K ∈ L2(M) is orthogonal to E , then K = 0.Assume that K ∈ L2(M) is orthogonal to E , and define, for everyt ≥ 0,

Xt =

∫ t

0Kud〈M,M〉u.



Proposition 5.3




Xt =

∫ t

0Kud〈M,M〉u.



Proposition 5.3




Xt =

∫ t

0Kud〈M,M〉u.


Proof of Proposition 5.3 (cont)

To see that the integral in the right-hand side makes sense, anddefines a finite variation process (Xt )t≥0, we use the Cauchy-Schwarzinequality to get that

E

[∫ t

0|Ku|d〈M,M〉u

]≤

(E

[∫ t

0K 2

u d〈M,M〉u

])1/2

· (E[〈M,M〉∞])1/2.

The right-hand side is finite since M ∈ H2 and K ∈ L2(M), and thuswe have in particular

a.s. ∀t ≥ 0,∫ t

0|Ku|d〈M,M〉u <∞.

Thus (Xt )t≥0 is well defined as a finite variation process. Thepreceding bound also shows that Xt ∈ L1 for every t ≥ 0.



Let 0 ≤ s < t , let F be a bounded Fs-measurable random-variable,and let H ∈ E be the elementary process defined byHr (ω) = F (ω)1(s,t](r). Since (H,K )L2(M) = 0, we have

E

[F∫ t

sKud〈M,M〉u

]= 0.

It follows that E[F (Xt − Xs)] = 0 for every s < t and every boundedFs-measurable random-variable F . Since the process X is adaptedand we know that Xt ∈ L1 for every t ≥ 0, this implies that X is a(continuous) martingale. On the other hand, X is also a finitevariation process and, by Theorem 4.8, this is only possible if X = 0.



We have thus proved that∫ t

0Kud〈M,M〉u = 0, ∀t ≥ 0,a.s.

which implies that, a.s., the signed measure having density Ku withrespect to d〈M,M〉u is the zero measure, which is only possible if

Ku = 0, d〈M,M〉u − a.e. a.s.

or, equivalently K = 0 in L2(M).

Recall that, for stopping time T , X T stands for the stopped processX T

t = Xt∧T . If M ∈ H2, the fact that 〈MT ,MT 〉∞ = 〈M,M〉Timmediately implies that MT ∈ H2. Furthermore, if H ∈ L2(M), theprocess 1[0,T ]H defined by (1[0,T ]H)s(ω) = 10≤s≤T (ω)Hs(ω) alsobelongs to L2(M).



We have thus proved that∫ t

0Kud〈M,M〉u = 0, ∀t ≥ 0,a.s.

which implies that, a.s., the signed measure having density Ku withrespect to d〈M,M〉u is the zero measure, which is only possible if

Ku = 0, d〈M,M〉u − a.e. a.s.

or, equivalently K = 0 in L2(M).

Recall that, for stopping time T , X T stands for the stopped processX T

t = Xt∧T . If M ∈ H2, the fact that 〈MT ,MT 〉∞ = 〈M,M〉Timmediately implies that MT ∈ H2. Furthermore, if H ∈ L2(M), theprocess 1[0,T ]H defined by (1[0,T ]H)s(ω) = 10≤s≤T (ω)Hs(ω) alsobelongs to L2(M).


Theorem 5.4

Let M ∈ H2. For every H ∈ E of the form

Hs(ω) =

p−1∑j=0

H(j)(ω)1(tj ,tj+1](s).

the formula

(H ·M)t =

p−1∑j=0

H(j)(Mtj+1∧t −Mtj∧t )

defines a process H ·M ∈ H2. The mapping H 7→ H ·M extends to anisometry from L2(M) to H2. Furthermore, H ·M is the uniquemartingale of H2 with the property

〈H ·M,N〉 = H · 〈M,N〉, ∀N ∈ H2. (1)

If T is a stopping time, we have

(1[0,T ]H) ·M = (H ·M)T = H ·MT . (2)


We often use the notation

(H ·M)t =

∫ t

0HsdMs

and call H ·M the stochastic integral of H with respect to M.

The quantity H · 〈M,N〉 in the right-hand side of (1) is an integral withrespect to a finite variation process. The fact that we use a similarnotation H · A and H ·M for the integrals with respect to a finitevariation process A and with respect to a martingale M creates noambiguity since these two classes of processes are essentiallydisjoint.


We often use the notation

(H ·M)t =

∫ t

0HsdMs

and call H ·M the stochastic integral of H with respect to M.

The quantity H · 〈M,N〉 in the right-hand side of (1) is an integral withrespect to a finite variation process. The fact that we use a similarnotation H · A and H ·M for the integrals with respect to a finitevariation process A and with respect to a martingale M creates noambiguity since these two classes of processes are essentiallydisjoint.


Proof of Theorem 5.4First note that the definition of H ·M does not depend on the

decomposition chosen for H in the first display of the theorem. Usingthis remark, one then checks that the mapping H 7→ H ·M is linear.

Now we verify that this mapping is an isometry from E (viewed as asubspace of L2(M)) into H2.

Fix H ∈ E of the form given in the theorem, and for everyj ∈ 0,1, . . . ,p − 1, define

M jt = H(j)(Mtj+1∧t −Mtj∧t ),

for every t ≥ 0. It is easy to check that M j is a martingale and thismartingale belongs to H2. It follows that H ·M =

∑p−1j=0 M j is also a

martingale in H2. Note that the continuous martingales M j areorthogonal, and their respective quadratic variations are given by

〈M j ,M j〉t = H2(j)(〈M,M〉tj+1∧t − 〈M,M〉tj∧t ).


Proof of Theorem 5.4 (cont)

Thus

〈H ·M,H ·M〉t =

p−1∑j=0

H2(j)(〈M,M〉tj+1∧t − 〈M,M〉tj∧t ) =

∫ t

0H2

s d〈M,M〉s.

Consequently,

‖H ·M‖2H2 = E[〈H ·M,H ·M〉∞] = E [

∫ ∞0

H2s d〈M,M〉s] = ‖H‖2

L2(M).

By linearity, this implies that H ·M = H ′ ·M if H ′ is another elementaryprocess that is identified with H in L2(M). Therefore the mappingH 7→ H ·M makes sense from E (viewed as a subspace of L2(M)) intoH2. This latter mapping is linear, and, since it preserves the norm, itis an isometry from E (viewed as a subspace of L2(M)) into H2. SinceE is dense in L2(M) and H2 is a Hilbert space, this mapping can beextended in a unique way to an isometry from L2(M) into H2.



Let us now prove (1). Fix N ∈ H2. We first note that, if H ∈ L2(M), theKunita-Watanabe inequality shows that

E[∫ ∞

0|Hs||d〈M,N〉s|

]≤ ‖H‖L2(M)‖N‖H2 <∞

and thus the variable∫∞

0 Hsd〈M,N〉s = (H · 〈M,N〉)∞ is well definedand in L1. Consider first the case where H is an elementary processof the form given in the theorem, and define the continuousmartingales M j , j = 0, . . .p − 1, as before. Then,

〈H ·M,N〉 =

p−1∑j=0

〈M j ,N〉

and for every j = 0, . . .p − 1,

〈M j ,N〉t = H(j)(〈M,N〉tj+1∧t − 〈M,N〉tj∧t ).



It follows that

〈H ·M,N〉t =

p−1∑j=0

H(j)(〈M,N〉tj+1∧t − 〈M,N〉tj∧t ) =

∫ t

0Hsd〈M,N〉s.

which gives (1) when H ∈ E . We then observe that the linear mappingX 7→ 〈X ,N〉∞ is continuous from H2 into L1. Indeed, by theKunita-Watanabe inequality,

E[〈X ,N〉∞] ≤ E[〈X ,X 〉∞]1/2E[〈N,N〉∞]1/2 = ‖X‖H2‖N‖H2 .

If (Hn)n≥1 is a sequence in E such that Hn → H in L2(M), we havetherefore

〈H ·M,N〉∞ = limn→∞〈Hn ·M,N〉∞ = lim

n→∞(Hn ·〈M,N〉)∞ = (H ·〈M,N〉)∞.

where the convergences hold in L1, and the last equality again followsfrom the Kunita-Watanabe inequality by writing



E[∫ ∞

0(Hn

s − Hs)d〈M,N〉s]≤ E[〈N,N〉∞]1/2‖Hn − H‖L2(M).

We have thus obtained the identity 〈H ·M,N〉∞ = 〈H ·M,N〉∞, butreplacing N by the stopped martingale N t in this identity also gives〈H ·M,N〉t = 〈H ·M,N〉t , which completes the proof of (1).

It is easy to see that (1) characterizes H ·M among the martingalesof H2. Indeed, if X is another martingale of H2 hat satisfies the sameidentity, we get, for every N ∈ H2

〈H ·M − X ,N〉 = 0.

Taking N = H ·M − X and using Proposition 4.12 we obtain thatX = H ·M.



It remains to verify (2). Using the properties of the bracket of twocontinuous local martingales, we observe that, if N ∈ H2,

〈H ·MT ,N〉t = 〈H ·M,N〉t∧T = (H · 〈M,N〉)t∧T = (1[0,T ]H · 〈M,N〉)t

which shows that the stopped martingale (H ·M)T satisfies thecharacteristic property of the stochastic integral (1[0,T ]H) ·M. The firstequality in (2) follows. The second one is proved analogously, writing

〈H ·MT ,N〉 = H · 〈MT ,N〉 = H · 〈M,N〉T = 1[0,T ]H · 〈M,N〉.

This completes the proof of the theorem.


RemarkWe could have used (1) to define the stochastic integral H ·M,observing that the mapping N 7→ E[(H · 〈M,N〉)∞] yields a continuouslinear map on H2, and thus there exists a unique martingale H ·M inH2 such that

E[(H · 〈M,N〉)∞] = (H ·M,N)H2 = E[(〈H ·M,N〉)∞].

Using the notation introduced after Theorem 5.4, we can rewrite (1) inthe form

〈∫ ·

0HsdMs,N〉t =

∫ t

0Hsd〈M,N〉s

We interpret this by saying that the stochastic integral “commutes”with the bracket. Let us immediately mention a very importantconsequence.


RemarkWe could have used (1) to define the stochastic integral H ·M,observing that the mapping N 7→ E[(H · 〈M,N〉)∞] yields a continuouslinear map on H2, and thus there exists a unique martingale H ·M inH2 such that

E[(H · 〈M,N〉)∞] = (H ·M,N)H2 = E[(〈H ·M,N〉)∞].

Using the notation introduced after Theorem 5.4, we can rewrite (1) inthe form

〈∫ ·

0HsdMs,N〉t =

∫ t

0Hsd〈M,N〉s

We interpret this by saying that the stochastic integral “commutes”with the bracket. Let us immediately mention a very importantconsequence.


If M ∈ H2 and H ∈ L2(M), two successive applications of (1) give

〈H ·M,H ·M〉 = H · (H · 〈M,M〉) = H2 · 〈M,M〉

using the “associativity property” integrals with respect to finitevariation processes. Put differently, the quadratic variation of thecontinuous martingale H ·M is

〈∫ ·

0HsdMs,

∫ ·0

HsdMs〉t =

∫ t

0H2

s d〈M,M〉s.

More generally, if N is another martingale of H2 and K ∈ L2(N), thesame argument gives

〈∫ ·

0HsdMs,

∫ ·0

KsdNs〉t =

∫ t

0HsKsd〈M,N〉s.

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Math 562 Fall 2020rsong/562f20/562-10-07.pdf · General Info 5.1 The Construction of Stochastic Integrals We often use the notation (H M) t = Z t 0 H sdM s and call H M the stochastic