Math 562 Fall 2020rsong/562f20/562-10...General Info 5.1 The Construction of Stochastic Integrals Math 562 Fall 2020 Renming Song University of Illinois at Urbana-Champaign October

General Info 5.1 The Construction of Stochastic Integrals

Math 562 Fall 2020

Renming Song

University of Illinois at Urbana-Champaign

October 09, 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.


Outline

1 General Info

2 5.1 The Construction of Stochastic Integrals


Proposition 5.5

Let M ∈ H2 and H ∈ L2(M). If K is a progressive process, we haveKH ∈ L2(M) if and only if K ∈ L2(H ·M). If the latter properties hold,

(KH) ·M = K · (H ·M).

Proof of Proposition 5.5

By the associative property of stochastic integral with bracket, wehave

E[∫ ∞

0K 2

s H2s d〈M,M〉s

]= E

[∫ ∞0

K 2s d〈H ·M,H ·M〉s

],

which gives the first assertion. For the second one, we write forN ∈ H2,

〈(KH) ·M,N〉 = KH · 〈M,N〉 = K · (H · 〈M,N〉) = K · 〈H ·M,N〉

and, by the uniqueness statement in (5.2), this implies that(KH) ·M = K · (H ·M).


Proposition 5.5

Let M ∈ H2 and H ∈ L2(M). If K is a progressive process, we haveKH ∈ L2(M) if and only if K ∈ L2(H ·M). If the latter properties hold,

(KH) ·M = K · (H ·M).

Proof of Proposition 5.5

By the associative property of stochastic integral with bracket, wehave

E[∫ ∞

0K 2

s H2s d〈M,M〉s

]= E

[∫ ∞0

K 2s d〈H ·M,H ·M〉s

],

which gives the first assertion. For the second one, we write forN ∈ H2,

〈(KH) ·M,N〉 = KH · 〈M,N〉 = K · (H · 〈M,N〉) = K · 〈H ·M,N〉

and, by the uniqueness statement in (5.2), this implies that(KH) ·M = K · (H ·M).


Moments of stochastic integrals

Let M,N ∈ H2, H ∈ L2(M) and K ∈ L2(N). Since H ·M,K · N ∈ H2,we have, for every t ∈ [0,∞),

E[

∫ t

0HsdMs] = 0 (1)

E

[(∫ t

0HsdMs

)(∫ t

0KsdNs

)]= E

[∫ t

0HsKsd〈M,N〉s

](2)

using Proposition 4.15 (v) and the associative property of stochasticintegral and bracket. In particular,

E

(∫ t

0HsdMs

)2 = E

[∫ t

0H2

s d〈M,M〉s

]. (3)


Furthermore, since H ·M is a (true) martingale, we also have forevery 0 ≤ s < t ≤ ∞,

E[

∫ t

0Hr dMr |Fs] =

∫ s

0Hr dMr (4)

or equivalently

E[

∫ t

sHr dMr |Fs] = 0.

It is important to observe that these formulas (and particularly (1) and(3)) may no longer hold for the extensions of stochastic integrals thatwe will now describe.


We now extend the definition of H ·M to an arbitrary continuous localmartingale. If M is a continuous local martingale, we write L2

loc(M)(resp. L2(M)) for the collection of all progressive processes H suchthat∫ t

0H2

s d〈M,M〉s <∞, ∀t ≥ 0 a.s.(

resp. E[

∫ ∞0

H2s d〈M,M〉s] <∞

).

Note that L2(M) (with the same identifications as in the case whereM ∈ H2) can again be viewed as an “ordinary” L2-space and thus hasa Hilbert space structure.


Theorem 5.6

Let M be a continuous local martingale. For every H ∈ L2loc(M), there

exists a unique continuous local martingale with initial value 0, whichis denoted by H ·M, such that, for every continuous local martingaleN,

〈H ·M,N〉 = H · 〈M,N〉. (5)

If T is a stopping time, we have

(1[0,T ]H) ·M = (H ·M)T = H ·MT . (6)

If H ∈ L2loc(M), and K is a progressive process, we have

K ∈ L2loc(H ·M) if and only if HK ∈ L2

loc(M) and then

K · (H ·M) = HK ·M. (7)

Finally, if M ∈ H2 and H ∈ L2(M), the definition of H ·M is consistentwith that of Theorem 5.4.


Proof of Theorem 5.6We may assume that M0 = 0 (in the general case, we write

M = M0 + M ′ and set H ·M = H ·M ′, noting that 〈M,N〉 = 〈M ′,N〉 forevery continuous local martingale N). Also we may assume that theproperty

∫ t0 H2

s d〈M,M〉s <∞ for every t ≥ 0 holds for every ω ∈ Ω(on the negligible set where this fails we may replace H by 0).

F or every n ≥ 1, define

Tn = inft ≥ 0 :

∫ t

0(1 + H2

s )d〈M,M〉s ≥ n,

so that (Tn)n≥1 is a sequence of stopping times that increase to∞.Since

〈MTn ,MTn〉t = 〈M,M〉t∧Tn ≤ n,

the stopped martingale MTn is in H2. Furthermore, we also have


Proof of Theorem 5.6 (cont)∫ ∞0

H2s d〈MTn ,MTn〉s =

∫ Tn

0H2

s d〈M,M〉s ≤ n.

Hence H ∈ L2(MTn ), and the definition of H ·MTn makes sense byTheorem 5.4. Moreover, by property (5.3), we have, if m > n,

H ·MTn = (H ·MTm )Tn .

It follows that there exists a unique process denoted by H ·M suchthat, for every n,

(H ·M)Tn = H ·MTn .

Clearly H ·M has continuous sample paths and is also adapted since(H ·M)t = limn→∞(H ·MTn )t . Since the processes H ·MTn aremartingales in H2, we get that H ·M is a continuous local martingale.


Proof of Theorem 5.6 (cont)

To verify (5), we may assume that N is a continuous local martingalesuch that N0 = 0. For every n ≥ 1, define T ′n = inft ≥ 0 : |Nt | ≥ nand Sn = Tn ∧ T ′n. Then, noting that NT ′

n ∈ H2, we have

〈H ·M,N〉Sn = 〈(H ·M)Tn ,NT ′n 〉 = 〈H ·MTn ,NT ′

n 〉

= H · 〈MTn ,NT ′n 〉 = H · 〈M,N〉Sn = (H · 〈M,N〉)Sn ,

which gives the equality 〈H ·M,N〉 = H · 〈M,N〉, since Sn ↑ ∞ ast ↑ ∞. The fact that this equality (written for every continuous localmartingale N) characterizes H ·M among continuous localmartingales with initial value 0 is derived from Proposition 4.12 as inthe proof of Theorem 5.4.



(6) is then obtained by the very same arguments as in the proof ofproperty (5.3) in Theorem 5.4 (these arguments only depended onthe characteristic property (5.2) which we have just extended in (5).Similarly, the proof of (7) is analogous to the proof of Proposition 5.5.

Finally, if M ∈ H2 and H ∈ L2(M), the equality〈H ·M,H ·M〉 = H2 · 〈M,M〉 follows from (5), and implies thatH ·M ∈ H2. Then the characteristic property (5.2) shows that thedefinitions of Theorems 5.4 and 5.6 are consistent.

In the setting of Theorem 5.6, we will again write

(H ·M)t =

∫ t

0HsdMs.

The formulas (5.4) and (5.5) remain valid when M and N arecontinuous local martingales and H ∈ L2

loc(M) and K ∈ L2loc(N).

Indeed, these formulas immediately follow from (5).



(6) is then obtained by the very same arguments as in the proof ofproperty (5.3) in Theorem 5.4 (these arguments only depended onthe characteristic property (5.2) which we have just extended in (5).Similarly, the proof of (7) is analogous to the proof of Proposition 5.5.

Finally, if M ∈ H2 and H ∈ L2(M), the equality〈H ·M,H ·M〉 = H2 · 〈M,M〉 follows from (5), and implies thatH ·M ∈ H2. Then the characteristic property (5.2) shows that thedefinitions of Theorems 5.4 and 5.6 are consistent.

In the setting of Theorem 5.6, we will again write

(H ·M)t =

∫ t

0HsdMs.

The formulas (5.4) and (5.5) remain valid when M and N arecontinuous local martingales and H ∈ L2

loc(M) and K ∈ L2loc(N).

Indeed, these formulas immediately follow from (5).


Connection with Wiener integral

Suppose that B is an (Ft )-Brownian motion, andh ∈ L2(R+,B(R+),dt) is a deterministic square integrable function.We can then define the Wiener integral

∫ t0 h(s)dBs = G(h1[0,t]),

where G is the Gaussian white noise associated with B. It is easy toverify that this integral coincides with the stochastic integral (h · B)twhich makes sense by viewing h as a (deterministic) progressiveprocess. This is immediate when h is a simple function, and thegeneral case follows from a density argument.

Let us now discuss the extension of the moment formulas that westated above in the setting of Theorem 5.4. Let M be a continuouslocal martingale, H ∈ L2

loc(M) and t ∈ [0,∞]. Then, under thecondition

E

[∫ t

0H2

s d〈M,M〉s

]<∞, (8)

we can apply Theorem 4.13 to (H ·M)t and get that (H ·M)t is amartingale in H2. It follows that properties (5.6) and (5.8) still hold:


Connection with Wiener integral

Suppose that B is an (Ft )-Brownian motion, andh ∈ L2(R+,B(R+),dt) is a deterministic square integrable function.We can then define the Wiener integral

∫ t0 h(s)dBs = G(h1[0,t]),

where G is the Gaussian white noise associated with B. It is easy toverify that this integral coincides with the stochastic integral (h · B)twhich makes sense by viewing h as a (deterministic) progressiveprocess. This is immediate when h is a simple function, and thegeneral case follows from a density argument.

Let us now discuss the extension of the moment formulas that westated above in the setting of Theorem 5.4. Let M be a continuouslocal martingale, H ∈ L2

loc(M) and t ∈ [0,∞]. Then, under thecondition

E

[∫ t

0H2

s d〈M,M〉s

]<∞, (8)

we can apply Theorem 4.13 to (H ·M)t and get that (H ·M)t is amartingale in H2. It follows that properties (5.6) and (5.8) still hold:


E

[∫ t

0HsdMs

]= 0, E

(∫ t

0HsdMs

)2 = E

[∫ t

0H2

s d〈M,M〉s

],

and similarly (5.9) is valid for 0 ≤ s ≤ t . In particular (case t =∞), ifH ∈ L2(M), the continuous local martingale H ·M is in H2 and itsterminal value satisfies

E

[(∫ ∞0

HsdMs

)2]

= E[∫ ∞

0H2

s d〈M,M〉s].

If the condition (8) does not hold, the previous formulas may fail.However, we always have the bound

E

(∫ t

0HsdMs

)2 ≤ E

[∫ t

0H2

s d〈M,M〉s

]


We now extend the definition of stochastic integrals to continuoussemimartingales. We say that a progressive process H is locallybounded if

∀t ≥ 0, sups≤t|Hs| <∞, a.s.

In particular, any adapted process with continuous sample paths is alocally bounded progressive process. If H is (progressive and) locallybounded, then for every finite variation process V , we have

∀t ≥ 0,∫ t

0|Hs||dVs| <∞, a.s.

and similarly H ∈ L2loc(M) for every continuous local martingale M.


Definition 5.7Let X be a continuous semimartingale and let X = M + V be itscanonical decomposition. If H is a locally bounded progressiveprocess, the stochastic integral H · X is the continuoussemimartingale with canonical decomposition

H · X = H ·M + H · V

and we write

(H · X )t =

∫ t

0HsdXs.

Properties

(i) The mapping (H,X ) 7→ H · X is bilinear.(ii) H · (K · X ) = (HK ) · X if H and K are progressive and locally

bounded.


Definition 5.7Let X be a continuous semimartingale and let X = M + V be itscanonical decomposition. If H is a locally bounded progressiveprocess, the stochastic integral H · X is the continuoussemimartingale with canonical decomposition

H · X = H ·M + H · V

and we write

(H · X )t =

∫ t

0HsdXs.

Properties

(i) The mapping (H,X ) 7→ H · X is bilinear.(ii) H · (K · X ) = (HK ) · X if H and K are progressive and locally

bounded.


Properties (cont)

(iii) For every stopping time T , (H · X )T = H1[0,T ] · X = H · X T .(iv) If X is a continuous local martingale, resp. if X is a finite variation

process, then the same holds for H · X .

(v) If H is of the form Hs(ω) =∑p−1

j=0 H(j)(ω)1(tj ,tj+1](s), where0 = t0 < t1 < · · · < tp, and, for every j = 0, . . . ,p − 1, H(j) isFtj -measurable, then

(H · X )t =

p−1∑j=0

H(j)(Xtj+1∧t − Xtj∧t ).

Property (ii) can be restated as, if Yt =∫ t

0 KsdXs then∫ t

0HsdYs =

∫ t

0HsKsdXs.


Properties (cont)

(iii) For every stopping time T , (H · X )T = H1[0,T ] · X = H · X T .(iv) If X is a continuous local martingale, resp. if X is a finite variation

process, then the same holds for H · X .

(v) If H is of the form Hs(ω) =∑p−1

j=0 H(j)(ω)1(tj ,tj+1](s), where0 = t0 < t1 < · · · < tp, and, for every j = 0, . . . ,p − 1, H(j) isFtj -measurable, then

(H · X )t =

p−1∑j=0

H(j)(Xtj+1∧t − Xtj∧t ).

Property (ii) can be restated as, if Yt =∫ t

0 KsdXs then∫ t

0HsdYs =

∫ t

0HsKsdXs.


Properties (i)–(iv) easily follow from the results obtained when X is acontinuous local martingale, resp. a finite variation process. As forproperty (v), we first note that it is enough to consider the case whereX = M is a continuous local martingale with M0 = 0, and by stoppingM at suitable stopping times, we can even assume that M is in H2.There is a minor difficulty coming from the fact that the variables H(j)are not assumed to be bounded (and therefore we cannot directly usethe construction of the integral of elementary processes). Tocircumvent this difficulty, we set, for every n ≥ 1,

Tn = inft ≥ 0 : |Ht | ≥ n = inftj : |H(j)| ≥ n.

It is easy to verify that Tn is a stopping time, and we have Tn ↑ ∞ asn ↑ ∞.


Furthermore, we have for every n,

Hs1[0,Tn](s) =

p−1∑j=0

Hn(j)1(tj ,tj+1](s)

where the random variables Hn(j) = H(j)1Tn>tj satisfy the same

properties as the H(j)’s and additionally are bounded by n. HenceH1[0,Tn] is an elementary process, and by the very definition of thestochastic integral with respect to a martingale of H2, we have

(H ·M)t∧Tn = (H1[0,Tn] ·M)t =

p−1∑j=0

Hn(j)(Mtj+1∧t −Mtj∧t )

The desired result now follows by letting n tend to infinity.

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Math 562 Fall 2020rsong/562f20/562-10...General Info 5.1 The Construction of Stochastic Integrals Math 562 Fall 2020 Renming Song University of Illinois at Urbana-Champaign October