Evans L.C. An Introduction to Stochastic Differential Equations

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Example 2. Again take X(·)=W (·), u(x, t)=e

λx−

, F ≡ 0, G ≡ 1. Then



λW (t)−





−

λW (t)−



dt + λe

λW (t)−

by Itˆo’s formula. Thus



dY = λY dW

Y (0) = 1.

This is a stochastic diﬀerential equation, about which more in Chapters 5 and 6. 

In the Itˆo stochastic calculus the expression e

λW (t)−

plays the role that e

λt

plays in

ordinary calculus. We build upon this observation:

Example 3. For n =0, 1,..., deﬁne

(x, t):=

(−t)

/2t



−x

/2t



the n-th Hermite polynomial. Then

(x, t)=1,h

(x, t)=x

(x, t)=

−

(x, t)=

−

(x, t)=

−

, etc.

THEOREM (Stochastic calculus with Hermite polynomials). We have



(W, s) dW = h

n+1

(W (t),t) for t ≥ 0and n =0, 1,...;

that is,

n+1

(W, t)=h

(W, t) dW.

Consequently in the Itˆo stochastic calculus the expression h

(W (t),t) plays the role

that

plays in ordinary calculus.

Proof. (from McKean [McK]) Since

dλ

−

(x−λt)

λ=0

=(−t)

−x

/2t

we have

dλ

λx−

λ=0

=(−t)

/2t

−x

/2t

)

= n!h

(x, t).

Hence

λx−

∞



n=0

(x, t),

and so

Y (t)=e

λW (t)−

∞



n=0

(W (t),t).

But Y (·) solves



dY = λY dW

Y (0) = 1;

that is,

Y (t)=1+λ



YdW for all t ≥ 0.

Plug in the expansion above for Y (t):

∞



n=0

(W (t),t)=1+λ



∞



n=0

(W (s),s) dW

=1+

∞



n=1



n−1

(W (s),s) dW.

This identity holds for all λ and so the coeﬃcients of λ

on both sides are equal. 

PROOF OF IT

O’S FORMULA. We now begin the proof of Itˆo’s formula, by veri-

fying directly two important special cases:

LEMMA (Two simple stochastic diﬀerentials). We have

(i) d(W

)=2WdW + dt,

and

(ii ) d(tW )=Wdt+ tdW .

Proof. We have already established formula (i). To verify (ii), note that



tdW = lim

n→∞

−1



k=0

(W (t

k+1

) − W (t

)),

where P

= {0=t

< ··· <t

= r} is a sequence of partitions of [0,r], with

|→0. The limit above is taken in L

(Ω).

Similarly, since t → W (t) is continuous a.s.,



Wdt= lim

n→∞

−1



k=0

W (t

k+1

)(t

k+1

− t

since for amost every ω the sum is an ordinary Riemann sum approximation and for this

we can take the right-hand endpoint t

k+1

at which to evaluate the continuous integrand.

We add these formulas to obtain



tdW +



Wdt= rW (r).

These integral identities for all r ≥ 0 are abbreviated d(tW )=tdW + Wdt. 

These special cases in hand, we now prove:

THEOREM (Itˆo product rule). Suppose

= F

dt + G

= F

dt + G

(0 ≤ t ≤ T ),

for F

∈ L

(0,T), G

∈ L

(0,T)(i =1, 2). Then

(4) d(X

)=X

+ X

+ G

dt.

Remarks. (i) The expression G

dt here is the Itˆo correction term. The integrated

version of the product rule is the Itˆo integration-by-parts formula:

(5)



= X

(r)X

(r) − X

(s)X

(s) −



−



dt.

(ii) If either G

or G

is identically equal to 0, we get the ordinary calculus integration-

by-parts formula. This conﬁrms that the Paley–Wiener–Zygmund deﬁnition



gdW = −





W dt,

for deterministic C

functions g, with g(0) = g(1) = 0, agrees with the Itˆo deﬁnition. 

Proof. 1. Choose 0 ≤ r ≤ T .

First of all, assume for simplicity that X

(0) = X

(0)=0,F

(t) ≡ F

, G

(t) ≡ G

where F

are time–independent, F(0)-measurable random variables (i =1, 2). Then

(t)=F

t + G

W (t)(t ≥ 0,i=1, 2).

Thus



+ X

+ G



+ X

dt +



+ X



t + G

W )F

+(F

t + G

W )F



t + G

W )G

+(F

t + G

W )G

dW + G

= F

+(G

+ G

)



Wdt+



tdW

+2G



WdW+ G

We now use the Lemma above to compute 2



WdW= W

(r)−r and



Wdt+



tdW =

rW(r). Employing these identities, we deduce:



+ X

+ G

= F

+(G

+ G

)rW(r)+G

(r)

= X

(r)X

(r).

This is formula (5) for the special circumstance that s =0,X

(0) = 0, and F

time–

independent random variables.

The case that s ≥ 0, X

(s),X

(s) are arbitrary, and F

are constant F(s)-measurable

random variables has a similar proof.

2. If F

are step processes, we apply Step 1 on each subinterval [t

k+1

) on which

and G

are constant random variables, and add the resulting integral expressions.

3. In the general situation, we select step processes F

∈ L

(0,T), G

∈ L

(0,T), with



− F

|dt) → 0



− G

)

dt) → 0

as n →∞, i =1, 2.

Deﬁne

(t):=X

(0) +



ds +



dW (i =1, 2).

We apply Step 2 to X

(·)on(s, r) and pass to limits, to obtain the formula

(r)X

(r)=X

(s)X

(s)+



+ X

+ G

dt.



Now we are ready for

CONCLUSION OF THE PROOF OF IT

O’S FORMULA. Suppose dX = Fdt+

GdW , with F ∈ L

(0,T), G ∈ L

(0,T).

1. We start with the case u(x)=x

, m =0, 1,..., and ﬁrst of all claim that

(6) d(X

)=mX

m−1

dX +

m(m − 1)X

m−2

dt.

This is clear for m =0, 1, and the case m = 2 follows from the Itˆo product formula. Now

assume the stated formula for m − 1:

d(X

m−1

)=(m − 1)X

m−2

dX +

(m − 1)(m − 2)X

m−3

=(m − 1)X

m−2

(Fdt+ GdW )+

(m − 1)(m − 2)X

m−3

dt,

and we prove it for m:

d(X

)=d(XX

m−1

)

= Xd(X

m−1

)+X

m−1

dX +(m − 1)X

m−2

(by the product rule)

= X



(m − 1)X

m−2

dX +

(m − 1)(m − 2)X

m−3



+(m − 1)X

m−2

dt + X

m−1

= mX

m−1

dX +

m(m − 1)X

m−2

dt,

because m − 1+

(m − 1)(m − 2) =

m(m − 1). This proves (6).

Since Itˆo’s formula thus holds for the functions u(x)=x

, m =0, 1,... and since the

operator “d” is linear, Itˆo’s formula is valid for all polynomials u in the variable x.

2. Suppose now u(x, t)=f(x)g(t), where f and g are polynomials. Then

d(u(X, t)) = d(f(X)g)

= f(X)dg + gdf(X)

= f(X)g



dt + g[f



(X)dX +



(X)G

dt]

∂u

∂t

dt +

∂u

∂x

dX +

∂

∂x

dt.

This calculation conﬁrms Itˆo’s formula for u(x, t)=f(x)g(t), where f and g are polyno-

mials. Thus it is true as well for any function u having the form

u(x, t)=



i=1

(x)g

(t),

where f

and g

polynomials. That is, Itˆo’s formula is valid for all polynomial functions u

of the variables x, t.

3. Given u as in Itˆo’s formula, there exists a sequence of polynomials u

such that

→ u,

∂u

∂t

→

∂u

∂t

∂u

∂x

→

∂u

∂x

∂

∂x

→

∂

∂x

uniformly on compact subsets of R×[0,T]. Invoking Step 2, we know that for all 0 ≤ r ≤ T ,

(X(r),r) − u

(X(0), 0) =



∂u

∂t

∂u

∂x

F +

∂

∂x



∂u

∂x

GdW almost surely;

the argument of the partial derivatives of u

is (X(t),t).

We may pass to limits as n →∞in this expression, thereby proving Itˆo’s formula in

general. 

A similar proof gives this:

GENERALIZED IT

O FORMULA. Suppose dX

= F

dt + G

dW , with for F

∈

(0,T), G

∈ L

(0,T), for i =1,...,n.

If u : R

×[0,T] → R is continuous, with continuous partial derivatives

∂u

∂t

∂u

∂x

∂

∂x

(i, j =1,...,n), then

d(u(X

,...,X

,t)) =

∂u

∂t

dt +



i=1

∂u

∂x



i,j=1

∂

∂x

dt.



E. IT

O’S INTEGRAL IN HIGHER DIMENSIONS.

Notation. (i) Let W(·)=(W

(·),...,W

(·)) be an m-dimensional Brownian motion.

(ii) We assume F(·) is a family of nonanticipating σ-algebras, meaning that

(a) F(t) ⊇F(s) for all t ≥ s ≥ 0

(b) F(t) ⊇W(t)=U(W(s) |0 ≤ s ≤ t)

(t):=U(W(s) − W(t) |t ≤ s<∞).

DEFINITIONS. (i) An M

n×m

-valued stochastic process G =((G

)) belongs to L

n×m

(0,T)

∈ L

(0,T)(i =1,...n; j =1,...m).

(ii) An R

-valued stochastic process F =(F

,...,F

) belongs to L

(0,T)if

∈ L

(0,T)(i =1,...n).

DEFINITION. If G ∈ L

n×m

(0,T), then



G dW

is an R

–valued random variable, whose i-th component is



j=1



(i =1,...,n).



Approximating by step processes as before, we can establish this

LEMMA. If G ∈ L

n×m

(0,T), then





G dW



=0,

and









G dW







= E





|G|



where |G|



1≤i≤n

1≤j≤m

DEFINITION. If X(·)=(X

(·),...,X

(·)) is an R

-valued stochastic process such that

X(r)=X(s)+



F dt +



G dW

for some F ∈ L

(0,T), G ∈ L

n×m

(0,T) and all 0 ≤ s ≤ r ≤ T ,wesayX(·) has the

stochastic diﬀerential

dX = Fdt + GdW.

This means that

= F

dt +



j=1

for i =1,...,n.

THEOREM (Itˆo’s formula in n-dimensions). Suppose that dX = Fdt + GdW,as

above. Let u : R

×[0,T] be continuous, with continuous partial derivatives

∂u

∂t

∂u

∂x

∂

∂x

(i, j =1,...,n). Then

(5)

d(u(X(t),t))=

∂u

∂t

dt +



i=1

∂u

∂x



i,j=1

∂

∂x



l=1

dt,

where the argument of the partial derivatives of u is (X(t),t).

An outline of the proof follows some preliminary results:

LEMMA (Another simple stochastic diﬀerential). Let W (·) and

W (·) be indepen-

dent 1-dimensional Brownian motions. Then

d(W

W )=Wd

W +

WdW.

Compare this to the case W =

W . There is no correction term “dt” here, since W,

are independent.

Proof. 1. To begin, set X(t):=

W (t)+

W (t)

√

We claim that X(·) is a 1-dimensional Brownian motion. To see this, note ﬁrstly that

X(0) = 0 a.s. and X(·) has independent increments. Next observe that since X is the

sum of two independent, N(0,

) random variables, X(t)isN(0,t). A similar observation

shows that X(t) − X(s)isN(0,t− s) for t ≥ s. This establishes the claim.

2. From the 1-dimensional Itˆo calculus, we know











d(X

)=2XdX + dt,

d(W

)=2WdW + dt,

)=2

W + dt.

Thus

d(W

W )=d



−



=2XdX + dt −

(2WdW + dt)

−

W + dt)

=(W +

W )(dW + d

W ) − WdW −

= Wd

W +

WdW.



We will also need the following modiﬁcation of the product rule:

LEMMA (Itˆo product rule with several Brownian motions). Suppose

= F

dt +



k=1

and

= F

dt +



l=1

where F

∈ L

(0,T) and G

∈ L

(0,T) for i =1, 2; k =1,...,m. Then

d(X

)=X

+ X



k=1

dt.

The proof is a modiﬁcation of that for the one–dimensional Itˆo product rule, as before,

with the new feature that

d(W

)=W

+ W

+ δ

dt,

according to the Lemma above.

The Itˆo formula in n-dimensions can now be proved by a suitable modiﬁcation of the

one-dimensional proof. We ﬁrst establish the formula for a multinomials u = u(x)=

...x

, proving this by an induction on k

,...,k

, using the Lemma above. This done,

the formula follows easily for polynomials u = u(x, t) in the variables x =(x

,...,x

) and

t, and then, after an approximation, for all functions u as stated.

CLOSING REMARKS.

1. ALTERNATIVE NOTATION. When

dX = Fdt + GdW,

we sometimes write



k=1

Then Itˆo’s formula reads

du(X,t)=



∂u

∂t

+ F · Du +

H : D



dt + Du · GdW,

where Du =



∂u

∂x

,...,

∂u

∂x



is the gradient of u in the x-variables, D

u =



∂

∂x



the Hessian matrix, and

F · Du =



i=1

∂u

∂x

H : D

u =



i,j=1

∂

∂x

Du · GdW =



i=1



k=1

∂u

∂x

2. HOW TO REMEMBER IT

O’S FORMULA.

We may symbolically compute

d(u(X,t)) =

∂u

∂t

dt +



i=1

∂u

∂x



i,j=1

∂

∂x

and then simplify the term “dX

” by expanding it out and using the formal multipli-

cation rules

(dt)

=0, dtdW

=0,dW

= δ

dt (k, l =1,...,m).

The foregoing theory provides a rigorous meaning for all this.