Martingales in continuous time

6 Martingales in continuous time

Just as in discrete time, the notion of a martingale will play a key r^ole in our continuous time models.

Recall that in discrete time, a sequence X 0

;X 1

;:::;X

nfor which E[jX

r

j]<1

for each r is a martingale if

E[X r

jF r,1] =

X r,1

:

The sequence fF r

g n

r=0 is called a ltration. In order to avoid abstract measure

theory, we omit a detailed discussion of ltrations in continuous time. Instead we content ourselves with the following `working denition'.

Denition 6.1

The symbolF X

t denotes the `information generated by the

stochas-tic process X on the interval [0;t]'. If, based upon observations of the trajectory fX(s);0stg, it is possible to decide whether a given event A has occured or

not, then we write this as

A2F X t

:

If the value of a stochastic variable can be completely determined given observa-tions of the trajectory fX(s);0s tg then we also write

Z 2F X t

:

If Y is a stochastic process such that we have Y(t) 2 F X

t for all

t 0, then we

say that Y is adapted to the ltration fF X t

g t0.

This denition is only intended to have intuitive content. Nevertheless, it is rather simple to use.

Example 6.2

1. Dene A by

A=fX(s)3:14;8s18g:

ThenA2 F X 18 but

A2= F X 17.

2. For the event A=fX(10) >8g, A2F X

s if and only if

s10.

3. The stochastic variable

Z(s) = Z

5

0

X(s)ds

is in F X

s if and only if

s 5.

4. If B

t is Brownian motion and M

t = max0st B

s, then

M is adapted to the

Brownian ltration. 5. If B

t is Brownian motion and ~ M

t = max0st+1 B

s, then ~

M is not adapted

to the Brownian ltration.

Denition 6.3

Consider a probability space (;P) and a ltration fF t

g t0 on

this space. An adapted family fM t

g

t0 of random variables on this space with E[jM

t

j]<1 for all t0 is a martingale if, for any s t, E

P[ M

t jF

s] = M

Remarks:

1. Often we shall be sloppy about specifying the ltration. In all of our exam-ples there will be a Brownian motion around and it will be implicit that the ltration is that generated by the Brownian motion.

2. Many stochastic calculus texts specify a probability space as (;fF t

g t0

;P),

thereby specifying the ltration explicitly from the very beginning. 3. The natural ltration is the name given to the ltration for whichF

tconsists

of those sets that can be decided by observing trajectories of the specied process up to time t. There are other possibilities, but we always use the

natural ltration corresponding to Brownian motion in our examples.

Lemma 6.4

If fB t

g

t0 is a standard Brownian motion generating the ltration fF

t g

t0, then

1. B t is an

F

t-martingale.

2. B 2 t

,t is an F

t-martingale.

3.

exp

B

t ,

2

2 t

is an F

t-martingale (called an exponential martingale).

Proof:

The proofs are all rather similar. For example, consider M t =

B 2 t

,t.

EvidentlyE[jM t

j]<1. Now

E

B 2 t

,B 2 s

F s

= E

(B t

,B s)

2

+ 2B s(

B t

,B s)

F s

= E

(B t

,B s)

2

F s

+ 2B s

E[(B t

,B s)

jF s]

= t,s:

Thus

E

B 2 t

,t

F s

=E

B 2 t

,B 2 s +

B 2 s

,(t,s),s

F s

= (t,s) +B 2 s

,(t,s),s=B 2 s

,s:

2

Theorem 6.5 (Optional Sampling Theorem)

IffM t

g

t0is a continuous

mar-tingale with respect to the ltrationfF t

g

t0 and if

1 and

2 are two stopping times

such that 1

2

K where K is a nite real number, then M

2 is integrable

(that is has nite expectation) and

E[M 2

jF 1] =

M 1

; P,a:s:

Remarks:

2. Notice in particular that if is a bounded stopping time thenE[M ] =

E[M 0].

3. The name Optional Sampling Theorem is used because stopping times are sometimes also called optional times.

We illustrate the application of this by calculating the moment generating function for the hitting timeT

a of level

a by Brownian motion.

Proposition 6.6

Let B

t be a Brownian motion and let T

a= inf

fs0 :B s =

ag

(or innity if that set is empty). Then

E

e ,Ta

=e ,

p 2 jaj

:

Proof:

We assume thata 0. (The case a<0 follows by symmetry.) We apply

the Optional Sampling Theorem to the martingale

M

t = exp

B t

,

1 2

2 t

:

We cannot apply it directly to T

a as it may not be bounded. Instead we take

1 = 0 and

2 = T

a

^n. This gives us that

E[M T

a

^n] = 1 :

Now

0M

Ta^n = exp

B Ta^n

,

1 2

2( T

a ^n)

exp (a):

On the other hand, ifT a

<1, lim n!1

M Ta^n =

M

Ta, and if T

a = 1, B

t

a for

allt and so lim n!1

M

Ta^n = 0. From the Dominated Convergence Theorem,

E

Ta<1exp

,

1 2

2 T

a+ a

= lim

n!1 E[M

Ta^n] = 1 :

Taking 2 = 2

completes the proof. 2

Warning:

It was essential that there was a dominating random variable here. In this setting, the Dominated Convergence Theorem says that for a sequence of random variables Z

n, with limn!1 Z

n =

Z, if there is a random variable Y with jZ

n

jY for alln andE[Y]<1, then we may deduce thatE[Z] = lim n!1

E[Z n].

In this example, the dominating random variable is just the constant e a.

7 Stochastic integration and It^o's formula

We already saw (in Lemma 6.4) a number of random variables that are martingales with respect to the same probability measure and adapted to the same ltration. All those martingales were expressed in terms of the underlying Brownian motion. In this section, we shall see that this situation is generic: if there is a measure

Q under which the process M t is an

F

t-martingale, then any other

Q-martingale

adapted to the same ltrationF

t can be expressed in terms of M

technical assumptions). To understand this representation, we must rst under-stand the notion of stochastic integral.

Processes that model stock prices are usually functions of one or more Brow-nian motions. Here, for simplicity, we restrict ourselves to functions of just one Brownian motion. The rst thing that we should like to do is to write down a dierential equation for the way in which the stock price evolves. The diculty is that Brownian motion is `too rough' for the familiar Newtonian calculus to be any help to us.

Suppose that the stock price is of the formS t=

f(B

t). Formally, using Taylor's

Theorem (and assuming thatf at least is nice),

f(B t+t)

,f(B

t) = ( B

t+t ,B

t) f

0

(B

t) (4)

+ 12! (B t+t

,B t)

2 f

00( B

t) + :

Now in our usual derivation of the chain rule, whenB

t is replaced by a Lipschitz

function, the second term on the right hand side is order O(t

2). However, for

Brownian motion, we know that E[(B t+t

,B t)

2] is

t. Consequently we cannot

ignore the term involving the second derivative. Of course, now we have a problem, because we must interpret the term involving the rst derivative. If (B

t+t ,B

t) 2

is O(t), then (B t+t

, B

t) should be O(

p

t), which could lead to unbounded

changes inyover a bounded time interval. However, things are not hopeless. The

expected value of B t+t

,B

t is zero, and the uctuations around zero are on the

order of p

t. By comparison with the Central Limit Theorem, we see that it is

possible that S t

,S

0 is a bounded random variable. Assuming that we can make

this rigorous, the dierential equation governingS t =

f(B

t) will take the form

dS t=

f 0

(B t)

dB t+ 12

f 00

(B t)

dt:

It is convenient to write this in integrated form,

S t =

S 0+

Z t

0 f

0

(B s)

dB s+

Z t

0

1 2f

00

(B s)

ds: (5)

We have to make rigorous mathematical sense of the stochastic integral (that is, the rst integral) on the right hand side of this equation. The key is the following fact.

Brownian motion has nite quadratic variation.

Before proving this we dene total variation and quadratic variation. For a func-tion f : [0;T]!R, its variation is dened in terms of partitions.

Denition 7.1

Let be a partition of [0;T], N() the number of intervals that

make up and () be the mesh of (that is the length of the largest interval

in the partition). Write t i

;t

i+1, for the endpoints of a generic interval of the

partition. Then the variation off is

lim

!0 8 < :

sup

:()= N()

X jf(t

j+1) ,f(t

j) j

If a function is `nice', for example dierentiable, then it has bounded variation. Brownian motion has unbounded variation.

Denition 7.2

The quadratic variation of a function f is dened as

q:v:(f) = lim !0

8 < :

sup

:()= N() X

1 jf(t

j+1) ,f(t

j)) j

2 9 = ; :

Notice that quadratic variation will be nite for functions that are much rougher than those for which the variation is bounded. Roughly speaking, nite quadratic variation will follow if the uctuation of the function over an interval of length is order

p .

We can now be more precise about the quadratic variation of Brownian motion.

Theorem 7.3

Let B

t denote Brownian motion and for a partition

of [0;T]

dene

S() = N()

X

j=1

B t

j ,B

t j,1

2

:

Let

n be a sequence of partitions with (

n)

!0. Then E

jS(

n) ,Tj

2

!0 as n!1:

Proof.

We expand the expression inside the expectation and make use of our knowl-edge of the normal distribution. So, rst observe that

jS( n)

,Tj 2

=

N(n) X

j=1 n

B t

n;j ,B

t n;j,1

2

,(t n;j

,t n;j,1)

o 2

:

Write n;j for

B t

n;j ,B

t n;j,1

2

,(t n;j

,t

n;j,1). Then

jS( n)

,Tj 2

=N(n) X

j=1

2 n;j+ 2

X

j<k

n;j

n;k :

Note that since Brownian motion has independent increments,

E[ n;j

n;k] =

E[ n;j]

E[

n;k] = 0 if

j 6=k;

and

E

2 n;j

=E h

B t

n;j ,B

t n;j,1

4

,2

B t

n;j ,B

t n;j,1

2

(t n;j

,t

n;j,1) + ( t

n;j ,t

n;j,1) 2

i :

Now for a normally distributed random variable X with mean zero and variance , it is easy to check thatE[jXj

4] = 3

2, so that

E

2 n;j

= 3(t n;j

,t n;j,1)

2 ,2(t

n;j ,t

n;j,1) 2

+ (t n;j

,t n;j,1)

2

= 2(t n;j

,t n;j,1)

2 2(

n)( t

n;j ,t

Summing overj

E

jS( n)

,Tj 2

2 N(

n ) X

j=1 (

n)( t

n;j ,t

n;j,1)

= 2( n)

T !0 as n!1:

2

This result is not enough to dene the integralR f(B

s) dB

sin the classical way, but

it is enough to allow us to essentially mimic the construction of the (Lebesgue) in-tegral, at least for functions for whichE[f

2( B

)] 2L

1[0

;T]. However, although the

construction of the integral may look familiar, its behaviour is far from familiar. We rst illustrate this by deningR

T 0

B s

dB s.

From classical integration theory we are used to the idea that

Z T

0 f(x

s) dx

s = lim ()!0

N(),1 X

j=0 f(x

t j)

, x

t j+1

,x t

j

: (6)

Let us dene the stochastic integral in the same way, that is

Z T

0 B

s dB

s= lim ()!0

N(),1 X

j=0 B

t j

, B

t j+1

,B t

j

: (7)

Consider again the quantityS() of Theorem 7.3.

S() = N() X

j=1 ,

B t

j ,B

t j,1

2

=

N() X

j=1 n

B 2 tj

,B 2 tj,1

,2B

t j,1

, B

t j

,B t;j,1

o

= B 2 T

,B 2 0

,2 N(),1

X

j=0 B

t j

, B

t j+1

,B t

j

:

The left hand side isT (by Theorem 7.3) and so letting()!0 and rearranging

we obtain Z

T

0 B

s dB

s= ( B

2 T

,B 2 0

,T)

2

Remark.

Notice that this is not what one would have predicted from classical integration theory. The extra term in the stochastic integral corresponds toS().

In equation (6), we use f(x

tj) to approximate the value of

f on the interval

(t j

;t

j+1), but in the classical theory we could equally have taken any other point

in the interval in place oft

j and, in the limit, the result would have been the same.

1. The limit as()!0 of

N(),1 X

j=0 B

tj+1 ,

B tj+1

,B tj

:

2.

lim

()!0 N(),1

X

j=0

B t

j + B

t j+1

2

,

B t

j+1 ,B

t j

:

By choosing dierent points within each subinterval of the partition with which to approximatef over the subinterval we obtain dierent integrals. The It^o integral

is dened as

Z T

0 f(B

s) dB

s= lim ()!0

N() X

j=1 f(B

t j)

, B

t j+1

,B t

j

:

The Stratonovich integral is dened as

Z T

0 f(B

s) dB

s = lim ()!0

N() X

j=1

f(B t

j) + f(B

t j+1)

2

,

B t

j+1 ,B

t j

:

The Stratonovich integral has the advantage from the calculational point of view that the rules of Newtonian calculus hold good. From a modelling point of view, at least for our purposes, it is the wrong choice. To see why, think of what is happening over an innitesimal time interval. We might be modelling, for example, the value of a portfolio. We readjust our portfolio at the beginning of the time interval and its change in value over the innitesimal tick of the clock is beyond our control. A Stratonovich model would allow us to change our model nowon the basis of the average of the value corresponding to current stock prices and the value corresponding to prices after the next tick. We don't have that information when we make our investment decisions.

Consider then the It^o integral. We have evaluated it in just one special case. We increase our repertoire in the same way as in the classical setting by rst considering the value on simple functions.

Denition 7.4

A simple function is one of the form

f(B s) =

n X

i=1 a

i( B

s)

I i(

s);

where I i = [

s i

;s i+1),

[ n i=1

I i = [0

;T), I i

\I j =

f;g if i 6= j and the functions a i

satisfy E[a i(

B s)

2] <1.

By our denition,

Z T

0 f(B

s) dB

s= n X

a i(

B s)

, B

s i+1

,B s

Now, just as for regular integration, we approximate more general functions by simple functions and pass to a limit. We have to be sure, however, that the integrals converge when we pass to such a limit. This will not be true for all functions that can be approximated by simple functions. The next Lemma helps identify the space of functions for which we can reasonably expect a nice limit.

Lemma 7.5

Suppose that f is a simple function, then

1. R t 0

f s(

B s)

dB

s is a continuous F

t-martingale.

2.

E "

Z T

0 f(B

s) dB

s

2 #

=

Z T

0 E

f(B

s) 2

ds:

3.

E "

sup

tT Z

T

0 f(B

s) dB

s

2 #

4 Z

T

0 E

f(B

s) 2

ds:

Remark:

The second assertion is the famous It^o isometry. It suggests that we should be able to extend our denition of the integral to functions such that

R t 0

E[f s(

B s)

2]

ds < 1. Moreover, for such functions, all three assertions should

remain true. In fact one can extend the denition a little further, but the integral may then fail to be a martingale and this property will be important to us.

Proof.

The third assertion follows from the second by an application of a famous result of Doob:

Theorem 7.6 (Doob's inequality)

If fM t

g

0tT is a continuous martingale,

then

E

sup

0tT M

2 t

4E

M

2 T

:

We omit the proof of this remarkable Theorem.

We conne ourselves to proving the second assertion of Lemma 7.5. By our denition we have

Z T

0 f(B

s) dB

s= n X

i=1 a

i( B

s) ,

B s

i+1 ,B

s

;

and so

E "

Z T

0 f(B

s) dB

s

2 #

= E 2 4

n X

i=1 a

i( B

si) ,

B si+1

,B si

!

2 3 5

= E "

n X

i=1 a

2 i(

B s

i) ,

B s

i+1 ,B

s i

2

#

+2E "

X a

i( B

s i)

a j(

B s

j) ,

B s

i+1 ,B

s i

, B

s j+1

,B s

j

Suppose that j >i, then

E

a i(

B si)

a j(

B sj)

, B

si+1 ,B

si ,

B sj+1

,B sj

=E

a i(

B s

i) a

j( B

s j)

, B

s i+1

,B s

i

E ,

B s

j+1 ,B

s j

F s

j

= 0 Moreover,

E h

a 2 i(

B si)

, B

si+1 ,B

si

2 i

= E h

a 2 i(

B si)

E h

, B

si+1 ,B

si

2

F si

ii

= E

a 2 i(

B s

i)

(s i+1

,s i)

:

Substituting we obtain

E "

Z T

0 f(B

s) dB

s

2 #

= n X

i=1 E

a

2 i(

B si)

(s i+1

,s i)

=

Z T

0 E

f(B

s) 2

ds:

2

Notation:

We write

H =ff :R +

R !R : Z

T

0 E

f

s( B

s) 2

ds <1g:

Theorem 7.7

LetF

t denote the natural ltration generated by Brownian motion.

There exists a unique linear mapping, J, from H to the space of continuous F t

-martingales dened on [0;T] such that

1. If f is simple,

J(f) t=

Z t

0 f

s( B

s) dB

s ;

2. If tT,

E

J(f) 2 t

=

Z t

0 E

f

s( B

s) 2

ds;

3.

E

sup

0tT J(f)

2 t

4 Z

T

0 E

f

s( B

s) 2

ds:

Sketch of proof:

The last part follows from Doob's inequality once we know that J(f) is a

martingale.

The second assertion follows almost as our proof of existence of Brownian motion. We rst take a sequence of simple functions such that

E Z

t jf

s ,f

(n) s

j 2

ds

One then checks that (with probability one) the uniform limit ofJ(f

(n)) exists on

[0;T]. (This is an easy consequence of Lemma 7.5.) 2

We write

J(f) t=

Z t

0 f

s( B

s) dB

s :

Having made some sense of the stochastic integral, we are now in a position to try to make sense of the chain rule for stochastic calculus.

Theorem 7.8 (It^o's formula)

Forf such that @f @x

2H,

f(t;B t)

,f(0;B 0) =

Z t

0 @f @x

(s;B s)

dB s+

Z t

0 @f @s

(s;B s)

ds+ 12 Z

t

0 @

2 f @x

2( s;B

s) ds:

Notation:

Often one writes this in dierential notation as

df t=

f 0 t

dB t+ _

f t

dt+ 12f 00 t

dt:

Outline of proof:

To simplify notation, suppose that _f t

0. The formula then becomes

f(B t)

,f(B 0) =

Z t

0 @f @x

(B s)

dB s+ 12

Z t

0 @

2 f @x

2( B

s) ds:

Let be a partition of [0;t] with mesh . Then

f(B t)

,f(B 0) =

N(),1 X

0 ,

f(B tj+1)

,f(B tj)

:

We apply Taylor's Theorem on each interval of the partition.

f(B t)

,f(B 0) =

N(),1 X

0 f

0

(B t

j) ,

B t

j+1 ,B

t j

+ 12

N(),1 X

0 f

00

(B t

j) ,

B t

j+1 ,B

t j

2

+ 13!

N(),1 X

0 f

000

( j)

, B

t j+1

,B t

j

3

for some points j

2 [t j

;t

j+1]. If f

000 is uniformly bounded, then the third term

tends to zero in probability (Exercise). The rst converges to the It^o integral and the second, by Theorem 7.3, converges to 1=2

R t 0

f 00(

B s)

ds. 2

Example 7.9

Use It^o's formula to compute E[B 4 t].

Let us deneZ t =

B 4

t. Then by It^o's formula dZ

t= 4 B

3 t

dB t+ 6

B 2 t

dt;

and, of course,Z

0 = 0. In integrated form,

Z t

,Z 0 =

Z t

4B 3 s

dB s+

Z t

6B 2 s

Taking expectations, the expectation of the stochastic integral vanishes (by the martingale property) and so

E[Z t] =

Z t

0

6E[B 2 s]

ds = Z

t

0

6sds= 3t 2

:

2

The most common model of stock price movements is given by geometric Brownian motion, dened by

S

t= exp(

t+B t)

:

Applying It^o's formula,

dS

t = S

t dB

t+ ,

+ 1 2

2

S t

dt S

0 = 1 :

This expression is called the stochastic dierential equation forS

t. It is common to

write such symbolic equations even though it is the integral equation that makes sense.

Writing=+ 2

=2, geometric Brownian motion is a martingale if and only

if = 0 and E[S

t] = exp( t).

It is convenient to have a version of It^o's formula that allows us to work directly with S

t (that is to write down a stochastic dierential equation for f(S

t) for

example). We now know how to make our original heuristic calculations rigorous, so with a clear conscience we proceed as follows:

f(S t+t)

,f(S t)

f 0(

S t)(

S t+t

,S t) + 12

f 00(

S t)(

S t+t

,S t)

2

f 0(

S t)

dS t+ 12

f 00(

S t)

2 S

2 t

dB 2 t +

2

S 2 t

dt 2+ 2

S 2 t

dB t

dt

=f 0(

S t)

dS t+ 12

f 00(

S t)

2

S 2 t

dt:

(We have used that t(B t+t

,B t) =

o(t) which is usually written symbolically

asdtdB

t = 0.) As before, allowing

f to also depend on tintroduces an extra term

_

f(S t)

dt. Writing this version of It^o's formula in integrated form gives then:

f(t;S t)

,f(0;S 0) =

Z t

0 @f @x

(u;S u)

dS u+

Z t

0 @f @u

(u;S u)

du+12 Z

t

0 @

2 f @x

2( u;S

u)

2 S

2 u

du

=

Z t

0 @f @x

(u;S u)

S u

dB u+

Z t

0 @f @x

(u;S u)

S u

du

+

Z t

0 @f @u

(u;S u)

du+ 12 Z

t

0 @

2 f @x

2( u;S

u)

2 S

2 u

du

Warning:

Be aware that the stochastic integral with respect to S will not be a

martingale with respect to the probability under which B

t is a martingale except

in the special case when = 0. To actually calculate it is often wise to separate

Theorem 7.10

If Y

t satises dY

t = a(Y

t) dB

t+ b(Y

t) dt;

and

Z t =

f(t;Y t)

;

then

dZ t=

f 0(

t;Y t)

dY t+ _

f(t;Y t)

dt+ 12f 00(

t;Y t)

a(Y t)

2 dt:

Remark:

Notice that

M t=

Y t

,Y 0

, Z

t

0 b(Y

s) ds

is a martingale with mean zero. From the It^o isometry, we know that the variance is

E[M 2 t] =

E Z

t

0 a(Y

s) 2

ds

:

The expression R t 0

a(Y s)

2

ds is the quadratic variation of M

t, often denoted hMi

t

or [M] t.

Suppose now that we have two stochastic dierential equations,

dY t =

a(Y t)

dB t+

b(Y t)

dt;

dZ t= ~

a(Z t)

dB t+ ~

b(Z t)

dt:

Write

M Y t =

Z t

0 a(Y

s) dB

s

and

M Z t =

Z t

0

~

a(Z s)

dB s

:

Then the covariance is given by

E

M Y t

M Z t

= 14E h

, M

Y t +

M Z t

2

, ,

M Y t

,M Z t

2

i

= E Z

t

0 a(Y

s)~ a(Z

s) ds

The quantity R t 0

a(Y s)~

a(Z s)

ds, often denoted hM Y

;M Z

i t or [

M Y

M Z]

t, is called

the covariation of M Y and

M Z.

Notice that

M 2 t

,hMi t

is an F

t-martingale and so is M

Y t

M Z t

,hM Y

;M Z

i t

:

Theorem 7.11 (Integration by parts)

LetX t =

Y t

Z twith

Y;Z as above, then

dX t=

Y t

dZ t+

Z t

dY t+

dhM Y

;M Z

i t

:

Proof:

We apply the It^o formula to (Y t+

Z t)

2 and Y

2 t and

Z 2

t, and subtract the

second two from the rst to obtain

Y t

Z t

,Y 0

Z 0 =

Z t

0 Y

s dZ

s+ Z

t

0 Z

s dY

s+ Z

t

0 a(Y

s)~ a(Z

s) ds;

which is the integrated form of the result. 2

We now have a very large number of continuous time martingales in our hands. For any reasonable functionf,

M t=

Z

f(s;B s)

dB s

is a martingale with respect to the Brownian probability and adapted to the Brownian ltration. It is natural to ask if there are any others. The answer is provided by the martingale representation theorem which says, essentially, no.

Theorem 7.12 (Brownian martingale representation theorem)

LetfF t

g 0tT

denote the natural ltration of Brownian motion. Let fM t

g

0tT be a

square-integrable F

t-martingale. Then there exists an F

t-adapted process

s such that

with probability one,

M t=

M 0+

Z t

0

s dB

s :

The process

s is essentially unique which leads, with a little work, to

Theorem 7.13 (Levy's characterisation of Brownian motion)

If M t is a

continous (local) martingale with quadratic variation hMi t =

t (with probability

one), then M

t is a standard Brownian motion.

We'll use this to `prove' one more important result. Recall that in the discrete setting we were able to reduce pricing options to calculating expectations once we had found a probability measure under which the discounted stock price was a martingale. The same will be true in the continuous world, but it will no longer be possible to nd the martingale measure by linear algebra. The key now will be Girsanov's Theorem.

Theorem 7.14 (Girsanov's Theorem)

Suppose that B t is a

P-Brownian

mo-tion with the natural ltramo-tion F

t. Suppose that

t is an F

t-adapted process such

that

E

exp

1 2

Z T

0

2 t

dt

<1:

Dene

L

t= exp

, Z

t

0

s dB

s ,

1 2

Z t

0

2 s

and let P

(L) be the probability measure dened by

P (L)(

A) = Z

A L

t(

!)P(d!):

Then under the probability measure P

(L), the process fW

t g

0tT, dened by

W t=

B t+

Z t

0

s ds;

is a standard Brownian motion.

Notation:

We write

dP (L) dP

Ft

=L t

:

(L

t is the Radon-Nikodym derivative of P

(L) with respect to P.)

Remarks:

1. The condition

E

exp

1 2

Z T

0

2 t

dt

<1

is enough to guarantee thatL

tis a martingale. It is clearly positive and has

expectation one so thatP

(L) really does dene a probability measure.

2. Just as in the discrete world, two probability measures are equivalent if they have the same sets of probability zero. EvidentlyP and P

(L) are equivalent.

3. If we wish to calculate an expectation with respect toP

(L) we have

E (L)[

t] =

E [ t

L t]

:

This will be fundamental in option pricing.

Outline of proof:

We have already said thatL

t is a martingale. We don't prove this in full, but

we nd supporting evidence by nding the stochastic dierential equation satised by L

t. We do this in two stages. First, dene

Z t =

, Z

t

0

s dB

s ,

1 2

Z t

0

2 s

ds:

Then

dZ t=

, t

dB t

,

1 2

2 t

dt:

Now we use Theorem 7.9 applied toL

t= exp( Z

t).

dL

t = exp( Z

t) dZ

t+ 12 exp( Z

t)

2 t

dt

= ,

texp( Z

t) dB

t = ,

t L

t dB

Now we integrate by parts (using Theorem 7.10) to nd the stochastic dierential equation forW

t L

t. Since

dW t=

dB t+

t

dt;

d(W t

L t) =

W t

dL t+

L t

dW t+

dhM W

;M L

i t

= W t

dL t+

L t

dB t+

L t

t

dt, t

L t

dt

= (L t

, t

L t

W t)

dB t

:

Granted enough boundedness (which is guaranteed by our assumptions), W t

L t is

then a martingale and has expectation zero. Thus, under the measure P (L),

W t is

a martingale.

Now the quadratic variation of W

t is the same as that of B

t, and we proved

in Theorem 7.3 that with P-probability one, the quadratic variation of B

t is just t. Now P and P

(L) are equivalent and so have the same sets of probability one.

Therefore W

t also has quadratic variation

t with P

(L)-probability one. Finally, by

Levy's characterisation of Brownian motion (Theorem 7.12) we have thatW t is a P

(L)-Brownian motion as required.

2

We now try this in practice

Example 7.15

Let X

t be the drifting Brownian motion process X

t= B

t+ t;

where B t is a

P-Brownian motion and and are constants. Then taking = =, under P

(L) of Theorem 7.13 we have that W

t = B

t+

t= is a Brownian

motion, and X t =

W

t is then a scaled Brownian motion.

Notice that, for example,

E P

X

2 t

=E P

2 B

2 t + 2

tB t+

2

t 2

= 2

t+ 2

t 2

;

whereas

E P

(L)

X 2 t

=E P

(L)

2

W 2 t

= 2

t: