Exercise 10.2. Construct the stochastic integral with respect to the ( ; F ; F ; P ) martingale f M

Stochastic Integral Exercises

Exercise 10.1. Show that the de…nition of the stochastic integral for simple processes does not depend on the chosen representation.

Exercise 10.2. Construct the stochastic integral with respect to the ( ; F ; F ; P ) martingale f M

: t 0 g such that E

[M

] < 1 , 8 t 0:

Exercise 10.3. Stochastic integral with respect to the Poisson process.

To interpret the results in a …nancial context, assume the Poisson process N represents the number of dividend payments: at time t, there are N (t) payments.

a) Let X be the basis process

X

(!) = C (!) I

(t) (1)

where a < b 2 R and C is a square-integrable (E

[C

] < 1 ) F

measurable random variable. X

is the amount of dividend at time t, if there is a dividend payment. In the case of a basis process, some of the dividend payment may be nil!

Mimicking the construction of Itô’s integral with respect to the Brownian motion, con- struct the integral with respect to the Poisson process and interpret it in the …nancial context.

b) Is nR

0

X

dN

: t 0 o

adapted ? a martingale ? Justify.

c) Justify that for basis processes, Z

0

X

dN

= C the number of events that occurred during (a; b]

d) Let X be the simple process f X

: t 0 g where X

X

i=1

C

I

(t) :

Construct the stochastic integral for simple processes with respect to the Poisson process.

e) Let 0 a

< b

a

< b

::: a

< b

. Interpret Z

0

X

i=1

C

I

(s)

! dN

:

f ) Let X be a predictable process. Justify, intuitively in a few words, Z

0

X

dN

X

Tn=1n t

X

:

1

A rigorous proof goes above the course requirement. Interpret this integral in the context of dividends payments.

Exercise 10.5. Let f M

: t 0 g and f N

: t 0 g be two martingales construct on the

…ltered probability space ( ; F ; fF

: t 0 g ; P ). If a and b are constant and Z

= aM

+ bN

, t 0;

then show that Z

X

dZ

= a Z

0

X

dM

+ b Z

0

X

dN

whenever

a) f X

: t 0 g is a basis process.

b) f X

: t 0 g is a simple process.

c) f X

: t 0 g is a predictable

Exercise 10.6. Let M be a ( ; F ; fF

: t 0 g ; P ) stochastic process and Y

=

Z

H

dM

where H is a predictable process

a) Show that if X and H are basis processes, then Z

0

X

dY

= Z

0

X

H

dM

b) Extend the result to the case where X and H are simple processes.

c) Use the limit for the case where X and H are predictable processes.

Exercise 10.7. Use exercise 10.6 to show that if Y

=

Z

K

ds + Z

0

H

dW

where W is a ( ; F ; fF

: t 0 g ; P ) Brownian motion, then

Z

X

dY

= Z

0

X

K

ds + Z

0

X

H

dW

and Z

0

X

d h Y i

= Z

0

X

H

ds where X is a predictable process.

2

Solutions

1 Exercise 10.3

a) Z

0

X

dN

= C (N

N

) This is the amount of dividend received during (0; t].

b) No, it is not a martingale. Indeed, let t > b. In this case E

Z

X

dN

= E [C (N

N

)]

= E [C (N

N

)]

= E [CE [N

N

jF

]]

= E [CE [N

N

]]

= E [CE [N

]]

= E [C] (b a)

6

= 0

= E Z

0

X

dN

c) Z

0

X

dN

= C (N

N

)

| {z }

nb of events during(a;b]

d) Z

0

X

dN

= X

i=1

C

(N

N

)

e) Z

0

X

dN

= X

i=1

C

(N

N

)

| {z }

nb of events during(T^ai;T^bi]

Since C

represents the amount of dividend paid each time during (a

; b

], R

0

X

dN

is the total amount during (0; T ).

3

f) Consider a sequence of simple processes such that b

a

tends to 0 and n ! 1 . In the short time interval of length b

a

, there is 0 or 1 event (dividend payment). The probability to have two or more is negligible. In this case, N

N

= 0 or 1:

4