Springer Finance

11.4 Jump Processes and Their Integrals 473

m = 1, ... , M. Because the processes Q and Q have the same distribution,

these statements must also be true for the total number of jumps and the sizes

of the jumps of the process Q of (11.3.6), which is what the theorem asserts.

0

The substance of Theorem 11.3.3 is that there are two equivalent ways of

regarding a compound Poisson process that has only finitely many possible

jump sizes. It can be thought of as a single Poisson process in which the

size-one jumps are replaced by jumps of random size. Alternatively, it can be

regarded as a sum of independent Poisson processes in each of which the size­

one jumps are replaced by jumps of a fixed size. We restate Theorem 11.3.3

in a way designed to make this more clear.

Corollary 11.3.4. Let Y 1 , ... , YM be a finite set of nonzero numbers, and

let p(y 1 ), •.• ,p(yM) be positive numbers that sum to 1. Let Y 1 , Y 2 , ... be a

sequence of independent, identically distributed random variables with IP'{Yi =

Ym} = p(ym), m = 1, ... , M. Let N(t) be a Poisson process and define the

compound Poisson pr ocess

N(t)

Q(t) = L }i.

i=l

For m= (^1) , ... , M, let Nm(t) denote the number of jumps in Q of size Ym up
to and including time t. Then
M M
N(t) = L Nm(t) and Q(t) = L YmNm(t).
m=l m=l
The processes N 1 , ... , N M defined this way are independent Poisson processes,
and each Nm has intensity .>..p(ym)·

11.4 Jump Processes and Their Integrals

In this section, we introduce the stochastic integral when the integrator is a

process with jumps, and we develop properties of this integral. We shall have

a Brownian motion and Poisson and compound Poisson processes. There will

always be a single filtratoin associated with all of them, in the sense of the

following definition.

Definition 11.4.1. Let (!I, F, IP') be a probability space, and let F(t), t 2:: 0,
be a filtration on this space. We say that a Brownian motion W is a Brownian
motion relative to this filtration if W(t) is F(t)-measurable for every t and
for every u > t the increment W(u) - W(t) is independent of F(t). Similarly,
we say that a Poisson process N is a Poisson process relative to this filtration
if N(t) is F(t)-measurable for every t and for every u > t the increment