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474 11 Introduction to Jump Processes

N(u) - N(t) is independent of J="(t). Finally, we say that a compound Poisson
process Q is a compound Poisson process relaative to this filtration if Q(t) is
J="(t)-measumble for every t and for every u > t the increment Q(u)- Q(t) is
independent of J="(t).

11.4.1 Jump Processes

We wish to define the stochastic integral

lot !P(s) dX(s),


where the integrator X can have jumps. Let ( n, J=", P) be a p robability space

on which is given a filtration J="(t), t � 0. All processes will be adapted to
this filtration. Furthermore, the integrators we consider in this section will be
right-continuous and of the form

X(t) = X(O) + I(t) + R(t) + J(t). (11.4.1)


In (11.4.1), X(O) is a nonrandom initial condition. The process

I(t) =lot F(s) dW(s) {11.4.2)


is an Ito integml of an adapted process F(s) with respect to a Brownian


motion relative to the filtration. We shall call I(t) the ItO integml part of X.

The process R(t) in (11. 4.1) is a Riemann integml^1


R(t) = 1 t e(s) ds (11.4.3)


for some adapted process e(t). We shall call R(t) the Riemann integml part


of X. The continuous part of X(t) is defined to be

xc(t) = X(O) + I(t) + R(t) = X(O) + 1t F(s) dW(s) +lot e(s) ds.


The quadratic variation of this process is


an equation that we write in differential form as


1 One usually takes this to be a Lebesgue integral with respect to dt, but for all the
cases we consider, the Riemann integral is defined and agrees with the Lebesgue
integral.
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