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.