Springer Finance

11.4 Jump Processes and Their Integrals 475

Finally, in (11.4.1), J(t) is an adapted, right-continuous pure jump process
with J(O) = 0. By right-continuous, we mean that J(t) = lims.j.t J(s) for all
t ;::: 0. The left- continuous version of such a process will be denoted J(t-). In
other words, if J has a jump at timet, then J(t) is the value of J immediately
after the jump, and J(t-) is its value immediately before the jump. We assume
that J does not jump at time zero, has only finitely many jumps on each finite
time interval (0, T], and is constant between jumps. The constancy between
jumps is what justifies calling J(t) a pure jump process. A Poisson process
and a compound Poisson process have this property. A compensated Poisson
process does not because it decreases between jump s. We shall call J(t) the
pure jump part of X.

Definition 11.4.2. A process X(t) of the fo rm {11.4.1}, with Ito integral part
I(t), Riemann integral part R(t), and pure jump part J(t) as described above,
will be called a jump process. The continuous part of this process is xc(t) =
X(O) + I(t) + R(t).

A jump process in this book is not the most general possible because
we permit only finitely many jumps in finite time. For many applications,
these processes are sufficient. Furthermore, the stochastic calculus for these
processes gives a good indication of how the stochastic calculus works for the
more general case.
A jump process X(t) is right-continuous and adapted. Because both I(t)
and R(t) are continuous, the left-continuous version of X(t) is

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

The jump size of X at time t is denoted

LlX(t) = X(t) - X(t-).

If X is continuous at t, then LlX(t) = 0. If X has a jump at time t, then
LlX(t) is the size of this jump, which is also LlJ(t) = J(t) - J(t-), the size
of the jump in J. Whenever X(O-) appears in the formulas below, we mean
it to be X(O). In particular, LlX(O) = 0; there is no jump at time zero.

Definition 11.4.3. Let X(t) be a jump process of the form {1 1.4.1}-{11.4.3}
and let q';( s) be an adapted process. The stochastic integral of q; with respect
to X is defined to be

1

t

q'j(s) dX(s) = 1

t

q'j(s)F(s) dW(s) + 1

t

q'j(s)8(s) ds + L q'j(s)LlJ(s).

(^0 0 0) O<s:<:::t
(11.4.4)
In differential notation,