Stochastic Calculus for Jump-Diffusions
This appendix gives an overview of some basic tools involving processes with jumps.
The reader is assumed to be familiar with diffusion processes and the Itô formula for
such processes. This appendix gives the extension of Itô’s formula to processes with
a finite number of jumps on finite intervals. It also describes how the distribution of
jump-diffusions can change under an equivalent change of measure. We define the
concepts in a framework which is more general, namely within a class of processes
known as special semimartingales. Working within this class gives us a way of
defining what we mean by a jump-diffusion and it also carries additional dividends.
For example, special semimartingales represent the most general class of processes
within which we can formulate “beta” relationships, i.e. equations which describe
the excess returns on processes in terms of their local covariation with a state price
density process (which then in turn can be related to the return on a market portfolio).
We look at how this can be formulated.
D.1 The Poisson Process
The Poisson process is the fundamental building block of jump processes. It is just as
essential to modeling jump processes as Brownian motion is to diffusion modeling.
Given a filtered probability space (Ω, F, P, F
), a Poisson process N
with intensity λ
≥ 0 is the unique process satisfying the following properties:
|i. ||N0 = 0;|
|ii. ||N has independent increments; and|
|iii. ||P(Nt–Ns = k) = ((λ(t–s))k/k!) exp(–λ(t–s)) for k ∈ N0 and t > s ≥ 0|
Just as one needs to prove the existence of a Brownian motion (after all, the defining
properties could be self-contradictory), one also has to “construct” a Poisson process.
This is a lot easier than for Brownian motion. The construction is simply done
by letting ε1, ε2, … be a sequence of independent, identically, and exponentially
distributed random variables with