A Poisson process with intensity parameter , where  is a deterministic function of time, is a stochastic process  with independent increments such that  and

for all . If the intensity parameter  itself is stochastic, the corresponding process is called a doubly stochastic Poisson process or Cox process.

A compound Poisson process is a stochastic process  of the form , where  is a standard Poisson process and  are independent and identically distributed random variables. A compound Cox process is defined in a similar way.

The parameter lambda is the intensity. It can be constant or time-dependent. It can also be a function of other stochastic variables, in which case the so-called doubly stochastic Poisson process (or Cox process) will be created.

The parameter X is the jump size distribution. The value of X can be a distribution, a random variable or any algebraic expression involving random variables.

If called with one parameter, the PoissonProcess command creates a standard Poisson or Cox process with the specified intensity parameter.

The Finance[PoissonProcess] command was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.

Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.