A Poisson process with intensity parameter $0<\mathrm{\lambda }\left(t\right)$, where $\mathrm{\lambda }\left(t\right)$ is a deterministic function of time, is a stochastic process $N$ with independent increments such that $N\left(0\right)=0$ and

for all $0\le t$. If the intensity parameter $\mathrm{\lambda }\left(t\right)$ itself is stochastic, the corresponding process is called a doubly stochastic Poisson process or Cox process.

A compound Poisson process is a stochastic process $J\left(t\right)$ of the form $J\left(t\right)=\sum _{i=1}^{N\left(t\right)}{Y}_i$, where $N\left(t\right)$ is a standard Poisson process and $Y_i$ 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.

$\mathrm{with}\left(\mathrm{Finance}\right)\&colon;$

$J≔\mathrm{PoissonProcess}\left(1.0\right)\&colon;$

$\mathrm{PathPlot}\left(J\left(t\right)\&comma;t=0..3\&comma;\mathrm{timesteps}=50\&comma;\mathrm{replications}=20\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\&comma;\mathrm{markers}=\mathrm{false}\right)$

$W≔\mathrm{WienerProcess}\left(J\right)\&colon;$

$\mathrm{PathPlot}\left(W\left(t\right)\&comma;t=0..3\&comma;\mathrm{timesteps}=20\&comma;\mathrm{replications}=10\&comma;\mathrm{markers}=\mathrm{false}\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{thickness}=3\&comma;\mathrm{gridlines}=\mathrm{true}\&comma;\mathrm{axes}=\mathrm{BOXED}\right)$

$Y≔\mathrm{Statistics}\left[\mathrm{RandomVariable}\right]\left(\mathrm{Normal}\left(0.3\&comma;0.5\right)\right)\&colon;$

$\mathrm{\lambda }≔0.5$

$\mathrm{\lambda }≔0.5$

$X≔\mathrm{PoissonProcess}\left(\mathrm{\lambda }\&comma;Y\right)\&colon;$

$\mathrm{PathPlot}\left(X\left(t\right)\&comma;t=0..3\&comma;\mathrm{timesteps}=20\&comma;\mathrm{replications}=10\&comma;\mathrm{markers}=\mathrm{false}\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{thickness}=3\&comma;\mathrm{gridlines}=\mathrm{true}\&comma;\mathrm{axes}=\mathrm{BOXED}\right)$

$T≔3$

$T≔3$

$\mathrm{ExpectedValue}\left(X\left(T\right)\&comma;\mathrm{replications}={10}^4\&comma;\mathrm{timesteps}=100\right)$

$\left[\mathrm{value}=0.4435146732\&comma;\mathrm{standarderror}=0.007164725012\right]$

$\mathrm{\lambda }T\mathrm{Statistics}\left[\mathrm{ExpectedValue}\right]\left(Y\right)$

$0.45$

$\mathrm{\kappa }≔0.354201$

$\mathrm{\kappa }≔0.354201$

$\mathrm{\mu }≔1.21853$

$\mathrm{\mu }≔1.21853$

$\mathrm{\nu }≔0.538186$

$\mathrm{\nu }≔0.538186$

$\mathrm{y0}≔1.81$

$\mathrm{y0}≔1.81$

$y≔\mathrm{SquareRootDiffusion}\left(\mathrm{y0}\&comma;\mathrm{\kappa }\&comma;\mathrm{\mu }\&comma;\mathrm{\nu }\right)\&colon;$

$J≔\mathrm{PoissonProcess}\left(y\left(t\right)\right)\&colon;$

$\mathrm{PathPlot}\left(y\left(t\right)\&comma;t=0..3\&comma;\mathrm{timesteps}=100\&comma;\mathrm{replications}=10\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

$\mathrm{PathPlot}\left(J\left(t\right)\&comma;t=0..3\&comma;\mathrm{timesteps}=100\&comma;\mathrm{replications}=10\&comma;\mathrm{thickness}=3\&comma;\mathrm{color}=\mathrm{red}..\mathrm{blue}\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

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

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

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