The GammaProcess command creates a Gamma process with the specified parameters. The Gamma process $G\left(t\right)$ with mean parameter mu and variance parameter sigma is a continuous-time process with stationary, independent gamma increments such that for any $0<h$, $G\left(t+h\right)-G\left(t\right)$ has a Gamma distribution with shape parameter $\frac{{\mathrm{\mu }}^{2}h}{\mathrm{\sigma }}$ and scale parameter $\frac{\mathrm{\sigma }}{\mathrm{\mu }}$.

The parameter mu is the mean. The parameter sigma is the variance.

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

$\mathrm{\mu }≔1\&colon;$$\mathrm{\sigma }≔3\&colon;$

$G≔\mathrm{GammaProcess}\left(\mathrm{\mu }\&comma;\mathrm{\sigma }\right)\&colon;$

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

$\mathrm{ExpectedValue}\left(G\left(3\right)\&comma;\mathrm{replications}={10}^4\right)$

$\left[\mathrm{value}=2.376349494\&comma;\mathrm{standarderror}=0.01792180394\right]$

$S≔\mathrm{SampleValues}\left(G\left(2\right)-G\left(1.98\right)\&comma;\mathrm{timesteps}={10}^2\&comma;\mathrm{replications}={10}^3\right)$

$S≔\left[0.\&comma;0.\&comma;7.35991702799366\times {10}^{\mathrm{-6}}\&comma;0.\&comma;0.\&comma;7.35602354562381\times {10}^{\mathrm{-6}}\&comma;2.22044604925031\times {10}^{\mathrm{-16}}\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;2.68673971959288\times {10}^{\mathrm{-14}}\&comma;0.\&comma;0.\&comma;1.35483735341779\times {10}^{\mathrm{-10}}\&comma;1.40403509477430\times {10}^{\mathrm{-6}}\&comma;0.000394374611912163\&comma;0.\&comma;0.0154969225911339\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;7.51650208741239\times {10}^{\mathrm{-9}}\&comma;0.\&comma;0.\&comma;2.63580268722308\times {10}^{\mathrm{-11}}\&comma;0.\&comma;0.0971434323130915\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.0413351186423117\&comma;0.\&comma;0.0000276411228146145\&comma;0.\&comma;0.\&comma;0.\&comma;5.41153927224869\times {10}^{\mathrm{-13}}\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.0198811462516445\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.000210839334942481\&comma;0.\&comma;1.81071369009800\times {10}^{\mathrm{-9}}\&comma;5.21804821573824\times {10}^{\mathrm{-14}}\&comma;0.\&comma;0.\&comma;0.195217033877362\&comma;1.61692881306408\times {10}^{\mathrm{-12}}\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;1.01821726494755\times {10}^{\mathrm{-7}}\&comma;0.\&comma;0.\&comma;0.\&comma;0.\&comma;0.00408745142293243\&comma;1.31116352972340\&comma;5.16159894009549\times {10}^{\mathrm{-6}}\&comma;0.\&comma;\dots \&comma;\text{⋯ 900 Array entries not shown}\right]$

$\mathrm{Xu}≔\mathrm{GammaProcess}\left(1\&comma;3\right)\&colon;$

$\mathrm{Xd}≔\mathrm{GammaProcess}\left(0.9\&comma;3\right)\&colon;$

$X≔t\mapsto \mathrm{Xu}\left(t\right)-\mathrm{Xd}\left(t\right)$

$X≔t\mapsto \mathrm{Xu}\left(t\right)-\mathrm{Xd}\left(t\right)$

$\mathrm{PathPlot}\left(X\left(t\right)\&comma;t=0..3\&comma;\mathrm{timesteps}=20\&comma;\mathrm{replications}=5\&comma;\mathrm{thickness}=2\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}=\mathrm{true}\right)$

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

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

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