The WienerProcess command creates a new Wiener process. If called with no arguments, the WienerProcess command creates a new standard Wiener process, $W\left(t\right)$, that is a Gaussian process with independent increments such that $W\left(0\right)=0$ with probability $1$, $E\left(W\left(t\right)\right)=0$ and $\mathrm{Var}\left(W\left(t\right)-W\left(s\right)\right)=t-s$ for all $0\le s\le t$.

The WienerProcess(Sigma) calling sequence creates a Wiener process with covariance matrix Sigma. The matrix Sigma must be a positive semi-definite square matrix. The dimension of the generated process will be equal to the dimension of the matrix Sigma.

If an optional parameter J is passed, the WienerProcess command creates a process of the form $W\left(J\left(t\right)\right)$, where $W\left(t\right)$ is the standard Wiener process. Note that the subordinator $J\left(t\right)$ must be an increasing process with non-negative, homogeneous, and independent increments. This can be either another stochastic process such as a Poisson process or a Gamma process, a procedure, or an algebraic expression.

$\mathrm{with}\left(\mathrm{Finance}\right)\:$

$W≔\mathrm{WienerProcess}\left(\right)\:$

$P≔\mathrm{PathPlot}\left(W\left(t\right)\,t=0..3\,\mathrm{timesteps}=50\,\mathrm{replications}=20\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{red}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)\:$$P$

$T≔3$

$T≔3$

$\mathrm{ExpectedValue}\left(\exp \left(0.05T+0.3W\left(T\right)\right)\,\mathrm{replications}={10}^5\right)$

$\left[\mathrm{value}=1.331151006\,\mathrm{standarderror}=0.002335223363\right]$

$Y≔t\mapsto \exp \left(\mathrm{\mu }\cdot t+\mathrm{\sigma }\cdot W\left(t\right)\right)\:$$Y\left(t\right)$

${\ⅇ}^{\mathrm{\mu }t+\mathrm{\sigma }\mathrm{\_W}\left(t\right)}$

$\mathrm{subs}\left(Y\left(t\right)='Y'\left(t\right)\,\mathrm{Drift}\left(Y\left(t\right)\right)\right)$

$\mathrm{\mu }Y\left(t\right)+\frac{{\mathrm{\sigma }}^{2}Y\left(t\right)}{2}$

$\mathrm{subs}\left(Y\left(t\right)='Y'\left(t\right)\,\mathrm{Diffusion}\left(Y\left(t\right)\right)\right)$

$\mathrm{\sigma }Y\left(t\right)$

$J≔\mathrm{PoissonProcess}\left(0.3\right)$

$J≔\mathrm{\_P}$

$\mathrm{W2}≔\mathrm{WienerProcess}\left(J\right)$

$\mathrm{W2}≔\mathrm{\_W0}$

$\mathrm{P2}≔\mathrm{PathPlot}\left(\mathrm{W2}\left(t\right)\,t=0..3\,\mathrm{timesteps}=50\,\mathrm{replications}=20\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{blue}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)\:$$\mathrm{P2}$

$\mathrm{plots}\left[\mathrm{display}\right]\left(P\,\mathrm{P2}\right)$

$\mathrm{\kappa }≔0.1\:$$\mathrm{\sigma }≔0.3\:$$\mathrm{\theta }≔0.5\:$$\mathrm{R0}≔0.02\:$

$\mathrm{\tau }≔\frac{\exp \left(2\mathrm{\kappa }t\right)-1}{2\mathrm{\kappa }}$

$\mathrm{\tau }≔5.000000000{\ⅇ}^{0.2t}-5.000000000$

$\mathrm{W3}≔\mathrm{WienerProcess}\left(\mathrm{\tau }\right)$

$\mathrm{W3}≔\mathrm{\_W1}$

$R≔t\mapsto \mathrm{R0}\cdot \exp \left(-\mathrm{\kappa }\cdot t\right)+\mathrm{\theta }\cdot \left(1-\exp \left(-\mathrm{\kappa }\cdot t\right)\right)+\mathrm{\sigma }\cdot \exp \left(-\mathrm{\kappa }\cdot t\right)\cdot \mathrm{W3}\left(t\right)\:$$R\left(t\right)$

$-0.48{\ⅇ}^{-0.1t}+0.5+0.3{\ⅇ}^{-0.1t}\mathrm{\_W1}\left(t\right)$

$\mathrm{P3}≔\mathrm{PathPlot}\left(R\left(t\right)\,t=0..3\,\mathrm{timesteps}=50\,\mathrm{replications}=20\,\mathrm{thickness}=3\,\mathrm{color}=\mathrm{blue}\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)\:$$\mathrm{P3}$

$\mathrm{ExpectedValue}\left(R\left(3\right)\,\mathrm{timesteps}=100\,\mathrm{replications}={10}^5\right)$

$\left[\mathrm{value}=0.1517355448\,\mathrm{standarderror}=0.009505937484\right]$

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

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