scheme = unbiased or Euler -- This option specifies which discretization scheme should be used for simulating this process.

The BrownianMotion($x_0$, mu, sigma) and BrownianMotion($x_0$, mu, sigma, t) commands create a new one-dimensional Brownian motion process. This is a stochastic process $X\left(t\right)$, which is governed by the stochastic differential equation (SDE)

$\mathrm{\mu }=\mathrm{\mu }\left(t\right)$ is the drift,

$\mathrm{\sigma }=\mathrm{\sigma }\left(t\right)$ is the volatility,

$W\left(t\right)$ is the standard Wiener process.

The parameter $x_0$ defines the initial value of the underlying stochastic process. It must be a real constant.

In the simplest case of a constant drift, mu is a real number (i.e. any expression of type $\mathrm{realcons}$). Time-dependent drift can be given either as an algebraic expression or as a Maple procedure. If mu is given as an algebraic expression, then the parameter $t$ must be passed to specify which variable in mu should be used as a time variable. Maple procedure defining a time-dependent drift must accept one parameter (the time) and return the corresponding value for the drift.

Similar to the drift, the volatility parameter can be constant or time dependent. In addition to this, the volatility can involve other (one-dimensional) stochastic variables. Note that stochastic drift is not supported.

The BrownianMotion($X_0$, Mu, Sigma) defines an $n$-dimensional Brownian motion with drift Mu and covariance Sigma. This process is defined by the SDE

$\mathrm{{\rm M}}$ is a vector of size $n$,

$B$ is a $n\times m$-matrix $B{B}^T=\mathrm{\Sigma }$,

$W\left(t\right)$ is the standard $m$-dimensional Wiener process.

In this case the drift vector and the covariance matrix are time-independent. The drift parameter Mu must be given as a Vector and the covariance matrix Sigma must be a given as a symmetric matrix (see Matrix).

The scheme option specifies the discretization scheme used for simulation of this process. By default the standard Euler scheme is used. When scheme is set to unbiased the transition density will be used to simulate a value $X\left(t+\mathrm{dt}\right)$ given $X\left(t\right)$. This scheme is appropriate in the case of a time-dependent drift and/or volatility.

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

$T≔2.0$

$T≔2.0$

$X≔\mathrm{BrownianMotion}\left(0.\,0.02\,0.3\right)\:$

$A≔\mathrm{SamplePath}\left(X\left(t\right)\,t=0..T\,\mathrm{timesteps}=100\,\mathrm{replications}=10\right)$

$\mathrm{PathPlot}\left(X\left(t\right)\,t=0..T\,\mathrm{timesteps}=100\,\mathrm{replications}=10\,\mathrm{axes}=\mathrm{BOXED}\,\mathrm{gridlines}=\mathrm{true}\right)$

$W≔\mathrm{BrownianMotion}\left(1\,\mathrm{\mu }\left(t\right)\,\mathrm{\sigma }\left(t\right)\,t\right)$

$W≔\mathrm{\_X1}$

$\mathrm{Drift}\left(W\left(u\right)\right)$

$\mathrm{\mu }\left(u\right)$

$\mathrm{Diffusion}\left(W\left(u\right)\right)$

$\mathrm{\sigma }\left(u\right)$

$\mathrm{Drift}\left(\exp \left(W\left(u\right)\right)\right)$

$\mathrm{\mu }\left(u\right){\ⅇ}^{\mathrm{\_X1}\left(u\right)}+\frac{{\mathrm{\sigma }\left(u\right)}^2{\ⅇ}^{\mathrm{\_X1}\left(u\right)}}{2}$

$\mathrm{\mu }≔\mathrm{piecewise}\left(t<1\&comma;0.02\&comma;\frac{0.02}{t}\right)$

$\mathrm{\mu }≔\left\{\begin{array}{cc}0.02& t<1\\ \frac{0.02}{t}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{\sigma }≔\mathrm{evalf}\left[3\right]\left(\mathrm{CurveFitting}:-\mathrm{Spline}\left(\left[\left[0\&comma;0.5\right]\&comma;\left[1\&comma;0.3\right]\&comma;\left[1.5\&comma;1.5\right]\&comma;\left[2\&comma;1.0\right]\right]\&comma;t\&comma;\mathrm{degree}=3\right)\right)$

$\mathrm{\sigma }≔\left\{\begin{array}{cc}0.500-1.40t+1.20t^3& t<1.\\ -1.90+2.20t+3.60{\left(t-1.\right)}^2-6.40{\left(t-1.\right)}^3& t<1.5\\ 1.00t-6.00{\left(t-1.5\right)}^2+4.00{\left(t-1.5\right)}^3& \mathrm{otherwise}\end{array}\right.$

$Y≔\mathrm{BrownianMotion}\left(0.\&comma;\mathrm{\mu }\&comma;\mathrm{\sigma }\&comma;t\right)\&colon;$

$S≔\mathrm{SamplePath}\left(Y\left(t\right)\&comma;t=0..T\&comma;\mathrm{replications}={10}^6\right)\&colon;$

$\mathrm{\mu }≔\mathrm{unapply}\left(\mathrm{\mu }\&comma;t\right)$

$\mathrm{\mu }≔t\mapsto \left\{\begin{array}{cc}0.02& t<1\\ \frac{0.02}{t}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{\sigma }≔\mathrm{unapply}\left(\mathrm{\sigma }\&comma;t\right)$

$\mathrm{\sigma }≔t\mapsto \left\{\begin{array}{cc}0.500-1.40\cdot t+1.20\cdot t^3& t<1.\\ -1.90+2.20\cdot t+3.60\cdot {\left(t-1.\right)}^2-6.40\cdot {\left(t-1.\right)}^3& t<1.5\\ 1.00\cdot t-6.00\cdot {\left(t-1.5\right)}^2+4.00\cdot {\left(t-1.5\right)}^3& \mathrm{otherwise}\end{array}\right.$

$Z≔\mathrm{BrownianMotion}\left(0.\&comma;\mathrm{\mu }\&comma;\mathrm{\sigma }\right)\&colon;$

$S≔\mathrm{PathPlot}\left(Z\left(t\right)\&comma;t=0..T\&comma;\mathrm{replications}=10\&comma;\mathrm{axes}=\mathrm{BOXED}\right)\&colon;$

$\mathrm{X0}≔\mathrm{Vector}\left(\left[0.\&comma;0.\right]\right)$

$\mathrm{X0}≔\left[\begin{array}{c}0.\\ 0.\end{array}\right]$

$\mathrm{{\rm M}}≔\mathrm{Vector}\left(\left[0.2\&comma;0.3\right]\right)$

$\mathrm{{\rm M}}≔\left[\begin{array}{c}0.2\\ 0.3\end{array}\right]$

$\mathrm{\Sigma }≔\mathrm{Matrix}\left(\left[\left[1.0\&comma;0.05\right]\&comma;\left[0.5\&comma;1.0\right]\right]\right)$

$\mathrm{\Sigma }≔\left[\begin{array}{cc}1.0& 0.05\\ 0.5& 1.0\end{array}\right]$

$W≔\mathrm{BrownianMotion}\left(\mathrm{X0}\&comma;\mathrm{{\rm M}}\&comma;\mathrm{\Sigma }\right)$

$W≔\mathrm{\_X4}$

$\mathrm{SamplePath}\left(W\left(t\right)\&comma;t=0..T\&comma;\mathrm{replications}={10}^5\right)$

$V≔\mathrm{SampleValues}\left(X\mapsto X\left[1,2\right]+X\left[2,2\right]\&comma;W\&comma;0..T\&comma;\mathrm{replications}={10}^5\right)\&colon;$

$\mathrm{Statistics}\left[\mathrm{DataSummary}\right]\left(V\right)$

$\left[\begin{array}{c}\mathrm{mean}=0.988749213777756\\ \mathrm{standarddeviation}=2.04881151111321\\ \mathrm{skewness}=0.00386575235144395\\ \mathrm{kurtosis}=2.99903802870409\\ \mathrm{minimum}=\mathrm{-7.10227512307098}\\ \mathrm{maximum}=9.60382569520030\\ \mathrm{cumulativeweight}=100000.\end{array}\right]$

$\mathrm{Statistics}\left[\mathrm{Histogram}\right]\left(V\&comma;\mathrm{bincount}=100\right)$

$R≔\mathrm{OrnsteinUhlenbeckProcess}\left(0.5\&comma;1.0\&comma;0.5\&comma;0.3\right)$

$R≔\mathrm{\_X5}$

$Y≔\mathrm{BrownianMotion}\left(0\&comma;0.\&comma;{R\left(t\right)}^2\&comma;t\right)$

$Y≔\mathrm{\_X7}$

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

$f≔\sin \left(t\right)$

$f≔\sin \left(t\right)$

$\mathrm{W1}≔\mathrm{BrownianMotion}\left(0\&comma;f\&comma;0.001\&comma;t\right)$

$\mathrm{W1}≔\mathrm{\_X8}$

$\mathrm{W2}≔\mathrm{BrownianMotion}\left(0\&comma;f\&comma;0.001\&comma;t\&comma;\mathrm{scheme}=\mathrm{unbiased}\right)$

$\mathrm{W2}≔\mathrm{\_X9}$

$\mathrm{W3}≔\mathrm{eval}\left(y\left(t\right)\&comma;\mathrm{dsolve}\left(\left\{\mathrm{diff}\left(y\left(t\right)\&comma;t\right)=\sin \left(t\right)\&comma;y\left(0\right)=0\right\}\right)\right)$

$\mathrm{W3}≔-\cos \left(t\right)+1$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{plot}\left(\mathrm{W3}\&comma;t=0..3\right)\&comma;\mathrm{PathPlot}\left(\mathrm{W1}\left(t\right)\&comma;t=0..3\&comma;\mathrm{timesteps}=10\&comma;\mathrm{replications}=10\right)\&comma;\mathrm{thickness}=3\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}\right)$

$\mathrm{plots}\left[\mathrm{display}\right]\left(\mathrm{plot}\left(\mathrm{W3}\&comma;t=0..3\right)\&comma;\mathrm{PathPlot}\left(\mathrm{W2}\left(t\right)\&comma;t=0..3\&comma;\mathrm{timesteps}=10\&comma;\mathrm{replications}=10\right)\&comma;\mathrm{thickness}=3\&comma;\mathrm{axes}=\mathrm{BOXED}\&comma;\mathrm{gridlines}\right)$

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

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

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

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