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Finance[WienerProcess] - create new Wiener process
Calling Sequence
WienerProcess(J)
WienerProcess(Sigma)
Parameters
J
-
(optional) stochastic process with non-negative increments, or a deterministic function of time; subordinator
Sigma
Matrix; covariance matrix
Description
The WienerProcess command creates a new Wiener process. If called with no arguments, the WienerProcess command creates a new standard Wiener process, , that is a Gaussian process with independent increments such that with probability , and for all .
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 , where is the standard Wiener process. Note that the subordinator 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.
Compatibility
The Finance[WienerProcess] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
Examples
First create a standard Wiener process and generate replications of the sample path and plot the result.
Define another stochastic variable as an expression involving . You can compute the expected value of using Monte Carlo simulation with the specified number of replications of the sample path.
Define another stochastic variable , which also depends on but uses symbolic coefficients. Note that is an Ito process, so it is governed by the stochastic differential equation (SDE) . You can use the Drift and Diffusion commands to compute and .
Create a subordinated Wiener process that uses a Poisson process with intensity parameter as subordinator.
Here is a representation of the Ornstein-Uhlenbeck process in terms of or a subordinated Wiener process. In this case the subordinator is a deterministic process that can be specified as a Maple procedure or an algebraic expression.
See Also
Finance[BlackScholesProcess], Finance[CEVProcess], Finance[Diffusion], Finance[Drift], Finance[ExpectedValue], Finance[GeometricBrownianMotion], Finance[ItoProcess], Finance[PathPlot], Finance[SamplePath], Finance[SampleValues], Finance[StochasticProcesses], Finance[WienerProcess]
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