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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, Wt, that is a Gaussian process with independent increments such that W0=0 with probability 1, EWt=0 and VarWtWs=ts for all 0st.

• 

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 WJt, where Wt is the standard Wiener process. Note that the subordinator Jt 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.

Examples

withFinance:

First create a standard Wiener process and generate 50 replications of the sample path and plot the result.

W:=WienerProcess:

P:=PathPlotWt,t=0..3,timesteps=50,replications=20,thickness=3,color=red,axes=BOXED,gridlines=true:P

Define another stochastic variable as an expression involving W1. You can compute the expected value of X3 using Monte Carlo simulation with the specified number of replications of the sample path.

T:=3

T:=3

(1)

ExpectedValueⅇ0.05T+0.3WT,replications=105

value=1.331151006,standarderror=0.002335223363

(2)

Define another stochastic variable Y, which also depends on W1 but uses symbolic coefficients. Note that Y is an Ito process, so it is governed by the stochastic differential equation (SDE) dYt=μYt,tdt+σYt,tdWt. You can use the Drift and Diffusion commands to compute μ and σ.

Y:=t→ⅇμt+σWt:Yt

ⅇμt+σ_Wt

(3)

subsYt='Y't,DriftYt

μYt+12σ2Yt

(4)

subsYt='Y't,DiffusionYt

σYt

(5)

Create a subordinated Wiener process that uses a Poisson process with intensity parameter λ=0.3 as subordinator.

J:=PoissonProcess0.3

J:=_P

(6)

W2:=WienerProcessJ

W2:=_W0

(7)

P2:=PathPlotW2t,t=0..3,timesteps=50,replications=20,thickness=3,color=blue,axes=BOXED,gridlines=true:P2

plots[display]P,P2

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.

κ:=0.1:σ:=0.3:θ:=0.5:R0:=0.02:

τ:=ⅇ2κt12κ

τ:=5.000000000ⅇ0.2t5.000000000

(8)

W3:=WienerProcessτ

W3:=_W1

(9)

R:=t→R0ⅇκt+θ1ⅇκt+σⅇκtW3t:Rt

0.48ⅇ0.1t+0.5+0.3ⅇ0.1t_W1t

(10)

P3:=PathPlotRt,t=0..3,timesteps=50,replications=20,thickness=3,color=blue,axes=BOXED,gridlines=true:P3

ExpectedValueR3,timesteps=100,replications=105

value=0.1517355448,standarderror=0.009505937484

(11)

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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