create new Wiener process
(optional) stochastic process with non-negative increments, or a deterministic function of time; subordinator
Matrix; covariance matrix
The WienerProcess command creates a new Wiener process. If called with no arguments, the WienerProcess command creates a new standard Wiener process, W⁡t, that is a Gaussian process with independent increments such that W⁡0=0 with probability 1, E⁡W⁡t=0 and Var⁡W⁡t−W⁡s=t−s for all 0≤s≤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⁡J⁡t, where W⁡t is the standard Wiener process. Note that the subordinator J⁡t 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.
First create a standard Wiener process and generate 50 replications of the sample path and plot the result.
W ≔ WienerProcess⁡:
P ≔ PathPlot⁡W⁡t,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 X⁡3 using Monte Carlo simulation with the specified number of replications of the sample path.
T ≔ 3
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) dY⁡t=μ⁡Y⁡t,t⁢dt+σ⁡Y⁡t,t⁢dW⁡t. You can use the Drift and Diffusion commands to compute μ and σ.
Y ≔ t→ⅇμ⁢t+σ⁢W⁡t:Y⁡t
Create a subordinated Wiener process that uses a Poisson process with intensity parameter λ=0.3 as subordinator.
J ≔ PoissonProcess⁡0.3
W2 ≔ WienerProcess⁡J
P2 ≔ PathPlot⁡W2⁡t,t=0..3,timesteps=50,replications=20,thickness=3,color=blue,axes=BOXED,gridlines=true: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 ≔ ⅇ2⁢κ⁢t−12⁢κ
W3 ≔ WienerProcess⁡τ
R ≔ t→R0⁢ⅇ−κ⁢t+θ⁢1−ⅇ−κ⁢t+σ⁢ⅇ−κ⁢t⁢W3⁡t:R⁡t
P3 ≔ PathPlot⁡R⁡t,t=0..3,timesteps=50,replications=20,thickness=3,color=blue,axes=BOXED,gridlines=true:P3
The Finance[WienerProcess] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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