# ItoProcess

ItoProcess[{a,b},x,t]

represents an Ito process , where .

ItoProcess[{a,b,c},x,t]

represents an Ito process , where .

ItoProcess[,{x,x0},{t,t0}]

uses initial condition .

ItoProcess[,,,Σ]

uses a Wiener process , with covariance Σ.

ItoProcess[proc]

converts proc to a standard Ito process whenever possible.

ItoProcess[sdeqns,expr,x,t,wdproc]

represents an Ito process specified by a stochastic differential equation sdeqns, output expression expr, with state x and time t, driven by w following the process dproc.

# Details and Options  • ItoProcess is also known as Ito diffusion or stochastic differential equation (SDE).
• ItoProcess is a continuous-time and continuous-state random process.
• If the drift a is an -dimensional vector and the diffusion b an × -dimensional matrix, the process is -dimensional and driven by an -dimensional WienerProcess.
• Common specifications for coefficients a and b include:
•  a scalar, b scalar a scalar, b vector a vector, b vector a vector, b matrix • A stochastic differential equation is sometimes written as an integral equation .
• The default initial time t0 is taken to be zero, and the default initial state x0 is zero.
• The default covariance Σ is the identity matrix.
• For a general covariance Σ, ItoProcess canonicalizes the process by converting the diffusion matrix b to b.Σ1/2, with Σ1/2 the lower Cholesky factor of Σ when possible.
• A standard Ito process has output , consisting of a subset of differential states .
• Processes proc that can be converted to standard ItoProcess form include OrnsteinUhlenbeckProcess, GeometricBrownianMotionProcess, StratonovichProcess, and ItoProcess.
• Converting an ItoProcess to standard form automatically makes use of Ito's lemma.
• The stochastic differential equations in sdeqns can be of the form , where is \[DifferentialD], which can be input using dd . The differentials and are taken to be Ito differentials.
• The output expression expr can be any expression involving x[t] and t.
• The driving process dproc can be any process that can be converted to a standard Ito process.
• Method settings in RandomFunction specific to ItoProcess include:
•  "EulerMaruyama" Euler–Maruyama (order 1/2, default) "KloedenPlatenSchurz" Kloeden–Platen–Schurz (order 3/2) "Milstein" Milstein (order 1) "StochasticRungeKutta" 3‐stage Rossler SRK scheme (order 1) "StochasticRungeKuttaScalarNoise" 3‐stage Rossler SRK scheme for scalar noise (order 3/2)
• ItoProcess can be used with such functions as RandomFunction, CovarianceFunction, PDF, and Expectation.

# Examples

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## Basic Examples(1)

Define a process by its stochastic differential equation:

 In:= Out= Simulate the process:

 In:= Out= In:= Out= Compute mean function:

 In:= Out= Compute covariance function:

 In:= Out= In:= Out= ## Possible Issues(2)

Introduced in 2012
(9.0)