StratonovichProcess

StratonovichProcess[{a,b},x,t]
represents a Stratonovich process , where .

StratonovichProcess[{a,b,c},x,t]
represents a Stratonovich process , where .

StratonovichProcess[,,{x,x0},{t,t0}]
represents a Stratonovich process with initial condition .

StratonovichProcess[,,,Σ]
uses a Wiener process , with covariance Σ.

StratonovichProcess[proc]
converts proc to a standard Stratonovich process whenever possible.

StratonovichProcess[sdeqns,expr,x,t,wdproc]
represents a Stratonovich 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 OptionsDetails and Options

  • StratonovichProcess is also known as Stratonovich diffusion or stochastic differential equation (SDE).
  • StratonovichProcess 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 is taken to be zero, and default initial state is zero.
  • The default covariance Σ is the identity matrix.
  • A standard Stratonovich process has output , consisting of a subset of differential states .
  • Processes proc that can be converted to standard StratonovichProcess form include OrnsteinUhlenbeckProcess, GeometricBrownianMotionProcess, ItoProcess, and StratonovichProcess.
  • The stochastic differential equations in sdeqns can be of the form , where is \[DifferentialD], which can be input using EscddEsc. The differentials and are taken to be Stratonovich differentials.
  • The output expression expr can be any expression involving and t.
  • The driving process dproc can be any process that can be converted to a standard Stratonovich process.
  • Method settings in RandomFunction specific to StratonovichProcess include:
  • "EulerMaruyama"EulerMaruyama (order 1/2, default)
    "KloedenPlatenSchurz"KloedenPlatenSchurz (order 3/2)
    "Milstein"Milstein (order 1)
    "StochasticRungeKutta"3stage Rossler SRK scheme (order 1)
    "StochasticRungeKuttaScalarNoise"3stage Rossler SRK scheme for scalar noise (order 3/2)
  • StratonovichProcess can be used with such functions as RandomFunction, CovarianceFunction, PDF, and Expectation.

ExamplesExamplesopen allclose all

Basic Examples  (1)Basic Examples  (1)

Define a process by its stochastic differential equation:

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Simulate the process:

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Compute the mean function:

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Compute the covariance function:

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Introduced in 2012
(9.0)