represents an Ito process , where .


represents an Ito process , where .


uses initial condition .


uses a Wiener process , with covariance Σ.


converts proc to a standard Ito process whenever possible.


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"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)
  • ItoProcess can be used with such functions as RandomFunction, CovarianceFunction, PDF, and Expectation.


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Basic Examples  (1)

Define a process by its stochastic differential equation:

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Simulate the process:

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Compute mean function:

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Compute covariance function:

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Scope  (16)

Applications  (5)

Properties & Relations  (2)

Possible Issues  (2)

See Also

WienerProcess  OrnsteinUhlenbeckProcess  GeometricBrownianMotionProcess  StratonovichProcess  TransformedProcess  AffineStateSpaceModel

Introduced in 2012