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

converts proc to a standard Ito process whenever possible.

ItoProcess[sdeqns, expr, x, t, wDistributeddproc]
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 OptionsDetails and Options

  • ItoProcess is also known as Ito diffusion.
  • 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 is taken to be zero, and the default initial state is zero.
  • The default covariance is the identity matrix.
  • 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 EscddEsc. The differentials and are taken to be Ito 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 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.

ExamplesExamplesopen allclose all

Basic Examples (1)Basic Examples (1)

Define a process by its stochastic differential equation:

Click for copyable input

Simulate the process:

Click for copyable input
Click for copyable input

Compute mean function:

Click for copyable input

Compute covariance function:

Click for copyable input
Click for copyable input
New in 9
New to Mathematica? Find your learning path »
Have a question? Ask support »