SOLUTIONS

BUILTIN MATHEMATICA SYMBOL
ItoProcess
ItoProcess[{a, b}, x, t]
represents an Ito process , where .
ItoProcess[{a, b, c}, x, t]
represents an Ito process , where .
ItoProcess[..., {x, x_{0}}, {t, t_{0}}]
uses initial condition .
ItoProcess[..., ..., ..., ]
uses a Wiener process , with covariance .
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 OptionsDetails and Options
 ItoProcess is also known as Ito diffusion.
 ItoProcess is a continuoustime and continuousstate 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" 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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