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
open allclose allBasic Examples (1)
Scope (16)
Basic Uses (9)
Define a Wiener process with drift and diffusion
from the stochastic differential equation (SDE)
:
Directly convert from the parametric process:
Use differential notation to define the same process:
Define a vector process with output
:
Define a vector process , where
:
Define a vector process where
:
Define a process driven by two correlated Wiener processes: »
The canonicalized process has diffusion matrix equal to , with
the diffusion matrix before canonicalization:
Define a scalar process corresponding to the SDE
:
Define vector process and
corresponding to the SDE
and
:
Define a process corresponding to the 2D correlated Wiener process:
Define vector process driven by correlated 2D Wiener process:
Special Ito Processes (5)
An Ito process corresponding to the WienerProcess:
An Ito process corresponding to the GeometricBrownianMotionProcess:
An Ito process corresponding to the BrownianBridgeProcess:
An Ito process corresponding to the OrnsteinUhlenbeckProcess:
An Ito process corresponding to the CoxIngersollRossProcess:
Process Slice Properties (2)
Define Jacobi diffusion process:
Compute low-order cumulants of time‐slice distribution:
Find the limit of infinite time horizon:
Compare with cumulants of the uniform distribution:
Define a vector process given by a system of linear SDEs:
Find the probability density function of the time‐slice distribution:
Applications (5)
Computing Properties (3)
Compute cross-covariance of the Ornstein–Uhlenbeck process and its underlying Wiener process
:
Compute moments of the process , where
is the standard Wiener process:
Vector Ito process driven by scalar noise (1D oscillator driven by white noise):
Compute mean and variance functions:
Plot mean function and the standard deviation band, together with generated paths:
Martingales (1)
Modeling (1)
The Gompertz curve is typically used in the modeling of a growth process, such as tumor growth. By assuming Gaussian noise in the logarithm of the growth process, you can write the model as a stochastic differential equation:
Mean of the process is the usual Gompertz curve:
Slice distribution of the process at time obeys LogNormalDistribution:
Simulate the process with , and
from
to
:
Generate a thousand samples with the same conditions, then visualize the paths and slice data at :
Properties & Relations (2)
Convert StratonovichProcess to ItoProcess:
Transformed Wiener processes are related to ItoProcess:
Possible Issues (2)
ItoProcess does not support random initial conditions, so cannot be represented:
But it supports processes with fixed initial condition:
Initial time of the driven process needs to match with ItoProcess:
Text
Wolfram Research (2012), ItoProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ItoProcess.html (updated 2016).
CMS
Wolfram Language. 2012. "ItoProcess." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/ItoProcess.html.
APA
Wolfram Language. (2012). ItoProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ItoProcess.html