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" 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.
Examplesopen allclose all
Basic Examples (1)
Basic Uses (9)
Define a process driven by two correlated Wiener processes: »
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)
Computing Properties (3)
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:
Slice distribution of the process at time obeys LogNormalDistribution:
Properties & Relations (2)
Wolfram Research (2012), ItoProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/ItoProcess.html (updated 2016).
Wolfram Language. 2012. "ItoProcess." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/ItoProcess.html.
Wolfram Language. (2012). ItoProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ItoProcess.html