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.
- Properties related to ItoProcess include:
"Drift" drift term "Diffusion" diffusion matrix "Output" output state "TimeVariable" time variable "TimeOrigin" origin of time variable "StateVariables" state variables "InitialState" initial state values "KolmogorovForwardEquation" Kolmogorov forward equation (Fokker-Planck equation) "KolmogorovBackwardEquation" Kolmogorov backward equation "Derivative" Ito derivative "FeynmanKacFormula" PDE obtained from Feynman-Kac formula
- 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)
Process Properties Extraction (2)
Additional arguments can be provided for the generalized situations. With an additional argument , the property "FeynmanKacFormula" gives a PDE whose solution satisfies the conditional expectation and the same terminal condition:
Define Heston model with ItoProcess:
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 dynamics of a free particle under the effect of thermal fluctuation can be modeled by the Langevin equation of motion, , where is the standard WienerProcess and is the strength of the thermal noise. Here it is assumed that can only depend on and focus on the equation of velocity. There are two common ways to integrate the equation of motion: Ito formulation and Stratonovich formulation. They can be defined via:
If is velocity dependent, then due to the nature of the WienerProcess, has nonzero quadratic variation and the two formulations lead to different results. Convert Stratonovich formulation to the equivalent Ito formulation:
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:
Ito Process Representations (3)
Visualize the dynamic of the solution with Animate:
Solve the Black–Scholes equation symbolically with DSolve:
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