Stochastic differential equations (sdes) occur where a system described by differential equations is influenced by random noise. Stochastic differential equations are used in finance (interest rate, stock prices, …), biology (population, epidemics, …), physics (particles in fluids, thermal noise, …), and control and signal processing (controller, filtering, …). The Wolfram Language provides common special sdes specified by a few parameters as well as general Ito and Stratonovich sdes and systems specified by their differential equations. The symbolic representation of sde processes allows a uniform way to compute a variety of properties, from simulation and mean and covariance functions to full state distributions at different times.

Special Diffusion Processes

WienerProcess — Wiener process or Brownian motion

OrnsteinUhlenbeckProcess — Ornstein–Uhlenbeck process

BrownianBridgeProcess  ▪  GeometricBrownianMotionProcess  ▪  CoxIngersollRossProcess

General Diffusion Processes

ItoProcess — Ito sde process

StratonovichProcess — Stratonovich sde process

Process Framework

RandomFunction — simulate an sde process (Euler–Muryama, stochastic Runge–Kutta, …)

SliceDistribution — distribution of states at particular times

CovarianceFunction  ▪  CorrelationFunction  ▪  AbsoluteCorrelationFunction