Parametric random processes are processes specified using a few parameters. They provide standard models for a variety of areas, including finance (interest rates, …), insurance (claims process, …), and physics (radioactive decay, …). The Wolfram Language provides complete support for working with parametric random processes. The symbolic representation of a process makes it easy to simulate its behavior, estimate parameters from data, and compute state probabilities at different times. There is a full suite of standard properties, including state distributions at different times as well as mean and covariance functions.

Discrete Time and Discrete State

RandomWalkProcess — random walk on infinite lattice

BernoulliProcess  ▪  WhiteNoiseProcess  ▪  BinomialProcess  ▪  RenewalProcess  ▪  CompoundRenewalProcess  ▪  TransformedProcess

Discrete Time and Continuous State

WhiteNoiseProcess  ▪  CompoundRenewalProcess  ▪  TransformedProcess

Continuous Time and Discrete State

PoissonProcess — Poisson counting process

InhomogeneousPoissonProcess  ▪  TelegraphProcess  ▪  CompoundPoissonProcess  ▪  RenewalProcess  ▪  CompoundRenewalProcess  ▪  TransformedProcess

Continuous Time and Continuous State

WienerProcess — integrated Gaussian white noise

OrnsteinUhlenbeckProcess  ▪  BrownianBridgeProcess  ▪  GeometricBrownianMotionProcess  ▪  CoxIngersollRossProcess  ▪  FractionalBrownianMotionProcess  ▪  FractionalGaussianNoiseProcess  ▪  CompoundPoissonProcess  ▪  CompoundRenewalProcess  ▪  TransformedProcess