A stochastic process describes the evolution of random variables/systems over time and/or space and/or any other index set. It has applications in areas such as econometrics, weather, signal processing, etc. Examples: Gaussian process, Markov process, etc.
A stochastic process is a collection of random variables ${\bf X} = \{ X_t : t \in T \}$ defined on a common probability space, taking values in a common set $S$ (state space), and indexed by set $T$, often either $\mathbb{N}$ or $[0, \infty)$, and thought of as time (either continuous or discrete). (Reference: Random Services)
Common examples of stochastic processes are:
- Random Walks
- Simple random walk: defined on the integers in discrete time, and is based on a Bernoulli process, where each $iid$ Bernoulli variable takes either the value $+1$ or $-1$.
- Bernoulli Process
- A sequence of $iid$ Bernoulli random variables, where each event is a Bernoulli trial.
- Wiener Process
- Stochastic process with stationary and independent increments whose size is normally distributed.
- Poisson Process
- Defined as a counting process, which is a process that represents the random number of points or events up to some time.
- The number of points of the process that are located in the interval from zero to some (given, non-random) time is a Poisson random variable.
- Markov Chains (both continuous and discrete time)
- The behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.
- Martingales
- A discrete-time or continuous-time stochastic processes with the property that the expectation of the next value of a martingale is equal to the current value given all the previous values of the process.
(Reference: Wikipedia)
