A stochastic process describes 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)

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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 positive one or negative one
  • Bernoulli Process
    • A sequence of $iid$ Bernoulli random variables, where each event is a Bernoulli trial.
  • Weiner Process
    • Stochastic process with stationary and independent increments that are normally distributed based on the size of the increments
  • 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, nonrandom) 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.

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