Stochastic Process

Definition

A stochastic process is a parameterized family of random variables

\fs4 \{X_t\}_{t\in T}

All random variables are defined on the probability space \fs4 (\Omega, \mathcal{F}, P). Typically \fs4 T = \{0, 1, 2, \ldots\} for discrete time processes, or \fs4 T = [0, \infty) for continuous time processes.

We can have three points of view for a stochastic process

  • If we fix time we have a random variable
\fs4 \omega \rightarrow X_t(\omega), \omega \in \Omega
  • If we fix a state of the world \fs4 \omega we have the trajectory or path
\fs4 t \rightarrow X_t(\omega), t \in T
  • We can also consider the stochastic process as a function of two variables
\fs4 X_t(\omega) = X(t, \omega)\text{ : }T\times \Omega \rightarrow \mathbb{R}^n

Although the probability measure is defined on the \fs4 \sigma-algebra \fs4 \mathcal{F}, we typically model the flow of information with a filtration \fs4 \{ \mathcal{F}_t \} for \fs4 t \in T (setting a filtered space).

Examples

The most fundamental example of a stochastic process is the Brownian motion.

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