Markov Property

Topics: Markov Process - Stochastic Process

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(definition)

A stochastic process has the Markov property if the probability of future states depends only upon the present state and not the other past states.

Any such stochastic process is also known as a Markov process. This property is called after the Russian mathematician Andrey Markov.

Formal Definition §

(definition)

More formally, given $\left\{X_t\mid t\in \mathbb{R}\right\}$ a stochastic process, we say it satisfies the Markov property if, for any $t$:

$$\begin{array}{rl}\mathbb{P}& (X_{t+1}=x_{t+1}\mid X_0=x_0,\dots ,X_{t-1}=x_{t-1},X_t=x_t)\\ & =\mathbb{P}(X_{t+1}=x_{t+1}\mid X_t=x_t)\end{array}$$

That is, it satisfies the Markov property if the conditional probability of any future result given the present result and every other past result is equal to the conditional probability of any future result given only the present result.