The Markov property defines a subclass of stochastic systems where evolution in time is determined only by the last state. When appropriate, this assumption greatly simplifies the mathematical formulation of the system by removing any conditioning on any other previous state.
| \( n \) | This symbol represents any given whole number, \( n \in \htmlClass{sdt-0000000014}{\mathbb{W}}\). |
| \( X \) | This symbol represents a random variable. It is a measurable function that maps a sample space of possible outcomes to a a measurable space. |
| \( \mathbf{x} \) | This symbol represents a state of the dynamical system at some time point. |
This is true by definition, if considering the system state at time point \( n \) to be a random variable \( \mathbf{X}_n \) with realizations of the form \( \htmlClass{sdt-0000000046}{\mathbf{x}}_n \in \htmlClass{sdt-0000000038}{\mathcal{X}} \).
For example, a game of chess obeys the Markov property. As long as you can observe the current board arrangement, it is possible to decide on a move for your player with the same accuracy as someone who knows the previous moves that have been made. You need no information prior to the current arrangement in order to reason about a move, and thus determine the next state.