---
title: "Markov process"
sort_title: "Markov process"
date: 2021-11-14
categories:
- Mathematics
- Stochastic analysis
layout: "concept"
---

Given a [stochastic process](/know/concept/stochastic-process/)
$$\{X_t : t \ge 0\}$$ on a filtered probability space
$$(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, P)$$,
it is said to be a **Markov process**
if it satisfies the following requirements:

1.  $$X_t$$ is $$\mathcal{F}_t$$-adapted,
    meaning that the current and all past values of $$X_t$$
    can be reconstructed from the filtration $$\mathcal{F}_t$$.
2.  For some function $$h(x)$$,
    the [conditional expectation](/know/concept/conditional-expectation/)
    $$\mathbf{E}[h(X_t) | \mathcal{F}_s] = \mathbf{E}[h(X_t) | X_s]$$,
    i.e. at time $$s \le t$$, the expectation of $$h(X_t)$$ depends only on the current $$X_s$$.
    Note that $$h$$ must be bounded and *Borel-measurable*,
    meaning $$\sigma(h(X_t)) \subseteq \mathcal{F}_t$$.

This last condition is called the **Markov property**,
and demands that the future of $$X_t$$ does not depend on the past,
but only on the present $$X_s$$.

If both $$t$$ and $$X_t$$ are taken to be discrete,
then $$X_t$$ is known as a **Markov chain**.
This brings us to the concept of the **transition probability**
$$P(X_t \in A | X_s = x)$$, which describes the probability that
$$X_t$$ will be in a given set $$A$$, if we know that currently $$X_s = x$$.

If $$t$$ and $$X_t$$ are continuous, we can often (but not always) express $$P$$
using a **transition density** $$p(s, x; t, y)$$,
which gives the probability density that the initial condition $$X_s = x$$
will evolve into the terminal condition $$X_t = y$$.
Then the transition probability $$P$$ can be calculated like so,
where $$B$$ is a given Borel set (see [$$\sigma$$-algebra](/know/concept/sigma-algebra/)):

$$\begin{aligned}
    P(X_t \in B | X_s = x)
    = \int_B p(s, x; t, y) \dd{y}
\end{aligned}$$

A prime examples of a continuous Markov process is
the [Wiener process](/know/concept/wiener-process/).
Note that this is also a [martingale](/know/concept/martingale/):
often, a Markov process happens to be a martingale, or vice versa.
However, those concepts are not to be confused:
the Markov property does not specify *what* the expected future must be,
and the martingale property says nothing about the history-dependence.



## References
1.  U.H. Thygesen,
    *Lecture notes on diffusions and stochastic differential equations*,
    2021, Polyteknisk Kompendie.