Categories: Mathematics, Stochastic analysis.

# Markov process

Given a 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:

- $X_t$ is $\mathcal{F}_t$-adapted, meaning that the current and all past values of $X_t$ can be reconstructed from the filtration $\mathcal{F}_t$.
- For some function $h(x)$,
the 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):

A prime examples of a continuous Markov process is
the Wiener process.
Note that this is also a 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

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