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+---
+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.