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Diffstat (limited to 'source/know/concept/markov-process')
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1 files changed, 23 insertions, 23 deletions
diff --git a/source/know/concept/markov-process/index.md b/source/know/concept/markov-process/index.md index fd6b076..c938866 100644 --- a/source/know/concept/markov-process/index.md +++ b/source/know/concept/markov-process/index.md @@ -9,37 +9,37 @@ layout: "concept" --- Given a [stochastic process](/know/concept/stochastic-process/) -$\{X_t : t \ge 0\}$ on a filtered probability space -$(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, P)$, +$$\{X_t : t \ge 0\}$$ 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 $h(x)$, +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 $\sigma(h(X_t)) \subseteq \mathcal{F}_t$. + $$\mathbf{E}[h(X_t) | \mathcal{F}_s] = \mathbf{E}[h(X_t) | X_s]$$, + 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 $X_s$. +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**. +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/)): +$$P(X_t \in A | X_s = x)$$, 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) |