Im getting a bit confused with the terminology of continous time markov processes. Now let $\{X_t | t \geq 0 \}$ be a stochastic Process with state space $\mathbb{R}$, and $(F_t)$ be its canonical Filtration My professor says X is a markov process, if:
$\mathbb{P}(X_t \in A | F_s)=\mathbb{P}(X_t \in A |X_s)$ for all $s \leq t$.

Now i was wondering what does this imply about the transition probability?
We can define functions $\mathbb{P}(X_t \in A | X_s=x)=\mathbb{P}_{X_t|X_s=x}(A)$. I dont know what these functions are called in english, maybe conditional distribution? If we fixed $t_1 < t_2 <....< t_n$ would the markov property given above imply:
$\mathbb{P}(X_{t_n} \in A | X_{t_{n-1}},...., X_{t_1})=\mathbb{P}(X_{t_n} \in A | X_{t_{n-1}})$ as functions in $x_1,...,x_{n-1}$?


Yes, it is true that $\mathbb{P}\{X_{t_{n}} \in A \, \mid \, X_{t_{n - 1}},\dots,X_{t_{1}}\} = \mathbb{P}\{X_{t_{n}} \in A \, \mid \, X_{t_{n}}\}$. This follows from the fact that $\sigma(X_{t_{n - 1}},\dots,X_{t_{1}}) \subseteq \mathcal{F}_{t_{n - 1}}$ and $\sigma(X_{t_{n}}) \subseteq \sigma(X_{t_{n - 1}},\dots,X_{t_{1}})$. In particular, if $A_{n - 1},\dots,A_{1}$ are Borel sets, then \begin{align*} \mathbb{P}\{X_{t_{n}} \in A, X_{t_{n - 1}} \in A_{n - 1},\dots,X_{t_{1}} \in A_{1}\} &= \mathbb{E}\left(\mathbb{P}\{X_{t_{n}} \in A \, \mid \, \mathcal{F}_{t_{n -1}}\} : X_{t_{n - 1}} \in A_{n - 1},\dots,X_{t_{1}} \in A_{1}\right) \\ &= \mathbb{E}\left(\mathbb{P}\{X_{t_{n}} \in A \, \mid \, X_{t_{n - 1}}\} : X_{t_{n - 1}} \in A_{n - 1},\dots,X_{t_{1}} \in A_{1}\right). \end{align*}
Since $\mathbb{P}\{X_{t_{n}} \in A \, \mid \, X_{t_{n - 1}}\}$ is $\sigma(X_{t_{n - 1}},\dots,X_{t_{1}})$-measurable, this proves that $\mathbb{P}\{X_{t_{n}} \in A \, \mid \, X_{t_{n - 1}}, \dots, X_{t_{1}}\} = \mathbb{P}\{X_{t_{n}} \in A \, \mid \, X_{t_{n - 1}}\}$ almost surely. More equivalent definitions of the Markov property can be found in the textbook by Chung and Walsh.

Your question regarding transition functions is a bit vague. Yes, what you wrote is referred to as the conditional distribution. If you're wondering if the Markov property implies the transition functions are time-homogeneous, then the answer is no.

  • $\begingroup$ Hello, thank you for your answer. My question regarding conditional distribution was as follows: Does for the conditional distributions hold: $\mathbb{P}_{X_{t_n}|X_{t_{n-1}},...., X_{t_1}}((x_1,....,x_{n-1}),A)$=$\mathbb{P}_{X_{t_n}|X_{t_{n-1}}}((x_{n-1}),A)$ $\endgroup$ – StefanWK Jun 19 '18 at 15:21
  • 1
    $\begingroup$ Yes, that's true by the definition of the conditional distribution and the fact that $\mathbb{P}\{X_{t_{n}} \in A \, \mid \, X_{t_{n - 1}},\dots,X_{t_{1}}\} = \mathbb{P}\{X_{t_{n}} \in A \, \mid \, X_{t_{n - 1}}\}$. $\endgroup$ – fourierwho Jun 19 '18 at 17:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.