2 equivalent definitions of a markov continuous time process

Question

I read many documents on markov processes and sometimes authors present a definition with conditional expectations and others define with simply a probability measure $\mathbb{P}$.

I want to understand the link between both representations. Is it an equivalence? Or only one sense is true? And why?

First one, $(V_t)_{t \geq 0}$

$\forall t \geq 0 \forall s \geq 0 $ $$\mathbb{E} [f(V_{t+s}) | \mathcal{F}_s ] = \mathbb{E}[ f(V_{t+s}) | V_s ] $$ $\forall$ borel function f.

Another classical definition:

$\forall A \in \mathcal{B}(\mathbb{R}) , \forall t \geq s$,

$$\mathbb{P} [ V_t \in A | \mathcal{F}_s] = \mathbb{P} [ V_t \in A | V_s ]$$

where $\mathcal{F}_s = \sigma ( V_u ; u \leq s )$

Thank you.

Answer 1

I believe that the Portmanteau's lemma is the result that you need to see that these definitions are equivalent, with $E$ the expectancy and $P$ the probability :

The Portmanteau lemma:

For any random vectors $X_n$ and $X$ the following statements are equivalent.

• $P \left( X_n \le x \right) \to P \left( X\le x \right)$ for all continuity points of $x \mapsto P \left( X \le x \right)$;
• $E f(X_n) \to E f(X)$ for all bounded, continuous functions $f$;
• $E f(X_n) \to E f(X)$ for all bounded, Lipschitz functions $f$;
• $\lim \inf E f(X_n) \ge E f(X)$ for all nonnegative, continuous functions $f$;
• $\lim \inf P (X_n \in G) \ge P (X \in G)$ for every open set $G$;
• $\lim \sup P \left( X_n \in F \right) \le P (X \in F) $ for every closed set $F$;
• $P \left( X_n \in B \right) \to P \left( X\in B \right)$ for all Borel sets $B$ with $P \left( X\in\delta B \right) = 0$, where $\delta B = \bar{B} - \mathring{B}$ is the boundary of $B$.

Source : Asymptotic Statistics, Van der Vaart.

I believe that you meant $\forall f$ continuous, and bounded.