Dear all,

[Bauer, Probability Theory, Exercise 2 of Chapter 39] -->

http://books.google.de/books?id=w76IHsPHybcC&pg=PA339#v=onepage&q&f=false

gives the following characterisation for the continuity of sample paths of a $\mathbb{R}^d$ valued stochastic process:

*A stochastic process $(\Omega,\mathcal{A},P,(X_t)_{t\geq0})$with state space $\mathbb{R}^d$has a continuous modification $(X'_t)_{t\geq0}$if and only if there is a countable dense subset $S$ of $\mathbb{R}_+$with the following properties:

• For every pair of positive realnumbers $\eta,k$ $$\lim_{\delta\downarrow 0}P(\bigcup_{s,t\in S,s,t\leq k,|s-t|\leq \delta}\{|X_s-X_t|\geq\eta\})=0$$
• For every $t\in\mathbb{R}_+$there is a sequence $(s_n)$ in $S$ converging to $t$ such that the sequence $(X_{s_n})$converges stochastically to $X_t$.*

My question now is: Does anyone know a reference for an analogous statement for processes with a general complete and separable metric state space $(E,d)$?

Of course, one then has to substitute $d(X_t,X_s)$ for $|X_s-X_t|$ in the first condition; but apart from that I do not see any reason why the result should not carry over to this more general setting. But I did not find any reference in the literature yet!

Thanks for answers! Kind regards

lpdbw

Dear all,

[Bauer, Probability Theory, Exercise 2 of Chapter 39] -->

http://books.google.de/books?id=w76IHsPHybcC&pg=PA339#v=onepage&q&f=false

gives the following characterisation for the continuity of sample paths of a $\mathbb{R}^d$ valued stochastic process:

*A stochastic process $(\Omega,\mathcal{A},P,(X_t)_{t\geq0})$with state space $\mathbb{R}^d$has a continuous modification $(X'_t)_{t\geq0}$if and only if there is a countable dense subset $S$ of $\mathbb{R}_+$with the following properties:

• For every pair of positive realnumbers $\eta,k$ $$\lim_{\delta\downarrow 0}P(\bigcup_{s,t\in S,s,t\leq k,|s-t|\leq \delta}\{|X_s-X_t|\geq\eta\})=0$$
• For every $t\in\mathbb{R}_+$there is a sequence $(s_n)$ in $S$ converging to $t$ such that the sequence $(X_{s_n})$converges stochastically to $X_t$.*

My question now is: Does anyone know a reference for an analogous statement for processes with a general complete and separable metric state space $(E,d)$?

Of course, one then has to substitute $d(X_t,X_s)$ for $|X_s-X_t|$ in the first condition; but apart from that I do not see any reason why the result should not carry over to this more general setting. But I did not find any reference in the literature yet!

Thanks for answers! Kind regards

lpdbw