Let $E$ be a locally compact separable metric space, $D([0,\infty),E)$ denote the Skorohod space, $C_0(E)$ denote the space of continuous functions $E\to\mathbb R$ vanishing at infinity and $E^\ast:=E\uplus\left\{\infty\right\}$ denote the Alexandroff one-point compactification.

I'm trying to understand the proof of Theorem 2.5 of Chapter 4 in the book of Ethier and Kurtz: Let

  • $(T(t))_{t\ge0}$ be a Feller (i.e. contractive, nonnegative and strongly continuous) semigroup on $C_0(E)$ with generator $(\mathcal D(A),A)$
  • $(T_n(t))_{t\ge0}$ be a uniformly continuous semigroup on the space $B(E)$ of bounded measurable functions $E\to\mathbb R$ (equipped with the supremum norm) with (bounded) generator $A_n$
  • $D$ be a core of $(\mathcal D(A),A)$
  • $(X_t)_{t\ge0}$ and $(X^n_t)_{t\ge0}$ be $E$-valued càdlàg strong Markov processes on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$\operatorname E\left[f(X_{s+t})\mid\mathcal(X_r)_{r\in[0,\:s]}\right]=(T(t)f)(X_s)\;\;\;\text{almost surely for all }f\in C_0(E)\tag1$$ and $$\operatorname E\left[f(X^n_{s+t})\mid\mathcal(X^n_r)_{r\in[0,\:s]}\right]=(T_n(t)f)(X^n_s)\;\;\;\text{almost surely for all }f\in B(E)\tag2$$ for all $s,t\ge0$ and $n\in\mathbb N$

Assume $$\left\|T_n(t)f-T(t)f\right\|_\infty\xrightarrow{n\to\infty}0\;\;\;\text{for all }f\in C_0(E)\text{ and }t\ge0\tag3,$$ which is known to be equivalent to the claim that for all $f\in D$, there is a $f_n\in B(E)$ for all $n\in\mathbb N$ with $$\left\|f_n-f\right\|_\infty+\left\|A_nf_n-Af\right\|_\infty\xrightarrow{n\to\infty}0.\tag4$$ Moreover, assume $$X_0^n\to X_0\tag5$$ in distribution.

I want to show that $(X^n_\ast\operatorname P)_{n\in\mathbb N}\subseteq\mathcal M_1(D([0,\infty),E))$ is relatively compact, where $\mathcal M_1(D([0,\infty),E))$ denotes the space of probability measures on $\mathcal B(D([0,\infty),E))$ equipped with the topology of weak convergence$^1$.

The proof in Ethier and Kurtz utilizes quite general results. I would like to proof the claim directly without using these general results. The guideline seems to be to firstly show that $(X^n_\ast\operatorname P)_{n\in\mathbb N}$ is relatively compact when considered as a sequence in $\mathcal M_1(D([0,\infty),E^\ast))$.

How can we do that?

(compare with If $X^n$ is a càdlàg process in a locally compact space $E$, show that $\{X^n\}$ is relatively compact as processes in the compactification of $E$).

EDIT: For example, in the used Theorem 9.4 of Chapter 3, we should be able to choose $C_a=C_0(E)$ and $D=\mathcal D(A)$, noting that $$f(X^n_t)-\int_0^t(A_nf)(X^n_s)\:{\rm d}s\tag6$$ is a martingale wrt the filtration $\mathcal F^{X_n}$ generated by $X^n$, and hence the conditions $(9.17)$ and $(9.18)$ should be trivially satisfied. This should allow a much simpler proof of that Theorem.


Your Answer

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

Browse other questions tagged or ask your own question.