Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in D$. Denote by $\mathcal M$ the set of probability measures $\mathbb P$ under which $X$ is a martingale. Clearly, for each probability $\mathbb P\in\mathcal M$, there exists a corresponding triplet

$$\big(B^{\mathbb P}:=(B^{\mathbb P}_t)_{t\in [0,1]}, C^{\mathbb P}:=(C^{\mathbb P}_t)_{t\in [0,1]}, v^{\mathbb P}:=(v^{\mathbb P}_t)_{t\in [0,1]}\big)$$

of predictable semimartingale characteristics, see e.g. Chapter 2 of Limit Theorems for Stochastic Processes. Roughly speaking, $B^{\mathbb P}$ describes the drift, $C^{\mathbb P}$ the continuous diffusion and $v^{\mathbb P}$ the jumps of X.

Let $\epsilon>0$ be fixed and consider the subset

$$\mathcal N:=\big\{\mathbb P\in \mathcal M: B^{\mathbb P}_t\ge \epsilon \text{ for all } t\in [0,1],~ \mathbb P-a.s.\big\}$$

My question is whether we may find some convergence of probabilities under which the set $\mathcal N$ is closed? Such as the weak convergence induced by Skorokhod topology, or S-topology introduced by Jakubowski, see http://citeseerx.ist.psu.edu/viewdoc/download?doi=

I'm very interested in this question but I didn't find any related reference about it. If someone knows the answer please let me know. Thanks a lot for the reply.


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