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Sep 13 
awarded  Yearling 
May 18 
answered  Existence of dominating measure for weak*compact set of measures 
Apr 12 
comment 
Absolute continuity of probabilities on Polish spaces and open sets.
No: If $P$ is Lebesgue measure on $[0,1]$, then $P(E) > 0$ for every open set $E$, and every probability measure $Q$ on $[0,1]$ vacuously satisfies your assumption. 
Mar 1 
comment 
Wasserstein distance between two diffusion processes.
Explicit formulas may be too much to hope for in general, but you can certainly bound the distance, at least if the $f_i,\sigma_i$ are Lipschitz. Solve the SDEs strongly, with respect to the same Brownian motion, and then use the inequality $W(P_1,P_2)^2 \le \mathbb{E}[\int_0^T(X_t  Y_t)^2dt]$ along with standard stability estimates. 
Feb 19 
comment 
Hausdorff distance and sum of independent variables
If $F$ degenerate and $N$ nondegenerate there is no convergence, since $\sigma(F)$ is trivial and $\sigma(F + \epsilon N) = \sigma(N)$ is not. 
Jan 30 
answered  Metrization of weak convergence of signed measures 
Jan 29 
answered  Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable? 
Dec 29 
comment 
When is a space of measures a measurable space?
@Gerald: Unless I'm missing something, the topology generated by the maps $\mu \mapsto \mu(A)$ for $A \in \Sigma$ is weaker than the total variation topology. In particular, it is not discrete, and its Borel $\sigma$algebra contains the cylindrical $\sigma$algebra proposed by Gerald. The inclusion will usually be strict when $\Sigma$ is infinite, as suggested by the nonseparability of the topology. Bogachev's Measure Theory Vol I discusses this topology, beginning on page 291. 
Dec 22 
awarded  Enthusiast 
Dec 16 
answered  Continuous Markov Process and Change of Measure 
Dec 7 
comment 
derivative of conditional expectation
If the random variables $\{\partial H/ \partial x(\cdot,x) : x \in X\}$ are uniformly integrable, then this follows quickly from the dominated convergence theorem for conditional expectations. The details are essentially the same as here: math.stackexchange.com/questions/94628/…

Dec 3 
revised 
total variation distance between two solutions of SDE
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Dec 2 
awarded  Commentator 
Dec 1 
revised 
total variation distance between two solutions of SDE
edited body; edited body 
Dec 1 
revised 
total variation distance between two solutions of SDE
added 297 characters in body 
Dec 1 
answered  total variation distance between two solutions of SDE 
Nov 19 
comment 
Properties of the Euler Discretization of a diffusion
I can't seem to find a reference for this type of convergence, except in some nonLipschitz cases. Maybe you can chase their references: hal.inria.fr/docs/00/05/42/25/PDF/RR5637V2.pdf and arxiv.org/pdf/1010.3756.pdf The rate should be at least $n^{1/2}$. When you say "convergence of optimal stopping", what do you mean exactly? Convergence of the value function, or convergence of the optimal time itself? Such a question may warrant a new post. 
Nov 18 
answered  Properties of the Euler Discretization of a diffusion 
Nov 2 
comment 
Stochastic optimal control with no diffusion
When the coefficients are random, the maximum principle is the best approach, since PDE methods won't work. In the case $\sigma \equiv 0$, the maximum principle (see, for example, H. Pham's book from 2009) still applies, but the adjoint equations will become more difficult. In particular, you will arrive at a forwardbackward SDE (FBSDE) with a degenerate volatility, and the best bet to solve it will probably be PengWu's result. If this doesn't sound familiar, I can elaborate later. 
Oct 28 
comment 
Bochner integral of stochastic process = path by path Lebesgue integral?
Of course, of course, hence the "linear". $X :[0,T] \rightarrow \mathcal{H}$ is certainly Bochner integrable, so if by $\int_0^tX(s)ds$ you mean the Bochner integral, then the answer is yes. But I suppose the point is that the Bochner integral may disagree with the pathwise Lebesgue or Riemann integral. 