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seen Jun 24 '13 at 11:39

Sep
13
awarded  Yearling
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
added 493 characters in body
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 non-Lipschitz cases. Maybe you can chase their references: hal.inria.fr/docs/00/05/42/25/PDF/RR-5637-V2.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 forward-backward SDE (FBSDE) with a degenerate volatility, and the best bet to solve it will probably be Peng-Wu's result. If this doesn't sound familiar, I can elaborate later.