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- 73 votes cast
Jun
25 |
awarded | Promoter |
Oct
24 |
answered | modification of Doob inequality |
Sep
23 |
awarded | Yearling |
May
23 |
comment |
Cover time and intersection time of random walks
Interesting question. Can you elaborate on why you think the inequality should hold? And tell us how you attempted to solve the problem and why it failed? |
May
21 |
comment |
Brownian motion & time shift
It is still does not seem correct with the correction, you should request that $A$ depends on $(B_t)_{t\ge\tau}$ only. Anyway, I think what you are trying to formulate is the \emph{Markov property} of Brownian motion, I therefore recommend you to inform yourself about the basics of Markov processes (or, to start with, about Markov chains). |
May
19 |
comment |
Properties preserved under passage to augmented filtration
Interesting question. In a slightly different direction: One problem with augmented filtrations is the inability to extend a compatible family of probability measures $\mathbb Q_t$ to $\mathcal A$, see for example arxiv.org/abs/0910.4959, where a weaker kind of augmentation is introduced and studied (introduced before also by Bichteler) |
Apr
28 |
comment |
A generalization of the Sanov Theorem
@tipanverella, sorry for the late answer, I wasn't notified of your comment, strangely. The Gartner-Ellis theorem is restricted to $\mathbb R^n$, as far as I know, and here we are considering the space of empirical measures on a measurable space $S$, which can be embedded into $\mathbb R^n$ only if $S$ is finite. Maybe one can work around it by discretizing the space, though... |
Apr
28 |
comment |
One point on $\phi$-irreducibility
@Ilya, thanks for accepting. I am sorry, but I don't know references for exercises by heart, and I haven't used these concepts much yet. But you may check in relevant books, e.g. the book by Meyn and Tweedie, or I suppose that in Kallenberg's "Foundations of Modern Probability" there might be exercises, at least there will be more references on this topic. Last but not least, Grimmett and Stirzaker might provide exercises. |
Apr
28 |
comment |
One point on $\phi$-irreducibility
@Ilya, feel free to accept the answer, if it correctly answered your question. |
Apr
28 |
comment |
One point on $\phi$-irreducibility
@Ilya, you are welcome. Yes, you can define $B$ like this. For the intuition: $\phi$ "gives" the set of all states that can be reached from any point. Now if from these states we take one step further, i.e. if we map $\phi$ to $\phi P$, then a fortiriori, we can reach these states from any point. |
Apr
27 |
answered | One point on $\phi$-irreducibility |
Apr
24 |
answered | Is the Simplex Method still polynomial when all inequalities are through the origin? |
Apr
18 |
comment |
A generalization of the Sanov Theorem
This works whenever the $X_n$ only take a finite number of values, but can it be made rigorous in the general case? |
Apr
5 |
answered | Convergence of stochastic process |
Mar
6 |
comment |
When is $\mathbf{E}^{x}[f(X_t)]$ a continuous function of $x$?
@ShawnD: Shame on me, I did not see the exponent $3$ in $\mathbb R^3$. Of course you are right. Sorry to both of you. If I am not mixing things up again, the sample path condition you are looking for is called regularity in one-dimensional diffusions. |
Mar
6 |
comment |
What is the structure of a space of $\sigma$-algebras?
@Tom: Thank you too for the nice reference. By the way, if you put @Pascal at the start of your comment, I get notified by MO (I just learned this reading the comments of this question: mathoverflow.net/questions/90218/… ) |
Mar
5 |
revised |
When is $\mathbf{E}^{x}[f(X_t)]$ a continuous function of $x$?
added 204 characters in body |
Mar
5 |
comment |
When is $\mathbf{E}^{x}[f(X_t)]$ a continuous function of $x$?
@ShawnD: Maybe you are right, because an Ito process is a time-inhomogeneous diffusion, so the Markov property should already be enough, I suppose. However, this does not answer your question about the Feller property. I think that George's example of a process is a diffusion according to Ito-McKean or Revuz-Yor (with generator $Lf(x) = -f'(x)1_{x<0}$), but it is not Feller. |
Mar
5 |
comment |
When is $\mathbf{E}^{x}[f(X_t)]$ a continuous function of $x$?
contd. Indeed, the Ito-McKean definition of a diffusion does not imply Feller. This explains why Rogers-Williams speak of "Feller-Dynkin diffusions" and assume the Feller property. I presume that if the diffusion coefficient does not vanish, then the diffusion is Feller, but I could not find a reference to that |
Mar
5 |
comment |
When is $\mathbf{E}^{x}[f(X_t)]$ a continuous function of $x$?
@George: Do you mean the example given by ShawnD? Maybe I misunderstood it, probably he meant 0 to be a shunt (in which case it would be a diffusion, however). I understood that when the process starts at x < 0 it gets reflected at 0, such that it always stays below 0. In this case, I think that the process is not strong Markov, because when stopping it at 0, one has to know whether it came from below or from above in order to continue. As for your example, you are right, this is a strong Markov process with continuous paths but not a Feller process. |