# Tagged Questions

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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### Factorization of the Fokker-Planck semigroup

I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...
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### methods to analyze martingale conditioned on return in the future

Consider a martingale $S_t$ on $\mathbb{Z}$ starting from 0. Assume that for any $t$, $Var[s_t\, | \, \mathcal{F}_{t-1}] < V$, where $V$ is some positive constant. Fix an $n$ and for $t \leq n$, ...
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### skorokhod integral as Weak Integral

Is it possible to express the skorokhod integral on a Banach space $B$ as a special case of the weak (or Pettis) integral over an appropriate Banach space $E$? For example if $E$ is the space of ...
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### Prediction with positive weights?

Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$\gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0.$$ It is guaranteed to be positive definite by Polya's criterion (...
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### Understanding the limits of the Ito Process Defintion

I would like to understand what kind of stochastic process are Ito Processes. According to Kuo[p. 102] an Ito Process is a stochastic process of the form $$dX_t=g(t)dt+f(t)dW(t),$$ where $W(t)$ is a ...
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### Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary finite-dimensional smooth manifold. For simplicity, let's assume that $M$ has no boundary. Does there always exist a gaussian random field with constant variance on $M$? If not,...
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### Recurrence of Poisson binomial distributed random walk

Let $X_n$ be the outcome of a Bernoulli trial where the probability of getting 1 is $p_n$ and the probability of getting 0 is $1-p_n$, and let $S_n = \sum_{i=1}^n \left(X_i - \textrm{E} X_i \right)$. ...
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### regularity of zero point

We consider 1-d process $X$ $$X(t) = b t + J_{t} + M_{t}$$ where $b$ is constant, $M$ is a continuous martingale process with $M(0) = 0$, and $J$ is a symmestric $\alpha$-stable process with its ...
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### Markov-semigroup sobolev inequality

I have a question about the following definition: A probability measure $\mu$, such that the Markov semi-group $e^{Lt} \in L(L^2)$ exists and is symmetric, satisfies the Sobolev inequality iff for ...
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### Exponential of approximate quadratic variation of Brownian motion

Let $X_t$ be a Brownian motion or a Brownian Bridge on a (\edit: compact) Riemannian manifold. Let $T>0$ be given. The question is: Does there exists a constant $C>0$ such that for all ...
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Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence). Is it true that the associated sigma-fields satisfy $F_{S_n}\... 1answer 151 views ### Large deviation for Brownian path on$[0,\infty)$It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path. If we equip the space of continuous function starting from$0$, ... 1answer 65 views ### Reference request: Urbanik's work on random integrals and Orlicz spaces Several important papers on Lévy processes are referring to the following paper: K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, ... 1answer 285 views ### Nearest neighbor for planar Poisson is normally distributed This was previously asked on MathSE, but was not answered. Answering a question, I realized that the nearest point for a planar Poisson point process (with constant intensity$\lambda>0$) is ... 0answers 47 views ###$X_t = B_t^q$,$X_t = (\sin B_t)^q$,$X_t = B_t^q (\sin B_t)^r$,$dM_t = R_t\,M_t\,dB_t$[closed] What are the SDE's satisfied by the following processes?$X_t = B_t^qX_t = (\sin B_t)^qX_t = B_t^q (\sin B_t)^r$Assume$B_t$is a standard Brownian motion with$B_0 > 0$and the ... 0answers 49 views ### Full distribution of FPTs in random walks on graphs There is a lot of published research on the mean passage passage time (FPT) for random walks on various types of graphs. How about the variance of the FPT and higher momenta? In fact, I would be ... 2answers 63 views ### Distribution of the RKHS norm of the posterior of a Gaussian process In a classical noisy regression setting, let$\big(f(x)\big)_{x\in\cal X}$be a centered Gaussian process of covariance$k$on a compact$\cal X$, and$\mathcal{F}_n$be the filtration generated by ... 0answers 204 views ### Example of an adapted measurable process which is not Progressively Measurable In this question Progressively measurable vs adapted, one finds a discussion on the subject of adapted processes versus progressively measurable processes. Counter-examples can be readily given. We ... 1answer 188 views ### Bound on expectation, not a really simple process, circumvent using Itō's lemma? Assume that$H_t$is a progressively measurable process such that with probability one$|H_t| \le k$for all$t$. Let$$Z_t = \int_0^t H_s\,dB_s.$$How do I see that for all$s < t$,$\lambda \in \...
Suppose you have $k$ black balls and $X\cdot k$ white balls. The procedure start with you having a bag containing $y\le k$ white balls (e.g. $k+1,\ldots k+y$). In every iteration: A single white ...