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

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How the diffusion in the unit ball induce the boundary process on the boundary directly?

This is Example 1.2.3 from Fukushima, Masatoshi, Oshima, Yoichi and Takeda, Masayoshi's book "Dirichlet Forms and Symmetric Markov Processes". In this example, we define a Dirichlet form in the unit ...
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45 views

Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ ist standard Wiener. This solution is ...
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1answer
110 views

Is it safe to work on a Cadlag modification of a Feller process?

Let $f$ be a continuous bounded function. $X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write $$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...
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1answer
70 views

Compactness of cadlag martingales w.r.t. to the point-wise topology

Given a sequence of cadlag (right-continuous with left limits) martingales $X^n=(X^n_t)_{0\le t\le 1}$, we may use the well known criteria to determine whether it is weakly convergent, i.e. subtract a ...
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1answer
50 views

Problem of random scheduling of queues of tasks

Consider $L$ queues in a discrete time system. At each time $n=0,1,2,\ldots$, one task would arrive at one of the queues with equal probability $\frac{1}{L}$. Immediately after that, a task scheduler ...
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144 views

Product and convex combination of two stochastic matrices

Let $K_1$ and $K_2$ be two $N \times N$ stochastic matrices (hence non-negative and rows adding up to one) with zeros on the diagonal. If $\alpha \in (0,1)$, is it possible to have $$K_1 K_2 = ...
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2answers
91 views

A question about Skorokhod embedding problem

The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the ...
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1answer
221 views

Blumenthal and Kolmogorov 0-1 law

Blumenthal's 0-1 law see theorem 5.8/5.9 tells us that an event in the germ $\sigma-$ algebra has either probability zero or one with respect to a measure induced by a Brownian motion starting in some ...
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1answer
105 views

Generalized Ito's lemma

I have the following quantity: $$ g(t)=(f(t))^{M_{t}}, $$ where $M_{t}$ is a jump process neither Markovian nor Levy and $f(t)$ is a positive, increasing but limited, right-continuous function. How ...
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54 views

Modify exponential family representation to a semimartingale

Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space. We ...
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2answers
205 views

Brownian motion in $n$ dimensions

Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in ...
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40 views

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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46 views

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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48 views

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 ...
4
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2answers
195 views

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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2answers
84 views

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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1answer
220 views

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 ...
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2answers
163 views

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 $S_n = \sum_{i=1}^n \left(X_i - \textrm{E} X_i \right)$. ...
4
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104 views

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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1answer
100 views

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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1answer
96 views

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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119 views

Stopping time sigma-fields

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 ...
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1answer
132 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 $0$, ...
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1answer
54 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, ...
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1answer
208 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 ...
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46 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? $X_t = B_t^q$ $X_t = (\sin B_t)^q$ $X_t = B_t^q (\sin B_t)^r$ Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the ...
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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 ...
4
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2answers
57 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 ...
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109 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 ...
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184 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 ...
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66 views

What is the success probability of this stochastic process?

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 ...
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55 views

Law of motion when initial condition is perturbed

We know how to find the law of motion (Ito process) of the value function: $$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$ such that ...
2
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1answer
84 views

Solve SDE $dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$

I am trying to solve the following SDE $$dX_t=(c+\sigma_\zeta W'_tX_t)dt + \sigma_\epsilon dW_t$$ $c\in \mathbb{R}$ is a constant, $X_t$ is a stochastic process, $\sigma_\zeta,\sigma_\epsilon \in ...
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1answer
79 views

Continuity of expected payoff from a diffusion

Fix a discount rate $r>0$, and let $m,v,f:\mathbb{R} \rightarrow \mathbb{R}$ be bounded measurable functions of locally bounded variation, with $v$ globally bounded below by some strictly positive ...
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37 views

How to implement conjugate gradient method to minimize this nonlinear action?

Given a 2D stochastic differential equation: \begin{align} \dot{x}_{i}=f_{i}(\textbf{x})+g_{ij}\xi_{j}(t), \end{align} where $i=2$, $g_{ij}g_{jk}=2\epsilon\delta_{ik}$ and ...
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1answer
279 views

Time Change of a Brownian motion

We know that for if $X$ is a stochastic integral of the form below - $X_t = \int_0^t v(s,\omega) db(s,\omega)$. then we can use time change formula to claim that $X_t = W_{\alpha(t)}$ where $W$ is ...
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2answers
171 views

The Levy measure of the compound Poisson distribution

The compound Poisson distribution is defined as(see Levy processes and infinitely divisible distributions page: 18): Let $c>0$ and $\sigma$ be a measure on $\mathbb{R}$ with $\sigma(\{0\})=0$, a ...
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82 views

strong law of large number for semimartingale

I just want to know if for semimartingale X we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{<X>_{t}}=0$ or when it is possible. i know it is true for brownian motion. Thanks
3
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1answer
129 views

Brownian bridge on a Lie group as a stochastic differential equation

Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation $$dg_t = dB_t \circ g_t$$ where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes ...
2
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0answers
52 views

Using compactness method to prove the existence of a pathwise solution to an SPDE

For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...
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59 views

Strong solution to an SDE with a discontinuous diffusion term

I am having an SDE for which I would be in trouble if there were no strong solution. The SDE is - $ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$ where $b_1$ and $b_2$ are two ...
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1answer
176 views

Optimisation of betting strategy

Consider integers $r \geq 1$ and $k \geq 1$ and consider the following game: We start with $r$ tokens and at each round we choose $i \in \{1,...,r\}$ tokens to bet (if we have $N<r$ tokens we ...
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1answer
118 views

Time for brownian motion to cross a coordinate plane

Can I get a reference or some insight into the following? Suppose a particle moves by Brownian motion, starting from a point $P$ in $\mathbf{R}^{n}$. What can we say about the distribution of the ...
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1answer
117 views

Tail bound for a martingale

The setup is as follows. We are given a martingale $X_0,X_1,...,X_k$. The difference $X_i-X_{i-1}$ is always between $[-1,1]$. Variance $D^2(X_i-X_{i-1}| X_{i-1})$ is something, but we can show that ...
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53 views

Convergence of approximate quadratic variation in $L^p$

For a diffusion $X_t$, I can set $$[X]^N_t = \sum_{j=1}^N \bigl(X_{t\frac{j}{N}}-X_{t\frac{j-1}{N}}\bigr)^2$$ Then it is well-known that the process $[X]^N_t$ tends to the quadratic variation $[X]_t$ ...
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57 views

Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
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172 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
2
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1answer
105 views

Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...
3
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36 views

Continuity of expected hitting value of diffusion

Let $W$ be a $d$-dimensional Brownian motion and $X$ the strong solution to $$\mathrm{d} X = \mu(X)\mathrm{d} t + \sigma(X)\mathrm{d} W,$$ starting from some $x$, where $\mu$ and $\sigma$ are ...
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37 views

Validating a probability density distribution forecast model for a Markov process

Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...