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

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106 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 ...
3
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1answer
106 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 ...
4
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1answer
108 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 ...
2
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0answers
124 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 $F_{S_n}\...
4
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1answer
150 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 $0$, ...
4
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1answer
64 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, ...
5
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1answer
273 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 ...
3
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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? $X_t = B_t^q$ $X_t = (\sin B_t)^q$ $X_t = B_t^q (\sin B_t)^r$ Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the ...
5
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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
votes
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 ...
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184 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 ...
6
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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 \...
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69 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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0answers
63 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 $$dX_t=\mu(t,X_t)...
3
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103 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 \...
1
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1answer
84 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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39 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 $f(\textbf{x})=-\nabla\phi(\...
0
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1answer
436 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
305 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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101 views

Strong law of large number for semimartingale

I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion. Thanks
3
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1answer
165 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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55 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 ...
2
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67 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 ...
5
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1answer
184 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 can'...
4
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1answer
126 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 ...
0
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1answer
125 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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60 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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58 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 $E=\...
2
votes
1answer
111 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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0answers
38 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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38 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 $P_t(x)$". ...
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2answers
97 views

Is zero a regular point for a drifted $\alpha$-stable process?

We consider 1-d process of the form $Y_{t} = bt + M_{t}^{\alpha}$, where $M_{t}^{\alpha}$ is $\alpha$-stable process for some $\alpha \in (0,2)$ with its levy symbol $\eta(u) = - |u|^{\alpha}.$, and $...
5
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2answers
246 views

Infimum of Gaussian process

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(...
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0answers
38 views

Uniform convergence problem of the iterative function series

A process $\{\theta_{t}\}_{t=1}^{\infty}$ with finitely continuous state space $\mathcal{S}=[\underline{\theta},\bar{\theta}]$.The transition density is $\phi(\theta_{t},\theta_{t+1})$.I have known ...
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0answers
44 views

Basic results for chi square processes

I could not find any introductory material with basic results regarding chi-square processes. Their definition from The Supremum of Chi-Square Processes is as a sum of $d$ squares of independent ...
3
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1answer
61 views

Number of samples needed as input to Bernoulli factory

Let $\{X_i\}$ denote an i.i.d. sequence of Bernoulli variables with parameter $p$. A Bernoulli factory is a procedure that generates events with probability $f(p)$ using the observations $\{X_i\}$, ...
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65 views

Markov chain matching local time

Let $\left(X_{t}\right)_{t\geq0}$ be a Markov process taking values in a finite state space $E$. Its local time at $y\in E$ started at $x\in E$ is defined as $$ l_{x}\left(t,y\right)=\int_{0}^{t}1_{...
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0answers
59 views

Stationary distribution of two-dimensional Markov Process?

A two-dimensinal Markov process $\{\theta_{t},S_{t}\}_{t=1}^{\infty}$ where $\theta_{t} \in \Theta$ and $S_{t} \in S$.$\Theta$ is a continuous state space and $S$ is a discrete state space. Suppose I ...
2
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70 views

integrability of Brownian motion stopped at some stopping time

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion starting at zero and denote by $S=(S_t)_{t\ge 0}$ its running maximum, i.e. $S_t=\sup_{0\le s\le t}B_s$. Given a fixed number $p>1$, define the ...
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41 views

Systems of stochastic differential equations with non-Lipschitz coefficients

I am looking for references to any literature which might consider the existence / behavior / regularity of solutions to systems of stochastic differential equations with non-Lipschitz coefficients. ...
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1answer
164 views

expected value of cosine wirh Gaussian phase

Is there a solution to the expected value/variance for a Gaussian with random phase: $$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$ ? For $t=0$, the solution is for example ...
4
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50 views

Existence of martingales given some constraint on laws

Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple ...
2
votes
2answers
306 views

Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
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40 views

Proof of Linear Stochastic Sate-Space Model is Gaussian Process

I would like to prove that the vector-linear stochastic state space model $$ \dot{x}(t)=A(t)x(t)+B(t)u(t)+G(t)q(t) \\ y(t)=C(t)x(t)+D(t)u(t)+F(t)r(t) $$ corresponds to a particular multi-output ...
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130 views

The existence of stationary measures for certain Markov process

My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...
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1answer
140 views

Stationary distribution with exponential transition density function

A Markov chain with continuous state space has a transition exponential density function: $$p(x_{t},x_{t+1})=\frac{1}{x_{t}}exp(-\frac{x_{t+1}}{x_{t}})$$ i.e. the realized value in period t is the ...
0
votes
1answer
118 views

Distribution of bounded summation of i.i.d random variables

We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ ...
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0answers
44 views

Construction of a path of quadratic variation

This question has been posted to Stack Exchange earlier, and no answer is available yet. Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) =...
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0answers
44 views

Ergodicity property for continuous-time Harris positive Markov process

I have posted this question on there, but got no answer. The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328: Theorem 13.3.3. If $\Phi$...