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

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Can we transform $\int_\rho^1 (W_t - W_{t-\rho}) dW_t$ to make its law $\rho$-invariant?

I just bumped into the stochastic integral $$ \int_\rho^1 (W_t - W_{t-\rho}) dW_t $$ where $0 < \rho < 1$ is a constant and $W$ is a standard Wiener process. It would be nice if we have a closed-...
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29 views

Continuous Local Martingales under time change under what conditions are they still local martingales?

This question is motivated by reading a section in Continuous Martingales and Brownian Motion by Daniel Revuz, Marc Yor. In Chapter V there is a section on time-change: Definition: A time change $C$...
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0answers
21 views

Two correlated time series processes

Consider two correlated time series processes $\{X_t\}$ and $\{Y_t\}$ and also suppose that I have "all necessary information" on $\{X_t\}$. Can that "information" be used to get better estimates/...
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1answer
50 views

Is the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ atomless?

Let $B_t$ denote a standard Brownian motion, and $0 < l < u$. I am wondering if the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ is atomless, that is, $\mathbb{P}\left(\sup_{l \leq t \...
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30 views

Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail. I ...
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0answers
31 views

Random solute transport equation

After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
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0answers
15 views

Generators and Covariance Operators of Diffusions

For a constant coefficient Ornstein-Uhlenbeck process, how should I think about the relationship between the infinitesimal generator of the process and the covariance operator of the process (or, ...
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0answers
27 views

Differentiability of a stochastic process depending on a spatial parameter

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $M:\Omega\times\overline I\times\mathbb R^d\to\mathbb R$ such that $M(\;\cdot\;,\;\cdot\;,x)$ is $\...
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0answers
48 views

Question about Protter's proof of the Ito's formula

The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential ...
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2answers
195 views

Counting Hamiltonian cycles in $n \times n$ square grid

I wonder if anyone has counted these curves, either exactly or asymptotically? Let $S_n$ be an $n \times n$ subset of $\mathbb{Z}^2$ consisting of $n^2$ lattice points: a lattice square. Define a ...
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0answers
39 views

Does completion of natural filtration preserve the Markov property?

Let $(\Omega, \mathcal{A}, P)$ be complete probability space and $(X_t)_{t \geq 0}$ a Markov process on a general measurable space $(E, \mathcal{E})$ meaning that the Markov property $P(X_{t+h} \in B ...
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0answers
34 views

Stationary recursive sequence and nonzero probabilities

A while ago I posted the following problem: Suppose I have a two sided stationary sequence of random variables $\ldots,X_{-1},X_0,X_1,\ldots$ such that all finite dimensional joint densities $f(x_1,\...
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0answers
36 views

Reflected SDE with vanishing diffusion term at the boundary

I have a reflected SDE of the form $dX_{t} = a(X_{t})dt + b(X_{t})dW_{t}+dL_{t}, \quad X_{0}=x_{0}\in (0,\infty) $ where $L$ keeps the process from going below zero. The coefficients $a$ and $b$ ...
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56 views

Why is the Jain Monrad condition the right condition on general Gaussian processes?

Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process. Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...
4
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1answer
57 views

Expected time of distinguishability of a series of Poisson processes bounded by each other

Consider a system of $n$ "bounded" Poisson processes over the integers, $X_1, \ldots X_n$, all incrementing at rate $\lambda$. Initially all the processes begin at $0$. The process $X_i$ is inactive ...
3
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1answer
134 views

Are Holder Condition and signal to noise ratio (SNR) related?

This question was posted in https://math.stackexchange.com but I got hardly any view. If posting here is an objection please let me know I would delete it immediately. This question has evolved from ...
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0answers
31 views

Reference request: semimarkov processes

What are some good modern introductions to the theory of semimarkov processes? To be clear, by a semimarkov processes, I mean a Markov chain, together with "waiting times" between transitions, the ...
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0answers
64 views

Hitting times for stochastic convolution integral

Let $N$ be a Poisson measure of intensity 1 on $\mathbb{R}$. To remind you, this is a counting measure on $\mathbb{R}$ such that $$ N(s,t]\sim \mathrm{Pois}(t-s)\quad \forall t>s\in \mathbb{R}$$ ...
2
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1answer
269 views

Is the ito integral $\int_0^t \operatorname{sign}(W_s)\mathrm{d}W_s$ a Brownian motion?

Consider the ito integral of the sign of the Brownian motion $W_s$ from $0$ to $t$: $$\int_0^t \operatorname{sign}(W_s)\,dW_s$$ This appears for instance in the Tanaka formula. I think this is a ...
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0answers
23 views

Semimartingale property up to a specified time

I have a question about semi-martingale property of stochastic processes. Let us consider the following cusp \begin{equation*} C=\{(x,y)\in \mathbb{R}^2 \mid |y| \le x^{\gamma}\},\quad 0<\gamma&...
2
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1answer
46 views

order of the singularity of a Green's function to the fractional Laplacian

I was looking at a problem which involves the Green's function of a fractional Poisson equation. To fix notation, let $D\subset \mathbb{R}^n$ very nice, i.e. a hypercube, and \begin{equation} \begin{...
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1answer
92 views

fractional Brownian Motion driven stochastic integrals

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\...
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0answers
14 views

variance of a recursive distribution

I have the following recursive equation, $\tilde{S}(k+1) = \beta \bar{r}(k) + (1-\beta)\tilde{S}(k)$ where $\bar{r}(k) \sim \mathcal{N}(0, I)$. How can I calculate the variance of $\tilde{S}(k)$? (...
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0answers
32 views

Ergodicity of differentiated processes

Let $S$ be a vector space, and $X$ a jointly-measurable random process/field with two parameters: $$ X: [0,\infty)\times\mathbb{R}\times\Omega\to S,$$ i.e. $X_{t,\theta}:\Omega\to S$ are random ...
3
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0answers
85 views

Regularity of martingales with respect to spatial parameters

In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...
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0answers
25 views

Are Matérn class kernels universal kernels or not?

This is a question that I can't find the solution. I don't know it is a open question or it is a well-known result that can be attained from several lemmas. Here are the definition of Matérn class ...
3
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1answer
65 views

Under what condition we get back path from signatures in rough path theory?

A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path It appears path and signatures has one to one mapping in many cases. I understand that the signature is not ...
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0answers
68 views

Meaning of $. \wedge t$ (. \wedge t) in stochastic analysis

In Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I they define (on page 4) a metric : $${\bf d}_\infty ((t,\omega),(t',\omega')) := |t-t'| + \|\omega_{.\wedge t}-{\omega'...
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1answer
35 views

Jumping times on Borel sets away from zero are stopping times

The following comes from some remarks of Philip Protter at page 26 of the book Stochastic integration and Differential equations that I have not been able to prove yet. Let $X$ a Levy process, under ...
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1answer
70 views

Obtaining the distribution of the First Hitting time of the Bessel Process

Let $(X_{t})_{t\geq 0}$ be a Bessel Process starting at $x>0$ of dimension $\delta>0$. Namely \begin{align*} X_{t}=x+W_{t}+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{X_{s}}\, ds. \end{align*} where ...
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0answers
116 views

CLT for random sums: Anscombe's Theorem vs. “classical” version

Given a compound Poisson distribution $$S(t):=\sum_{k=1}^{N(t)} X_{k}$$ with $N(t)\in\mathbb{N},\,t\geq0$ a Poisson process with rate $\lambda.$ $X_{k}\in L^{2}$ are iid random variables, i.e. $\...
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65 views

Bounded over time versus bounded over stopping times

Consider the expectation $E(G(v,X_v)|\mathcal{F}_t)$ for $t\leq v \leq T$ for a stochastic process $X_t$. We can impose one of the two following conditions : $E(G(v,X_v)|\mathcal{F}_t)$ has a ...
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1answer
88 views

Continuity w.r.t time vs Continuity w.r.t. stopping times

Several places in "Optimal Stopping and Free-Boundary Problems" Peskir and Shiryaev make the assumption that a (Markov) process $X = (X_t)_{t\geq 0}$ has sample paths which are right continuous and ...
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1answer
72 views

How to construct a Poisson process not based on Lebesgue measure?

I am not really a professional, but this question has been asked on Math.SE already and in spite of a bounty it was not answered. That made me decide to give it a try here, and I hope that is ...
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1answer
77 views

Limit distribution of Ornstein-Uhlenbeck equation

Let \begin{equation*} X_t=xe^{-\lambda t}+\sigma e^{-\lambda t}\int_0^t e^{\lambda s} dB_s \end{equation*} be the solution of Ornstein-Uhlenbeck equation where $B$ is Brownian motion, and $x,\sigma,\...
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0answers
57 views

Exit time of a stochastic process defined by a SDE

Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation \begin{align*} \...
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0answers
54 views

Transition probabilities for an approximating tree for a couple of correlated stochastic processes

i apologize if the question is not at research level, in that case i'll remove and post it elsewhere. First the motivation for the question. I have two continuous stochastic processes $x(t),y(t)$ (...
4
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1answer
188 views

Support of bivariate joint distribution of stationary and ergodic sequence

Let $\{X_t\}_{t\in \mathbb{N}}$ be a strictly stationary and ergodic sequence of real valued random variables and let the support of $X_1$ equal $[-1,1]$. Can the support of $(X_1,X_2)$ equal the unit ...
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0answers
39 views

Connection between deterministic and stochastic problems in PDEs

A few known problems in partial differential equations are: Deterministic Cauchy problem: $$(1) \hspace{1cm} \begin{cases} u_t+f(u)_x=0 \\[2ex] u(x,0)=u_0 (x) \end{cases} $$ Stochastic Cauchy ...
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0answers
39 views

Invariant measures for a renewal process driven by Interarrival times bounded away from zero

Good morning, I apologize in advance if my question sounds too basic but after some research I was unable to come up with satisfactory answers to my doubts. I am currently studying a model which ...
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0answers
54 views

Prove that a local martingale with spatial parameter is differentiable

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...
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0answers
135 views

Random averages over a Point process - Campbell's Theorem

Let $(N_t)_{t \geq 0}$ be a non-homogeneus Poisson process with intensity function $t \mapsto \lambda(t)$. Let $(T_n)_{n \in \mathbb{N}}$ be the corresponding simple point process (arrival times), ...
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1answer
102 views

Does sequence almost sure convergence imply almost sure convergence?

This is a cross-post of this and this questions from math.stackexchange.com since I have not received any response there. I would like to seek help here. Suppose $x(t,\omega): [0,T]\times\Omega\...
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1answer
62 views

Convergence of an integral with respect to the Wiener measure

Most probably this question should be well studied in the theory of stochastic processes, but I am not educated in that area. Sorry if this question is too elementary. Let $V\colon \mathbb{R}\to \...
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0answers
36 views

Literature request: Characterization of semimartingales

Are the results of the following two articles by Christophe Stricker (in French) covered in English in a book or in an article, and if so, could you please provide a reference? Quasimartingales, ...
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1answer
145 views

Covariance function of Brownian motion and the second derivative operator

I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me. Suppose $W$ is a Brownian motion, and we ...
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46 views

What is the Wiener measure of the curves with Hölder index $\frac 1 2$?

One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index ...
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59 views

Growth models with lookahead

Given a connected graph $G$ with a connected subgraph $H$, we can consider the uniform distribution on the set of all sequences $H_0, H_1, \dots, H_r$ where $r = |E(G) \setminus E(H)|$, $H_0 = H$, $...
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0answers
56 views

Ornstein-Uhlenbeck type process with thresholding

(Edited) I met a univariate Ornstein-Uhlenbeck type process but with self soft-thresholding: $$ dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\mu}\big]_+ dt + \sigma dB(t), \quad X(0)=0, $$ where $...
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votes
3answers
289 views

Perturbation of a stochastic differential equation

Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively \begin{align} dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\ dx &= -(k_0(t)+\epsilon ...