# Questions tagged [stochastic-processes]

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

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### Interpretation of Lévy process with signed Lévy measures

Suppose that I have a non-decreasing, pure jump Lévy process of finite variation $X$ with Lévy measure $\pi$. The Lévy measure is then supported on $(0,+\infty)$. Suppose that the Lévy measure is a ...

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### Questions about Lamperti's criteria for stochastic process recurrence

I'm working through Lamperti's 1960 paper "Criteria for the recurrence or transience of stochastic process. I" (J. Math. Anal. Appl. 1(3–4), 314–330. DOI: 10.1016/0022-247x(60)90005-6) as ...

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### Book recommendation in functional analysis and probability

I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend?
I'm looking for a book that has ...

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### Boundary-condition-changing Operators for Free Boson BCFT with Dirichlet Boundary Conditions (or more general BCFTs)?

(NOTE: This is a crosspost from this Physics.SE post)
Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. ...

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### Stochastic processes with variable dimensionalty

I am curious about stochastic processes where the dimensionality of the state space can grow over time which could have application to describe real world processes such evolution or technological ...

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### Pathwise Hölder continuity of Ito diffusions - is this result written anywhere?

Let $X$ be the solution to the multidimensional SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $W$ a Brownian motion, and $\mu, \sigma$ Lipschitz continuous with $\sigma$ nowhere zero. I'm ...

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### Construction of a "Dirac" jump process

We work on a probability space $(\Omega,\mathscr{F},\mathbb{P})$ endowed with a filtration $\mathbb{F}$, and consider the positive line $[0,\infty)$. I am wondering if one can make sense of the ...

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### Is the average of a $\alpha$-Hölder process Hölder continuous of every order less than $\alpha$?

Let $X_t$ be a stochastic process on $[0, 1]$ that is almost surely Hölder continuous of order $\alpha > 0$, and almost surely uniformly bounded by some deterministic constant. It is not hard to ...

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### Continuity dependence and convergence of the renormalized $\Phi^4_2$ model

This question is continuous for the one asked here: Local solutions of renormalized stochastic PDE but it was better to ask it separetely.
Again, we are interested in the local behavior of the $\Phi_2^...

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### Combination of the Dirichlet and Cauchy problems, find the PDE by which $\mathbb{E}_x M(X_{\tau_D \wedge t})$ is met

$X_t$ is an Itô diffusion process with continuous version, $\mathbb{L}_X$ is its generator. $D$ is a closed set in $\mathbb{R}$. The stopping time $\tau_D$ is the first entry time of $D$, that is $\...

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### Integral functional minimal value problems

\begin{align}
& F_n(\theta)=\int_0^T f_{n}(t,\theta(t)) \, dt \\[6pt]
& f_n(t,\xi)=\int_\Omega\mathcal{L}(X(t) + Z_n(t,\omega),Y(t),\xi (\omega)) + R_n(\xi(\omega)) Pd(\omega)
\end{align}
...

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### Simulation of Markov processes with exponential timestepping

Let $(Y_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. Numerical simulation of $(Y_t)_{t\ge0}$ can be done in the following way:
Choose an initial ...

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### Stability of Hölder constants of frozen Itô stochastic integrals

$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...

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### Holder-Besov space and time continuity

Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions.
We consider a dyadic partition of unity $(...

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### Simple version of fourth moment theorem of Nualart and Peccati

I'm trying to understand this theorem, which gives several conditions for a sequence of random variables in the $n$-th Weiner chaos to converge to a normal law. I am finding even the statements of the ...

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### Uniform distribution as argument for copula likelihood

I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...

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### Markov process on a torus with prescribed invariant distribution

In Euclidean space, $\mathbb R^d$, the Langevin diffusion $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1,$$ where $\sigma:\mathbb R^d\to\mathbb R^{d\times k}$, $$b:=\frac{\Sigma+U}2\nabla\ln p+...

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### When does a local supermartingale become a proper supermartingale?

This is a cross-post of my question on MSE.
Abstract: When a local supermartingale is bounded from below, is it a proper supermartingale?
Question: In remark 4.2 (p.16) of the lecture notes by Martin ...

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### Penalty shootout

Two teams are having an intense penalty shootout. The game ends when either team leads by a certain threshold, or once a certain number of rounds has passed, whichever comes first. Currently team $X$ ...

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### Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function

Let $dX_t=\sigma(X_t)\,dW_t+\mu(X_t)\,dt$ be an Ito diffusion with Lipschitz coefficients and $\sigma(x)>0$. Let $f(x)$ be a continuous and bounded and non decreasing function. Can we prove that ...

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### Uniform concentration bound (function-valued random variable / continuous stochastic process)

I'm trying to consider a probability space $\Omega$ and
$f(x,\xi):\mathcal{X}\times\Omega\to\mathbb{R}$ (stochastic process over space? or function-valued random variable?), where $\mathcal{X}\subset\...

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### Exact probability distribution for an alternating renewal counting process

Consider a scenario where a detector counts the number of photons incident on it's surface over a time interval of $[0,\tau]$. We suppose that the photons arrive at the detector's surface with ...

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### Is there a proof of the de Moivre-Laplace central limit theorem along these lines?

Let $X_1, X_2, \dots$ denote independent identically distributed random variable with, say, distribution given by $P(X_i= \pm 1)=1/2$. As usual, set $$S_n=X_1+ \cdots +X_n.$$
It follows from Skorokhod'...

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### Are speed, scale function and killing measures of Itô diffusion absolutely continuous respect to Lebesgue measure and do have smooth derivative?

In Borodin and Salminen's Handbook of Brownian motion (MR1912205, Zbl 1012.60003), pages 16–17, they mention the fact that if the three basic characteristics (speed measure, scale function and killing ...

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### Best textbooks/resources for "advanced" probability theory?

When I say "Advanced Probability", I mean for a person acquainted with the measure-theoretic foundations of probability theory, that wants to learn about Stochastic Processes from there, in ...

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### Is this predictable process left-continuous?

Let $X$ be a predictable process defined on some filtered probability space (as good as possible) such that
$$X_t \in \{0,1\},\quad \forall t\ge 0.$$
Does this imply the left continuity of $X$? If so, ...

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### Ornstein Uhlenbeck process with discontinuous drift

This question is a modified version of this unanswered question asked on MSE, which mainly concerns an Ornstein-Uhlenbeck process with discontinuous drift on $\mathbb R^n$(for simplicity let $n=2$ for ...

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### Stochastic Geometric Progression [closed]

Let $\mu_1, \mu_2, \ldots, \mu_n, \ldots \in \mathbb{R}$, let $\sigma_1, \sigma_2,
\ldots \in [0, \infty)$ be sequences of numbers.
Let $z_1, z_2, \ldots, z_n, \ldots$ be independent random variables ...

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### Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?

$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...

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### Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?

$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...

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### Does $X_t$ with $t>0$ admit a density?

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### Error in an argument using spectral theory

Let $Z=\sqrt{2}\mathbb Z$, and consider the sequence on $\mathbb{Z}$ $$\xi(k)= 1_{Z\cap [k,k+1]\neq\emptyset}.$$(remark that the intersection is either empty or with one point.)
Thanks to the comments,...

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### The equivalence of stochastic quantization and path integral quantization

I am looking for a reference in which the equivalence of stochastic quantization and
path integral quantization has been shown. It would be great if I can see such a relation for a Euclidean quantum ...

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### Random matrix with power law decay in eigenvalues

What positive semi-definite random matrices have (roughly) $n^{-\alpha}$ for $n^{th}$ singular value? The power law decay need not be exact.
I want to find random matrix ensembles that naturally ...

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### Conditional expectation w.r.t filtration of Brownian motion as a continuous map of its paths

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which we define Brownian motion $B$ and let us denote by $\mathcal{F}_t$ its natural filtration. Assume we have Ito process $dX_t = \...

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### References for a class of Banach space-valued Gaussian processes

Let $E$ be a separable Banach space, consider a centered $E$-valued Gaussian process $\{x_t,t\ge 0\}$ that satisfies
\begin{equation}
\mathbb{E}\phi(x_s)\psi(x_t)=R(s,t)K(\phi,\psi),\quad \phi,\psi\in ...

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### How does the first hitting time depend on the drift of drifted Brownian motion?

Let $W$ be a standard Brownian motion, and $a,b:\mathbb R_+\times \mathbb R\to\mathbb R$ be Lipschitz. Consider the stochastic differential equations:
$$X_t=1+\int_0^ta(s,X_s)ds + W_t,\quad\quad Y_t=1+...

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### Constructing Wiener process on a given probability space

This is just a short question, and may be to basic, but:
is there a way to construct a sequece of independent wiener processes on a given probability spaces?

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### Brownian bridge as a limit of SDEs

Let $B$ be a Brownian motion and with respect to some probability measure $\mathbf{P}$ and filtration $(\mathcal{F})_{t \geq 0}$ and let $S_\epsilon = \{B_1 \in (-\epsilon,\epsilon)\}$.
For every $t \...

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### Measurable Extension

Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...

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### Does the definition of mixing time work for general non-Markovian processes?

A definition of the mixing time for Markov chains is given by
\begin{equation}
\tau_{\text{mix}}\equiv\inf{\{t>0: \sup_i\left\vert \frac{\boldsymbol{p}(t|p_j(0)=\delta_{ij})}{\boldsymbol{\pi}}-\...

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### Definition of semi-metric for empirical process theory

In the following lecture notes on empirical processes (https://www.stat.columbia.edu/~bodhi/Talks/Emp-Proc-Lecture-Notes.pdf) a semi-metric space $(\Theta, d)$ is defined in the following way: for any ...

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### Bound from above and from below the probability that a 1-D centered random walk remains at each step inside a square root boundary

Let $W_n = \sum_{i = 1}^{n}X_i$ be a random walk on $\mathbb{R}$, where the increments $X_i$ are i.i.d., symmetric around the origin ($X\sim -X$), such that $-1\leq |X(\omega)| \leq 1$ $\forall\omega\...

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### Characterizing filtrations generated by a stopping time

Setup
Let $\Omega$ be the set of càdlàg functions $f : [0,\infty) \to \mathbb R^d$ equipped with the Skorokhod topology for any $d \geq 1$, and let $X_t(\omega) = \omega(t)$ for any $\omega \in \Omega,...

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### Can the optimal stopping problem be expressed in another form by strong Markov property?

$X_t$ is a strong Markov process in $(\Omega, \mathcal{F},\mathcal{F}_t,\mathbb{P})$. $\tau$ is a stopping time, $T>0, \mathbb{E}_x(\cdot)=\mathbb{E}(\cdot|X_0=x)$. By Markov property, $\mathop{\rm{...

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### MDP Average Reward independent of Initial State

Consider a Markov Decision Process where the state space $S$ and the action space $A$ are continuous and compact.
In state $s$, if action $a$ is chosen and the next state becomes $s'$, the ...

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### White noise: a tempered distribution version of the stochastic convolution

Let $\xi$ be a space-time white noise, that is a centered Gaussian process with covariance $E[\xi_{f}\xi_h]=\int_{\mathbb{R}_+ \times \mathbb{R}^d}fh,$ for $f,h\in L^2(\mathbb{R}_+ \times \mathbb{R}^d)...

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### Feynman–Kac formula for other operators

I recently came across the Feynman-Kac formula which states that given an open domain $\Omega\in\mathbb{R}^n$ and $f \in L^2(\Omega)$
where $x \in \Omega$ and $t > 0$, then
$e^{t\Delta_D}f(x) = ...

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### translation invariance of expectation value of hit counting variable for Lévy process

Let $(X_t)_{t \in [0, \infty)}$ a $\mathbb{R}$- valued
Markov process (in my question I'm primary interested in dealing with Lévy process), $s, a, u >0$,
$I(a) :=
\{[k \cdot a, (k+1) \cdot a] \ : \...

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### Reference request: "doubly empirical" measure associated to a random measure

I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In ...