Questions tagged [stochastic-processes]

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

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Lower bound of $\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]$ (without observing history)

Let $X$ be the solution to some stochastic differential equation $$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$ Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ ...
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Request for article in Rev. Roumaine Math. Pures Appl. (1981)

I am looking for the following article: Al-Hussaini, A. N. A projective limit view of $L_1$-bounded martingales. Rev. Roumaine Math. Pures Appl.26 (1981), no.1, 51–54, but I can't find it anywhere. Do ...
mathex's user avatar
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Stationary Distribution of Langevin Dynamics driven by Lévy Process

Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\...
Small Deviation's user avatar
5 votes
1 answer
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How can we prove that a stochastic process converges to a deterministic value?

As an illustrative example, consider a modified O-U process $dX_t = -X_tdt + \exp(-t)dW_t$. It is not too hard to understand that after a while the behaviour is dominated by the deterministic ...
Adrien Corenflos's user avatar
3 votes
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262 views

Version of Kolmogorov tightness criterion without moments

Kolmogorov tightness criterion says that if $X_N$ is a sequence of continuous process with $X_N(0)=0$ and $E[[X_N(t)-X_N(s)|^p]\leq C_p |t-s|^{1+\beta}$ then for all $\gamma\in (0,\beta/p)$ we have ...
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Hölder continuity of process from Donsker like theorem with Cauchy random variables

Let $X_k$ be i.i.d. Cauchy random variables with parameters $0,1$. For each $N$ define the process $Y_N$ by $$Y_N(t)=\frac{1}N\sum_{k=1}^{\lfloor tN\rfloor}X_k+\text{piecewise linear interpolation}.$$ ...
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Asymptotic behaviour of the solution to some delayed stochastic differential equation

Consider the delayed stochastic differential equation as below: $$dX_t^\theta=X_{(t-\theta)^+}^\theta(1-X_{(t-\theta)^+}^\theta)(dt+dW_t),\quad \forall t>0$$ $$dY_t^\theta=Y_{(t-\theta)^+}^\theta(1-...
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Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process

(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...
Tom's user avatar
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Convergence of stochastic linear recurrences

Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$). Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously ...
cfp's user avatar
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Can a diffusion process admit an invariant measure with a non-differentiable density?

The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
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Escaping probability of a Brownian particle in random enviroment

Let $\Omega\subset \mathbb R^d$ be a bounded open (and connected) set. Consider $E\subset \Omega$ and $x\in \Omega\setminus E$. Denote by $W^x$ the standard Brownian motion starting at $x$, i.e. $W^...
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A question about convergence of stochastic processes converging to a random walk

Consider the following random walk $(y_t)_{t \in \mathbb Z_+}$: $$y_t = y_{t-1} + u_t,\quad (u_t)_{t \in \mathbb Z_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z_+)$$ where $y_0, u_1, u_2,...$ ...
PSE's user avatar
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Garsia-Rodemich-Rumsey without Markov

Let $X$ be a $\mathbb R^d$ valued continuous stochastic process. I am interested in bounding $$P(\|X\|_\gamma>R).$$ The standard technique to do so, is to apply Markov inequality and then Garsia-...
user479223's user avatar
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SPDEs driven by fractional brownian noise

I am looking for some references for the following kind of SPDEs $$dX_t= AX_t\,\mathrm{d}t+BX_t\,\mathrm{d}W^H_t,$$ given $X(0)=X_0$, where $A$ and $B$ are operators and $W^H_t$ is the fractional ...
MathAnimal's user avatar
3 votes
1 answer
259 views

Request for references of random matrices

I need some good books aimed as a detailed and gentle introduction to random matrices, containing good discussion and derivation of Marchenko–Pastur distribution. Also, I request some other references ...
AgnostMystic's user avatar
2 votes
1 answer
237 views

Functional integral formulas for the wave equation and other hyperbolic PDEs

The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation \begin{align*} \partial_t u &= \frac{1}{2}\Delta_x u,\\ u(0,x) &= ...
crystalline cohomology's user avatar
2 votes
0 answers
114 views

Martingale regularization

Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$ I was wondering if there ...
mathex's user avatar
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Characteristic function of quadratic variation of compound Poisson process

If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the joint characteristic function of this process and its quadratic variation process? If not ...
Frimousse's user avatar
-1 votes
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131 views

joint density of two relevant random variables

It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
Wang Jing's user avatar
3 votes
1 answer
224 views

"Ergodic theorem" for Markov kernels

Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify ...
Francesco Bilotta's user avatar
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Step in the derivation of the total idle time distribution of an M/G/1 queue

I'm trying to work my way through the proof of Thm. 1.11 in Kyprianou's Introductory Lectures on Fluctuations of Levy Processes with Applications but really struggle to understand the following step. ...
Othman El Hammouchi's user avatar
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Maximise the probability that a drifted Brownian motion doesn't hit zero prior to $T$

Let $W=(W_t)_{t\ge 0}$ be a standard Brownian motion starting from zero and $Z>0$ be an independent random variable. Fix $T>0$ and $C>0$. Denote by $\mathcal A$ the set of progressively ...
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Characteristic function of a Dirichlet Process

Suppose $P \sim \text{DP}(\alpha,G) $ where $G \sim N(0,1)$ is the base measure and $\alpha > 0$ is the concentration parameter. The stick breaking representation says that $P$ can be expressed as \...
Zilch's user avatar
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Randomly chosen walk of fixed length

Let $G=(V, E)$ be the graph on vertices $V = \{0, \cdots, k\}^n$, where vertices $(v_1, \cdots, v_n)$ and $(w_1, \cdots, w_n)$ share an edge iff $\lvert v_i - w_i\rvert \leq 1$ for all $i$. A walk of ...
S. M. Roch's user avatar
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1 answer
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(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes

Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$, \begin{equation} \int_0^te^{-\lambda ...
Wenguang Zhao's user avatar
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1 answer
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Markov chain to solve a particle fusion problem

A sequence of elementary particles arrive at Poisson rate $r$ to a system. A pair of elementary particles can be fused into a level-$1$ particle. The fusion process succeeds with probability $p_0$. ...
lchen's user avatar
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How to find lower bounds of a modified mixing time (defined below) with respect to spectral of a finite Markov chain?

I am focused on a time-homogeneous continuous-time Markov chain with a finite state space $\mathcal{X}$, whose Markov kernel is $K$ and the corresponding semigroup is $H_t=e^{-t(I-K)}$. The invariant ...
Richard Ben's user avatar
3 votes
0 answers
164 views

Asymptotics for optimal survival time in a stochastic control problem

An individual, henceforth called the runner starts at the center of an open ball $\Omega_r \subset \mathbb R^2$ of radius $r > 1$. At each turn, a vector $x \in S^1$ is chosen uniformly at random, ...
Nate River's user avatar
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1 vote
1 answer
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Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials

This question is cross-posted from https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials I am trying to study the asymptotic behavior ...
Francesco Bilotta's user avatar
1 vote
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97 views

Concatenation of Markov processes and independence

In chapter 14 of Sharpe's General Theory of Markov Processes the concatenation of Markov processes $X^1$ and $X^2$ is described. I've posed the relevant part at the bottom of this post. It is rather ...
0xbadf00d's user avatar
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Dealing with noise that is white in time, colored in space numerically

I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...
Yhtomit's user avatar
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-2 votes
1 answer
125 views

Branching process with varying offspring distribution at each step

Consider a simple branching process $Z_0,Z_1,Z_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$. ...
stopro's user avatar
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Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution

Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
Math_Y's user avatar
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Can we use epsilon-net method on an open set?

Let $X_{t\in T}$ be a random function where $T$ is a subset of $\mathbb{R}^n$. Since $T$ has inifite points, we are not able to use union bound to estimate $\sup X_t$. Thus instead, when $T$ is ...
happyle's user avatar
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3 votes
1 answer
256 views

Strong blow up limits for SDE

Note: This is a strengthening of the following result, motivated by the need for strong convergence in applications. Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution ...
Nate River's user avatar
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1 vote
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Eigenvalues/eigenfunctions of a diffusion generator

Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by: $$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
Greyearl's user avatar
2 votes
0 answers
64 views

Regularity of a function depending on first exit time of martingale

Consider a parametrised martingale as follows : $$X^x_t := x+ \int_0^t\sqrt{2p_s} \, dW_s,$$ where $W$ is a standard Brownian motion and $(p_t)_{t\ge 0}$ is a locally square integrable process ...
Fawen90's user avatar
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5 votes
1 answer
270 views

Does the entropy of a SDE with nondegenerate noise always increase?

Let $W$ be a standard Brownian motion, and let $X$ be the solution to the one dimensional SDE $$dX_t = \sigma(t, X_t) \, dW_t$$ with initial condition $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. We ...
Nate River's user avatar
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2 votes
1 answer
316 views

Convergence of the quadratic variation process

Suppose we are given a sequence of stochastic processes $X^n, n\in\mathbb{N},$ with finite quadratic variations and a stochastic process $X$ such that for every $t\geq0$ $$ \lim_{n\to\infty}\mathbb{E}(...
El_mago's user avatar
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2 votes
0 answers
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Local martingale for a (two-dimensional) diffusion

Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda ...
Serguei Popov's user avatar
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68 views

Calculation of the difference of two Brownian bridges

I was told that the difference of two independent brownian bridge process is $\sqrt{2}$ times a brownian bridge process, i.e., $$B_{1t} - B_{2t} = \sqrt{2}B_t$$ where $B_{1t}$ and $B_{2t}$ are ...
John Smith's user avatar
4 votes
0 answers
200 views

A notion of SDE via the martingale representation theorem

$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
crystalline cohomology's user avatar
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0 answers
65 views

Probability distribution for a Bayesian Update

I am struggling with a process like this: $$X_t=\begin{cases} \frac{\alpha\omega_t}{\alpha\omega_t+\beta(1-\omega_t)} & \text{with prob } p\\ \frac{(1-\alpha)\omega_t}{(1-\alpha)\omega_t+(1-\beta)(...
DreDev's user avatar
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0 answers
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Convergence bounds for ergodic random walk

We are given a simple connected graph $G(V,E)$, where $V$ and $E$ denote the vertex and edge sets respectively. Let $G'(V,E')$ be the graph generated by $G$ by adding one self-loop edge for each ...
Penelope Benenati's user avatar
1 vote
1 answer
95 views

Phase space Brownian bridge

I understand the concept of the 1 dimensional Brownian bridge with the form of: $$dx_t=\frac{-1}{1-t}x_t \, dt + dw_t$$ s.t. $x_0=0$ and $x_1=0$ where $dw_t$ is a Wiener process. I am thinking about ...
BayesFans's user avatar
2 votes
0 answers
48 views

How to determine speed (rate) in large deviation principle for geometric Brownian motion

By reading Asymptotics for volatility derivatives in multi-factor rough volatility models by Lacombe, Muguruza and Stone, I am not familiar with the way they deduce the speed (or rate) when showing ...
Mili's user avatar
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7 votes
2 answers
395 views

Existence of solutions to the heat equation on nonsmooth domains

Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation $$ \begin{cases}...
Brazilian Cérebro's user avatar
4 votes
1 answer
208 views

Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?

Adapted from math stack exchange. Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel. My ...
Lance's user avatar
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7 votes
2 answers
380 views

Fractional Brownian motion of Riemann-Liouville type is not a semimartingale

Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
El_mago's user avatar
  • 99
2 votes
0 answers
74 views

Stochastic domination and coupling of point processes with random intensity

Suppose we have two (regular) point processes $N, N^*$ on the half real line (but more general spaces welcome). I will characterize these by their conditional intensity function (which uniquely ...
jdods's user avatar
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