Questions tagged [stochastic-processes]

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

Filter by
Sorted by
Tagged with
7 votes
2 answers
400 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
212 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
  • 203
7 votes
2 answers
388 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
75 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
  • 213
3 votes
0 answers
67 views

The boundary between infinite clusters connected by closed and open bonds

In the following, I'll heuristically describe a boundary between two infinite clusters arising in percolation on the triangular lattice. I expect this concept has been well-studied before. My hope is ...
user196574's user avatar
0 votes
0 answers
71 views

Expand White Noise and Brownian Motion in Haar basis: which version of Haar basis?

Start with the Haar basis of $L^2(\mathbb{R})$, namely, the functions $$ \chi(t-k) \text { and } 2^{j / 2} h\left(2^j t-k\right), j \geq 0, k \in \mathbb{Z}, \quad \quad \quad (1) $$ where $\chi(t)$ ...
Mark's user avatar
  • 297
2 votes
0 answers
50 views

Rates of convergence in the functional CLT/weak invariance principle for martingale triangular arrays

There are results for the rate of convergence of the functional CLT/weak invariance principle for martingales difference sequences, for example theorem 4.5 in the book Martingale theory and its ...
Aurelien's user avatar
  • 191
0 votes
1 answer
73 views

Reference request: $\mathbb{E}|X_t| \to \infty$ as $t \to \infty$ when $\{X_t\}_{t\geq 0}$ is a continuous-time (symmetric) random walk

Let $\{X_t\}_{t\geq 0}$ be a one dimensional continuous-time (symmetric) random walk on $\mathbb Z$ defined via $$X_t = X_0 + \sum_{i=1}^{N_t} Y_i,$$ where $X_0 \in \mathbb{Z}_+$ is a non-negative ...
Fei Cao's user avatar
  • 700
0 votes
0 answers
32 views

Sum of entries of $W^k$ in terms of limiting spectral density of $W$?

Suppose $h$ is spectrum of a random matrix $M$ and $e$ is a vector valued time-series in $\mathbb{R}^d$ with $d\approx \infty$, which starts with $(1,1,\ldots,1)$ and updates $i$'th component at each ...
Yaroslav Bulatov's user avatar
6 votes
1 answer
337 views

Probabilistic problem on random spanning trees

Let $G(V,E)$ be a connected simple graph, where $V$ and $E$ denote respectively its vertex and the edge set respectively. Let $f: V\to \{-1,1\}$ a function mapping each vertex to a value in $\{-1,1\}$....
Penelope Benenati's user avatar
1 vote
1 answer
168 views

concentration of random field to its expectation function

Question Given a random field $X(t)$ where the parameter space $T\subset\mathbb{R}_N$. Is there result regarding the concentration of the random field? For example $\mathbb{P}\{\|X(t)-\mathbb{E}\{X(t)\...
M.K's user avatar
  • 155
2 votes
1 answer
231 views

When does a solution to SDE have full support?

Suppose an $n$-dimensional process $(X_t)_{0 \leq t \leq 1}$ satisfies an SDE of the form: $$dX_t = u_t(X_t) \,dt + dB_t, ~~X_0 = 0$$ where $(B_t)_{t\geq 0}$ is a Brownian motion with $B_1 \sim N(0,K)$...
Tom's user avatar
  • 716
2 votes
0 answers
267 views

Identify two continuous martingales in law as time-changed Brownian motions

Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by $$X_t:...
Fawen90's user avatar
  • 975
1 vote
0 answers
70 views

Minus sign inside derivative operator, notation problem

Hello fellow mathematicians. Can anybody help me understand what the minus (-) sign in this derivative means? Its the usual d/dy but with a minus added d-/dy. I can't find references, the book cited ...
Comeberza's user avatar
1 vote
0 answers
76 views

Urn model with delayed replacement

Suppose I have x red and y blue balls. At each timestep I draw a ball with probability $$P(\text{red ball}) = (x/(x+y))^z, P(\text{blue ball}) = 1-P(\text{red ball})$$ where z is fixed. Each ball is ...
timbuktu's user avatar
1 vote
1 answer
117 views

About boundary local time on reflecting brownian motion

Definition: A bounded measurable function $u$ on $\bar{D}$ is called a weak solution of the Neumann problem $N(D ; q, \varphi)$ if, for all $x \in \bar{D}$, $$ M_{\varphi}^u(t)=u\left(X_t\right)-u\...
neveryield's user avatar
0 votes
2 answers
245 views

What mathematical formalism might be used to disprove natural selection, on the basis that there are too many independent genetic parameters? [closed]

I have nagging doubts that the random genetic mutation process of natural selection is sufficient to explain evolution, even when coupled with sexual selection (Darwin proposed that evolution is ...
user501885's user avatar
3 votes
2 answers
425 views

Random spanning trees probability problem

We are given a simple connected graph $G(V,E)$ with vertex and edge set $V$ and $E$ respectively. For any vertex $v\in V$, let $D_T(v)$ the degree of $v$ in a uniformly generated random spanning tree $...
Penelope Benenati's user avatar
2 votes
2 answers
504 views

The Borel-Cantelli lemma for random walks

I want to know whether the Borel-Cantelli lemma is true for a random walk. More precisely, this question can be described as follows. Let $X_1,X_2,\cdots$ be i.i.d. taking values in $\mathbb{R}^d$ ...
Skywalker's user avatar
2 votes
1 answer
235 views

Joint distribution for sticky Brownian motion

$\newcommand{\R}{\mathbb R}$The one-dimensional Sticky Brownian Motion (SBM in short) is an $\R$-valued Markov process given by \begin{gather*} dX_t=1_{[X_t\neq 0]}dB_t\\ L_t(X)=\int_0^t 1_{[X_s=0]}ds,...
leo monsaingeon's user avatar
5 votes
3 answers
898 views

"Practical" use of time-continuous stochastic processes like Wiener process or Poisson (point) process?

If one uses the Wiener process as an ingredient to model something, then for practical purposes one could just as well take a simple discrete random walk (with sufficiently fine scale). If one uses a ...
Mr H's user avatar
  • 59
0 votes
0 answers
61 views

Different measurability of Hilbert-space valued random variable

My question is motivated by this link. Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable. Now let $H$ be a ...
John's user avatar
  • 483
0 votes
0 answers
47 views

Alternating exponential distributions

Consider a process where objects arrive according to an exponential distribution with $rate=\lambda$. Let $X$ be the number that arrive over an interval of length $T$. Then the number that arrive is ...
ericf's user avatar
  • 614
2 votes
1 answer
332 views

Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions?

$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\ \mathrm d}$Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. $B=(B^1, \ldots, B^N)$ independent one-dimensional Brownian motions. $X=(X_0^...
Akira's user avatar
  • 815
4 votes
1 answer
242 views

When is $\prod_{i=0}^\infty (I-x_i x_i^T)=0$ for isotropic Gaussian $x_i$?

Suppose $x_i$ is sampled IID from isotropic zero-centered Gaussian random variable in $d$ dimensions with covariance $\Sigma=c*I$. When is the following true with probability 1? $$\prod_{i=0}^\infty (...
Yaroslav Bulatov's user avatar
2 votes
0 answers
224 views

Ito lemma for SDEs on a Lie group

I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group. In doing so i'm trying to define a formula ...
Marco's user avatar
  • 263
3 votes
1 answer
401 views

Positive definiteness of a matrix-valued function

This question is a repost from math.se, where I didn't receive an answer. Are there simple conditions on an $d \times d$ matrix B under which $$ f(t, s) = \begin{cases} \exp(-B |t - s|^\alpha), &...
tsnao's user avatar
  • 490
1 vote
2 answers
187 views

Converse Cameron-Martin theorem for shifts by adapted processes

Let $W$ be a standard one dimensional Brownian motion, $\mathcal F_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$. Given a $C[0, 1] $ valued random ...
Nate River's user avatar
  • 4,822
4 votes
1 answer
360 views

Derive the solution of the diffusion equation from the solution of a random walk

Summary The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
Sam's user avatar
  • 281
3 votes
1 answer
129 views

Are there any known results on the probability distributions of perpetuities with power law discount rates?

Currently I am working on studying stochastic integrals of the form: $$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$ where $S_t$ is a Compound-Poisson process with Exponentially-distributed ...
jam jelly's user avatar
2 votes
1 answer
307 views

On the mean value taken by Bernoulli random variables with joint distribution constraints

We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X_1, X_2, \ldots X_n$, such that the probability $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...
Penelope Benenati's user avatar
1 vote
1 answer
82 views

Brownian motion hitting open set starting from its boundary

Let $\{W(t),\,t \in [0,1]\}$ be a standard Brownian motion in $\mathbb{R}^d$, starting from $0$. Let $U$ be a non-empty open set such that $0 \in \partial U$. Which conditions on $U$ are necessary and ...
ssss nnnn's user avatar
4 votes
2 answers
323 views

Rate of convergence of sample maximum, $\Big|\max_{j \leq n} |f(U_j)| - \|f\|_\infty\Big|$

Suppose that $f \colon [0, 1] \to \mathbb{R}$ is a $1$-Lipschitz function. Define the uniform norm $\|f\|_\infty = \sup_{x} |f(x)|$. Given $\{U_j\}_{j=1}^\infty$ independent and identically ...
Drew Brady's user avatar
1 vote
2 answers
272 views

Convergence in sup norm of elementary integrals to the Itô integral process

Let $W$ be a standard one dimensional Brownian motion, and $X$ a continuous process adapted to $W$ such that $\int_0^T X^2 \, ds < \infty$ almost surely for some $T > 0$. Define for any sequence ...
Nate River's user avatar
  • 4,822
1 vote
0 answers
196 views

CLT for dependant random variables

I define a distribution of probability $L$ on $C:=C_0([0,1],\mathbb{R})$ the set of continuous functions $f$ on $[0,1]$ such that $f(0)=f(1)=0$. I suppose that $L$ is centered and has a covariance ...
Charles's user avatar
  • 21
1 vote
0 answers
68 views

Does weak convergence of filtrations preserve progressive measurability?

Suppose on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ I have a sequence of filtrations $\mathbb{F}^n =(\mathcal{F}^n_t)_{t \geq 0}$ generated by Brownian motions $W^n$ for each $n$, ...
PeterGoGo's user avatar
1 vote
1 answer
90 views

How to obtain this differential relation about moments of a stochastic process?

$\newcommand{\Ex}{\mathbb E}$ I'm reading an argument in the proof of Proposition 3.8. in the paper Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos. ...
Akira's user avatar
  • 815
7 votes
1 answer
635 views

How is the Gronwall lemma used in this paper?

Let $(X_t, t \ge 0)$ be a $\mathbb R^d$-valued stochastic process. Let $\lambda>0$. Assume we have $\mathbb E [|X_0|^2] < \infty$ and $$ \mathbb E [|X_t|^2] - \mathbb E [|X_0|^2] \le -2 \lambda \...
Akira's user avatar
  • 815
5 votes
1 answer
470 views

Maximum distance from origin of simple random walk

Let $\epsilon_1, \dots, \epsilon_n$ be random signs, equiprobably in $\{-1, 1\}$, independently. Let $S_k = \sum_{j=1}^k \epsilon_j$. I am wondering what is known about the expectation $$ \mathbb{E}\...
Drew Brady's user avatar
0 votes
0 answers
29 views

How can I obtain a SDE with an advection function that contains the difference in covariates?

Suppose that $\mathbf{s}(t)\in S$ denotes the spatial location of a process at time $t$. Further, let $\mathbf{x}(\mathbf{s}(t))$ denote covariates at the location $\mathbf{s}(t)$. My goal is to write ...
Ron Snow's user avatar
  • 101
0 votes
0 answers
162 views

A variant of Dubins–Schwarz's theorem

Let $W$ be a Brownian motion and $\alpha$, $\beta$ be two progressively measurable processes taking values in $\mathbb R_+$ s.t. $\alpha_t\le \beta_t$ for all $t\ge 0$. Define respectively $X$, $Y$ by ...
Fawen90's user avatar
  • 975
1 vote
1 answer
280 views

Existence of linear stochastic differential equation given solution

Normally if you have a linear SDE given such as $dx_t = (A(t)x_t + a(t))dt + \sigma(t) dW_t$, we want to find $x_t$, more precisely we want to find the mean and variance of $x_t$ at each timestep $t$. ...
mathemagier's user avatar
2 votes
0 answers
110 views

What is the variance of a multi-type branching process?

Problem I have a multi-type branching process with $S$ types, where an individual of type $i$ generates $i$ offsprings. The probability that an offspring is of type $j$ is $p_j$, regardless of its ...
stopro's user avatar
  • 67
0 votes
0 answers
91 views

On a stochastic control problem

Let $(\Omega,\mathcal F, \mathbb P)$ be a probability space on which a Brownian motion $W$ is defined, and $\mathcal U$ be the set of progressively measurable (w.r.t. the Brownian filtration) ...
Fawen90's user avatar
  • 975
1 vote
0 answers
53 views

Bounding difference of characteristic functions with mixing coefficients

Setting Let $X = \{ X(t), t \in \mathbb{R} \}$ be a stationary, $\alpha$-mixing stochastic process and define $$ \begin{align} I_{n, l} &= \{ (l - 1)b_{n} + 1 + r_{n}, \ldots, lb_{n} \}, \quad l = ...
AlbertRapp's user avatar
0 votes
1 answer
211 views

Can we define the divergence of a stochastic process?

Suppose I have a stochastic process $(X_t)_{t\in \mathbb{R}^d}$ with infinitesimal generator $\mathcal{A}$, for example $\mathcal{A}f(X) = -\mu f'(X) + \frac{1}{2}\sigma^2f''(X)+\lambda \int (f(X')-f(...
David's user avatar
  • 1
1 vote
0 answers
136 views

Interpretation of the Lévy measure of an infinitely divisible random vector

We know that a random vector $X$ is infinitely divisible (ID) if for all $n \in \mathbb N$, there exist $X_1^n,..., X_{n}^n$ i.i.d. random vectors such that: \begin{equation} X = X_1^n + ...+ X_n^...
PSE's user avatar
  • 13
8 votes
2 answers
379 views

Regularity of translations for Brownian motion

Let $B_t$ be the classic Brownian motion. I understand that, if $s>1/2$, almost surely $B_t$ is nowhere $s$-Hölder continuous i.e. almost surely for no point $x$ it happens that $B_t\in C^s(x)$. ...
pipenauss's user avatar
  • 297
2 votes
0 answers
231 views

KL Divergence between the solution to two SDEs

What is the KL divergence between the laws of solutions to SDEs? That is, let \begin{align*} dX^1&=b_1(X^1,t) \, dt+\sigma(X^1,t) \, dB\\ dX^2&=b_2(X^2,t) \, dt+\sigma(X^2,t) \, dB \end{align*}...
user499216's user avatar
2 votes
1 answer
216 views

When is a stationary measure of a Markov chain "exponentially localized"?

Here exponentially localized can be thought in a non-rigorous manner as a measure that is mostly supported on a sparse number of nodes. Some intuition can gained by thinking about a diffusion process, ...
Piyush Grover's user avatar

1
3 4
5
6 7
48