Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
2,351
questions
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
$$
\begin{cases}...
4
votes
1
answer
212
views
Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?
Adapted from math stack exchange.
Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel.
My ...
7
votes
2
answers
388
views
Fractional Brownian motion of Riemann-Liouville type is not a semimartingale
Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
2
votes
0
answers
75
views
Stochastic domination and coupling of point processes with random intensity
Suppose we have two (regular) point processes $N, N^*$ on the half real line (but more general spaces welcome). I will characterize these by their conditional intensity function (which uniquely ...
3
votes
0
answers
67
views
The boundary between infinite clusters connected by closed and open bonds
In the following, I'll heuristically describe a boundary between two infinite clusters arising in percolation on the triangular lattice. I expect this concept has been well-studied before. My hope is ...
0
votes
0
answers
71
views
Expand White Noise and Brownian Motion in Haar basis: which version of Haar basis?
Start with the Haar basis of $L^2(\mathbb{R})$, namely, the functions
$$
\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)
$$
where $\chi(t)$ ...
2
votes
0
answers
50
views
Rates of convergence in the functional CLT/weak invariance principle for martingale triangular arrays
There are results for the rate of convergence of the functional CLT/weak invariance principle for martingales difference sequences, for example theorem 4.5 in the book Martingale theory and its ...
0
votes
1
answer
73
views
Reference request: $\mathbb{E}|X_t| \to \infty$ as $t \to \infty$ when $\{X_t\}_{t\geq 0}$ is a continuous-time (symmetric) random walk
Let $\{X_t\}_{t\geq 0}$ be a one dimensional continuous-time (symmetric) random walk on $\mathbb Z$ defined via $$X_t = X_0 + \sum_{i=1}^{N_t} Y_i,$$ where $X_0 \in \mathbb{Z}_+$ is a non-negative ...
0
votes
0
answers
32
views
Sum of entries of $W^k$ in terms of limiting spectral density of $W$?
Suppose $h$ is spectrum of a random matrix $M$ and $e$ is a vector valued time-series in $\mathbb{R}^d$ with $d\approx \infty$, which starts with $(1,1,\ldots,1)$ and updates $i$'th component at each ...
6
votes
1
answer
337
views
Probabilistic problem on random spanning trees
Let $G(V,E)$ be a connected simple graph, where $V$ and $E$ denote respectively its vertex and the edge set respectively. Let $f: V\to \{-1,1\}$ a function mapping each vertex to a value in $\{-1,1\}$....
1
vote
1
answer
168
views
concentration of random field to its expectation function
Question
Given a random field $X(t)$ where the parameter space $T\subset\mathbb{R}_N$. Is there result regarding the concentration of the random field? For example
$\mathbb{P}\{\|X(t)-\mathbb{E}\{X(t)\...
2
votes
1
answer
231
views
When does a solution to SDE have full support?
Suppose an $n$-dimensional process $(X_t)_{0 \leq t \leq 1}$ satisfies an SDE of the form:
$$dX_t = u_t(X_t) \,dt + dB_t, ~~X_0 = 0$$
where $(B_t)_{t\geq 0}$ is a Brownian motion with $B_1 \sim N(0,K)$...
2
votes
0
answers
267
views
Identify two continuous martingales in law as time-changed Brownian motions
Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by
$$X_t:...
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 ...
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 ...
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\...
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 ...
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 $...
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$ ...
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,...
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 ...
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 ...
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 ...
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^...
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 (...
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 ...
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
$$
f(t, s)
=
\begin{cases}
\exp(-B |t - s|^\alpha), &...
1
vote
2
answers
187
views
Converse Cameron-Martin theorem for shifts by adapted processes
Let $W$ be a standard one dimensional Brownian motion, $\mathcal F_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$.
Given a $C[0, 1] $ valued random ...
4
votes
1
answer
360
views
Derive the solution of the diffusion equation from the solution of a random walk
Summary
The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
3
votes
1
answer
129
views
Are there any known results on the probability distributions of perpetuities with power law discount rates?
Currently I am working on studying stochastic integrals of the form: $$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$
where $S_t$ is a Compound-Poisson process with Exponentially-distributed ...
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)$...
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 ...
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 ...
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 ...
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 ...
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$, ...
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.
...
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 \...
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
$$
\mathbb{E}\...
0
votes
0
answers
29
views
How can I obtain a SDE with an advection function that contains the difference in covariates?
Suppose that $\mathbf{s}(t)\in S$ denotes the spatial location of a process at time $t$. Further, let $\mathbf{x}(\mathbf{s}(t))$ denote covariates at the location $\mathbf{s}(t)$. My goal is to write ...
0
votes
0
answers
162
views
A variant of Dubins–Schwarz's theorem
Let $W$ be a Brownian motion and $\alpha$, $\beta$ be two progressively measurable processes taking values in $\mathbb R_+$ s.t. $\alpha_t\le \beta_t$ for all $t\ge 0$. Define respectively $X$, $Y$ by
...
1
vote
1
answer
280
views
Existence of linear stochastic differential equation given solution
Normally if you have a linear SDE given such as
$dx_t = (A(t)x_t + a(t))dt + \sigma(t) dW_t$, we want to find $x_t$, more precisely we want to find the mean and variance of $x_t$ at each timestep $t$. ...
2
votes
0
answers
110
views
What is the variance of a multi-type branching process?
Problem
I have a multi-type branching process with $S$ types, where an individual of type $i$ generates $i$ offsprings. The probability that an offspring is of type $j$ is $p_j$, regardless of its ...
0
votes
0
answers
91
views
On a stochastic control problem
Let $(\Omega,\mathcal F, \mathbb P)$ be a probability space on which a Brownian motion $W$ is defined, and $\mathcal U$ be the set of progressively measurable (w.r.t. the Brownian filtration) ...
1
vote
0
answers
53
views
Bounding difference of characteristic functions with mixing coefficients
Setting
Let $X = \{ X(t), t \in \mathbb{R} \}$ be a stationary, $\alpha$-mixing stochastic process and define
$$
\begin{align}
I_{n, l} &= \{ (l - 1)b_{n} + 1 + r_{n}, \ldots, lb_{n} \}, \quad l = ...
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(...
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^...
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 $B_t\in C^s(x)$.
...
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*}...
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, ...