# Questions tagged [stochastic-processes]

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

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### 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 ...

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### Eigenvalues/eigenfunctions of a diffusion generator

Consider the following symmetric second order diffusion operator:
$$L := \lambda_1 R_1 \partial_{R_1} - \lambda_2 (\sigma^2 - R_2)\partial_{R_2} + \frac{1}{2}\lambda_1^2\sigma^2 \partial^2_{R_1, R_1}$...

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### 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 ...

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### 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 ...

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### 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}(...

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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 ...

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### Reference request: Gaussian estimates for SDE with discontinuous diffusion coefficient

Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R_+ \times \mathbb R^d \to \mathcal M_{d \times d}^{\text{sym}} (\mathbb R)$ be bounded measurable where $\sigma$ is ...

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### 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 ...

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### 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 ...

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### 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)(...

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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 ...

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### 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 ...

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### 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 ...

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### 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}...

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### 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 ...

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### 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 $...

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### 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 ...

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### Bound the variance of the largest order statistics among discrete random variables

I have a question about bounding the variance of order statistics.
Given that for $i \in \{1,\cdots,\lambda\}$, denote $Bin(s,\frac{1}{n})$ to be a binomial random variable with success probability $\...

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### 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 ...

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### 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 $\chi(t)$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\{-1,1\}$....

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### 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)\...

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### 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 $B_1 \sim N(0,K)$...

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### 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:...

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### 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 ...

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### 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 ...

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### 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\...

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### 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 ...

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### 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 $...

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### 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 $\mathbb{R}^d$ ...

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### 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,...

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### "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 ...

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### 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 ...

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### Asymptotic distribution of counting process when we include higher moments (thus not Gaussian)

Question
Consider a counting process $(N(t))_{t \ge 0}$ such that the time $\tau$ between arrivals is I.I.D with some probability distribution $f$. Call $\langle \tau^n\rangle$ the $n$th moment of $$, ...

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### 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 ...

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### 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^...

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### 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 (...

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### 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 ...

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### 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), &...

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### 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 ...

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### 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 ...

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### 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 ...

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1
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### 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 $\mathbb{P}(X_1=X_2=\ldots=X_n=0)$...

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### 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 ...

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### 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 ...

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1
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### 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 ...

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### 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 ...