Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
2,460 questions
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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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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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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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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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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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Does $X_t$ with $t>0$ admit a density?
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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.?
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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?
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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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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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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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Rademacher complexity of function class $(x,y) \mapsto 1[|yf(x)-\alpha| \ge \beta]$ in terms of $\alpha$, $\beta$, and Rademacher complexity of $F$
Let $X$ be a measurable space and let $P$ be a probability distribution on $X \times \{\pm 1\}$. Let $F$ be a function class on $X$, i.e., a collection of (measurable) functions from $X$ to $\mathbb R$...
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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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Hunting an invisible target
An invisible target on the integer line starts at $0$. On each round it either stays put, moves to the left or moves to the right by $1$ with probability $\frac{1}{3}$ each. You are then asked to ...
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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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Bounding random process
Def
$\{X_t\}_{t\in T}$ is called Lipschitz for metric $d$ on $T$ if there exists a random variable $C$ such that
$$|X_t-X_s|\leq Cd(t,s),\text{ for all }t,s\in T.$$
Lemma
Suppose $\{X_t\}_{t\in T}$ is ...
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Does this filtration have a name?
In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
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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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What is the expected remaining life duration of a cell in the $t\to\infty$ limit?
Consider the following population model: We start with a population of a single cell at time $t=0$. Each cell divides into $k$ new cells at random times $T$ distributed according to a probability ...
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What is the convergence rate of this "infinite monkey"-type probability?
Cross-posted from Math Stack Exchange, where it hasn’t received an answer yet:
Let $S$ be a finite set and $n,m\in\mathbb N$. Consider the process $R=(R_i)_{i\in\mathbb N}$ where all $R_i$ are iid ...
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Progressively measurable vs adapted
I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...
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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 ...
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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] \ : \...
- 128k
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Equivalence vs modification
Suppose that two processes have the same finite-dimensional distributions. Does there exist a coupling of them such that they are modifications of each other?
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PDE for the probability of Brownian motion staying in an area (reference request)
I am looking for a (preferably some monograph) reference on the following fact:
$$
u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \}
$$
satisfies the heat equation
$$
\frac{\...
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The limit ratio of two Markov Chain Probability
Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
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Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables
Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
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Liverani's CLT (a question)
Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to L^{...
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Optimally betting a beta-biased coin
This question is inspired by How to optimally bet on a biased coin? by Nate River but generalized slightly. I decided the generalization might be interesting enough to be its own question.
A number $p$...
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Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined
Remark: I've asked this question on MSE as well.
Let
$T>0$
$I:=[0,T]$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...
CommunityBot
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A coupon collector-ish question
Imagine we are in the coupon collector setting: every time step we get independently one coupon out of $n$ coupons uniformly at random. However, unlike the coupon collector problem, we stop the at the ...
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A question about one Malliavin derivative calculation
Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
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Expectation of $B_u \operatorname{argmax}_t B_t$
This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here.
Yesterday I asked a question about the joint law of ...
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Stochastic analysis on nuclear Fréchet spaces
This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise.
A lot of the time in infinite-...
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Concentration result for self-normalized empirical process
In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....
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The drunken blind man’s walk
Consider a drunk, blind man starting in the middle of the two dimensional open unit ball. At each turn, the man chooses a direction to move a step of size $\delta > 0$ in. Unfortunately, he is very ...
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Example of $F\in W_0^{1,2}$ a.s. so that the law of $F+B$ is equivalent to that of $B$ but DD exponential isn't integrable?
Is there an explicit example of progressively measurable $F=\int_0^\cdot f(s) ds\in W_0^{1,2}(0,1)$ a.s. so that the law of $F+B$ on $(0,1)$ is equivalent to that of a Brownian motion $B$ on $(0,1)$ ...
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Derivative with respect to initial condition for the solution of an SDE
Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution):
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
and define its ...
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Including fixed-time transitions into a continuous time Markov chain system
I have system which is mostly described by a CTMC (Continuous-time Markov chain) with a single absorbing state and a large but tractable and sparse transition matrix. However, at a fixed set of "...
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Explicit example of drift $F$ so that the law of $F+B$ is not absolutely continuous with respect to $B$
Let $\mu_0$ be the law of Brownian motion on the space of continuous functions. If $\mu\sim\mu_0$ agrees on null sets then there is some progressively measurable $F\in W_0^{1,2}$ a.s. so that $\mu$ is ...
- 1,904
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Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$
I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
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