Questions tagged [stochastic-processes]

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

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Tail for the integral of a diffusion process

I would like to compute the following tail, $$ \mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right), $$ assuming $$ \mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x), $$ and $X$ is a diffusion ...
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A simplified MCMC / MH algorithm. Are there known convergence results?

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
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Sufficiency of stationary policy for negative stochastic dynamic programming

Consider a Markov Decision Process with Borel state space $X$ and Borel action space $U$, like the one defined in the book "Stochastic Optimal Control: Discrete-time case" by Bertsekas and Shreve. All ...
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Some constants in Martingale Stein inequality

Dear all, the following is a special case of Stein inequalities for martingales. $\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a filtration of ...
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The spectrum of a Markov Operator and Invariant Measures

Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
Jeremy Voltz's user avatar
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A balls into bins problem with combinatorial constraints

We are given $m$ balls and $n$ bins, with $m \ge n$. Each bin can contain at most $c$ balls (we assume that $c$ is an even integer). In a sequential fashion, at each time step, one ball is placed into ...
Penelope Benenati's user avatar
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A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal

I apologize if this is too elementary a question, but I have not been able to make much progress. Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
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Norm estimate for parabolic SPDE solution

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
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Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set

Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE \begin{equation} dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,. \end{equation} Let $f(x,t|x_0,0)$ denote its transition density function. ...
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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 ...
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Flow property for semimartingale driven SDE at a stopping time

Let $S$ be an $n$-dimensional semimartingale such that the SDE $$dX_t = \sigma(X_t, t) \, dS_t$$ with $\sigma$ Lipschitz continuous admits a globally defined unique strong solution on $[0, T]$. For $t ...
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Stochastic braids

I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
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Multi-type Galton-Watson-like process where only majority-type is allowed to reproduce

Are you aware of any research papers that have explored a multi-type Galton-Watson process in which only particles of the majority type are permitted to reproduce in each generation? I've been unable ...
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Algebra core for generator of Dirichlet form

This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
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Inverse comparison principle for stochastic differential equations

Consider two SDEs (stochastic differential equations) as follows: $$dX_t=b^-(t,X_t) \, dt+a(t,X_t) \, dW_t;\quad dY_t = b^+(t,Y_t)\,dt+a(t,Y_t)\,dW_t,$$ where $b^-,b^+,a$ are Lipschitz such that $b^-&...
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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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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, ...
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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 $...
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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 ...
user196574's user avatar
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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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Example of an optional non-predictable process

To clarify better the notions of predictable and optional processes, I am looking for a simple example of a process that is optional, but not predictable. I found out something useful here, however, I ...
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Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?

In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
Martin Geller's user avatar
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Probabilistic optimization problem on tree vertex selection without replacement proportional to the degree

We are given a tree $T(V,E)$ with $|V|=n$ vertices, where $V=\{v_1,v_2,\ldots, v_n\}$. We denote by $d_i$ the degree of vertex $v_i$ for all $i\in\{1,2,\ldots,n\}$. In a sequential fashion, we select ...
Penelope Benenati's user avatar
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Martingale problem for regime switching SDE

Let $\mu$ be a continuous time Markov process switching between two possible values $a, b \in \mathbb R$ with time homogeneous transition rates. Consider the one dimensional SDE $$dX_t = \mu \, dt + ...
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Math videos featuring interesting data animations

I am looking for interesting videos featuring pure data animations (not someone talking about math, but a video featuring some math phenomenon). I am interested in videos that tell a story, rather ...
Vincent Granville's user avatar
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The distribution of collision stopping time in 2D random walk

Assume two particles A at $(0, 0)$ and B at $(a, b)$ in 2D discrete grid, both of them have the same possibility of $\frac{1}{4}$ for moving up/down/left/right (i.e. 2D random walk). We define the ...
Chenggang Zhao's user avatar
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Conditions of parameters to have bounded solution of Dynkin's equation in exit problem

Consider the following Dynkin’s equation in exit problem defined on unit disk $D_1(0)$ \begin{align} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma-1}{r} \sin\psi \frac{\partial}{\partial\...
GilbertDu's user avatar
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On the property of a nonnegative stochastic process "attracted" near zero

Let $\{X_k\}$ be a nonnegative stochastic process satisfying $$E\left[ X_{k+1} \mid \mathcal{F}_k \right] \leq \rho X_k + c,$$ where $0 < \rho < 1, c>0$. Intuitively, the process is likely to ...
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Regularity of harmonic functions for a degenerate elliptic operator

This is a question on a degenerate elliptic operator. Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
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Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix

In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
Tom's user avatar
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Continuous time Markov chains and invariance principle

This question may be elementary for experts Let $\{\xi_n\}_{n=1}^{\infty}$ be an i.i.d random variables on a probability space $(\Omega,\mathcal{F},P)$. We assume that the mean of $\xi_n$ is zero, and ...
sharpe's user avatar
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Probability of a particle surviving forever

Consider a particle whose position is driven by the following equation: $$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$ where $y>0$, $0<C<1$...
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Optimal rule for multiple stopping times for defect finding

Suppose a quality inspector is inspecting $b$ black amongst which $d_B$ are known to be defective and $w$ white gadgets amongst which $d_W$ are known to be defective. The gadgets come down along an ...
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Random walk in a switching scenery

For each $x \in \mathbf{Z}$ let $(\eta_t(x))_{t\geq0}$ denote independent copies of a process $(\eta_t(0))_{t\geq0}$ defined as follows. The process $\eta_t(0)$ takes values in $\{-1,1\}$, where $-1$ ...
as1's user avatar
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An integral involving Levy process with no positive jumps

Let $L_t$ be a Levy process that has no positive jumps, but is not strictly decreasing, i.e $$ L_t = \gamma t + \sigma B_t + J_t, $$ where $B_t$ is a Brownian motion, $J_t$ is a pure jump process with ...
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The domains of generators of Markov processes and moment of hitting times

This is a question on domains of generators and moment of hitting times. Let $M$ be a locally compact separable metric space and $X=(\{X_t\}_{t \ge 0}, \{P_x\}_{x \in M})$ a diffusion process on $M$ (...
sharpe's user avatar
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How does one define the gradient of a Markov semigroup?

In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper: ...
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Correlation of stopping times for integral of Brownian motion increment

Let $\mu(x):=\int_{\epsilon}^{x}\exp\{B_{s+\epsilon}-B_{s-\epsilon}\}ds$, where $(B_{s})_{s\geq 0}$ is a Brownian motion (starting at $B_{0}=0$) and epsilon is small $0<\epsilon\ll 1 $. Consider ...
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Markov semigroups and resolvents, difference of continuity

Let $(E,d)$ be a locally compact separable metric space. We have a Markov process $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ on $E$. For bounded measurable function $f$ on $E$, we define \begin{align*} ...
sharpe's user avatar
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Have stick-breaking priors with non-iid atoms been considered, and if not, why not?

Roughly speaking, a stick-breaking prior is a random discrete probability measure $P$ on a measurable space $\mathcal X$ of the form $$P=\sum_{j\ge1}w_j\delta_{\theta_j}$$ where $(w_j)_{j\ge1}$ is a ...
mathducky's user avatar
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Beta distribution and Wiener process

Suppose $W(t)$ is the standard Wiener process on $[0; 1]$ and $\{T_x\}_{x \in \mathbb{R}}$ is a collection of random variables defined by the following relation: $$T_x = \mu(\{t \in [0;1] | W(t) > ...
Chain Markov's user avatar
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Random Two-Player Asymmetric Game

About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have ...
DreamConspiracy's user avatar
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Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion

I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...
E_Wijler's user avatar
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484 views

Domain of the Generator of a Bessel process

Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$ \begin{align} \rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t} \end{align} where $(W_{t})_{t\geq ...
fast_and_fourier's user avatar
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Mutual dependencies of BSDE solutions with markovian drivers with different starting points

Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$. Let ...
Kolodez's user avatar
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Random walk on $\mathbb{R}$ with "sticky" origin

Let $P_i$, $N_i$, and $Z_i$, $i\in\mathbb{N}$ be r.v.'s with the $P_i$, $N_i$, and $Z_i$ being identically distributed with known pdf's $f_P$, $f_N$, and $f_Z$, respectively; and with no dependence ...
Nick Broderick's user avatar
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172 views

product of right continuous filtrations is right-continuous?

Let $\mathcal{G} = \sigma \lbrace G_{1},..., G_{n} \rbrace$ where $G_{1},..., G_{n}$ are subsets of $\Omega_{1}$ and $(\mathcal{F}_{t})$ is a right-continuous filtration on $\Omega_{2}$. Is $(\mathcal{...
Ivan's user avatar
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Galton Watson tree with various kinds of offspring

As far as I understood, for the Galton-Watson tree process, the offspring are of one type. I am thinking of the case where we have offspring of different types. I have illustrated this in the example ...
Wiliam's user avatar
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Generators and Covariance Operators of Diffusions

For a constant coefficient Ornstein-Uhlenbeck process, how should I think about the relationship between the infinitesimal generator of the process and the covariance operator of the process (or, ...
user2379888's user avatar
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Why is the Jain Monrad condition the right condition on general Gaussian processes?

Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process. Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...
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