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

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Distribution of the stopping time of an autoregressive sequence

Consider $e_t$ be i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $1$, where $$ W_T = \...
4
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2answers
1k views

Random walk to stay in an interval forever

Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities. To ...
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16 views

Modified Bernoulli trials [on hold]

Consider the modified version of i.i.d. Bernoulli trials where the first success in a success run is converted into a failure, e.g. 'FFSSSSFFSSS' $\rightarrow$ 'FFFSSSFFFSS'. Let the original success ...
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0answers
36 views

Power spectrum of the difference of two Poisson processes with equal rates

I am studying the asymptotic properties of a dynamic a model involving the difference of two balanced Poisson processes (i.e., $\lambda_1 = \lambda_2$). I recently discovered the Skellam Distribution ...
4
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1answer
138 views

Can all Local Martingales Be Represented using Only Brownian Motion and Finite Variation Processes?

This is a cross-post of my unanswered (more than a week) question on Math.SE. Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to try to ...
3
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50 views

Basic Definition and Notations in RWRE [closed]

From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ ...
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0answers
23 views

Closed form formula for fill rate given a discrete distribution? [closed]

I'm wondering whether there is a closed form way to obtain good estimates for fill rate given a discrete distribution of demand. I created a simple monte carlo simulation to see if I could see any ...
4
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62 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
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0answers
22 views

Reference Request: $M_t/M_t/1/K$ queue length distributions

I am investigating functionals defined over sequences of discrete probability distributions related to dynamical/stochastic system performance. As an initial step, I am searching for references that ...
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0answers
16 views

Nonparametric estimation in diffusion

Fan and Wang In the above paper, the Authors provide estimators for the squared spot volatility process $\left(\sigma^{2}_{t}\right)_{t\geq 0}$. My question is how to find estimators for the process ...
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0answers
45 views

How much can the integrability at zero tell about the decay rate around zero? [migrated]

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
3
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1answer
98 views

On the spectrum of stationary Gaussian process

What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum? In a similar way let us consider a stationary vector valued Gaussian ...
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1answer
58 views

Uniqueness of invariant measure for equivalent transition probabilities

Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \...
3
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1answer
148 views

Collecting stones in n buckets

There are $n$ stones distributed in $n$ buckets (initially one stone in each bucket). At each step the content of each bucket is put in a random bucket, chosen independently out of a set of $n$ new ...
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48 views

invariant measure for piecewise deterministic Markov process with only measurable switching intensity

Let $L$ denote the extended generator of a Markov process $(X_t)$ on a locally compact space with domain $\mathcal D(L)$. This means that for all $f \in \mathcal D(L)$, the process $$ f(X_t) - f(X_0) -...
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31 views

information about composite random process

I have a following composite random process $$X_j = v_0 + 1/j^2 + Y_j + Z_j$$ where $v_0$ is a constant, $Y_j \rightarrow 0$ almost surely as $j\rightarrow \infty$ and $Z_j \sim N\big(0, \frac{a^{2j}...
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33 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
3
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2answers
172 views

Lévy measure and Lévy process

A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying $$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$ A Lévy process can be characterized by triples $(b, A, \...
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0answers
95 views

Radon-Nikodym for continuous time processes

Likelihood theory for statistical inference concerning stochastic processes in continuous time are well used. How ever i've found no real literature concerning the fundamentals. What is know from ...
5
votes
1answer
132 views

Contraction of probability measures

Notations. Let $(X, \mathcal{B})$ be a separable Banach space, with its Borel sigma-algebra, $\|\cdot\|$ stands for the norm in $X$, $\mathcal{P}(X)$ - the set of all probability measures on $X$. Let ...
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23 views

Dependency of the error term on the states, in the definition of the transition rates of a continuous time Markov chain

I think this is certainly not a research or graduate level question. But I didn't get any answer from math.stackexchange.com. I'm studying G.F.Lawler's stochastic process book. There he defines the ...
4
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1answer
52 views

Uniform convergence of action of Feller semigroup with $1$ variable

Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous ...
5
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0answers
77 views

Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs. I'm reading Stochastic Differential Equations in ...
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1answer
65 views

The (infinite) invariant measure of an SPDE

Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type: \begin{equation}\left\{ \begin{aligned} &\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...
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124 views

concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
4
votes
1answer
121 views

What is the stationary distribution for the contact process on the half line?

The contact process is a well-studied Markov process. I'm just concerned with the one-dimensional nearest-neighbor version here. The state space is $\eta\in\{0,1\}^\mathbb Z$, and for state $\eta$ at ...
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1answer
72 views

Limit of first passage time

I have a conjecture that seems rather obvious but the proof seems elusive. Consider a diffusion given by, $dX_t = \mu(X_t) dt + \sigma(X_t) db_t$ where $b_t$ is a standard Brownian motion. $\mu,\...
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1answer
40 views

Existence of strong solution in SDEs and continuity in the time variable

I recently come across some literature in stochastic analysis that uses the following result: Consider the one-dimensional SDE $$dX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$ where $a, ...
4
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2answers
173 views

A Stochastic Taylor Expansion/Asymptotics

Question: Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth ...
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1answer
72 views

Computing transition operators for Markov processes

Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt$$ (or given by ...
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0answers
34 views

Karhunen-Loeve expansion convergence rate for Gaussian Proccess

Consider A Gaussian Procces $X(t):\mathbb{R} \to \mathbb{R}$ with and $\mathbb{E} \left[ X \right] = 0$. Consider also its KL expansion $X(t) = \sum\limits_{k=0}^{\infty} Z_k e_k (t)$, with $Z_k$ ...
2
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1answer
68 views

Optimal control / Portoflio optimization: Maximize expected utility of total consumption

I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is ...
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3answers
183 views

A question about intuition of fluid limit in queuing system

This is a question about intuition in understanding the fluid limit queuing system. Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit ...
3
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1answer
141 views

Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ where the coefficients are assumed to be Lipschitz continuous. I hope to ...
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70 views

Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and continuous)....
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10 views

Residual life distribution for renewal process after time T

Suppose we have a renewal process with inter-arrival times $\boldsymbol X=\{X_1, X_2, ...\}$, where $X_i$ are i.i.d variables. Assume that the CDF and PDF for $X_i$ are $F(x)$ and $f(x)$. 1) Let $A_t$...
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20 views

Explicit u-excessive function

Let $E$ be $\mathbb{R}^d$ for $d\geq 1$. Let $A \subset E$. Let $X$ be a Feller process en $E$, and let $L$ be its infinitesimal generator. I want to prove that $A$ is absorbing. I know that it is ...
4
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0answers
149 views

Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let $d\in\left\{2,3\right\}$ $\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
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145 views

Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
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1answer
32 views

Analyzing a multiple-queue single-server model

Consider the following multiple-queue single-server model of a packet network problem. At each discrete time $t=0,1,\ldots,n$, a packet may arrive at the server R with probability $1-\epsilon_1$. The ...
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27 views

Feller property for Ito diffusion with Lipschitz coefficients

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded ...
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126 views

markov processes and ergodic theory

For an ergodic Markov Chain $$ \frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f] $$ where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
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0answers
68 views

formula for density of maximal Poisson disk sampling of radius 1?

Maximal Poisson disk sampling of radius r, applied to a finite planar region, is defined by successively choosing sample points uniformly randomly from the part of the region that is not within ...
13
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1answer
180 views

Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
2
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171 views

Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process $$ dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0 $$ where $a\in (-\infty,+\infty), b&...
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0answers
37 views

Quadratic characteristic and constancy

Consider a change of measure on $\mathcal{F}_{t}$ defined by the restriction of two probability measures of the form \begin{align} \frac{dQ_{t}(\theta)}{dP_{t}}=\exp^{ \theta A_{t}-\kappa(\theta) S_{t}...
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0answers
105 views

Question about continuity in the “complete Skorohod Topology”?

I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process" https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf In one of the exercises, exercise 8.9 ...
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1answer
99 views

Predictable quadratic Variation <.> has same intervals of constancy as the process

From Revuz and Yor - Continuous Martingales and Brownian Motion 1999 Chapter IV Proposition 1.13 it is proven, that for a continuous local martingale $M_t$ the intervals of ...
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78 views

Construction of a random variable

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the statement:...
3
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0answers
59 views

“Local” functional central limit theorem for the empirical distribution function

This question is a repost from Mathematics Stack Exchange, where it did not receive any answer. Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[...