Questions tagged [stochastic-processes]

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

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Does stationary increments and self similarity imply Hölder?

Let $X(t)$ be a real valued continuous stochastic process. Suppose that $X(t)-X(s)=(t-s)^a X(1)$ in distribution for some $a\in (0,1)$. If $X$ has infinitely many moments then Kolmogorov continuity ...
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Existence of Brownian motion using Kolmogorov's extension theorem

When wishing to prove existence of Brownian motion, most authors take the route of defining the desired finite dimensional distributions, showing the family of defined finite dimensional distributions ...
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When are the transition densities of an SDE symmetric?

We fix $T>0$. Let $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ and $\sigma:[0, T] \times \mathbb{R}^d \rightarrow \mathcal{M}^\text{sym}_{d \times d}(\mathbb{R})$ be measurable and ...
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A complex question related to a certain convergence of Lévy measures

Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and: \begin{equation}\label{I}\tag{SP} X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
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Stochastic volatility model question

Let suppose that $S_t$ is a process defined as: $$ \begin{cases}dS_t = \mu S_t\,dt+m(v_t)\,dW^1_t\\ dv_t = \mu_v(v_t)\,dt + \sigma_v(v_t)\,dW^2_t\end{cases}$$ where the two Brownian motions have ...
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Large-deviation inequalities for a class of simple random multivariate polynomials

Let $N$ be a large positive integer and let $[N] := \{1,2,\ldots,N\}$. For any $k$, let $K_{N,k}$ denote the collection of $k$-element subsets of $[N]$. Let $x=(x_1,\ldots,x_N)$ be a uniformly random ...
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Markov process with time varying transition kernels

I cross post this question from StackExchange as it may be more appropriate. I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...
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Iteratively merging consecutive symbols of a discrete random process

Consider an i.i.d. source (more generally, an ergodic source) which generates symbols from a distribution $P$ over a finite alphabet $\mathcal{A}$. Consider the following stochastic process: At each ...
Television's user avatar
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Measurability of two hitting times at the stopped $\sigma$-algebra

Let $\mathcal{F}=(\mathcal{F}_t)_{t\ge 0}$ be the complete filtration generated by the Brownian motion $B $, and let $a<0<b$. Define the stopping times $\tau_a=\inf\{t\ge 0\mid B_t=a\}$ and $\...
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Example of random walk in a random environment (RWRE) saying things on the environment

I was wondering if anyone is aware of works/articles/examples where random walks in a random environment (RWRE) are actually used for obtaining information on the random environment. To clarify a bit, ...
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Reference request: showing that solution of an Ito SDE stays bounded with positive probability

Assume that we have a (well-posed) Ito SDE of the form $$\mathrm{d} X_t = b(X_t)\,\mathrm{d} t + \sigma(X_t)\,\mathrm{d}W_t \label{1}\tag{1},$$ where $b \colon \mathbb{R}^d \to \mathbb{R}^d$, $\sigma \...
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Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$

I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion. I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
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Freidlin Wentzell for stochastic differential inclusions

Consider the SDI $$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$ Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?
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Fractional power of a matrix with complex spectrum

I will use the notation I found here https://arxiv.org/abs/1812.01206. Please forgive me if this is a poorly stated question. I'm not sure of the things I wrote in parenthesis. The paper makes a claim ...
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How to show the joint weak convergence?

Given a $T>0$, let $\mathcal{C}[0,T]$ be the space of continuous functions on $[0,T]$. Let $Y_n(t)$ be stochastic processes in $\mathcal{C}[0,T]$. We define the weak convergence in the sense of ...
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Controlling the adjoint variables in a stochastically perturbed control problem

Suppose we have a deterministic control problem $$dX_t = b(X_t, u_t) \, dt$$ on a finite timeframe with no terminal cost; i.e. the objective functional to be maximised is $$\mathbb E \left [\int_{0}^T ...
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Expected number of steps until a queue of $n$ people has passed all $n$ ordered tests consecutively

We are given a queue of $n$ people $\{p_1, \ldots, p_n\}$. They each have to pass $n$ exams $\{t_1, \ldots, t_n\}$. For simplicity we can "draw" the setting in the following way: $$[t_n,t_{n-...
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A high probability bound for a Rademacher process

Let $\{x_i(t)\}_{i=1}^n$ be i.i.d. Gaussian processes for $t \in [0, T]$ with \begin{align*} \mathbb{E}[x_i(t)] & = 0, \quad \forall i \in [1 : n], \ t \in [0, T] \\ \mathbb{E}[x_i(s) x_i(t)] &...
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Joint tail for Brownian motion $P[B_{t_1}>g_1,...,B_{t_n}>g_n]$

Maybe not surprisingly there seems to be a lack of in-depth study of sharp estimates for the joint tail of Brownian motion over different times $$P[B_{t_1}>g_1,...,B_{t_n}>g_n]$$ for strictly ...
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Bounding expected maximum via adjacent differences

For $X_i$, $i\in[n]$ be a sequence of integrable random variables. Is there a universal constant $c>0$ such that $$\mathbb{E}\max_{i\in[n]}X_i \le c\left( \max_{i\in[n]}\mathbb{E}|X_i| + \mathbb{E}\...
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Rate of convergence to uniform distribution

Let $p=(p(1),\ldots,p(N))$ be a discrete distribution on $[N]:=\{1,2,\ldots,N\}$ with full support (i.e all the $p(i)$'s are strictly positive and sum to $1$). Let $i_1,i_2,\ldots,i_T$ be an iid ...
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Laplace transform of a stochastic process

Let $R := (R_1, R_2)$ be a two-dimensional diffusion process defined by the following SDE: $$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t} \, \mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \, \mathrm{d}W_t$...
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Embedded branching random walk converge to some random generalized function?

We know that two dimensional discrete GFF(2d-DGFF) on a box $V_N=N[0,1]^2\cap\mathbb{Z}$ with Dirichlet boundary condition will converge in distribution to the 2d continuum GFF with Dirichlet boundary ...
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References on estimates for suprema of uncentered Gaussian processes?

Let $X_t, t \in T$ denote a centered Gaussian process. Let $d(t, s) = \sqrt{\mathbb{E} (X_t - X_s)^2}$. Consider a mean function $t \mapsto \mu_t$. Define the expected supremum $$ S(T, \mu) = \mathbb{...
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Simplify nested sums - on the road of a stochastic problem (mixed replacement / kinda non-replacement)

In short, the expression I want to simplify is the following: $$ P(l) = \frac{P_0}{n^l} * \left(\sum_{p_l=1}^k p_l \sum_{p_{l-1}=p_l}^k p_{l-1} ... \sum_{p_1=p_2}^k p_1 \right) $$ or eventually easier ...
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If $u$ is harmonic, $\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x) \leq \alpha |x|+\beta,$ then $u$ is affine

We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$ Therefore $u-u(...
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Resources to understand Lebesgue measure of Brownian motion's path [closed]

[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47] Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
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Application of Ito's formula to Liouville's theorem

Liouville's theorem for bounded harmonic functions could be proved using Ito's formula, martingale convergence and Blumenthal's 0-1 law. I tried checking the classical books on Brownian motion and ...
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Stability results for general linear stochastic ODE

I am interested in the following time-invariant multivariate SDE: \begin{equation} dx_i = \sum_{j} a_{ij} x_j\,dt + \sum_{j,k} b_{ijk} x_k \, dW_j \end{equation} Despite its simplicity the general ...
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Bounding expectation of switching stochastic process

I am analyzing the behavior of an 1D stochastic dynamic system, where the state can vary randomly within a small magnitude. However, when the state deviates too much from zero, its expected magnitude ...
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Reference book for stochastic processes

I am looking for a good reference book for properties of stochastic processes for applied research. What I would like the reference to have is a collection of results on a large list of stochastic ...
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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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Ergodicity of linear dynamical systems and convergence of covariance matrices

Let $z(n+1)=Bz(n)+\xi(n+1)$ be an $N$-dimensional linear dynamical system with $\left(\xi(n)\right)_{n\in\mathbb{N}}$ being i.i.d. with $\xi(n)\sim\mathcal{N}(0,\Sigma_{\xi})$. Assumptions: a) The ...
Augusto Santos's user avatar
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Each diffusion SDE is associated to a *unique* family of transition kernels

I consider an SDE of the form $dX_t=b(X_t) \, dt + \sigma(X_t) \, dW_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$. How can I prove that there exists a unique family of transition ...
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Are the jumps of a càdlàg function "summable"?

This question is motivated by the question https://math.stackexchange.com/questions/4644235/ on Math Stack Exchange. First, I need to define a notion of transfinite summability that I have not seen ...
Julian Newman's user avatar
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Is a semimartingale that is continuous a continuous semimartingale?

Let $X$ be a centered semimartingale that has continuous sample paths almost surely. Is it then true that $X$ is a continuous semimartingale? Meaning that $X$ has a decomposition $X=M+V$ where $M$ is ...
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Locality and restriction properties for self-avoiding and loop-erasing random walks

This question has been cross-posted from math.stackexchange.com : https://math.stackexchange.com/questions/4742746/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa I ...
Testcase's user avatar
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Is deterministic evolution preserved under weak converge of stochastic processes?

Suppose you have a sequence of continuous stochastic processes $X_N$ with $X_N(0)=0$, and that $X_N$ converge weakly on the space of continuous functions, to a stochastic process $X$. Suppose $X_N$ ...
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How much is known about the action functional for small noise diffusions with general volatility coefficients?

Let $W$ be a d-dimensional Brownian motion, and for every $\varepsilon > 0$, let $X^\varepsilon$ be the solution to the SDE $$dX^\varepsilon_t = b(X^\varepsilon_t) \, dt + \varepsilon \sigma (X^\...
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A question related to the jumps of a Levy process

The Lévy–Khintchine formula says that any Lévy process, $X=(X(t), t \geq 0)$, has a specific form for its characteristic function. More precisely, for all $t \geq 0$, $u \in \mathbb R^d$: $$ \mathbb{E}...
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Possible covariance matrices of predictions of a stationary process

Let $X_t$ be a discrete time zero-mean real-valued stationary Gaussian process adapted to a $\sigma$-field ${F}_t$. Let us define $Z_{t,j} \equiv \mathbb{E}[X_{t+j}|{F}_t]$ I am interested in ...
Yashodhan Kanoria's user avatar
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Discrete approximation of continuous determinantal point processes

(throughout, "DPP" denotes "Determinantal Point Process") TL;DR: Discrete DPPs are straightforward to compute with, continuous DPPs less so. Can we approximate continuous DPPs well ...
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On a generator of a continuous-time Markov chain

Let $S$ be a countable set with discrete topology and let $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in S})$ be a continuous-time Markov chain on $S$. We assume that each $x \in S$ is a exponential holding ...
sharpe's user avatar
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1 answer
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Characteristic exponent after Girsanov transformation

Let $B$ be a standard Brownian motion. Its characteristic exponent (or Fourier transform) is easily calculated to be $$ \mathbb E [e^{ixB_t}] = e^{-\frac 12 x^2 t}. $$ Now I want to apply a Girsanov ...
Benjamin's user avatar
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1 answer
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Linear response for SDE

Consider a family of stochastic processes $dX^h_t=(g(X^h_t)+h(s))\,dt+dW_t$ and a functional $I_f:h(s) \rightarrow E[f(X_t^h)] $. I would like to compute the kernel of the derivative of this ...
Vash's user avatar
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Lower bounding the infimum of a random process

Let $X_{t}=\sum_{i=1}^n(1+s\cdot w_i)t_i\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise. How ...
tony's user avatar
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2 answers
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the infimum of a random process

Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $...
tony's user avatar
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Lipschitz maximal inequality for random process

I am confused on the Lemma 5.7 (Lipschitz maximal inequality) here. Let me first restate the definition and the lemma: Def $\{X_t\}_{t\in T}$ is called Lipschitz for metric $d$ on $T$ if there exists ...
tony's user avatar
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3 votes
2 answers
816 views

Can independent Brownian motions hit zero at the same time?

Consider for $i=1,\ldots, N\ge2$ $$X^i_t=x_i+W^i_t,\quad \forall t\ge 0,$$ where $x_1,\ldots, x_N\in (0,\infty)$ and $W^1,\ldots, W^N$ are independent Brownian motions. Denote by $\tau_i$ the first ...
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Ergodicity of the solution to some SDE

Consider the SDE (stochastic differential equation) as follows: $$dX_t=X_t\big(b(X_t)dt+a(X_t)dW_t\big)$$ where $b,a:\mathbb R\to\mathbb R$ are Lipschitz and bounded and $W$ is a real-valued Brownian ...
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