Questions tagged [stochastic-processes]

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

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7 votes
1 answer
701 views

Does the strong Markov property imply the "really strong Markov" property?

Let $\mathbf{\Omega}=(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space satisfying the Usual Conditions. Let $P \colon [0,\infty) \times \mathbb{R} \...
-1 votes
1 answer
210 views

Definition: Grigelionis Process? [closed]

Background I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and ...
3 votes
1 answer
155 views

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 ...
33 votes
4 answers
9k views

A Markov process which is not a strong markov process?

Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in ...
1 vote
1 answer
154 views

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 $X_N$ ...
1 vote
0 answers
46 views

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 ...
2 votes
0 answers
90 views

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^\...
5 votes
0 answers
729 views

Link between Fokker-Planck equation and Feynman-Kac formula

According to the Feynman-Kac formula, we know the solution of the partial differential equation: $${\frac {\partial u}{\partial t}}(x,t)+\mu (x,t){\frac {\partial u}{\partial x}}(x,t)+{\tfrac {1}{2}}\...
0 votes
0 answers
50 views

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 ...
2 votes
0 answers
84 views

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}...
1 vote
1 answer
96 views

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 ...
2 votes
0 answers
37 views

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 ...
1 vote
0 answers
37 views

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 ...
1 vote
1 answer
80 views

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 ...
1 vote
1 answer
61 views

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 ...
1 vote
2 answers
99 views

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 $...
0 votes
1 answer
114 views

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 ...
2 votes
0 answers
135 views

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 ...
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 ...
3 votes
0 answers
65 views

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^-&...
2 votes
0 answers
97 views

Bounding from below the distance between SDE started from different initial conditions

Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE $$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$ with $\mu, \sigma: \mathbb R \to \mathbb R$ Lipschitz ...
1 vote
0 answers
104 views

Lower bound of $\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]$ (without observing history)

Let $X$ be the solution to some stochastic differential equation $$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$ Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ ...
3 votes
1 answer
145 views

Recurrence of Drifted Brownian Motion Conditioned to not hit Moving Barrier

Suppose we have a Brownian motion $X$ with $X_0>0$ and drift $\mu$ conditioned to be less than a barrier $R$ which has behaviour $R_0 = r$, $dR_s = \nu \, ds$, where $\mu > \nu > 0$. Can we ...
3 votes
0 answers
133 views

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 ...
1 vote
0 answers
57 views

Convergence of stochastic linear recurrences

Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$). Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously ...
1 vote
0 answers
172 views

Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process

(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...
3 votes
1 answer
265 views

Version of Kolmogorov tightness criterion without moments

Kolmogorov tightness criterion says that if $X_N$ is a sequence of continuous process with $X_N(0)=0$ and $E[[X_N(t)-X_N(s)|^p]\leq C_p |t-s|^{1+\beta}$ then for all $\gamma\in (0,\beta/p)$ we have ...
2 votes
1 answer
273 views

Stationary Distribution of Langevin Dynamics driven by Lévy Process

Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\...
2 votes
0 answers
118 views

Can a diffusion process admit an invariant measure with a non-differentiable density?

The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
4 votes
1 answer
181 views

Hölder continuity of process from Donsker like theorem with Cauchy random variables

Let $X_k$ be i.i.d. Cauchy random variables with parameters $0,1$. For each $N$ define the process $Y_N$ by $$Y_N(t)=\frac{1}N\sum_{k=1}^{\lfloor tN\rfloor}X_k+\text{piecewise linear interpolation}.$$ ...
2 votes
1 answer
159 views

A question about convergence of stochastic processes converging to a random walk

Consider the following random walk $(y_t)_{t \in \mathbb Z_+}$: $$y_t = y_{t-1} + u_t,\quad (u_t)_{t \in \mathbb Z_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z_+)$$ where $y_0, u_1, u_2,...$ ...
2 votes
0 answers
43 views

Asymptotic behaviour of the solution to some delayed stochastic differential equation

Consider the delayed stochastic differential equation as below: $$dX_t^\theta=X_{(t-\theta)^+}^\theta(1-X_{(t-\theta)^+}^\theta)(dt+dW_t),\quad \forall t>0$$ $$dY_t^\theta=Y_{(t-\theta)^+}^\theta(1-...
1 vote
0 answers
34 views

Escaping probability of a Brownian particle in random enviroment

Let $\Omega\subset \mathbb R^d$ be a bounded open (and connected) set. Consider $E\subset \Omega$ and $x\in \Omega\setminus E$. Denote by $W^x$ the standard Brownian motion starting at $x$, i.e. $W^...
0 votes
1 answer
173 views

Characteristic function of quadratic variation of compound Poisson process

If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the joint characteristic function of this process and its quadratic variation process? If not ...
1 vote
1 answer
173 views

Garsia-Rodemich-Rumsey without Markov

Let $X$ be a $\mathbb R^d$ valued continuous stochastic process. I am interested in bounding $$P(\|X\|_\gamma>R).$$ The standard technique to do so, is to apply Markov inequality and then Garsia-...
2 votes
1 answer
89 views

SPDEs driven by fractional brownian noise

I am looking for some references for the following kind of SPDEs $$dX_t= AX_t\,\mathrm{d}t+BX_t\,\mathrm{d}W^H_t,$$ given $X(0)=X_0$, where $A$ and $B$ are operators and $W^H_t$ is the fractional ...
3 votes
1 answer
260 views

Request for references of random matrices

I need some good books aimed as a detailed and gentle introduction to random matrices, containing good discussion and derivation of Marchenko–Pastur distribution. Also, I request some other references ...
2 votes
0 answers
114 views

Martingale regularization

Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$ I was wondering if there ...
7 votes
2 answers
400 views

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}...
-1 votes
1 answer
131 views

joint density of two relevant random variables

It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
3 votes
0 answers
164 views

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, ...
1 vote
1 answer
182 views

Martingale representation of time-changed Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion. Let $\phi: [0,1)\to [0,\infty)$ be defined by $ \phi(t):=t/(1-t)$. Then $(M_t)_{0\le t<1}$ is a continuous Markov martingale with $M_t:=B_{\phi(...
3 votes
1 answer
226 views

"Ergodic theorem" for Markov kernels

Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify ...
-1 votes
1 answer
87 views

(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes

Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$, \begin{equation} \int_0^te^{-\lambda ...
0 votes
0 answers
63 views

Step in the derivation of the total idle time distribution of an M/G/1 queue

I'm trying to work my way through the proof of Thm. 1.11 in Kyprianou's Introductory Lectures on Fluctuations of Levy Processes with Applications but really struggle to understand the following step. ...
1 vote
0 answers
203 views

Two increasingly correlated Brownian motions and Williams decomposition

The Williams decomposition is Let $(B_t-\nu t)_{t\geq 0}$ be a Brownian motion with negative drift $\nu>0$ and let $M_\infty^{-\nu}:=\sup_{t\in [0,\infty]}(B_t-\nu t)$. Then conditionally on $M_\...
4 votes
1 answer
806 views

Correlated Brownian motions across different times and representation with independent processes

This is a more wide-net question of Two increasingly correlated Brownian motions and Williams decomposition. In our problem we have two correlated Brownian motions $B^1,B^2$ (starting at time $t=0$ ...
0 votes
1 answer
67 views

Maximise the probability that a drifted Brownian motion doesn't hit zero prior to $T$

Let $W=(W_t)_{t\ge 0}$ be a standard Brownian motion starting from zero and $Z>0$ be an independent random variable. Fix $T>0$ and $C>0$. Denote by $\mathcal A$ the set of progressively ...
2 votes
1 answer
97 views

Randomly chosen walk of fixed length

Let $G=(V, E)$ be the graph on vertices $V = \{0, \cdots, k\}^n$, where vertices $(v_1, \cdots, v_n)$ and $(w_1, \cdots, w_n)$ share an edge iff $\lvert v_i - w_i\rvert \leq 1$ for all $i$. A walk of ...
1 vote
0 answers
37 views

Characteristic function of a Dirichlet Process

Suppose $P \sim \text{DP}(\alpha,G) $ where $G \sim N(0,1)$ is the base measure and $\alpha > 0$ is the concentration parameter. The stick breaking representation says that $P$ can be expressed as \...

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