Questions tagged [stochastic-processes]

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

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Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$

Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...
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On almost sure convergence of conditional martingales

Let $X$ be a stochastic process with natural filtration $\mathcal F_t$, and $\mathcal G_t$ another filtration. Suppose that $X$ is a conditional martingale relative to $\mathcal G_t$, in the sense ...
Nate River's user avatar
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Autocovariance of harmonic oscillator in fluid (Langevin Equation)

I am looking to work out an analytical solution (if it is known) for the autocovariance $Cov[X_s,X_t]$ of a particle which behaves according to the Langevin equation for a Harmonic Oscillator in a ...
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Are the paths of the Brownian motion contained in a suitable RKHS?

Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$. But is ...
Mueller's user avatar
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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 ...
Nate River's user avatar
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Can we construct close discrete martingales if their terminal marginal laws are close?

As no answer or comment to Can we construct close martingales if their terminal marginal laws are close? we consider a simplified version (discrete-time) as below: Let $M=(M_k)_{0\le k\le n}$ be a ...
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Piecewise Ornstein-Uhlenbeck process time integral

Let $X_t$ be a piecewise Ornstein-Uhlenbeck process with infinitesimal variance $\sigma^2$ and (piecewise) infinitesimal mean $\theta_1$ for $x<c$ where $c$ is a constant and $\theta_2$ for $x\geq ...
17miles's user avatar
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Martingale property and martingale property in law

Let $(\Omega,\mathcal F, (\mathcal F_t)_{t \in T}, P)$, $\, T \subseteq \mathbb R$, be a filtered probability space. A stochastic process $X=(X_t)_{t\geq 0}$ adapted to $\mathcal F_t$ is an $\mathcal ...
Mr_3_7's user avatar
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Weak uniqueness of an SDE with locally Lipschitz drift and additive noise?

Consider the $d$-dimensional SDE, $d > 1$, $$dX_t = b(X_t) \, dt + \sqrt 2 \, dW_t$$ where $b$ is locally Lipschitz such that $|b(x)| \le C |x|^2$ for $x \in \mathbb R^d$. $W$ is a standard $d$-...
Akira's user avatar
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Is a martingale conditioned to be large a submartingale?

Let $X$ be a continuous time martingale such that $X_\infty := \lim_{t \to \infty} X_t$ exists almost surely. Let $x \in \mathbb R$ be such that $\mathbb P(X_\infty \geq x) > 0$, and define the ...
Nate River's user avatar
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On the convergence of a martingale

Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by : $$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$ and for $t\geq 0$, we ...
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Can we construct close martingales if their terminal marginal laws are close?

Let $M=(M_t)_{0\le t\le 1}$ be a real-valued continuous martingale. Let $\mu := {\rm Law}(M_1)$ and $\varepsilon \in (0,1)$. For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct ...
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On martingale convergence

Let $(X_t)_{t\ge0}$ be a martingale with continuous paths. It was previously shown here and here that then it is impossible that $X_t\to\infty$ almost surely as $t\to\infty$. Is it possible that there ...
Iosif Pinelis's user avatar
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Another curious martingale

This is a natural follow up question to A curious martingale. Does there exist an almost surely continuous martingale that converges in probability to $+\infty$? Note: We say a process $X_t$ converges ...
Nate River's user avatar
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What phenomena are better modelled by SDE instead of ODE?

Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...
Nate River's user avatar
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7 votes
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A curious martingale

Does there exist an almost surely continuous martingale $X$ with $X_t \to +\infty$ almost surely? Remark: Note that such a martingale exists in discrete time, or equivalently in continuous time if the ...
Nate River's user avatar
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1 vote
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A representation formula for the expected value of a stochastic process at a random time

Let $X$ be a continuous stochastic process, and $\tau$ an almost surely positive random variable, not necessarily a stopping time with respect to the natural filtration $\mathcal F_t$ of $X$. We write ...
Nate River's user avatar
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4 votes
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Paper request : “A random integral and Orlicz spaces” from K. Urbanick

I tried all my methods to find the paper : “K. Urbanik and WA Woyczynski, A random integral and Orlicz spaces, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, ...
Stochastic Student's user avatar
2 votes
1 answer
103 views

Thinning of (mixed) binomial point process

Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...
mariob6's user avatar
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Stochastic representation of Laplace equation with Neumann boundary condition

Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$. What if ...
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2 votes
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Macroscopic sets - a notion of largeness for Lebesgue null sets

Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
Nate River's user avatar
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3 votes
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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 "...
Andrea Marino's user avatar
4 votes
1 answer
100 views

Reflecting Brownian motion in disk

What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it? The transition density ...
Focus's user avatar
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Marcus-SDE to Itô-SDE

In the field of stochastic calculus, everyone knows the Itô and Stratonovich integrals, as well as the conversion from Stratonovich to Itô SDEs. The Stratonovich integration has the particularity of ...
Sofiane's user avatar
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How far does a random walker travel before returning to the origin?

Consider a (continuous time) simple symmetric random walk on $\mathbb Z$, starting from the origin. Let us denote this random walk by $\{X_t: t\geq 0\} $. It is well known that this random walk is ...
Tiago's user avatar
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Continuous-time Wold decomposition

I'm looking for a reference for the Wold–Zasukhin decomposition in continuous time for stationary random processes on the real line. I am aware of the classic result in the book from Rozanov, which ...
arknas's user avatar
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Calculating the expected hitting time of a specific birth and death chain

I am working with a specific birth and death chain, defined as follows. Consider a set of states $X = \{0,1,2,...,n\}$, where $x^* \in (0,n)$ is a recurrent state. Transition probabilities are defined ...
Roberto Rozzi's user avatar
2 votes
0 answers
134 views

Time reversal of infinite-dimensional SDE

Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
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Integration with respect to $B_H(t) B_H(s) - \mathbb{E} \{ B_H ( t ) \, B_H ( s) \}$

The time-derivative $\frac{dB_H}{dt}$ of the fractional Brownian motion may be interpreted as a random Schwartz distribution acting on a test function by $$ \left\langle \frac{dB_H}{dt}, f \right\...
tsnao's user avatar
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8 votes
1 answer
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Is there an infinite dimensional Stein's lemma?

Classical Stein's lemma says that if $\mathbf{X}$ is a centered Gaussian random vector and $g$ is a function which is nice enough, we have $$ \mathbb{E} \, X_i \, g ( \mathbf{X} ) = \sum_k \...
tsnao's user avatar
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A question on the convex hull of independent random walks

Consider $m$ independent random walks $X^1_n, \dots, X^m_n$ driven by a probability measure $\mu$ in $ \mathbb{Z}^d$. Assume that the $\mu$ has no drift, that is, the expected value of a $\mu$-...
Keivan Karai's user avatar
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2 votes
2 answers
315 views

SDE driven by fractional Brownian motion

Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below: $$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$ I am looking for references that ...
GJC20's user avatar
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2 votes
1 answer
162 views

Law of iterated logarithm for quadratic variation of Brownian motion

Let $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space supporting a standard Brownian motion $B$. Let $\Pi=\{\pi_n : n \ge 0\}$ denote the sequence of dyadic uniform partitions of the ...
user6384's user avatar
4 votes
0 answers
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Small angles between independent centred random walks in $ \mathbb{Z}^d$

Let $W_n$ and $W'_n$ denote two independent random walks in $ \mathbb{Z}^d$ defined using a finitely supported centred (mean zero) probability measure on $\mathbb{Z}^d$. For $N \ge 1$, let $\theta_n$ ...
Keivan Karai's user avatar
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0 votes
1 answer
124 views

Non-negativity of stochastic integral with indicator, Meyer-Tanaka Local Time

Consider the following stochastic integral: $$ X_t := \int_0^t \mathbb{I}_{ \{ W_s \geq 0 \}}\, dW_s. $$ Is $X_t$ almost-surely non-negative? Using this answer, it seems that $$ X_t = \max( W_t, 0) - \...
oswinso's user avatar
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1 answer
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Sign of error in the central limit theorem

Let $X_n$ and $Y_n$ be independent copies of two random variables $X$ and $Y$ with domain $\{-1,0,1\}$ for $n\in \mathbb{N}$. For a given $k\in \mathbb{N}$, I would like to find conditions on $X$ and $...
Flo Dorner's user avatar
2 votes
1 answer
116 views

A concentration inequality related to suprema of sub-Gaussian processes

Let $x_1,\dots,x_n$ be deterministic points in some space $X$ and consider a class of real-valued functions $\mathcal G$ on $X$. We further assume that for any $g \in \mathcal G$, $$ \Bigl(\frac1n \...
passerby51's user avatar
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2 votes
0 answers
168 views

Regularity of linear Bellman equation

Let $f(x,t):B_1 \times [0,1] \to \mathbb{R}$ be Lipschitz function on both $x$ and $t$, $\varphi$ be Lipschitz function on both $x$ and $t$ on the parabolic boundary of $B_1 \times [0,1]$. Let $A$ be ...
mnmn1993's user avatar
1 vote
1 answer
129 views

Generating function of the stopped simple random walk

Let $\varepsilon_i$ be independent random variables such that $\mathbb{P}(\varepsilon_i = \pm 1)= 1/2$ and denote $W_n = \sum_{i=1}^{n}\varepsilon_i$. That is, $W_n$ is the simple random walk on $\...
Ddzin's user avatar
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2 votes
1 answer
139 views

Estimates on perturbation of drift of SDEs

Let $\mu_1,\mu_2:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $\sigma:\mathbb{R}^n\rightarrow \mathbb{R}^{n\times n}$ be Lipschitz functions, of at-most linear growth; i.e. $\|\sigma(x)\|\lesssim \|x\|,\|...
Math_Newbie's user avatar
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0 answers
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A few questions on Feller processes

Update. Most of my questions have been answered in the comments. I am adding these answers to the post. There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
tsnao's user avatar
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4 votes
1 answer
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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 ...
DeepC's user avatar
  • 53
2 votes
1 answer
106 views

Can convergence in distribution necessarily be realised by almost-sure convergence?

Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each ...
Julian Newman's user avatar
2 votes
1 answer
216 views

Explicit solution to linear SDE with correlated Brownian motions

Let $W$ and $B$ be correlated one dimensional Brownian motions with constant correlation coefficient $r \in (-1, 1)$, that is, we have $d\langle W, B \rangle_t = r \, dt.$ We assume we have $B_0 = v$ ...
Nate River's user avatar
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1 vote
0 answers
116 views

Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1

Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
arrhhh's user avatar
  • 21
1 vote
1 answer
220 views

Convergence of concave/convex function

Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
NancyBoy's user avatar
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3 votes
0 answers
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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 ...
Eubos's user avatar
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1 vote
1 answer
176 views

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 ...
user479223's user avatar
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2 votes
2 answers
719 views

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 ...
Emmet's user avatar
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4 votes
1 answer
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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 ...
Akira's user avatar
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