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 ...
2
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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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1
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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 ...
3
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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 ...
1
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0
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123
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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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100
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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 ...
0
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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 ...
4
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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 $d$-...
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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 ...
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1
answer
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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 ...
4
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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 ...
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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 ...
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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 ...
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1
answer
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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 ...
4
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2
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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, ...
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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, \...
3
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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 ...
2
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1
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179
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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 ...
3
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121
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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 "...
4
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1
answer
100
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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 ...
1
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0
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137
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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 ...
0
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1
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207
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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 ...
2
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0
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54
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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 ...
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108
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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 ...
2
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134
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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 $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
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0
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Integration with respect to $B_H(t) B_H(s) - \mathbb{E} \{ B_H ( t ) \, B_H ( s) \}$
The time-derivative $\frac{dB_H}{dt}$ of the fractional Brownian motion may be interpreted as a random Schwartz distribution acting on a test function by
$$
\left\langle \frac{dB_H}{dt}, f \right\...
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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
$$
\mathbb{E} \, X_i \, g ( \mathbf{X} )
= \sum_k \...
2
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0
answers
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A question on the convex hull of independent random walks
Consider $m$ independent random walks $X^1_n, \dots, X^m_n$ driven by a probability measure $\mu$ in $ \mathbb{Z}^d$. Assume that the $\mu$ has no drift, that is, the expected value of a $\mu$-...
2
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2
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SDE driven by fractional Brownian motion
Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below:
$$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$
I am looking for references that ...
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1
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Law of iterated logarithm for quadratic variation of Brownian motion
Let $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space supporting a standard Brownian motion $B$. Let $\Pi=\{\pi_n : n \ge 0\}$ denote the sequence of dyadic uniform partitions of the ...
4
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Small angles between independent centred random walks in $ \mathbb{Z}^d$
Let $W_n$ and $W'_n$ denote two independent random walks in $ \mathbb{Z}^d$ defined using a finitely supported centred (mean zero) probability measure on $\mathbb{Z}^d$.
For $N \ge 1$, let
$\theta_n$ ...
0
votes
1
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124
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Non-negativity of stochastic integral with indicator, Meyer-Tanaka Local Time
Consider the following stochastic integral:
$$
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) - \...
4
votes
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 $...
2
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1
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116
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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 \...
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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 ...
1
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1
answer
129
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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 $\...
2
votes
1
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139
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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\|,\|...
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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 \...
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1
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250
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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 ...
2
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1
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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 ...
2
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1
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216
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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$ ...
1
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0
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116
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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 ...
1
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1
answer
220
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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 $x_n\in\mathbb{R}$ ...
3
votes
0
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51
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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 ...
1
vote
1
answer
176
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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 ...
2
votes
2
answers
719
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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 ...
4
votes
1
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350
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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 ...