Questions tagged [stochastic-processes]

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

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Comparison between the expected values of the inverse of the CDF of binomial-distributed random variables

Let us denote with $F(x;j,\mu)$ the cdf of a Binomial distributed random variable with $j$ trial with success probability $\mu$ considered in $x$, and let $f(x;j,\mu)$ be the pmf. Defining $0\leq \...
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What is the expected remaining life duration of a cell in the $t\to\infty$ limit?

Consider the following population model: We start with a population of a single cell at time $t=0$. Each cell divides into $k$ new cells at random times $T$ distributed according to a probability ...
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A question about one Malliavin derivative calculation

Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
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Expectation of $B_u \operatorname{argmax}_t B_t$

This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here. Yesterday I asked a question about the joint law of ...
tsnao's user avatar
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Stochastic Dynamics in Concentric Circles: Probabilistic Analysis and Mathematical Modeling [closed]

Question: A state-of-the-art laboratory investigates the stochastic behavior of a ball navigating within a highly structured environment comprising orthogonal circular ensembles. The experimental ...
Mathgrad's user avatar
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Concentration result for self-normalized empirical process

In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....
WeakLearner's user avatar
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Example of $F\in W_0^{1,2}$ a.s. so that the law of $F+B$ is equivalent to that of $B$ but DD exponential isn't integrable?

Is there an explicit example of progressively measurable $F=\int_0^\cdot f(s) ds\in W_0^{1,2}(0,1)$ a.s. so that the law of $F+B$ on $(0,1)$ is equivalent to that of a Brownian motion $B$ on $(0,1)$ ...
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Stochastic analysis on nuclear Fréchet spaces

This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise. A lot of the time in infinite-...
J_P's user avatar
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Derivative with respect to initial condition for the solution of an SDE

Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution): \begin{align} dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t \end{align} and define its ...
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Including fixed-time transitions into a continuous time Markov chain system

I have system which is mostly described by a CTMC (Continuous-time Markov chain) with a single absorbing state and a large but tractable and sparse transition matrix. However, at a fixed set of "...
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the distribution of a stopping time of a Brownian motion [closed]

Is there example of a stopping time of a standard brownian motion which has discontinuous distribution? is there any general result for such stopping time?
user524762's user avatar
10 votes
1 answer
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The drunken blind man’s walk

Consider a drunk, blind man starting in the middle of the two dimensional open unit ball. At each turn, the man chooses a direction to move a step of size $\delta > 0$ in. Unfortunately, he is very ...
Nate River's user avatar
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Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$

I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...
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Does this filtration have a name?

In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
Mushu Nrek's user avatar
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Expected value of a Stochastic process

Consider a discrete stochastic process $\{X_t\}_{t \in T}$ with the following properties. Each $t \in T$ has a value $v(t) \in \mathbb{R}_{+}$ and the value is added to the overall value conditioned ...
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Ask assistance for finding K. Sato - Lévy Processes on the Euclidean Spaces

The paper me and my professor want is called K. Sato (1995) Lévy Processes on the Euclidean Spaces, Lecture Notes, Institute of Mathematics, University of Zurich. I tried to find the paper on the ...
Zoël Li's user avatar
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Asymptotic stochastic ordering for weighted sum of i.i.d. random variables

Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$, \begin{equation} a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
Ben's user avatar
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1 answer
149 views

Construction of random tempered distributions

Let $(\xi_\phi)_{\phi \in L^2(\mathbb{R}_+ \times \mathbb{R}^d,\lambda_d)}$ be a collection of centered Gaussian processes on a probability space $(\Omega,\mathcal{F},P)$ such that $$\forall \phi \in ...
mathex's user avatar
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2 votes
1 answer
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Decay estimate of moment of an SDE

We consider an SDE $$ d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t, $$ where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...
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Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)

Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
ABIM's user avatar
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A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal

I apologize if this is too elementary a question, but I have not been able to make much progress. Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
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Does point process ordering ever imply conditional intensity ordering?

Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
jdods's user avatar
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2 votes
1 answer
231 views

If Kolmogorov continuity criterion gives the optimal Hölder regularity then does the process have all moments?

Although very useful in the Gaussian (or other infinite moment) setting, Kolmogorov continuity criterion is non optimal in the finite moment setting. For example, let $X(t)=Zt$ where $Z$ is a random ...
user479223's user avatar
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3 votes
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Norm estimate for parabolic SPDE solution

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
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How to modeling continuous batching in large-scale inference with queuing theory approach?

I want to model continuous batching in large model inference problems, but my knowledge in data theory is insufficient, and I haven't found an appropriate queuing theory model to use for modeling. So, ...
oleotiger's user avatar
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Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set

Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE \begin{equation} dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,. \end{equation} Let $f(x,t|x_0,0)$ denote its transition density function. ...
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Does domination of stochastic processes imply the domination can always be realized by the coupling temporally/incrementally?

Suppose we have two stochastic processes $X=(X_t)_{t\in[0,\infty)}$ and $Y=(Y_t)_{t\in[0,\infty)}$. Assume that we have all the necessary structure to make sense of stochastic domination in the ...
jdods's user avatar
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Explicit example of drift $F$ so that the law of $F+B$ is not absolutely continuous with respect to $B$

Let $\mu_0$ be the law of Brownian motion on the space of continuous functions. If $\mu\sim\mu_0$ agrees on null sets then there is some progressively measurable $F\in W_0^{1,2}$ a.s. so that $\mu$ is ...
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Brownian motion reflected at a trailing barrier

Let $X_t$ be a Brownian motion with positive drift starting at 0. The process with reflection at fixed barrier $b<0$ (sometimes called a "regulated Brownian motion") is: \begin{equation} \...
Dale123's user avatar
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Do we need to assume that $y$ is bounded or subgaussian?

Suppose that $X_1,\dots, X_n$ are iid $P$ on $\mathcal{X}$. The empirical measure $\mathbb{P}_n$ is defined by $$\mathbb{P}_n:=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$ For a real-valued function $f$ on $\...
Hermi's user avatar
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8 votes
2 answers
395 views

Optimally betting a beta-biased coin

This question is inspired by How to optimally bet on a biased coin? by Nate River but generalized slightly. I decided the generalization might be interesting enough to be its own question. A number $p$...
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Random walks on groups

I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a ...
Dimitri's user avatar
2 votes
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157 views

Hunting an invisible target

An invisible target on the integer line starts at $0$. On each round it either stays put, moves to the left or moves to the right by $1$ with probability $\frac{1}{3}$ each. You are then asked to ...
Nate River's user avatar
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20 votes
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How to optimally bet on a biased coin?

A number $p$ is drawn uniformly at random from $[0, 1]$. You are then given a biased coin that turns up heads with probability $p$, but the number $p$ is not known to you. You start with a total ...
Nate River's user avatar
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2 votes
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Asymptotic Independence of random walks from increments?

Suppose we have two random walks $(S_n:n\geq 1)$ and $(T_n:n\geq 1)$ building from independent identically distributed increment vectors $\{(X_k,Y_k):k\geq 1\}$, i.e. $S_n=\sum_{k=1}^n X_k, T_n=\sum_{...
MikeG's user avatar
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1 vote
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81 views

Gluing theorem for martingales

Let $M=(M_t)_{1\le t\le 2}$ be a continuous (resp. right-continuous) martingale. Denote $x:=\mathbb E[M_1]\in\mathbb R$. Can we construct on some probability space a continuous (resp. right-continuous)...
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Why shocks are independent with weighted sum of normal process

I am doing a problem and got stuck by the definition of "normal process". The problem is stated as follows: Suppose $e_t = \sum_{j}^{\infty}\theta^j Y_{t - j} $ and assume that $Y_t$ is a ...
tobinz's user avatar
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SDE driven by Lévy processes

Consider a stochastic differential equation (SDE) on some filtered probability space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$ : for all $t>0$ $$dX_t = u_tf(X_{t-})dt+ u_t g(X_{t-})dW_t + u_t\...
Fawen90's user avatar
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3 votes
1 answer
139 views

How many Uniform(L, H) RVs can be added up until their sum reaches a certain value?

I want to know how many consecutive i.i.d. RVs with: $$X_{i} \sim\text{Uniform}(L, H)$$ can be added until the sum of them is greater than or equal to a certain value ($r$). I'm calculating this for a ...
Rezvan's user avatar
  • 41
3 votes
1 answer
157 views

Simple linear asymptotics for leaving time of particle in open-boundary TASEP

EDIT: It appears the hypothesis may not be true - I am not sure. I therefore changed my question. ORIGINAL QUESTION: Consider a system $n$ linked discrete cells numbered $1 \ldots n$. Particles are ...
aellab's user avatar
  • 133
1 vote
1 answer
91 views

Interchange the deterministic and stochastic integrals

We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \...
Analyst's user avatar
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2 votes
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Embedding a Markov chain in a Markov process

Let $X_{t\ge 0}$ be a Markov process with values in a metric space $(\mathcal{X},d)$ defined on a probabiltiy space $(\Omega,\mathcal{F},\mathbb{P})$ and let $(\tau_n)_{n=1}^{\infty}$ be a sequence of ...
user521485's user avatar
2 votes
0 answers
69 views

Assumptions for uniform measure of SDE on manifolds

Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
optimal_transport_fan's user avatar
3 votes
1 answer
181 views

Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process

Consider the modified Ornstein–Uhlenbeck process $$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$ for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
Jean Daviau's user avatar
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79 views

Elliptic PDEs in BSDEs and in optimal control

This soft/reference question is related to this MO post of a similar nature. What are some examples of elliptic PDEs appearing in control and BSDEs?
ABIM's user avatar
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1 vote
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Regularity of Feynman-Kac formula for a simple diffusion

Let consider the diffusion process given by: $$dX_t = \alpha(X_t) dW_t$$ where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
NancyBoy's user avatar
  • 175
5 votes
0 answers
234 views

How to play golf in one dimension?

One-dimensional golf is a function $g$ on $\mathbb R$ such that $g(x)= 1+\min_\mu E[g(x+N(\mu,c\mu^2))]$ if $|x|>1$ and 0 if $|x|\le 1.$ Here $N$ is the normal distribution, whose mean $\mu$ you ...
domotorp's user avatar
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2 votes
0 answers
188 views

If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality

Related: On a deceptively tricky calculus problem. The way that Leonard Gross proves the log Sobolev inequality is in the following stages: He proves that for any operator $B$ that satisfies the log ...
matilda's user avatar
  • 90
3 votes
1 answer
183 views

Laplace transform of Brownian motion functional

Let $(B_r,r\geq 0)$ be a standard Brownian motion on $\mathbb{R}$ started at $0$. I am interested in the quantity $$g(s,t) = \mathbb{E}_0\left[ \exp \left(- \beta \int_s^t \left\vert \frac{B_r}{r}\...
David Geldbach's user avatar
1 vote
0 answers
55 views

One challenge encountered when dealing with the convergence of the AdaGrad-norm algorithm

Given $\{X_{n},\mathcal{F}_{n}\}$ is an adapted process satisfying the following conditions: $X_{n}>0,\ \forall\ n>0.$ There exists $ \epsilon>0,\ \sigma>0,$ such that $\mathbb{E}(X_{n}^...
金睿楠's user avatar

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