Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
2,349
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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....
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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-...
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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 ...
2
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35
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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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42
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the distribution of a stopping time of a Brownian motion
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?
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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 ...
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92
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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 ...
2
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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 ...
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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 ...
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Accuracy of the definition of the space-time white noise
We have seen that there exists $\overline{\xi}:\Omega \to \mathcal{E}'$ ($\mathcal{E}'$ is the space of tempered distributions) which usually replaces the space-time white noise (indexed by $L^2(\...
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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+...
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1
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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 $$\forall \phi \in ...
2
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1
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188
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Decay estimate of moment of an SDE
We consider an SDE
$$
d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t,
$$
where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...
6
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Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
3
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A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal
I apologize if this is too elementary a question, but I have not been able to make much progress.
Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
0
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1
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55
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Does point process ordering ever imply conditional intensity ordering?
Let $N$ and $N'$ be regular/non-explosive point processes on $[0,\infty)$. I will take the view that these are collections of random arrival times: $N=(t_n)_{n\in\mathbb N}$ and $N'=(t_n')_{n\in\...
2
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1
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If Kolmogorov continuity criterion gives the optimal Hölder regularity then does the process have all moments?
Although very useful in the Gaussian (or other infinite moment) setting, Kolmogorov continuity criterion is non optimal in the finite moment setting. For example, let $X(t)=Zt$ where $Z$ is a random ...
3
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Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\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, ...
3
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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. ...
-1
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0
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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 ...
3
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88
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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 ...
2
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78
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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}
\...
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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 $\...
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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 $p$...
0
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0
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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 ...
2
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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 ...
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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 ...
2
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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_{...
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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 ...
2
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57
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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 $t>0$
$$dX_t = u_tf(X_{t-})dt+ u_t g(X_{t-})dW_t + u_t\...
3
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1
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How many Uniform(L, H) RVs can be added up until their sum reaches a certain value?
I want to know how many consecutive i.i.d. RVs with:
$$X_{i} \sim\text{Uniform}(L, H)$$
can be added until the sum of them is greater than or equal to a certain value ($r$).
I'm calculating this for a ...
3
votes
1
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Simple linear asymptotics for leaving time of particle in open-boundary TASEP
EDIT: It appears the hypothesis may not be true - I am not sure. I therefore changed my question.
ORIGINAL QUESTION:
Consider a system $n$ linked discrete cells numbered $1 \ldots n$. Particles are ...
1
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1
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91
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Interchange the deterministic and stochastic integrals
We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \...
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Embedding a Markov chain in a Markov process
Let $X_{t\ge 0}$ be a Markov process with values in a metric space $(\mathcal{X},d)$ defined on a probabiltiy space $(\Omega,\mathcal{F},\mathbb{P})$ and let $(\tau_n)_{n=1}^{\infty}$ be a sequence of ...
2
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Assumptions for uniform measure of SDE on manifolds
Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
3
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1
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Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process
Consider the modified Ornstein–Uhlenbeck process
$$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$
for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...
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Elliptic PDEs in BSDEs and in optimal control
This soft/reference question is related to this MO post of a similar nature.
What are some examples of elliptic PDEs appearing in control and BSDEs?
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Regularity of Feynman-Kac formula for a simple diffusion
Let consider the diffusion process given by:
$$dX_t = \alpha(X_t) dW_t$$
where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
5
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How to play golf in one dimension?
One-dimensional golf is a function $g$ on $\mathbb R$ such that
$g(x)= 1+\min_\mu E[g(x+N(\mu,c\mu^2))]$ if $|x|>1$ and 0 if $|x|\le 1.$
Here $N$ is the normal distribution, whose mean $\mu$ you ...
2
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If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dotsb+ B_n'$ also satisfies the same inequality
Related: On a deceptively tricky calculus problem.
The way that Leonard Gross proves the log Sobolev inequality is in the following stages:
He proves that for any operator $B$ that satisfies the log ...
3
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1
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Laplace transform of Brownian motion functional
Let $(B_r,r\geq 0)$ be a standard Brownian motion on $\mathbb{R}$ started at $0$. I am interested in the quantity
$$g(s,t) = \mathbb{E}_0\left[ \exp \left(- \beta \int_s^t \left\vert \frac{B_r}{r}\...
1
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0
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55
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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}^...
3
votes
2
answers
176
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Can any right-continuous martingale be approximated by continuous ones?
It is known that any function that is right-continuous with left limits (càdlàg as a French abbreviation) can be approximated by continuous ones (under e.g. Skorokhod topology). Let $M=(M_t:0\le t\le ...
2
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0
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82
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How to estimate the difference between two Ito diffusions?
Suppose $𝑏:\mathbb R^d \to \mathbb R^d, \sigma:\mathbb R^d \to \mathbb R^{d\times d}$ are measurable functions and satisfy
\begin{equation*} 2\langle 𝑥−𝑦,𝑏(𝑥)−𝑏(𝑦)\rangle +\|\sigma(𝑥)−\sigma(�...
1
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0
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67
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On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$
Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...
3
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1
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290
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Trajectory regularity of conditional expectation with additional randomness
Consider a probability space that support a standard Brownian motion $W=(W_t)$ and a random variable $Z$ that is independent of $W$. Denote by $\mathbb F^W=(\mathcal F^W_t)_t$ the natural filtration ...