Questions tagged [stochastic-processes]

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

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Alternating colors on a line: infinitely often or converge?

Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / …). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...
Ngoc Mai Tran's user avatar
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Why, and how badly, does the proof of "no percolation at the critical point in half-spaces" fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...
Louigi Addario-Berry's user avatar
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Among regular graphs, do cliques have the highest infection rate?

Consider a graph $G$ with a particular node $i$ labeled as “infected”. Other nodes start uninfected, and will become infected over time according to the following process: To each edge of the graph, ...
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Monotonicity of log determinant of Gaussian kernel matrix

Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation} be a squared-exponential (Gaussian) kernel, with $\sigma,\vartheta>0$. Consider, for a set of $N$ ...
Heinrich A's user avatar
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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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Random walk on $\mathbf{Z}_d$ with Jacobi $\theta$ transition probabilities

In the context of a finite-dimensional quantum mechanical problem, I was led to study the random walk on $\mathbf{Z}_d$ (i.e the integers modulo $d$), $d$ odd with transition probabilities given by: $...
IchKenneDeinenNamen's user avatar
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Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let $d\in\left\{2,3\right\}$ $\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
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Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance $$ \mathbb{E} [\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)} \widehat{g}(\xi)}{|\xi|^{d-2[\...
Abdelmalek Abdesselam's user avatar
7 votes
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Do isonormal Gaussian processes have measurable sample paths?

Let $H$ be a real separable Hilbert space. Let $W=\{W(h):h\in H\}$ be a real-valued stochastic process defined on a complete probability space $(\Omega,\mathcal{F},P)$. Assume that $W$ is a centered ...
Jason Swanson's user avatar
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When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb R^...
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Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence, a noise is black if it is not isomorphic to standard Gaussian white noise. Tsirelson showed the existence of black noise through the scaling limit ...
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Does the law of a Feller Process on a non-locally-compact Polish space depend continuously on the initial condition (in Skorohod path-space)?

I am sure this is written down somewhere but cannot find it. Consider a Polish space $E$ and a strong Markov process $(X_t)_{t\ge 0}$ with values in $E$ and cadlag paths. More precisely, we have a ...
Wolfgang Loehr's user avatar
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planar mappings that preserve elliptic measure

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction $f:...
HMPanzo's user avatar
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Is there a continuous-time version of Kingman's subadditive decomposition theorem?

Kingman's subadditive ergodic theorem (see this article) states that if $x_{m,n}$ is a real valued process indexed on the set of pairs of non-negative integers $m < n$ satisfying: $x_{l,n} \le x_{...
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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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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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Running maximum/supremum of Brownian motion: add information to make it a Markov process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it ...
Ziv's user avatar
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Delayed Pólya's urn process

The standard Pólya's urn process can be stated as follows: You have an urn with red and green balls. At any time unit you choose one ball at random, note the colour, and give the ball back. At the ...
Matjaž Krnc's user avatar
6 votes
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193 views

Existence of stick breaking representations for random measures

The Dirichlet process has a roughly size ordered representation in terms of beta random variables, called a stick-breaking representation (Sethuraman, 1994). Similar results hold for the beta process, ...
Shannon S.'s user avatar
6 votes
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180 views

Distribution of the stopping time of an autoregressive sequence

Consider $e_t$ being i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which $$...
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Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs. I'm reading Stochastic Differential Equations in ...
0xbadf00d's user avatar
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Sum of the entries of the inverse covariance matrix

Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = \left[sinc\left(\frac{T\left(r-s\right)}{n}\right)\right]^n_{r,s=...
Student's user avatar
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Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular, Have the numerical implications, if any, of the results in ...
Sergio Almada's user avatar
6 votes
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219 views

Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
r_faszanatas's user avatar
6 votes
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182 views

Local structure in the stochastic sandpile model

Here's a question that came up at the recent AIM conference on chip-firing and generalizations. The stochastic sandpile model, I think originally due to Manna, is a stochastic process that (in one ...
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When is $\mathbf{E}^{x}[f(X_t)]$ a continuous function of $x$?

Let $E$ be a locally compact Hausdorff space with countable base and $X_t$ be a stochastic process taking values in the one-point compactification of $E$ (with the Borel $\sigma$-algebra). Let $f$ be ...
ShawnD's user avatar
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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 ...
domotorp's user avatar
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Uniform bound for the occupation time of a diffusion

Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$. Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions. Suppose the ...
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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}}\...
Carl's user avatar
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129 views

Random process on a sequence of rolls of an $n$-sided die

Let $\ X:=X_{k\,n}\ $ be a random variable of a $n$-sided die where $\Pr(X=i)=\frac{1}{n}$ for each $i\in\{1,2,\ldots,n\},\ $ where $\ k\in\{1, 2, \ldots,n\}\ $ and $\ n\ $ are fixed. Let $t$ be a ...
Let101's user avatar
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How many balls should we throw into $m$ bins so that at least $k$ bins get at least $r$ balls, with probability $1-\delta$?

Let $m,k,r\in\mathbb N$ and $\delta\in(0,1)$, such that $k\le m$. Suppose that we throw balls uniformly and independently into $m$ bins. I am looking for an upper bound $N_{m,k,r,\delta}$ on the ...
Don C's user avatar
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5 votes
1 answer
715 views

Meaningful interpretation for fixed point of a probability generating function

Suppose $f$ is the probability generating function for the Galton-Watson branching process. What intuition makes the fact that $f(s) = s$ is the probability of extinction obvious? Moreover, can one ...
libby's user avatar
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Are Stochastic Process Characterized by Their conditional Moments

Suppose that $X_t$ is a real-valued stochastic process. Then is $X_t$ characterized by it's conditional moments? In the sence that, if $Y_t$ is another process, such that $$ \mathbb{E}\left[\int_s^T\...
ABIM's user avatar
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Asymptotic behavior of row sums in 2-d array of random variables

Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables: $B^m_{1,1}$ $B^...
cosmo-grant's user avatar
5 votes
0 answers
226 views

Spectral gap for the Brownian motion with drift on a compact manifold

Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
Pierre PC's user avatar
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Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail. ...
Ceeerson's user avatar
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A conjecture in rate distortion theory and stochastic filtering

Let $(X_t)_{t\in T}$ be a stationary random process with known and fixed law $P_X$ describing a dynamic source. This source is to be encoded real-time by an encoder $e$ into an encoded message $E_t$ ...
S.Surace's user avatar
  • 1,675
5 votes
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388 views

Distribution of the inner product of Gaussian processes

Suppose that $X(s)$ and $Y(s)$ are second order real Gaussian processes (independent or not), respectively $X\sim \mathcal{N}(\mu_1,\mathcal{K}_1)$ and $Y\sim \mathcal{N}(\mu_2,\mathcal{K}_2)$, and $...
icurays1's user avatar
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0 answers
504 views

How to obtain the probability distribution of a sum of dependent discrete random variables more efficiently

I hope you are well. Here is my problem. Let $\{s_0,\,s_1,\ldots,\,s_T\}$ be a sequence of discrete random variables and denote $S_t=s_0+s_1+\cdots+s_t$, with $S_0=0$ and $S_T\leq M$, where $M$ and $T$...
Student1981's user avatar
5 votes
0 answers
394 views

Can we prove nowhere differentiability of Brownian path via Karhunen–Loève coefficient?

This post is partly inspired by Fourier Coefficients and Hölder Continuity. Typical proofs of the nowhere differentiability of Brownian paths is by contradiction using binary expansion from real ...
Henry.L's user avatar
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138 views

Law of Large Numbers for the Tasep from a Bernoulli Configuration (Rost's Theorem)

Let $(\eta_{t}^{\rho})_{t\geq 0}$ be a totally asymmetric simple exclusion process (TASEP) from an initial configuration distributed according to the Bernoulli measure $\nu_{\rho}$ on $\{0,1\}^{\...
Nahuel Albarracín's user avatar
5 votes
0 answers
631 views

Explicit martingale representation for a Brownian bridge

Let $W$ denote a Wiener process, $\displaystyle M_t = \max_{0 \le s \le t} W_s$ its running maximum. The martingale representation of $M$ is known explicitly: $$M_T = \sqrt{\frac{2T} \pi} + \int_0^T ...
Tartrate's user avatar
  • 341
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Skorokhod' representation theorem: What is (are) possible filtration(s) on the common probability space?

I asked this question on math.stackexchange at https://math.stackexchange.com/questions/1941142/skorokhods-representation-theorem-what-is-the-filtration-on-the-common-probabi and haven't received ...
Kratos1808's user avatar
5 votes
0 answers
224 views

Second order calculus and rough paths

In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form $$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$ where $X$ is a semimartingale on a manifold $M$...
Matthias Ludewig's user avatar
5 votes
0 answers
469 views

Hierarchical Random Walk (also known as Hierarchical Hidden Markov Model)

Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...
Minkov's user avatar
  • 1,117
5 votes
0 answers
93 views

Most visited vertex in a random walk with place dependent drift

Consider the following Markov chain on $\mathbb{Z}$: $$ P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}} $$ Do there exist constants $c,C>0$ such that $$ c\cdot P^t(z,z) \...
Snoop Catt's user avatar
5 votes
0 answers
214 views

Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help .... Any pointers or suggestions are appreicated! ...
Chee's user avatar
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5 votes
0 answers
65 views

Full distribution of FPTs in random walks on graphs

There is a lot of published research on the mean passage passage time (FPT) for random walks on various types of graphs. How about the variance of the FPT and higher momenta? In fact, I would be ...
Sandra Wollish's user avatar
5 votes
0 answers
176 views

Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mathscr{P}$ be a probability ...
Goulifet's user avatar
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5 votes
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588 views

Skorohod theorem (weak convergence) on a discrete setting

I have a question about the application of Skorohod representation theorem. The questions arises in this paper about robust hedging in mathematical finance. It is about the very last equation on page ...
hulik's user avatar
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