Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
901 questions with no upvoted or accepted answers
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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 ...
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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 ...
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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 $N$ ...
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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:
$...
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304
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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
$$
\mathbb{E}
[\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)}
\widehat{g}(\xi)}{|\xi|^{d-2[\...
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Do isonormal Gaussian processes have measurable sample paths?
Let $H$ be a real separable Hilbert space. Let $W=\{W(h):h\in H\}$ be a real-valued stochastic process defined on a complete probability space $(\Omega,\mathcal{F},P)$. Assume that $W$ is a centered ...
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When is an ODE a good approximation to an SDE?
Suppose $X_t$ is a weak solution to a stochastic differential equation in the form
$$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 ...
user24451
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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 ...
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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:...
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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$-...
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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 ...
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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 ...
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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, ...
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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,$$
where $X$ is a semimartingale on a manifold $M$...
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183
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Distribution of the stopping time of an autoregressive sequence
Consider $e_t$ being i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which
$$...
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774
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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 ...
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721
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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=...
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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 ...
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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 ...
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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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Integrating a Bessel Bridge
Preliminaries
An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. ...
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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 ...
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Sharpness of Doob's upcrossing inequality
Provided a martingale $X$, discrete-time $X=(X_n, n\in\mathbb N)$ or continuous-time $X=(X_t, t\ge 0)$, Doob's upcrosssing inequality states that :
If $U_N(a,b)$ denotes the number of up-crossings of $...
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112
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Discrete random walk in an expanding cage (i.e. in a growing domain)
In the book "A guide to First-Passage Processes" by Sidney Redner, a section is dedicated to the survival probability of a random walker in a growing domain.
For a fixed-length interval $[0,...
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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 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 ...
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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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897
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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}}\...
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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 ...
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Question/References on the Skorokhod M1 topology
Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
user128095
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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 ...
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Left passage probability of $SLE_8$?
Schramm's formula on left passage probabilities of $SLE_k$ is stated for $k \in [0,8)$ in theorem 2 here. However, after the statement he remarks that the formula simplifies to $1/2$ for $k = 8$. It ...
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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 ...
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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\...
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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^...
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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 $L=\Delta +X$. ...
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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.
...
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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 $E_t$ ...
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412
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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 $...
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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 $T$...
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426
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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 ...
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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\}^{\...
5
votes
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answers
653
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 ...
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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 ...
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485
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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/...
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