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

learn more… | top users | synonyms

2
votes
0answers
63 views
+50

Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ where the coefficients are assumed to be Lipschitz continuous. I hope to ...
0
votes
1answer
413 views

Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
8
votes
4answers
716 views

A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability. A quick google search gave a lot of references on SLE ...
1
vote
1answer
124 views

Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots, S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin. Let $\tau_{N}$ be the first time $S_{n}$ exits ...
1
vote
0answers
63 views

Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and ...
1
vote
1answer
45 views

Is there an easy way to convert a non-deterministic optimal policy to a deterministic optimal policy for a given MDP?

For a MDP (Markov Decision Process) is there an easy way to convert a non-deterministic optimal policy into a deterministic optimal policy? The trivial way will take ...
4
votes
0answers
110 views
+100

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 ...
0
votes
0answers
10 views

Residual life distribution for renewal process after time T

Suppose we have a renewal process with inter-arrival times $\boldsymbol X=\{X_1, X_2, ...\}$, where $X_i$ are i.i.d variables. Assume that the CDF and PDF for $X_i$ are $F(x)$ and $f(x)$. 1) Let ...
0
votes
0answers
18 views

Explicit u-excessive function

Let $E$ be $\mathbb{R}^d$ for $d\geq 1$. Let $A \subset E$. Let $X$ be a Feller process en $E$, and let $L$ be its infinitesimal generator. I want to prove that $A$ is absorbing. I know that it is ...
0
votes
0answers
127 views

Hadamard product (Schur product) in $L^2[0,1]$

Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
0
votes
1answer
25 views

Analyzing a multiple-queue single-server model

Consider the following multiple-queue single-server model of a packet network problem. At each discrete time $t=0,1,\ldots,n$, a packet may arrive at the server R with probability $1-\epsilon_1$. The ...
0
votes
0answers
26 views

Feller property for Ito diffusion with Lipschitz coefficients

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x\in \mathbb{R}^n,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded ...
2
votes
0answers
97 views

markov processes and ergodic theory

For an ergodic Markov Chain $$ \frac{1}{N}\sum_{i=1}^n f(X_i) \rightarrow E_\pi[f] $$ where $\pi$ is the invariant distribution. I am also dealing with a Markovian process (a state space model to ...
0
votes
1answer
230 views

Generalized Ito's lemma

I have the following quantity: $$ g(t)=(f(t))^{M_{t}}, $$ where $M_{t}$ is a jump process which is neither Markovian nor Levy, and $f(t)$ is a positive, increasing but limited, right-continuous ...
1
vote
0answers
62 views

formula for density of maximal Poisson disk sampling of radius 1?

Maximal Poisson disk sampling of radius r, applied to a finite planar region, is defined by successively choosing sample points uniformly randomly from the part of the region that is not within ...
4
votes
1answer
465 views

weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
2
votes
2answers
143 views

The necessary sufficient condition for recurrence of a Markovian random walk

Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk. I want to figure out the necessary ...
1
vote
1answer
77 views

Problem of random scheduling of queues of tasks

Consider $L$ queues in a discrete time system. At each time $n=0,1,2,\ldots$, one task would arrive at one of the queues with equal probability $\frac{1}{L}$. Immediately after that, a task scheduler ...
9
votes
2answers
9k views

Coin Pusher Game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins). Essentially, one has a distribution of ...
1
vote
1answer
102 views

A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ...
2
votes
0answers
160 views

Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process $$ dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0 $$ where $a\in (-\infty,+\infty), ...
0
votes
0answers
37 views

Quadratic characteristic and constancy

Consider a change of measure on $\mathcal{F}_{t}$ defined by the restriction of two probability measures of the form \begin{align} \frac{dQ_{t}(\theta)}{dP_{t}}=\exp^{ \theta A_{t}-\kappa(\theta) ...
0
votes
1answer
82 views

Predictable quadratic Variation <.> has same intervals of constancy as the process

From Revuz and Yor - Continuous Martingales and Brownian Motion 1999 Chapter IV Proposition 1.13 it is proven, that for a continuous local martingale $M_t$ the intervals of ...
1
vote
1answer
212 views

Supremum of a martingale

Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length: $$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window: $$R_n = ...
2
votes
0answers
848 views

Moments of function of Poisson process

(I'm new to Poisson processes, so please edit if my terminology is incorrect.) Edit: per comments, here is a (more) general version of the originally posted problem (which is now at the bottom, below ...
2
votes
0answers
99 views

Question about continuity in the “complete Skorohod Topology”?

I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process" https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf In one of the exercises, exercise 8.9 ...
4
votes
1answer
414 views

Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution. Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...
3
votes
2answers
117 views

Extreme couplings

Let $X,Y$ be Polish spaces, and $\mu$ and $\nu$ are probability measures on $X$ and $Y$ respectively. We say that $M$ is a coupling of $\mu$ and $\nu$ if it is a probability measure on $X\times Y$, ...
1
vote
0answers
75 views

Construction of a random variable

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the ...
3
votes
1answer
128 views

How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here. Say we have a stochastic process described by a ...
3
votes
0answers
56 views

“Local” functional central limit theorem for the empirical distribution function

This question is a repost from Mathematics Stack Exchange, where it did not receive any answer. Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb ...
13
votes
1answer
1k views

surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
2
votes
0answers
133 views

Infinitesimal generator and stationarity

The following question is bothering me. I think it is probably known but I cannot find any reference... Let $(X_t)$, $(Y_t)$, $(Z_t)$ denote 3 Feller processes with respective infinitesimal ...
3
votes
0answers
41 views

Feynman-Kac formula and time-ordering for vector bundles

Let $M$ be a compact Riemannian manifold and let $\mathrm{d}\mathbb{W}^{yx;T}(\gamma)$ denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from $x$ to $y$ in time $T$ ...
1
vote
0answers
52 views

A problem on Markov chains and Dirichlet forms

Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for ...
2
votes
0answers
47 views

Modify Process to a Semimartingale

The original post is from mathstackexchange According to some difficulties, i decided to ask here again. Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a ...
1
vote
0answers
101 views

Limit theorem : reproduce a proof with an adaption from discrete to continuous time

Im considering Theorem 5.2.2 in M. Sørensen "Exponential Families of stochastic processes". The setup is as follows: We have a Levy-Process $X_t$ fullfilling the CLT \begin{align} ...
3
votes
0answers
144 views

Self-adjusting random walk

Let $X_t$ be a random process such that \begin{eqnarray} X_1 &=& 0\\ X_t &=& X_{t-1} + \left\{\begin{array}{ll} A_t, & X_{t-1} \geq 0\\ B_t, & X_{t-1} < 0 ...
1
vote
1answer
51 views

A diagonalisation argument applied to density functions

There is a claim from a paper which I do not understand: Let $D$ be a domain in $\mathbb{R}^d$. Let $(p^{\eta})_{\eta >0}$ be a family of densities for random variables on $(C[0,T], ...
1
vote
1answer
91 views

Find a square, stochastic matrix (w/ non-neg entries) of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists. Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory). Preferably the ...
2
votes
0answers
40 views

A question about probabilistic graphical models

Say one is given a probabilistic graphical model and a cut of the underlying graph. Do we know any statements about when and how can one or many of the marginals (of the sources) or the conditionals ...
2
votes
1answer
139 views

Functional limit theorem under random change of time

Given a Levy-Process $U_t$ (cadlag-paths) with $E(|U_t|)<\infty$ and finite variance and $Var(X_1)=\sigma^{2}$ for which the limit theorem holds: \begin{align} ...
1
vote
1answer
96 views

Limit (Convergence) of stopping times

Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
4
votes
1answer
219 views

Cramer-Rao type bound for absolute estimation error

Let $\{X_1, X_2, \ldots, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...
1
vote
0answers
37 views

Alternative to generic chaining bounds for a particular family of stochastic processes

Generic chaining provides a general but rather abstract framework to bound suprema of stochastic processes. In many applications, however, we know more about the expression of the stochastic process. ...
1
vote
0answers
37 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
5
votes
4answers
686 views

What would be a fractional Poisson Process like

Hi all, I think that the definition of fractional Brownian Motion is widely known (for example as a Gaussian Process with particular variance covariance stucture parametrized by the so-called Hurst ...
0
votes
0answers
38 views

regularity of the conditional expectation: from Markov to Non-Markov

Let $B=(B_t)_{0\le t\le T}$ be a standard Brownian motion and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Let $\xi=\xi(B)$ be a bounded measurable functional. Now let's ...
0
votes
0answers
41 views

Brunett Derrida behaviour for the branching brownian motion with selection

In this paper Berard and Gouéré proved that for a binary branching random walk with selection of the N rightmost particles the cloud of particles moves asymptotically at a deterministic velocity ...
2
votes
2answers
299 views

Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...