6
votes
1answer
119 views

Properties of the time integral of Wiener process

Let $W_t$ be a Wiener process and consider the time integral $$ X_T:= \int_0^T W_t dt $$ It is often mentionend in literature that $X_T$ is a Gaussian with mean 0 and variance $T^3/6$. I am ...
3
votes
1answer
93 views

The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$. This property is called the ...
1
vote
2answers
121 views

On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients

Consider the SPDE $$\frac{\partial}{\partial t}u_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u_t(x) + u_t(x)(K-u_t(x)) + \sigma u_t(x) \xi(t,x),$$ where $(t,x)\in {\mathbb R}_+\times ...
5
votes
1answer
97 views

A question about extensions of Markov semigroups

I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. ...
2
votes
1answer
292 views

Dynamics of Master Equation

I'm going to do research on dynamics of master equation of $n$ states $$\dot p_i=A_{ij}p_j\qquad i=1\ldots n$$ where $p_i$ is the $i$-th component of probability vector and $A_{ij}$ is transition rate ...
0
votes
0answers
77 views

Master Equation to Fokker-Planck for a Jump-Diffusion

Does anyone know if there is a derivation of the Master Equation approximation by a Kolmogorov backward equation (Fokker-Planck) to a jump-diffusion with a compensated Poissonian integral? If not, can ...
4
votes
1answer
174 views

Coupling of non-probability/sub-probability measures

A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such ...
4
votes
0answers
87 views

How fast is discrete-time diffusion on a continuous set?

This question is inspired by Joseph O'Rourke's beautiful answer to my previous question. Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...
2
votes
0answers
86 views

A simplified MCMC / MH algorithm. Are there known convergence results?

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
1
vote
0answers
148 views

What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?

There is a theorem as follows: Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto ...
2
votes
1answer
79 views

Maximal probability of “infinitely often” over MDP

Let us consider a Markov Decision Process (MDP) with a Borel state space $X$. Often, the optimization problems over MDP involve optimization of some objectives dependent on the reward function $$ ...
1
vote
1answer
133 views

First moment of a function of a normally distributed random variable

I'm trying to find the first moment of the following function: $f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the ...
5
votes
0answers
168 views

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 ...
1
vote
1answer
152 views

Tail of solutions of a stochastic differential equation

As we know, solution to $dX_t=\mu dt+\sigma dW_t$ is normal distributed and is light tailed; solution to $dX_t=\mu X_tdt+\sigma X_t dW_t$ is log-normal distributed and is heavy tailed. Is there any ...
1
vote
0answers
99 views

time derivative of the median of a stochastic process

Suppose you have a cumulative distribution that is changing with time, namely $ P_t(x) $. Assume $ P_t $ is monotone increasing and smooth enough so that we can define $ x_t(P) = P_t^{-1} $. We want ...
9
votes
4answers
723 views

Optimal pebble-packing shape

Suppose you throw many ($n$) congruent convex bodies (in $\mathbb{R}^3$) of unit volume (or of unit area in $\mathbb{R}^2$) into a large container, and shake it until little else changes. Q. ...
32
votes
6answers
962 views

Tetris-like falling sticky disks

Suppose unit-radius disks fall vertically from $y=+\infty$, one by one, and create a random jumble of disks above the $x$-axis. When a falling disk hits another, it stops and sticks there. Otherwise, ...
2
votes
2answers
477 views

Midpoint lattice polygons

Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood. I am considering a variant, which I call midpoint lattice polygons. Start with a sequence of ...
3
votes
2answers
507 views

Finite time hitting probabilities for Brownian motion in the plane

Consider a Brownian particle in the plane with a circular trap at the origin. If we give the particle enough time it falls into the trap (since Brownian motion is space filling in 2D). However, ...
3
votes
3answers
263 views

Asking for a Fourier inverse transform, which is related to stable laws

Dear friends, Denote the function $$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$ It is well known that if $a\in ]0,2]$, ...
5
votes
1answer
919 views

Gluing Markov processes

I am looking for a reference on the gluing together of strong Markov processes to get a new one. Here is an example of what I have in mind. Let $B^1, B^2, \ldots $ be independent one-dimensional ...
8
votes
1answer
233 views

Mixing time of unitary Brownian motion

Let $B_t$ be the unitary Brownian motion, i.e. Brownian motion on the unitary group $U(N)$. What is known about the mixing time of $B_t$, that is, how fast does the measure $B_t(\delta_{\{Id\}})$ ...
2
votes
1answer
199 views

Applications of this project

Hi Guys, Just wondering if you could suggest applications of distribution of the supremum of a fractional Brownian motion process with a drift ? Also if you could possibly recommend how to approach ...
1
vote
0answers
278 views

Tanaka stochastic differential equation and Kolmogorov equation

Given Tanaka sde $$dX_t=[a{\rm sign}(X_t)+b]dW_t$$ is there associated a diffusion process and so a Kolmogorov (Fokker-Planck) equation? What is this equation? References answering the question are ...
3
votes
3answers
529 views

Is there a fair coin?

I attended a course on stochastic processes a few years ago. During the course the lecturer mentioned that there is a mathematical proof (with some assumptions, naturally) of non-existence of a fair ...
0
votes
0answers
218 views

Passage Time Distributions for Poisson processes.

Let $(X_t)_{t \geq 0}$ be a standard Poisson process with intensity $\mu$. Let $\tau_b = \inf ( t>0 : X_t= at + b)$, where $a>0$ and $b<0$, and let $\sigma = \inf (t>0 : X_t \geq at)$. ...
17
votes
3answers
737 views

Growing random trees on a lattice $\rightarrow$ Voronoi diagrams

Imagine growing trees from $k$ seeds on a square $n \times n$ region of $\mathbb{Z}^2$. At each step, a unit-length edge $e$ between two points of $\mathbb{Z}^2$ is added. The edge $e$ is chosen ...
9
votes
1answer
2k 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 ...
8
votes
1answer
301 views

Extending state space to make a process Feller

Let $X$ be a locally compact Hausdorff space, and let $Y_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$. Let $T_t$ be the ...
4
votes
2answers
480 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 ...
3
votes
4answers
827 views

Time integrals of diffusion processes

I was wondering if someone could recommend a reference that deals with time integrals of diffusion processes. Suppose $X$ is an Ito diffusion process with dynamics $dX_t = \mu(X_t)dt + ...
0
votes
2answers
909 views

Two dimensional brownian motion first passage time

Hello, I am looking for information on how to solve/compute first passage time for two dimensional Brownian motion. any papers, references, books or web links for study will be helpful. thanks ...
11
votes
1answer
488 views

“continuous” and “discontinuous” phase transitions in branching processes.

Consider a Galton-Watson branching process, with offspring distribution $\mathbf{p}=(p_0, p_1, \dots, p_n, \dots)$. Let $O$ be the root of the branching process. Write $\eta=P(\text{process survives ...