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

**7**

votes

**2**answers

138 views

### Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$.
If $L=M$ ($2$-dimensional square lattice), it is known (e.g., by Peierls' argument or Onsager's ...

**1**

vote

**0**answers

52 views

### “Bad” lower functions for a Bessel process?

Let $(X_t, t \ge 0)$ be a Bessel($\delta$) process, for some dimension $\delta > 2$, starting, say, from $1$.
Let $f: \mathbb{R}_+ \to \mathbb{R}_+$ be an upper semicontinuous function; assume ...

**2**

votes

**1**answer

114 views

### The uniform integrability of exponential of Poisson process

Let $\left\{N_t,\mathcal{F}_t\right\}_{t\ge0}$ be a Poisson process with intensity $\lambda>0$. Define
$$X_t=\exp{\left[N_t-\lambda t(e-1)\right]}$$
I can show that $\{X_t,\mathcal{F}_t\}_{t\ge0}$ ...

**1**

vote

**0**answers

109 views

### RKHS norm and posterior of Gaussian process

In Srinivas et al (2010) [appendix B], the authors claim the following "easy to see" property relating the norm of a function in a RKHS induced by a kernel $k(\cdot,\cdot)$, and its norm in the RKHS ...

**3**

votes

**1**answer

130 views

### Conditional Form of Rosenthal's Inequality

Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following:
If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and ...

**1**

vote

**1**answer

89 views

### Is it true that all stationary measurable stochastic processes are “measurably stationary”?

(Philosophically, the following question is of a similar flavour to A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary, but more "advanced".)
Let ...

**3**

votes

**2**answers

267 views

### Ito diffusion with highly oscillatory diffusion coefficient

Consider the stochastic differential equation on $\mathbb R$
$$
dx_t = f(x_t) dt + g(\omega t)\, dW_t
$$
with $W_t$ a standard Brownian motion, $f:\mathbb R \to \mathbb R$ a smooth function, and ...

**2**

votes

**0**answers

159 views

### Generalization of Ito's formula

If $f:R\to R$ is a convex function then we have Ito-Tanaka formula. Now my question is that if we are given a function $u: R\times R_+\to R$ such that $u(s,\cdot)$ is smooth for every $s\in R$ and ...

**2**

votes

**0**answers

54 views

### The distribution of maximum of fraction Brownian motion over finite time interval

Suppose that $\{B_t^H,\ t\geq 0\}$ is a fractional Brownian motion with Hurst exponent $H$, I wonder if there are explicit expressions for the joint distribution of
$(\sup_{0\leq t\leq ...

**3**

votes

**1**answer

205 views

### Unusual augmentation of a filtration

consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...

**1**

vote

**1**answer

80 views

### Reference request: seminal paper on the Blumenthal-Getoor index

Numerous papers are referring to the following one
R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech. 10 (1961), ...

**0**

votes

**0**answers

158 views

### Repeatedly changing queue behavior

I'm not sure if this question is suited to MO. I will happily delete if not.
Situation
Consider a general queueing system $\mathscr{S}$, whose customer arrival times are independent, and whose ...

**10**

votes

**1**answer

286 views

### Trapping a particle

A particle starts a brownian walk in the middle of a long tunnel in the plane, at one end of the tunnel is a region Y of given area A.
Does the shape of region Y affect average time for the particle ...

**1**

vote

**0**answers

85 views

### Conditional probabilities in epidemic model

I was contemplating an epidemic model where infection and recovery rates are determined by links. Here node $i$ is infected first and recovers at a rate $\mu_i$. For all other nodes, the recovery is ...

**1**

vote

**1**answer

174 views

### Markov chain Monte Carlo: why is non-reversible MC MC not as popular?

I am new to methods for simulating Markov chains in order to sample from the target, unknown distribution. After a couple days of reading, I found out that even though people have realized that ...

**3**

votes

**1**answer

90 views

### Regularity of finite variation kernels in the (intersection) of the semimartingale spaces $H^p$

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...

**3**

votes

**1**answer

185 views

### Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...

**2**

votes

**1**answer

122 views

### Fractional Brownian motion via Hilbert space

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:
Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...

**6**

votes

**2**answers

421 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**4**

votes

**1**answer

190 views

### Stability of convergence in distribution under randomization

Suppose you have a sequence of non-negative stochastic processes $(X^n)_{t \in \mathbb{R}}$, $n \geq 1$, with continuous paths and continuous in $t$ such that
$$\int_{-\infty}^{\infty} X^n_t \, ...

**4**

votes

**1**answer

199 views

### Can't figure out “standard application” of the Garsia-Rodemich-Rumsey Lemma

I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...

**2**

votes

**0**answers

56 views

### Almost sure transversality of smooth random maps

I still am novice as far as probability is concerned and after fruitlessly Googling for an answer for a few days I thought I might have a better chance with MO.
Let me first formulate the ...

**4**

votes

**0**answers

99 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 ...

**0**

votes

**1**answer

106 views

### Measurable functions lifted onto a space of point measures are measurable

I've been reading [1] and attempting to prove statements given without proof. In the paper the authors construct a measurable space of measures over a base space, and as an aside show an elegant way ...

**1**

vote

**2**answers

337 views

### Brownian motion of every point in the plane

Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable?
What proportion of the plane does ...

**0**

votes

**2**answers

116 views

### When does a stochastic process have its sample paths a.s. in the reproducing kernel hilbert space (RKHS) induced by its covariance function?

Let $T$ be a compact metrizable space. Consider a centered second order measurable process $(X_t\colon t\in T)$ with continuous covariance function $c(t,s):= \mathbb{E}X_t X_s$.
Are there any known ...

**0**

votes

**0**answers

67 views

### Reference for “Newtonian capacity estimates probability that A is hit by a Brownian motion”

I am looking for the following statement
"In fact, the Newtonian (logarithmic) capacity gives an estimate, up to a constant factor, the probability that A is hit by a Brownian motion started, say, ...

**44**

votes

**5**answers

3k views

### Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...

**0**

votes

**0**answers

101 views

### Probability that d-Brownian Motion ,$d\geq 3$, avoids a fixed set A

In other words, the probability that Brownian motion stays within $A^{c}$.
What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...

**2**

votes

**1**answer

116 views

### Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Denote by $p(x)$ the input probability distribution.
The mutual information between the input and the output of ...

**3**

votes

**2**answers

94 views

### Is this generating family of a measurable space of point measures a pi-system?

I'm learning some probability and measure theory and working my way through the first few paragraphs of [1]. My question is perhaps too basic for Math Overflow, but I hope it is welcome here.
Point ...

**4**

votes

**0**answers

767 views

### Progressively measurable vs adapted

I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...

**0**

votes

**0**answers

29 views

### Explicit construction of transition semigroup from generator for completely independent spin system (Feller process)

As an example of how to obtain the transition semigroup from the probability generator for a Feller process, I am looking at the easiest spin system, namely with all sites independent.
Notationwise, I ...

**1**

vote

**0**answers

56 views

### Jumps of jump diffusions

Let $W$ be a Brownian motion and $N$ a Poisson random measure defined on $\mathbb R_+ \times \mathbb R_0^n$ ($\mathbb R_0^n:=\mathbb R^n-\{0\}$) with compensator $\tilde N(dt,dz):= N(dt,dz) - dt ...

**3**

votes

**1**answer

128 views

### Wiener measure of hitting sets A,B but not C (or easier hitting A but not C)

I am trying to formulate the measure of event
$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,
where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise ...

**3**

votes

**1**answer

174 views

### An efficient method to find the MLE of the combination of two point processes

I have a point process defined in two parts as follows. Consider first the main process which we call $A$ which is homogeneous Poisson process with conditional intensity
$$\lambda(t) = \mu$$
For ...

**3**

votes

**1**answer

292 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 $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

**0**

votes

**0**answers

46 views

### Variance Gamma Distribution and Process

I have read that a variance gamma process $X_t=\theta G_t+\sigma W_{G_t}$ is such that $X_1\sim Variance Gamma(\theta,\sigma,\nu)$ but the variance gamma distribution has 4 parameters: $\mu$, ...

**0**

votes

**1**answer

47 views

### DTMC random walk model [closed]

For a discrete Markov chain random walk with p < 0.5 with state space S= {0,1,2..}
What is the stationary distribution?
I could use any help.
Thank you

**1**

vote

**1**answer

116 views

### Intuition for the definition of a probability generator of a Feller process

I am working with the definition of a probability generator of a Feller process as stated in Liggett's book, "Continuous time Markov processes":
Let $S$ be a compact state space and denote by $C(S)$ ...

**3**

votes

**1**answer

153 views

### Approximating Markov chains by Brownian motion

I would like a result along the following lines to be true, but haven't been able to locate it in the literature; pointers would be welcome.
Let $X_t$ be a finite-state, irreducible, aperiodic Markov ...

**0**

votes

**1**answer

102 views

### Combine results with different veracity [closed]

I have 3 neural networks processing 3 different vectors of values. Each NN processes a sample of it's vector and gives binary result (y/n) that is correct with given probability. All 3 NNs give answer ...

**1**

vote

**1**answer

169 views

### Branching Brownian Motion and the KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...

**0**

votes

**1**answer

220 views

### On the superior of generalized Ornstein-Uhlenbeck process

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde:
$dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $
$\partial_x X_t(0) = \partial_x X_t(1) = 0, $
$X_0 = 0, $
...

**1**

vote

**0**answers

78 views

### Variance of continuous stochastic process

In the paper "Directed Information, Causal Estimation, and Communication in Continuous Time" the author show an example of continuous Gaussian Channel:
Let $\{B_t\}$ be a standard Brownian motion and ...

**1**

vote

**1**answer

82 views

### Numerical computation of Skorokhod integral

How can I numerically compute the Skorokhod integral of a non-adapted process? If it is adapted, that is easy since the integral is just an Ito integral.
I have found that computing the Malliavin ...

**1**

vote

**0**answers

314 views

### random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...

**1**

vote

**0**answers

234 views

### Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...

**4**

votes

**3**answers

606 views

### How to explain “Feller process” to an undergraduate student?

I had to explain in informal terms what a Feller process was, to undergraduate students who understand Markov property, Poisson processes and such. It was easy to define Levy process as generalisation ...

**7**

votes

**1**answer

510 views

### Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.
Here is what I mean exactly. ...