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

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Ising model: probability of a long path of minus under plus boundary conditions

Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions. Low temperature, one minus spin. With a Peierls argument one can prove that, given a ...
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37 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
2
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1answer
129 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 $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
7
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1answer
374 views

Joint law of the time integral of Brownian motion and its maximum

Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively: $$M_t=\max_{0\leq s\leq t}\,W_s$$ ...
3
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1answer
146 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 \, ...
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73 views

Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
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26 views

How to prove a relation holds almost surely? [closed]

Let's assume x(t) and y(t) are to random processes. How should we prove a relation say x(t)>y(t) holds almost surely? Thanks in advance.
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25 views

Question about Skorokhod embedding problem

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion on some probability space. Now for every centered probability distribution $\mu$ on $R$, i.e. $\int_{R}|x|d\mu(x)<+\infty$ and ...
2
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0answers
34 views

Steady state of a dynamical equation

Suppose we have the following dynamical equation: $P(k+1)=A\bigg(P(k) - P(k)H^T(k)\big(H(k)P(k)H^T(k)+Z\big)^{-1}H(k)P(k)\bigg)A^T+W$ with $P(0)=0$, where $P$, $A$, $H$, $Z$, $W$ are all $N\times N$ ...
3
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2answers
625 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, ...
2
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0answers
37 views

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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32 views

Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit: $$ dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC $$ where $W$ is a standard Brownian motion ...
2
votes
1answer
59 views

Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...
2
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0answers
153 views

probability interpretation of Sturm-Liouville eigenvalue problem

For a diffusion process , Feymann Kac formula connects the second order elliptic equation. Look at the eigenvalue problem $u''-xu'=-\lambda u $, $ u(0)=0,u(1)=0 $. for some discrete eigenvalues of $ ...
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1answer
49 views

Could somebody recomends a good book or article about numerical methods for Stochastic Partial Differential Equations

Could somebody recomend a good book or article about numerical methods for Stochastic Partial Differential Equations. I'm looking for a good introductory material thanks.
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1answer
74 views

Diffusion processes with different diffusion coefficients and absolute continuity

I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level. My initial situation is the following. Consider two SDEs: \begin{align} ...
3
votes
1answer
113 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 ...
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0answers
12 views

Karhunen Loeve expansion of $cos(\theta)$ where $\theta$ is a Gaussian random process or Uniform distribution in $[0,\pi/2]$]

I want to expand the random process $\theta$ using KL expansion for uncertainty quantification using stochastic FEM. But my random variable is function of cosine. i.e. $cos(\theta)$. My pde has ...
12
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4answers
2k views

A Markov process which is not a strong markov process?

Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in ...
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0answers
15 views

Second Moment of Intensity Function of Stochastic Process [migrated]

I'm trying to compute the 1st and 2nd moment of the intensity function of a Hawkes Process. The intensity function is of the form $$\lambda(t)=\lambda_0+\int_{- inf}^tv(t-s)dN_s$$ where $\lambda_0$ ...
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5answers
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Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related? I'm a graduate student doing a crash course in ...
2
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1answer
155 views

A calculation involving a uniform random variable quantile

THE PROBLEM: Let $U$ be a uniform distribution and $U_{n}$ be its nth empirical distribution. Suppose $t\in (0,1)$ and $n\in \mathbb{N}$ are constants. What's the explicit expression to ...
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1answer
201 views

question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see: http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...
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1answer
1k 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 ...
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1answer
457 views

On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)

Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...
6
votes
1answer
271 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. ...
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3answers
235 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 ...
2
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0answers
34 views

Random matrices whose limit gives exact Wigner surmise

Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write ...
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1answer
158 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 ...
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2answers
519 views

References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...
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43 views

Models for events where position and time are correlated

Apologies in advance if this question is not sufficiently research-level: What are the standard models that are used to describe phenomena in which events that occur at the same time are likely to be ...
2
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1answer
102 views

explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition: $$V_t=V_0+\int_0^tH_sdX_s-K_t$$ where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...
2
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2answers
198 views

How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?

Recently I was stumped by the calculation of the probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where $A_i \sim \text{exp}(\lambda), S_i \sim ...
6
votes
2answers
110 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 ...
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2answers
195 views

Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$, the sequence $(M^n)_n$ converges. Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices ...
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0answers
37 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 ...
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1answer
62 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 $\{X_t,\mathcal{F}_t\}_{t\ge0}$ ...
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4answers
2k views

Correlated Brownian motion and Poisson process

Is there an (easy) way to construct, on the same filtered probability space,a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ? I first asked this question ...
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0answers
53 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
1answer
107 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 ...
3
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1answer
242 views

Law of Iterated Logarithm for autoregressive process

Suppose that $\{X_i\}$ is an $\mathrm{AR}(r)$, defined by: $X_{i}= h(i) + \varepsilon_i $, $h(i)=\alpha_1 X_{i-1} + \dots + \alpha_{r} X_{i-r}$ where $\{\varepsilon_i\}$ are i.i.d. ${\cal ...
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1answer
145 views

Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived. Consider a ...
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1answer
60 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 ...
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1answer
196 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 ...
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1answer
170 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, $ ...
2
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2answers
189 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 ...
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1answer
86 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| ...
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1answer
270 views

Comparing the expected stopping times of two stochastically ordered random processes (Added:(14.05.2014))

Information: a-) $X$ and $Y$ are two continuous random variables on $\mathbb{R}$ having continuous distribution functions $F$ and $G$ with $G(y)\geq F(y)$ for all $y$. b-) $S^X_n=\sum_{i=1}^n X_i$, ...
3
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1answer
178 views

Quantiles moments and Convergence

QUESTION: Let $F$ be an absolutely continuous distribution function with density $f$, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
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32 views

Under what conditions the limit process of a sequence of Markov processes is Markov?

Given a sequence of Markov process $X_t^n$, if $X^n_t$ convergences to $X_t$ in the sense of some topology, under which condition can we prove that the limit process $X_t$ is Markov? Please give some ...