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

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Regularity of the entrance measure of SRW

Let $S(n)$ be the discrete sphere of radius $n$ (i.e., the internal boundary of the Euclidean discrete ball $B(n)$) centered in the origin, and consider a simple random walk starting at some $x\in\...
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98 views

Is a local martingale with constant expectation necessarily a martingale?

Suppose $X\in \mathbb R$ is a weak solution to the SDE $dX_t = \sigma(X_t)dW_t$, in which $W$ is a one-dimensional Brownian motion, and $\sigma$ is Borel measurable so that a weak solution exists and ...
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175 views

Brownian motion in $\mathbb{R}^n$, probability of hitting a set

Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...
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84 views

Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable? This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion http://www.springer.com/gb/book/...
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66 views

Order statistic of Markov chain sample path and related probabilities

Consider a 1D sample path, denoted as $\{X(1), ..., X(t), ..., X(n)\}$, generated from a discrete time finite state (time homogeneous) Markov chain over states $\{1,...,m\}$, with transition ...
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138 views

Total absolute variation of brownian motion, with different sampling rates

Let $(B_t)$ be a brownian motion on [0,1]. For the following, let $\omega$ be fixed. Let's compute the total absolute variation when sampling period = $\delta$ is fixed: $$V(\delta) = \sum_{i=0}^{N-...
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31 views

Comparison between the entrance measure and the harmonic measure

Consider the standard two-dimensional Brownian motion, and define $\tau(A)$ to be the hitting time of $A\subset \mathbb{R}^2$. Let $hm_A$ be the harmonic measure (from infinity) on $A$. Let $B(r)$ be ...
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1answer
123 views

Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...
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1answer
58 views

“Convergence speed” results for the Langevin process

The Langevin process is defined by the following stochastic differential equation: $$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$ Its equilibrium distribution is the following: $$ p_\infty (x) \propto ...
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99 views

Full version of Soucaliuc's research announcement “Réflexion entre deux diffusions conjuguées”

Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes: [1] F. Soucaliuc, Réflexion entre deux diffusions ...
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111 views

Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
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1answer
110 views

Feller processes / probability generators

I am looking for a example of a function in $C_0(\mathbb{R})$ such that $f',f'' \,\text{and}\, f''' \in C_0(\mathbb{R})$ with $$ \inf f < \inf (f-a*f''')$$ for some $a>0$, but I couldn't find ...
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108 views

Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
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32 views

steady state distribution for a jump Markov chain

Consider a queueing process with the following transition matrix: $\mathbf{P}=\left( \begin{smallmatrix} 1-\lambda & \lambda & & & & & & &\\ \mu & 1-\...
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1answer
169 views

Random walk with continuously distributed steps on [-1,1]

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability $$P(S_n \textrm{ reaches } a \textrm{ before} -b) =...
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1answer
87 views

Finite hitting time implies hits at any finite time?

I was wondering about the following problem: Assume we have a state space $S:=\mathbb{Z}$ and a Markov chain, such that we can go from any state $x$ to some state $y$ with positive probabilities, i.e....
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1answer
85 views

Weak existence for modified Tanaka SDE

Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE $dX_t = dW_t + dL_t^0(X_t)$, where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at $0$....
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1answer
111 views

Markov chain with Feller property

Does anybody know whether there is an analysis of when the monotone decreasing chain has the Feller-property? The monotone decreasing is defined as a chain on $\mathbb{N}$ and the rate of going down $...
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53 views

hitting time of an Ornstein-Ulhenbeck

If we consider a nice Ornstein Uhlenbeck process $d x (t) = - \gamma x(t) dt + \sigma d w (t)$ with $x(0) = x_0 \in (-L,L)$. Here $\gamma, \sigma$ are positive constant and $w(t)$ is a Wiener process....
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154 views

approximate stationary distributions of a doubly stochastic matrix and its supports

Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...
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168 views

Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help .... Any pointers or suggestions are appreicated! ...
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59 views

Reference for branching processes

A popular model of a continuous time branching process was introduced around 1970, which is now called the Crump-Mode-Jagers branching process, was introduced here: A General Age-Dependent Branching ...
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1answer
159 views

using Feynman-Kac formula

I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...
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39 views

Memorylessness of residence times for a Markov process

I'm stuck on the trivial problem of showing memorylessness of holding (residence) times for a continuous time homogeneous Markov chain on finite state space. I have a homogeneous Markov process $x(t),...
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67 views

Random process & probability problem met in wireless communication

A random process r obeys the following distribution: $p(r,ṙ)=rb_0\exp(−\frac{r^2}{2b_0})\sqrt{2πb_2}exp(−\frac{\dot{r}^2}{2b_2})$, where $\dot{r}$ is the derivative of r in the time domain. You can ...
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1answer
188 views

General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations $$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
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19 views

Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let $...
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116 views

Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info For standard vector-valued diffusion processes the following result is well-known: Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by \begin{align*} ...
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1answer
198 views

Intuition about Skorohod integral

I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on. In particular ...
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331 views

What is the derivative of this integral?

I have asked this question here http://math.stackexchange.com/questions/1536018/how-to-find-derivative-of-this-intergral but still has no response. Might I ask it here ? Let $\alpha(t)\in\{0,1\}: ...
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1answer
115 views

Differentiability of stochastic process

Is it possible to construct a stochastic process $X_t$ where the limit $\lim_{\Delta \rightarrow 0} \rm{Var}\left(\frac{X_{t_0+\Delta}-X_{t_0}}{\Delta}\right)$ does not exist but the sample paths ...
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72 views

Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ is standard Wiener. This solution is ...
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1answer
142 views

Is it safe to work on a Cadlag modification of a Feller process?

Let $f$ be a continuous bounded function. $X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write $$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...
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1answer
82 views

Compactness of cadlag martingales w.r.t. to the point-wise topology

Given a sequence of cadlag (right-continuous with left limits) martingales $X^n=(X^n_t)_{0\le t\le 1}$, we may use the well known criteria to determine whether it is weakly convergent, i.e. subtract a ...
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1answer
87 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 ...
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164 views

Product and convex combination of two stochastic matrices

Let $K_1$ and $K_2$ be two $N \times N$ stochastic matrices (hence non-negative and rows adding up to one) with zeros on the diagonal. If $\alpha \in (0,1)$, is it possible to have $$K_1 K_2 = \...
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2answers
105 views

A question about Skorokhod embedding problem

The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the ...
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1answer
360 views

Blumenthal and Kolmogorov 0-1 law

Blumenthal's 0-1 law see theorem 5.8/5.9 tells us that an event in the germ $\sigma-$ algebra has either probability zero or one with respect to a measure induced by a Brownian motion starting in some ...
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1answer
252 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 ...
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58 views

Modify exponential family representation to a semimartingale

Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a class of probability measures $P=\{P_{\theta}:\theta \in \Theta\}$ definied on the filtered space. We ...
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2answers
243 views

Brownian motion in $n$ dimensions

Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in \...
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47 views

Factorization of the Fokker-Planck semigroup

I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...
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49 views

methods to analyze martingale conditioned on return in the future

Consider a martingale $S_t$ on $\mathbb{Z}$ starting from 0. Assume that for any $t$, $Var[s_t\, | \, \mathcal{F}_{t-1}] < V$, where $V$ is some positive constant. Fix an $n$ and for $t \leq n$, ...
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55 views

skorokhod integral as Weak Integral

Is it possible to express the skorokhod integral on a Banach space $B$ as a special case of the weak (or Pettis) integral over an appropriate Banach space $E$? For example if $E$ is the space of ...
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199 views

Prediction with positive weights?

Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$ \gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0. $$ It is guaranteed to be positive definite by Polya's criterion (...
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Understanding the limits of the Ito Process Defintion

I would like to understand what kind of stochastic process are Ito Processes. According to Kuo[p. 102] an Ito Process is a stochastic process of the form $$dX_t=g(t)dt+f(t)dW(t),$$ where $W(t)$ is a ...
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1answer
229 views

Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary finite-dimensional smooth manifold. For simplicity, let's assume that $M$ has no boundary. Does there always exist a gaussian random field with constant variance on $M$? If not,...
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177 views

Recurrence of Poisson binomial distributed random walk

Let $X_n$ be the outcome of a Bernoulli trial where the probability of getting 1 is $p_n$ and the probability of getting 0 is $1-p_n$, and let $S_n = \sum_{i=1}^n \left(X_i - \textrm{E} X_i \right)$. ...
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106 views

regularity of zero point

We consider 1-d process $X$ $$ X(t) = b t + J_{t} + M_{t}$$ where $b$ is constant, $M$ is a continuous martingale process with $M(0) = 0$, and $J$ is a symmestric $\alpha$-stable process with its ...
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1answer
106 views

Markov-semigroup sobolev inequality

I have a question about the following definition: A probability measure $\mu$, such that the Markov semi-group $e^{Lt} \in L(L^2)$ exists and is symmetric, satisfies the Sobolev inequality iff for ...