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

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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), ...
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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) ...
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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 ...
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70 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 ...
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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 ...
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“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 ...
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39 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 $T$ ...
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124 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 ...
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46 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 ...
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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 ...
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128 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 ...
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49 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], ...
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94 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} ...
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1answer
110 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 ...
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1answer
88 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 ...
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38 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 ...
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2answers
122 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 ...
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34 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. ...
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1answer
89 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 ...
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34 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 ...
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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 ...
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36 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 ...
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2answers
150 views

Do we have Karhunen–Loève expansion for White Noise?

Let $W$ be a random process (my White Noise) on $[-1,1]$ such that: $W(t)$ is a normal random variable with mean $0$ and standard deviation $1$ for all $t \in [-1,1]$ $E(W(t)W(s)) = 0$ for all $t, s ...
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37 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 ...
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151 views

Must rows of a transition matrix be distinct?

Is it true that for all continuous time Markov processes on a countable state space $S$, we have all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ? This ...
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58 views

Most visited vertex in a random walk with place dependent drift

Consider the following Markov chain on $\mathbb{Z}$: $$ P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}} $$ Do there exist constants $c,C>0$ such that $$ c\cdot P^t(z,z) ...
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34 views

Are the elementary predictable processes dense in $L^2([M])$ for $M$ a local martingale?

The question is the one from the title. I know this is true when $M$ is an $L^2$ bounded martingale (which is often used in the classical approach to the construction of the stochastic integral) but ...
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1answer
112 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} ...
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1answer
88 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 ...
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164 views

Expected visits to the origin by a symmetric random walk on the integers

Consider the first $2n$ steps of a simple random walk on the integers, starting at the origin. A simple binomial argument shows that regardless of $n$, the origin gets visited the most (in ...
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115 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 $X\times Y$, ...
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32 views

Strong Markov vector-valued process from component strong Markov process and independence

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...
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Bounds on Wasserstein (Kantorovich) distance

Let $X$ be a Polish space endowed with a bounded metric $\rho_X$. Let $\mu, \mu'$ be two probability measures, and $\kappa, \kappa'$ be two stochastic kernels on $X$. Assume that $\kappa, \kappa'$ are ...
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274 views

Berry-Esseen bound for martingale sequence with varying and dependent variances

Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e. $$ E[X_{k}|\mathcal{F}_{k-1}] = 0 $$ where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$. Let ...
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1answer
57 views

Brownian motion increments

We know that if $W_t$ is a Brownian motion, $W_{t+t_0}-W_{t_0}$ is one too. Does the "converse" holds : Let $t_0$ be a positive number. I have a Brownian motion $W_t$ and I seek another Brownian ...
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1answer
98 views

Random Walk 2D with dependent weights [closed]

I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated! Suppose I have a 3x3 grid as shown below. (3,1) (3,2) (3,3) (2,1) (2,2) ...
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1answer
162 views

What's the best betting strategy to double money if we have $\delta$ advantage?

Suppose that I am very skilled in a gambling game, and any day that I bet $x$, I get back $2x$ with probability $\frac 12+\delta$ (and nothing with probability $\frac 12-\delta$). My goal is to double ...
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103 views

Does the martingale property holds after changing filtration?

Let $\Omega$ be the space of continuous real-valued functions $\omega=(\omega_t)_{t\ge 0}$ starting at zero, i.e. $\omega_0=0$. Let $\Lambda=\Omega\times \mathbb R_+$ and denote by ...
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Characterisation of non-Gaussian stationary stochastic processes via auto-correlation functions

It is well-known that a centred stationary Gaussian stochastic process is characterised up to equivalence by its autocorrelation function. Wiener, in his Time Series, makes the off-hand remark that ...
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Construction of point process having same pair correlations as GUE

The distribution of the pair correlations of the eigenvalues of the GUE satisfies (in the limit, when being normalized appropriately) $$ g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + ...
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1answer
100 views

Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1 $$ ...
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132 views

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
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Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...
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32 views

Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
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71 views

Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, ...
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Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes". Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
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1answer
123 views

Transition probabilities for the symmetric random walk on the integers

I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...
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203 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 = ...
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126 views

Linking Wasserstein and total variation distances

I seek to bound the total-variation distance between two probability measures $p_1$ and $p_2$. It is extremely easy to build a parameter space where $p_1$ and $p_2$ are the marginals of some joint ...
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A question on improper Itô integrals and semimartingales

I am reading the article given in http://link.springer.com/chapter/10.1007/978-1-4614-5906-4_24#page-1. I have the following two questions: In which setting does one define improper integrals with ...