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

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Stopping times for Brownian motion

Let $B_t, t\geq 0$ be standard Brownian motion. Let $\big(\mathcal{G}_t, t\geq 0\big)$ be the natural filtration, defined by $\mathcal{G}_t=\sigma(B_s, 0\leq s\leq t)$. Define also a filtration $\...
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About Ito integral of power of brownian motion

Using Ito's lemma, one can get the following expression for Ito integral of monomials: $\int_0^TW(t)^ndW(t) = \frac{1}{n+1}W(t)^{n+1} - \frac{n}{2}\int_0^TW(t)^{n-1}dt.$ What can we say about the ...
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106 views

Hierarchical (Recursive) Random Walk Model

Consider the following hierarchical (recursive) random walk model. For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete) random walk. For the next level $\ell=2$, we ...
6
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97 views

Stochastic Analogue of Stokes Theorem

Dynkin's formula can be thought of as the stochastic version of the Fundamental Theorem of calculus, $$E^x[f(X_{\tau})]=f(x)+E^x\left[\int_0^{\tau}Af(X_u)du\right],$$ where $\tau$ is a first exit time ...
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33 views

Convergence of an inhomogeneous markov chain

A markov chain is defined as $X_t=F(X_{t-1})X_{t-1}$, where $X_t$ and $X_{t-1}$ are both vector. So the transition matrix depends on the current states. I want to show that for any given initial ...
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65 views

Version of Donsker-Invariance-Principle

Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$...
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Uniform Concentration Bounds on Weighted Sum of i.i.d. Bernoulli Random Variables

Let $\delta_1,...,\delta_n$ be $n$ independent identically distributed Bernoulli random variables with $\mathbb{P}(\delta_1=1)=p$. We consider a set $\Omega = \{\mathbf{a}:=(a_1,...,a_n)~|~a_i\in [0,c/...
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70 views

Variation on stones in buckets

This is a spinoff, see Collecting stones in n buckets. Frankly speaking my only motivation is that I became curious: what happens if one redistributes the stones into the same buckets? More ...
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42 views

Conditions for supremum and conditional Expectation to commute

I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and have (for simplicity) two stopping times $\tau_1$ and $\tau_2$ such that $\tau_2 \leq \tau_1$ and $U:\Bbb R\...
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56 views

Calculate Moments of SDE

I have posted a similar question on math.stackexchange (http://math.stackexchange.com/questions/1848492/calculate-mean-of-sde), but didn't find anyone who could help. I'm interested in the one-...
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65 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
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34 views

Boundary behavior for Ito diffusions

The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
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37 views

Probability for a SRW to be at some place in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
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66 views

Differentiability of a simple value function driven by a diffusion

Consider a diffusion given by, $d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$ $X_0 = x$. Suppose the functions $\mu$ and $\sigma$ are as follows - $f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
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259 views

Does random walk have more concentration surrounding the origin?

Consider a simple random walk $S_n$ on one dimension, starting at $0$. In this case, $S_n$ fluctuates between $-\infty$ and $\infty$, but intuition says that it might stay more often in an interval ...
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52 views

Stochastic process with discontinuous drift

While studying a portfolio optimization problem, I came across the process $$dX(t) = X(t)\,\Big(\,\big(\mu - \alpha\,1_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big)$$ which has a discountinuous ...
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171 views

Frequency of visiting states in Markov chains

Given a finite, ergodic Markov $\{X_i\}$, and two natural numbers $a>b$. Let $$p=P\left[\forall n, \sum_{k=n}^{n+a-1} \mathbf{1}_m(X_k)\leq b\right]$$ where $\mathbf{1}_m(X_k) =1$ if $X_k=m$ and 0 ...
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33 views

Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?

If this is too basic for MathOverflow... say the word and I shall move it to Math.SE First consider this system of ODEs. Say I have two variables $u$ and $a$, following $$ \dot u = -u + f(a) $$ $$ \...
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Distribution of the stopping time of an autoregressive sequence

Consider $e_t$ be i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which $$ ...
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2answers
1k views

Random walk to stay in an interval forever

Consider a random walk on the real time, starting from $0$. But this time assume that we can decide, for each step $i$, a step size $t_i>0$ to the left or the right with equal probabilities. To ...
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41 views

Power spectrum of the difference of two Poisson processes with equal rates

I am studying the asymptotic properties of a dynamic a model involving the difference of two balanced Poisson processes (i.e., $\lambda_1 = \lambda_2$). I recently discovered the Skellam Distribution ...
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1answer
149 views

Can all Local Martingales Be Represented using Only Brownian Motion and Finite Variation Processes?

This is a cross-post of my unanswered (more than a week) question on Math.SE. Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to try to ...
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109 views

Basic Definition and Notations in RWRE

From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ ...
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I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
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25 views

Reference Request: $M_t/M_t/1/K$ queue length distributions

I am investigating functionals defined over sequences of discrete probability distributions related to dynamical/stochastic system performance. As an initial step, I am searching for references that ...
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17 views

Nonparametric estimation in diffusion

Fan and Wang In the above paper, the Authors provide estimators for the squared spot volatility process $\left(\sigma^{2}_{t}\right)_{t\geq 0}$. My question is how to find estimators for the process ...
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101 views

On the spectrum of stationary Gaussian process

What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum? In a similar way let us consider a stationary vector valued Gaussian ...
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62 views

Uniqueness of invariant measure for equivalent transition probabilities

Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \...
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Collecting stones in n buckets

There are $n$ stones distributed in $n$ buckets (initially one stone in each bucket). At each step the content of each bucket is put in a random bucket, chosen independently out of a set of $n$ new ...
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55 views

invariant measure for piecewise deterministic Markov process with only measurable switching intensity

Let $L$ denote the extended generator of a Markov process $(X_t)$ on a locally compact space with domain $\mathcal D(L)$. This means that for all $f \in \mathcal D(L)$, the process $$ f(X_t) - f(X_0) -...
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information about composite random process

I have a following composite random process $$X_j = v_0 + 1/j^2 + Y_j + Z_j$$ where $v_0$ is a constant, $Y_j \rightarrow 0$ almost surely as $j\rightarrow \infty$ and $Z_j \sim N\big(0, \frac{a^{2j}...
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38 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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2answers
178 views

Lévy measure and Lévy process

A Lévy measure $\nu$ on $\mathbb R^{d}$ is a measure satisfying $$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.$$ A Lévy process can be characterized by triples $(b, A, \...
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99 views

Radon-Nikodym for continuous time processes

Likelihood theory for statistical inference concerning stochastic processes in continuous time are well used. How ever i've found no real literature concerning the fundamentals. What is know from ...
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134 views

Contraction of probability measures

Notations. Let $(X, \mathcal{B})$ be a separable Banach space, with its Borel sigma-algebra, $\|\cdot\|$ stands for the norm in $X$, $\mathcal{P}(X)$ - the set of all probability measures on $X$. Let ...
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26 views

Dependency of the error term on the states, in the definition of the transition rates of a continuous time Markov chain

I think this is certainly not a research or graduate level question. But I didn't get any answer from math.stackexchange.com. I'm studying G.F.Lawler's stochastic process book. There he defines the ...
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1answer
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Uniform convergence of action of Feller semigroup with $1$ variable

Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous ...
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Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs. I'm reading Stochastic Differential Equations in ...
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1answer
67 views

The (infinite) invariant measure of an SPDE

Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type: \begin{equation}\left\{ \begin{aligned} &\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(...
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concentration of functions of Gaussian processes

Let $\mathcal{C}\in\mathbb{R}^n$ be a subset of the unit ball. Also let $\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_m\in\mathbb{R}^n$ be i.i.d. random Gaussian vectors $\mathcal{N}(\mathbf{0},\mathbf{...
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What is the stationary distribution for the contact process on the half line?

The contact process is a well-studied Markov process. I'm just concerned with the one-dimensional nearest-neighbor version here. The state space is $\eta\in\{0,1\}^\mathbb Z$, and for state $\eta$ at ...
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1answer
91 views

Limit of first passage time

I have a conjecture that seems rather obvious but the proof seems elusive. Consider a diffusion given by, $dX_t = \mu(X_t) dt + \sigma(X_t) db_t$ where $b_t$ is a standard Brownian motion. $\mu,\...
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42 views

Existence of strong solution in SDEs and continuity in the time variable

I recently come across some literature in stochastic analysis that uses the following result: Consider the one-dimensional SDE $$dX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$ where $a, ...
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176 views

A Stochastic Taylor Expansion/Asymptotics

Question: Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth ...
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72 views

Computing transition operators for Markov processes

Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt$$ (or given by ...
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Karhunen-Loeve expansion convergence rate for Gaussian Proccess

Consider A Gaussian Procces $X(t):\mathbb{R} \to \mathbb{R}$ with and $\mathbb{E} \left[ X \right] = 0$. Consider also its KL expansion $X(t) = \sum\limits_{k=0}^{\infty} Z_k e_k (t)$, with $Z_k$ ...
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1answer
73 views

Optimal control / Portoflio optimization: Maximize expected utility of total consumption

I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is ...
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3answers
185 views

A question about intuition of fluid limit in queuing system

This is a question about intuition in understanding the fluid limit queuing system. Assume we have a sequence of queuing systems $\{S^N\}_{N=1}^{\infty}$ with N servers and each server has unit ...
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165 views

Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ where the coefficients are assumed to be Lipschitz continuous. I hope to ...
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Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and continuous)....