Tagged Questions

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

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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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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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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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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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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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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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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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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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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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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 ...
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 $$f(X_t) - f(X_0) -... 0answers 35 views information about composite random process I have a following composite random process$$X_j = v_0 + 1/j^2 + Y_j + Z_j$$where v_0 is a constant, Y_j \rightarrow 0 almost surely as j\rightarrow \infty and Z_j \sim N\big(0, \frac{a^{2j}... 0answers 38 views System of stochastic equations I want to know if this system of SDE:$$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$... 2answers 178 views Lévy measure and Lévy process A Lévy measure \nu on \mathbb R^{d} is a measure satisfying$$\nu\{0\} = 0, \ \int_{\mathbb R^{d}} (|y|^{2}\wedge 1) \nu(dy) <\infty.A Lévy process can be characterized by triples (b, A, \... 0answers 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 ... 1answer 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 ... 0answers 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 ... 1answer 59 views 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 ... 0answers 82 views 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 ... 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: \left\{ \begin{aligned} &\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(... 0answers 127 views 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{... 1answer 121 views 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 ... 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,\... 1answer 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 SDEdX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$where a, ... 2answers 176 views A Stochastic Taylor Expansion/Asymptotics Question: Let B(t) be the standard Brownian motion, \mu(t,x) and \sigma(t,x) are continuous functions, and$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$(\mu,\sigma) obeys the linear growth ... 1answer 72 views Computing transition operators for Markov processes Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion$$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt (or given by ...
Consider A Gaussian Procces $X(t):\mathbb{R}\times \Omega \to \mathbb{R}$ with $\Omega$ a probability space and $\mathbb{E} \left[ X_t \right] = 0$ for all $t\in \mathbb{R}$. Consider also its KL ...