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      Questions tagged [stochastic-processes]

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

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      votes
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      2k views

      Time integrals of diffusion processes

      I was wondering if someone could recommend a reference that deals with time integrals of diffusion processes. Suppose $X$ is an Ito diffusion process with dynamics $dX_t = \mu(X_t)dt + \sigma(X_t)...
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      votes
      1answer
      524 views

      Does generator of continuous time random walk map heat kernel from L^2 to L^2?

      Let $\Gamma = (G,E)$ be an undirected, infinite, connected graph with no multiple edges or loops. We equip $\Gamma$ with a set of edge weights $\pi_{xy}$, where, given $e=\{x,y\}\in E$, we write $\...
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      votes
      2answers
      805 views

      Counterexample Markov process

      Let $X$ be a homogeneous Markov process in a continuous time with value in the set $E$. Suppose that for some $T>0,x\in E, A\subset E$ we have $$ P_x[X_t\in A] = 0 $$ for all $t\in [0,T]$ but $$ ...
      1
      vote
      1answer
      983 views

      Autocorrelation of a ±1-valued random process with certain statistics

      Suppose $f(t)$ is a continuous-valued, zero-mean stochastic signal with Gaussian autocorrelation (with variance $\sigma^2$). Suppose I then pass this signal through a step function, producing a new $\...
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      vote
      2answers
      1k views

      Change of measure Markov process

      We begin with example. For the Poisson process with an intensity $\lambda_1$ there is an equivalent change of measure which makes it intensity to $\lambda_2$. I would like to find the conditions ...
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      votes
      2answers
      943 views

      Change of time or change of measure

      Consider simple diffusion $dX_t = \sigma dw_t$ and a parameter $a>0$ and $X_0=x$. Let us denote $Y_t = X_{at}$ - thus we made a change of time. Let us denote an original measure as $P$. How to find ...
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      votes
      1answer
      937 views

      Generalizations of a product formula for the gamma function

      Hello and Happy holidays. I am interested in generalizations of the following product formula for the gamma function $\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$: \begin{align} \...
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      votes
      1answer
      530 views

      Reachability for Markov process

      Let $X$ be a Markov process (in continuous or discrete time) and define an event $$ R(T,A) = (\exists t\leq T: X_t \in A). $$ I have seen in one paper that $$ \Pr[R(\infty,A)] = \sup\limits_{\tau} \...
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      votes
      2answers
      569 views

      Random walks on graphs: Cover time and blanket time

      Winkler and Zuckerman conjectured that the blanket time is within a constant factor of the cover time. The conjecture was recently proved. The cover time $C$ is the expectation of the first time $t$ ...
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      votes
      3answers
      582 views

      A type of stochastic jump process

      Let $X \geq 1$ be a integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we ...
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      votes
      1answer
      265 views

      A random variable in a game of knights and queens

      Suppose that a game is played on an $n \times n$ board as follows. There are two players, Player 1 has (only) $Q$ queens and Player 2 has only $K$ knights. Suppose that $Q, K \leq n/3$. The game is ...
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      2answers
      2k views

      Two dimensional brownian motion first passage time

      Hello, I am looking for information on how to solve/compute first passage time for two dimensional Brownian motion. any papers, references, books or web links for study will be helpful. thanks ...
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      votes
      2answers
      3k views

      Commuting supremum and expectation

      Given a one-parametric random function on a probability space $(\Omega,\mathcal F,\mathbb P)$: $X:U\times\Omega\to \mathbb R \text{ and } (a,w)\mapsto X(a,w), \text{ with } \sigma(X(a))\subseteq \...
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      vote
      1answer
      857 views

      Discounted total reward vs. Average total reward

      In a Markov Decision Process (MDP), the discounted total reward is defined as $\sum_{t=0}^\infty \gamma^tr_t$ where $r_t$ is the reward perceived at time $t$ and $\gamma$ is a real number $\in ]0, 1[$....
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      votes
      1answer
      516 views

      Stationary non-isotropic spatial stochastic processes

      I asked this question in math.stackexchange but got no response; Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic?...

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