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Questions tagged [stochastic-processes]

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

Most commonly studied classes of stochastic processes (if the indexed set is $\mathbb{R}^+$) encompasses processes that are diffusions, Markov, (sub-)(super-)martingales, Feller, but also semimartingales, Gaussian ...

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Brownian local time density

Dear All, I am not a mathematican, please be patient if I ask something in a not appropriate way! Let we suppose a Brownian motion with inital value of W(0)=0, and we look its possible realizations on time interval [0,T). MY QUESTION: What is the…
Tomi
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A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary

I asked this question on stats.stackexchange.com a little while back but didn't get an answer. It was suggested that I post it here at the time. There appears to be some migratory problems going on over there. Hopefully, this question is seen as…
Robby McKilliam
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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 and $A$ is the generator of the process…
Ali Fathi
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stopping time expectation for gambler's ruin

2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin. I let $X_n = x + \sum_{i=1}^{n}\xi_i$ where $\xi_i$ are i.i.d. with $P(\xi_i=\pm 1)=\frac{1}{2}$. Let $\tau_0 =…
Alan
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Hölder continuity of process from Donsker like theorem with Cauchy random variables

Let $X_k$ be i.i.d. Cauchy random variables with parameters $0,1$. For each $N$ define the process $Y_N$ by $$Y_N(t)=\frac{1}N\sum_{k=1}^{\lfloor tN\rfloor}X_k+\text{piecewise linear interpolation}.$$ Note that for each grid point, the sum of Cauchy…
user479223
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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 respect to Brownian motion? In the second page, I…
jumar
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Do we need Feller condition if the process jumps?

Consider the SDE: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} \end{equation} It describes a process $v_t$ which is a strictly positive if the drift is stronger enough, i.e. if drift parameters ($k$, the speed of…
Gabriele Pompa
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Correspondence between viscosity supersolution and supermartingale

Suppose $b : \mathbb{R} \to \mathbb{R}$ and $\sigma: \mathbb{R}\to \mathbb{R}$ are Lipschitz and that $(X_t)_{t\ge0}$ is a diffusion with $X_0 = x_0$ and $dX_t = b(X_t)dt + \sigma(X_t)dW_t$ . Consider the PDE $$b(x)v'(x) +…
Ben
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Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}$. We can interpret $\tau_{s,a,b}$ as the first time after time $s$ that the process hits $a$ or…
Ben
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Non-existence of such a continuous stochastic process

Below is actually a statement in textbook. But I don't have a good intuition of it. If we want a stochastic process $W_t$ to satisfy i). $s\neq t$ implies $W_s$ and $W_t$ are independent, ii). $\{W_t\}$ is stationary, iii). $E[W_t]=0$ for all…
pde_bk
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Why isn't conditional expectation called restricted random variable?

Definition of Condition Expectation( of random variable) can be seen here: http://en.wikipedia.org/wiki/Conditional_expectation My silly question is: Intuitively, conditional expectation seems to be a restriction of a measurable function on the…
pde_bk
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Infinite people on a road

I posted this on SE and did not get any replies. As a recap, there is a sequence of people on a line which has a infinite number of spots. People occupy one spot each. If a person is "clear" (which means that the person is at location x, and all…
picakhu
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How many Uniform(L, H) RVs can be added up until their sum reaches a certain value?

I want to know how many consecutive i.i.d. RVs with: $$X_{i} \sim\text{Uniform}(L, H)$$ can be added until the sum of them is greater than or equal to a certain value ($r$). I'm calculating this for a resource management algorithm and I want my…
Rezvan
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How to get/approximate the derivative of noisy time series?

I have a set of Langevin equations given by $${\mathbf{\dot{x}}} = \mathbf{-Q \,x} + \mathbf{\eta} \tag{1}$$ where $\eta$ is white Gaussian noise and $Q$ is not a function of $x$. Using Euler's method for SDE, I generated a time series of…
Bingkat
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When does a stochastic process have its sample paths a.s. in the reproducing kernel hilbert space (RKHS) induced by its covariance function?

Let $T$ be a compact metrizable space. Consider a centered second order measurable process $(X_t\colon t\in T)$ with continuous covariance function $c(t,s):= \mathbb{E}X_t X_s$. Are there any known sufficient conditions ensuring that $t \mapsto X_t$…
user127022
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