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Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Coin flips with any winning probability

Consider the following exercise from Lawler's Stochastic Calculus: Suppose two people want to play a game in which person A has probability 2/3 of winning. However, the only thing that they have is a fair coin which they can flip as many times…
J. D.
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On a functional of one-dimensional Brownian motion

I have a question about a positive functional of the one-dimensional Brownian motion. Let $(\{B_t\}_{t \ge 0},P_x)$ be the one-dimensional Brownian motion starting from $x >0$. I would like to know whether the following equation…
sharpe
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Transition matrix for a two state Markov Chain

What is the most general transition matrix for a two state Markov Chain? (both Markov and homogeneous). And show that any such chain has an equilibrium vector. Is it should be the following matrix? If it is, how could we show it alway exists an…
Jonny
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Questions about stationarity of a Markov process

I was wondering for a Markov process: Is it true that: it is a stationary process, iff its transition probability function is invariant to time-translation (i.e. it is homogeneous) and its initial distribution is invariant to …
Tim
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Poisson Process Question

Let $X(T)$ be a Poisson process. What is $$ \mathbb{P}(X(t) - X(s) = 1 \mid X(t) = 4) \,? $$ I split this up into $\mathbb{P}(X(s) = 3, X(t) - X(s) = 1)$ and found the answer. However, it did not work out to be the solution which is…
icobes
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Poisson Stochastic Process Question

I would really appreciate if you could help out with the following question Let $W_1,W_2,\ldots$ be the event times in a Poisson process $\{X(t);t\ge0\}$ of rate $\lambda$, and let $f(w)$ be an arbitrary function. Verify that …
icobes
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Is there a standard procedure for conditioning a stochastic process?

I've got a two-dimensional Markov stochastic process $(X_t, Y_t)$ that runs on time interval $[0, t_f]$. I know the transition function (or the infintesimal generator, if you like) of this process. I'd like to condition it such that $X_t \ge 0$ on…
GMB
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Gaussian white noise

I have two questions about a Gaussian white noise $\xi_t$: First of all, $\xi_t$ is a white noise. So, it's variance should be infinitive. But if the process is Gaussian then it has finite variance. How can Gaussian white noise exist? Say, we have…
Artem Zefirov
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Using the Fokker-Planck equation to describe a subdiffusive process

I am trying to understand whether a slightly modified version of the Fokker-Planck equation is suitable or not to study some subdiffusive processes. The original Fokker-Planck equation can be written as: $$\partial_t P(x,t)= D \left(\partial_x^2 -…
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Is it necessary to include the Gaussian condition in the definition of a Wiener process?

Please forgive me if this is a stupid question -- literally all I've done rigorously in stochastic processes so far is to look at the definitions ... but this Gaussian condition has me confused. Here's what I understand as the definition of a…
cantorhead
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Natural Increasing Integrable Process

An increasing integrable process $A_t$ is natural if $E\int_0^t m_s dA_s = E\int_0^t m_{s-} dA_s$ for every bounded right-continuous martingale. If both the Reimann-Stieltjes integrals $\int_0^t m_s dA_s, \int_0^t m_{s-} dA_s$ exist, then I think…
jpv
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Definition of Doob martingale

From Wikipedia A Doob martingale is a generic construction that is always a martingale. Specifically, consider any set of random variables $$ \vec{X}=X_1, X_2, ..., X_n $$ taking values in a set $A$ for which we are interested in the…
Tim
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Wiener process exercise

I have problems with one exercise connected with Wiener process. It can be really important exercise for me but I don't have any idea how to do it and even how to start We can define $\tau_a = \inf\{ t \geq 0 :X_t = a\}$ We have also $Y_a$ and $Y_b$…
ltrd
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Is every right continuous local martingale of finite variation constant?

I was reading a chapter in Dellacherie and Meyer. Suppose we have right continuous adapted processes $A$, $A'$ of finite variation. Both are null at zero and the difference is a local martingale. I know the following lemma: A continuous local…
user20869
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Expectation of expression involving Brownian motion

How do I compute $$E\left(tW_t - \int_0^t W_u du \Big| \mathcal{F}_s \right).$$ Given that $W_t$ is standard Brownian motion under the measure $P$ and $\{\mathcal{F}_t, t\ge 0\}$ denotes its standard filtration. could someone guide me on the…
Probabilityman
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