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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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Markovian and the Chapman–Kolmogorov equation

From Wikipedia In a Markov process, one assumes that $i_1 < \cdots < i_n$. Then, because of the Markov property, $$ p_{i_1,\ldots,i_n}(f_1,\ldots,f_n)=p_{i_1}(f_1)p_{i_2;i_1}(f_2\mid f_1)\cdots p_{i_n;i_{n-1}}(f_n\mid f_{n-1}), $$ where the…
Tim
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Using Central Limit Theorem

Can anyone help me with it: Using the central limit theorem for suitable Poisson random variables, prove that $$ \lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}=1/2$$ Thanks!
kira
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Jumps of cadlag processes

It is stated in a proof that there are only finitely many $s$ such that $|\Delta X_s|\geq\frac{1}{2}$ on each compact interval where $X$ is a càdlàg process. I thought of a process with sample path recursvly defined as $X_i(\omega)=1$ for…
peer
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Realization of a stochastic process

Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let's define a stochastic process as a function $$ S: \Omega \times \mathbb{R} \rightarrow \mathbb{R} \\ \omega \times t \mapsto S(\omega, t) \, . $$ It makes sense that at fixed $t$,…
dapias
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A strange-looking Feynman-Kac formula

I have a PDE which involves the predictable finite-variation parts of some semi-martingales and a quadratic-covariation process and I tried to derive a Feynman-Kac style expectation from the PDE. However, the result looks somewhat strange: it is in…
user126540
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Stationary distribution for Markov process with non-exponential waiting times

Stochastic processes often are described in terms of transition rates where the length of time waited before a transition occurs is an exponential random variable. For example: $0\rightarrow 1$ at rate $\alpha$ and $1\rightarrow0$ at rate $\beta$.…
jdods
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Gaussian process with deterministic bracket

Let $(M_t)_{t\geq 0}$ a continuous Gaussian process that is a martingale with $M_0=0$. Show that $\langle M,M \rangle _t=f(t)$ a.s. where $f$ is a continuous increasing function. In the particular case of Brownian motion we have $\langle M,M \rangle…
Patissot
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Hitting open sets

Let $(\Omega,\mathscr F,(\mathscr F_t)_{t\geq 0},\mathsf P)$ be a complete filtered probability space and $X = (X_t)_{t\geq 0}$ be a cadlag stochastic process with value in a Polish space $E$. Is it true that for any closed subset $A$ of $E$ the…
SBF
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Example of a predictable process that isn't caglad

The predictable processes are defined as the elements of the predictable sigma algebra, which is generated by the caglad processes. What kinds of extra processes do we get by extending from caglad to predictable processes? Specifically, are there…
mathrat
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Wiener Process and Random Walk

Quoted from Wikepedia: The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. I was wondering if the…
Tim
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Covariance between brownian bridge and its max.

Does anyone know how to compute $\text{Cov}[\max_{s\in [0,1]}B(s), B(t)]$ where $B(t)$ is the standard Brownian bridge on the interval $[0,1]$? Update. I have found a paper that solves the problem: On the maximum of the generalized brownian bridge…
Mark G.
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Cutting time out of a Poisson process

Suppose we have a Poisson process on the time interval $[0, \infty)$. Let $N(t)$ denote the number of arrival events up to epoch $t$, and let $S_n$ denote the epoch of the $n$th arrival. Suppose each sequence of arrival epochs $0 \le S_1 \le S_2 \le…
bryanj
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Question about non-positive Stochastic processes

This question will be a little out-of-character for me. I'm reading an evolutionary game theory book (which isn't for mathematicians), and I'm not sure of the mathematics involved. My definition of a stochastic matrix will be a square matrix with…
Nicole
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continuous semi-martingales as stochastic integrals of Brownian motion

What is an example of a continuous semi-martingale that cannot be written as a stochastic integral with respect to Brownian Motion?
random walk
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Average waiting time in a Poisson process

Sheldon Ross's Introduction to Probability Models, exercise 5.44.b: cars pass a certain street location according to a Poisson process with rate $\lambda$. A woman who wants to cross the street at that location waits until she can see that no cars…
Luke
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