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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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Is equality of processes stable to multiplication with an independent process.

Assume that all processes to be considered are regular (say cadlag). Assume $X^1$ and $X^2$ are stochastic processes such that $X^1_t = X^2_t$, that $Y^1$ and $Y^2$ are processes such that $Y^1_t=Y^2_t$ and that $S$ is a processes independent of…
htd
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What is the definition for the process $X_-$?

I often see for a stochastic process $X$ an new process denoted by $X_-$. I just know that this is left continuous, but I do not know the exact definition. Furthermore, if I have a RCLL process $X$ and looking at $X_-$ what can I say? I this process…
user20869
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Intutitive understanding of the law of a stochastic process

I'm trying to get my head around the notion of "Law of a stochastic process" intuitively. This is what I got for a Brownian motion: Denoting the law of a Brownian motion $\mathcal{L}_B:\mathcal{B}(C_\mathbb{R}{[0,1]})\rightarrow[0,1]$, given a set…
EZLearner
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$X_n$ and $Y_n$ are same irreducible, positive recurent, aperiodic markov chian and are independent. Is $(X_n,Y_n)$ positive recurrent?

Let $X_n,n \geq0$ be an irreducible, positive recurrent, aperiodic markov chain with state space $S$ and transition matrix $P$. $Y_n$ has the same $S$ and $P$ but independent with $X_n$. Then $\epsilon=(X_n,Y_n)$ becomes a new markov chian with…
abc1m2x3c
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Kolmogorov continuity theorem on Wikipedia.

I am wondering if the Kolmogorov continuity theorem on Wikipedia is wrong? : They say that the modification is sample continuous, and when we click on that link it says that it is a.s. continuous. However, I've seen the same theorem beeing…
user119615
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If $M_t$ is a martingale, is this process a martingale too?

If $M_t$ is a martingale, is this process a martingale too ? $X_t=\int_0^tY_sdM_s$ where $Y_t$ is some process that makes $\int_0^tY_sdM_s$ defined If not, what about the case $Y_t=M_t$ ?
mdrlol
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Lévy measure and compensation

In the Lévy-Itô decomposition it's necessary to compensate small jumps. That's clear. The small jumps are perhaps non-summable. But why are the jumps quared summalbe? In the "ordinary" proofs of the Lévy Khintchine Formula or the Lévy-Itô…
kuemmelsche
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Is 1-d Brownian filtration rich enough to admit a 2-d Brownian motion?

Let $B_t$ be the standard 1-d Brownian motion, and let $\mathcal F^B_t$ be the induced filtration. Is it possible to construct a 2-d Brownian motion adapted to $\mathcal F^B_t$? [EDIT] Come to think of it, I guess it is doable. By picking different…
Jay.H
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Question about Alternating Renewal Processes.

I am studying stochastic processes by myself with the textbook written by Sheldon M. Ross. Because of my short knowledge, I have been faced with some difficulties to understand.... My question is about the alternating renewal processes in the…
Danny_Kim
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Question about $X_n$ and $N(t)$ in Counting and Renewal process.

I am studying renewal theory. In the text book, $\{X_n, n=1, 2, \cdots\}$ denotes a sequence of non-negative i.i.d. with a common distribution $F$, and to avoid trivialities suppose that $F(0)<1$. We shall interpret $X_n$ as the time between the…
Danny_Kim
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$L^2$-stationary but not $L^2$-continuous process

I have to give a example of $L^2$-stationary (or also weakly stationary) but not $L^2$-continuous process. By definition, for a $X(.)$ $L^2$-stationary process, $EX(t):=m(t)=c$, for all $t\in R$ and some constant $c$ and the covariance function…
juliho
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Requirements for Ito's Lemma

Ito's Lemma is proved as Theorem 5 of his paper 'ON A FORMULA CONCERNING STOCHASTIC DIFFERENTIALS'. In his presentation it concerns a function $f$ of $t$ and a $n$-dimensional stochastic process $\xi$ with…
Andrew Kirk
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Sum of Gaussian processes

I would like to prove that the sum of Gaussian processes is also Gaussian, to be precise, $M_t=W_t+W_{t^2}$, where $W_t$ is standard Wiener process. That is kind of obvious, but I am looking for some more rigorous, as short as possible proof, other…
Julius
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Compound Poisson process with exponential distribution

Consider the following shock model. The count of shocks within a certain time $t$ is a Poisson process $N(t)$ with parameter $\lambda$, while every shock brings damage $Y_i$ to the subject, which is exponentially distributed with parameter $\mu$.…
Ziyuan
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Does a process with continuous sample path always has continuous variance

I thought there was a post asking the question in the title, but cannot find it anymore. It is not difficult to construct a process with countable many continuous sample paths, all starting from 0, but has discontinous variance at 0. So, I will…
Jay.H
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