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Mathematics Stack Exchange

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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$X$ is a Right-Continuous process iff $\mathcal{F}^X$ filtration is RC?

I have a doubt on this assertion : $X$ is a right-continuous adapted process $\iff$ $\mathcal{F}^X$ is right-continuous ? I have mainly a doubt on this direction $\Leftarrow$, I do not find a mean to prove this way.
Al Bundy
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Standard Wiener process continuity

This will be one big question. Basically, I had a lecturer who, to put it mildly, was not able to explain anything well. I am preparing for the exam now. One sample exam question is the following: Given the transition density function of a standard…
Naz
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Brownian Motion without path continuity assumption

I was reading this interesting discussion: Is the condition "sample paths are continuous" an appropriate part of the "characterization" of the Wiener process? and feel some follow up questions might be interesting. Basically, all the other…
Jay.H
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Any standard meaning of the term "bounded (stochastic) process"?

I always get confused when an author refers to a "bounded process", because it seems like it can take two meanings depending on the author. For example, in Rogers and Williams, it seems to mean $X(t, \omega) \leq K $, $\forall (t, \omega) $,…
Project Book
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Unique Levy Measure in Levy Khintchine Formula

The Levy-Khintchine formula gives a triple $(a,\sigma,\nu)$ for the characteristic exponent $\Psi(s)$ of an infinitely divisible random variable where $\Psi(s)=ias + \frac{1}{2}\sigma^2s^2 + \int_{\mathbb{R}}(1-e^{isx} +…
Fantastic Mr. Fox
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What is the limit as n goes to infinity of a Poisson process over time?

If $N(t)$ is a poisson process with rate $\lambda$, what is the meaning of $$ \lim_{t\rightarrow \infty} \frac{N(t)}{t}$$ In addition, what is the meaning of $$ \lim_{t\rightarrow 0} \frac{N(t)}{t}$$ and which would I use if I wanted the rate of…
Bianca
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Cubature on Wiener space

Suppose $(X_t)_{t\geq 0}$ diffuses as, $$ dX_t = \mu(X_t)\, dt + \sigma(X_t) \, dW_t $$ and, $$ g(t,x)=\mathbb{E}[g(T,X_T)\vert\mathcal{F}_t] $$ By Feynman-Kac we have, $$ \frac{\partial}{\partial…
AlmostSurely
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Dynamical programming principle - discrete case

Fix $\rho>0, C_2>C_1>0$ real numbers. Assume $dX_t=b(X_t,t)dt+\sigma(X_t,t)dW_t$ where $W_t$ is the standard Brownian motion, $X_0=x$, and $b_1\leq s_1\leq b_2\leq s_2\ldots\to\infty$ be a sequence of stopping times. And let…
Steven
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Inequality for certain stochastic process.

Let $b: \mathbb{R} \rightarrow \mathbb{R}$ be a Lipschitz-continuous function and let $X_t$ be a real valued stochastic process satisfying the stochastic differential equation $dX_t= b(X_t) dt+ dB_t$, $X_0=x$. Prove that for any $M> 0$, $t> 0$ and…
nick
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How to Derive Population Variance of AR(1) Process

If I have a process of the form $Y_t=\mu+\phi Y _{t-1} + \epsilon_t$, how is the population variance derived? Assuming that $\epsilon_t$ has a zero mean and a variance of 2.
Ethan
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Finding a stochastic differential equation as limit of a discrete stochastic process

I stumbled upon a problem that seems simple but I cannot tackle it. Let $X_n$ be a discrete process defined by the following algorithm. Choose $X_0\in[0,1]$, set $\kappa>0$ small enough and $X_{n+1}=X_n+\kappa(I_n-X_n)$ with $I_n=1$ with probability…
tst
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Filtered Poisson Process

I have a Poisson Process with rate $\lambda$ and also a filter which is applied on this process. After first event is issued, during time window $T$, all the following events are filtered. After the filter expires, one event could be issued again…
Wenjie
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Clark-Ocone Formula

Why is the Clark-Ocone formula: $F = E[F] + \int_0^T E[D_t F | F_t] dW_t$ important, besides its applications to finance. That is, can you give examples of any important pieces of pure theory where this formula is of central importance?
user30201
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restarting a Markov chain

I'm reading an article and having difficulty understanding some basic stochastic processes (I'm new to the subject so please pardon my wording of the question). Let $S$ be a set of states and $Q$ be a subset of $S$. Consider a Markov chain on $S$…
user144584
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A question about Convergence of a product in random variables

Let $\{U_k\}$ be a sequence of independent random variables, with each variable being uniformly distributed over the interval $[0,2]$, and let $X_n = U_1 U_2\cdots U_n$ for $n \geq 1$. (a) Determine in which of the senses (a.s., m.s., p., d.) the…
Youngsuk Andrew Oh
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