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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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Under which conditions the linear combination of two Wiener processes is still a Wiener process.

Let $\{W^{(1)}_t , t\ge0 \} $ and $\{W^{(2)}_t , t\ge0 \} $ be independent Wiener processes. Find all constants for which $\alpha W^{(1)}_t +\beta W^{(2)}_t, t\ge0$ is Wiener process. I already saw the answers to the questions linear combination of…
Helen
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When can I cover all the unique items?

I'm totally a newbie in this community. I would like to ask for help in a modeling question. Thanks for your time and patience in advance. Assume I have N unique items and I access T items per second. The access pattern follows some distribution…
Cheng Wang
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Kolmogorov's backward equation

In the book "Stochastic Differential Equations" by Bernt Øksendal, the Kolmogorov's backward equation is stated as following: Let $X_t$ be an Ito diffusion and $A$ is the generator of $X_t$. Define $u(t,x)=E^x(f(X_t))$ where $E^x$ is the expectation…
Q-Y
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Proof of thinning theorem

Thinning theorem If $N= (N_t)_{t\geq0} $ is a poisson process rate $\lambda$ and it is thinned by removing incidents with probability p independently of each other and the poisson process, then what remains (N~) is a poisson process rate $\lambda…
Rosie
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covariance function for Brownian motion

What is the covariance function for $U(t)$ if $U(t) = e^{-t}B(e^{2t})$ for $t \geq 0$ where $B(t)$ is standard Brownian motion. Thanks for the help!
icobes
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Chapman-Kolmogorov equation for conditional probabilities?

From Wikipedia (note that I have modified it from for a Markov process to for a general stochastic process): the conditional probability density $p_{i;j}(f_i\mid f_j)$ is the transition probability between the times $i>j$. So, the…
Tim
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Differentiate between linear death process and pure death process

I am having difficulty (and my textbook is of no help!) figuring out when to treat a problem as a linear death process with death parameters $\mu n$h vs. a pure death process with death parameter $\mu$h. Is there anything we need to look out for in…
icobes
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Modeling with pure death processes

A chemical solution contains $N$ molecules of type $\mathrm{A}$ and $M$ molecules of type $\mathrm{B}$. An irreversible reaction occurs between type $\mathrm{A}$ and type $\mathrm{B}$ molecules in which they bond to form a new compound…
icobes
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Explanation of Formal Definition of Dirichlet Process

I am reading about the Dirichlet process and I can understand the construction from Chinese restaurant process or stick-breaking process or Polya urn scheme. Now I am trying to understand why Dirichlet process is a distribution of distribution from…
DiveIntoML
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Predictable QV of Infinitesimal Generator Martingale

For a Markov process $X_{t}$ with $X_{0}=x$ and infinitesimal generator $\mathcal{L}$, we have the following martingale \begin{align*} M_{t} = f(X_{t})-f(x)-\int_{0}^{t}\mathcal{L}f(X_{s})ds \end{align*} for any suitable test function $f$. Without…
Mark Perlman
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Itô's lemma to solve the SDE

Given $dG_{t}=\alpha S_{t}dt+\upsilon S_{t}dW_{t}$ and $dS(t)={dG_{t}}-\epsilon_{t}dt$. How can I have $S_{t}=\mathbb{E}^{\mathbb{Q}}\left[\int_{t}^{+\infty}e^{-r(s-t)}\epsilon_{t}ds|\mathcal{F}_{t}\right]$ where…
Tony
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Mean Square Differentiability

If I want to show that a stochastic process is not mean square differentiable, is it enough to show, that the process $a.s.$ does not have differentiable sample paths?
Ichigo
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Markov Decision Process model

What does a deterministic Markov Decision Process (MDP) mean? Does it mean that the probability when going from one state to another is 1?
ella
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Is a reversible distribution unique?

For a time-homogeneous discrete time Markov chain, a reversible distribution of the chain is defined as $\pi$ that satisfies: $$ π_i p_{ij} = π_j p_{ji}, \forall i, j. $$ I was wondering if a reversible distribution is unique when…
Tim
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Limiting distribution and initial distribution of a Markov chain

For a Markov chain (can the following discussion be for either discrete time or continuous time, or just discrete time?), if for an initial distribution i.e. the distribution of $X_0$, there exists a limiting distribution for the distribution of…
Tim
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