Mathematics Stack Exchange
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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Defining stopping time without making use of filtrations

I am currently writing a document in which I want to make use of the concept of a "stopping time" but avoiding a digression into the concept of a "filtration." Is the following definition correct? Let $\{ X_t \}_{t \ge 0}$ be a stochastic process. A…
UtilityMaximiser
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Given a Ornstein-Uhlenbeck process conditionated on $V_s=v\in\mathbb R$

I'm given a Ornstein-Uhlenbeck proces $V=(V_t)_{t\geq 0}$, i.e., $$ V_t = \frac{\sigma}{\sqrt{2\beta}}e^{-\beta t}B_{e^{2\beta t}}. $$ I'm told to prove that $V_{s+t}$, conditionated on $V_s=v$, follows a $$ N(e^{-\beta…
R__
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Derivation of optimal replacement threshold

Bertsekas (1976) introduces a component replacement example in which the current state of a component $x\in [0,1]$ is determined at the beginning of each period, and the agent makes a decision whether or not to replace the component at cost $R>0$.…
Eliya
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Definition of a stochastic process

My understanding of a stochastic process is that it is a collection of random variables $\{X_t\}_{t \geq 0}$ that take a value $w$ from a sample space $\Omega$. It is said that we can see a stochastic process as a function of two variables $$(ω,t)…
Semmah
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Marginal default density with log-normal intensity

I'm interested in the stochastic process followed by the marginal default density $e^{-\int_0^th(s)ds}h(t)$ in the case where the default intensity $h(t)$ follows a log-normal process. Assuming $$\frac{dh(t)}{h(t)} = \mu(t)dt+\sigma(t)dB(t)$$ an…
Grant
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Process' mean, covariance and stationarity

Let $X_t=Yt+Zt^2$ be random process, where $Y$,$Z$ are uncorrelated random variables, with characteristics: $EY=3$, $EZ=0.5$, $DY=1$, $DZ=0.05$. Find $X_t$ mean and covariance and prove whether process is wide-sense…
atomoutside
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Martin Hairer's breakthrough math prize 2021

Wikipedia states the 2021 Breakthrough Prize in Mathematics announced in September 2020 was made to Martin Hairer - "For transformative contributions to the theory of stochastic analysis, particularly the theory of regularity structures in…
rupert
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Strong Markov property of the mirror coupling

Let $B=(B_t)_{t \ge 0}$ be a Brownian motion on $\mathbb{R}^d$ starting from $x \in \mathbb{R}^d$. Let $y \in \mathbb{R}^d$ with $y \neq x$, and $R$ be the mirror reflection with respect to the $(d-1)$-dimensional hyperplane $H:= \{z \in…
sharpe
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Construction binary tree

First let $\mu$ be the induced distribution of the random variable $X$ on $(\mathbb{R},\mathcal{B})$ and denote $EX=m$. We also define for all $A\in G_{n+1}$ and $\omega\in…
user79002
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Stochastik of triangle Angle

As an electrical engineer I am interested in the phase shift of two signals. At one specific moment I can identify the phasor of a signal as a vector in the complex plane. To transfer the problem into math: I've got two 2D-vectors $\vec{a}$ and…
user815227
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Are these two variants of a stochastic model functionally equivalent?

I'm trying to figure out if two models of human information processing might differ in their ability to fit different data, or whether they're functionally identical. The general framework is that information accumulates from a start point towards…
Mike Lawrence
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Values of a random variable $x(t)$

Suppose that we have a function of the form $$x(t)=e^{-b(W(t))^2}$$ where $W(t)$ is a Wiener process. Since the values that the Wiener process can take belong in $ [0,T] $, i assumed that the values that x can take belong in $ [1,e^{-bT^2}]$, but is…
Assassinos
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How would you go about modelling this as a Markov chain?

There is two machines which break down at different rates, µ$_A$ for machine A and µ$_B$ for machine B. When they break down, a machine can be fixed by one of two repairmen. Assume that two repairmen, X and Y, have different abilities, and they…
user851087
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Finding stationary Distribution

I need to know how to find the stationary distribution for this matrix: $$ Q= \begin{bmatrix} -2 & 2 & 0 & 0 \\ 1 & -2 & 1 & 0 \\ 0 & 1 & -2 &1\\ 0 & 0 & 2 & -2 \\ \end{bmatrix}$$ in three different ways. I only know one, which is to replace…
Akit
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Sum of hitting probabilities not equal to 1?

I have a Markov Chain with transition matrix \begin{equation} P= \begin{pmatrix} 0 & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4}\\ \frac{1}{5} & 0 & 0 & 0 & \frac{4}{5} \\ 0 & 0 & 1 & 0 & 0\\ \frac{5}{6} & 0 & \frac{1}{6} & 0 & 0\\ 0 & 0 &…
Clovis Nyu
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