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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How the Ornstein–Uhlenbeck process can be considered as the continuous-time analogue of the discrete-time AR(1) process?

Wikipedia says The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process. I was wondering how the Ornstein–Uhlenbeck process can be considered as the continuous-time analogue of the…
Tim
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Distribution of hitting time of line by Brownian motion

I came across the following question: Let $T_{a,b}$ denote the first hitting time of the line $a + bs$ by a standard Brownian motion, where $a > 0$ and $−\infty < b < \infty$ and let $T_a = T_{a,0}$ represent the first hitting time of the…
Ben Derrett
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Considering Brownian bridge as conditioned Brownian motion

Let $B$ be a standard Brownian motion. Define a Brownian bridge $b$ by $b_t=B_t-tB_1$. Let $\mathbb{W'}$ be the law of this process. According to Wikipedia, A Brownian bridge is a continuous-time stochastic process B(t) whose probability…
Ben Derrett
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equilibrium distribution, steady-state distribution, stationary distribution and limiting distribution

I was wondering if equilibrium distribution, steady-state distribution, stationary distribution and limiting distribution mean the same thing, or there are differences between them? I learned them in the context of Discrete-time Markov Chain, as far…
Tim
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Concept of Random Walk

Reading the Wikipedia page for Random Walk, I was wondering what is the definition for a general random walk as a random process, so that all the concepts such as random walk on $\mathbb{Z}^d$, Gaussian random walk and random walk on a graph can…
Tim
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Ant in a circle

An ant is sitting in the middle of a circle of radius 3 meters. Every minute, the ant picks a random direction and moves straight 1 meter. On average, how long does it take the ant to leave the circle? If it helps calculations, you can disregard…
Neil
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Relation between independent increments and Markov property

Independent increments and Markov property.do not imply each other. I was wondering if being one makes a process closer to being the other? if there are cases where one implies the other? Thanks and regards!
Tim
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A sequence of order statistics from an iid sequence

Given a sequence of iid random variables $X_i$ (without loss of generality from $U(0,1)$), an integer $k \ge 1$ and some $p \in (0,1)$, construct the sequence of random vectors $Z^{(j)}$, $j=0,1,...$ in the following way.…
Hans Engler
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How do we sample from a Gaussian process

I have one particular question on Gaussian processes. A Gaussian process is fully characterized by $\mu$ and $\Sigma$. However, I do not understand how can we sample a (random) function from the so defined Gaussian process. For example, in a…
Mou
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Distribution of compound Poisson process

Suppose a compound Poisson process is defined as $X_{t} = \sum_{n=1}^{N_t} Y_n$, where $\{Y_n\}$ are i.i.d. with some distribution $F_Y$, and $(N_t)$ is a Poisson process with parameter $\alpha$ and also independent from $\{Y_n\}$. Is it true that…
Tim
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Time integral of a stochastic process

It seems that the time integral of a stochastic process $X_t$ in the interval $[0,T]$ gives us a random variable. My question is how do we define/calculate such a time integral. For example $$\int_r^s \exp(B_t)dt,$$ where $B_t$ is standard Brownian…
user10248
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Is $(B_{t}+t)^{2}$ a Markov process?

Let $B_{t}$ be a Brownian motion relative to a filtration $F_{t}$, is $(B_{t}+t)^{2}$ a Markov process? Thanks!
user7762
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Why does a random walk on a graph G converge to a uniform distribution?

From Wikipedia: the conductance of a graph G=(V,E) measures how "well-knit" the graph is: it controls how fast a random walk on G converges to a uniform distribution. I was wondering why a random walk on a graph G converges to a…
Tim
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How can I spot positive recurrence?

Can someone please explain to me the intuition behind Positive recurrence. What does it mean and why is it different to normal recurrence?
Rosie
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moment generating function of exponential distribution

I have a question concerning the aforementioned topic :) So, with $f_X(t)={\lambda} e^{-\lambda t}$, we get: $$\phi_X(t)=\int_{0}^{\infty}e^{tX}\lambda e^{-\lambda X}dX =\lambda \int_{0}^{\infty}e^{(t-\lambda)X}dX =\lambda \frac{1}{t-\lambda}\left[…
Marie. P.
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