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.

16128 questions
1
vote
0 answers

what are conditions for satisfying $\sup_t M^2_t<\infty$ where $M_t$ is locally square integrable martingale?

what are conditions for satisfying $\sup_{t\in[0,T]} M^2_t<\infty$ a.s. where $M_t$ is locally square integrable martingale? I have no idea, besides using Doob's maximal inequalities, but they give bounds for expectation. may be some conditions…
c-walk
  • 447
1
vote
1 answer

How to solve forward equation for a continuous-time Markov chain?

Given the transition rate matrix of a CTMC as $G$, I was wondering how the forward equation $P'(t) = P(t) G, P(0)=I$ is usually solved for the transition matrix $P(t)$? Some book says the solution has the form $P(t) = exp\{tG\}$. Since exponential…
Tim
  • 47,382
1
vote
1 answer

Uniformly distributed variable

I've stumbled upon this correlation between two different variables thread and I can't understand this equation: $$\mathbb{E}\cos(2\Phi+\lambda \cdot (s+t)) = \frac{1}{2\pi} \int_0^{2\pi} \cos(2x+ \lambda \cdot (t+s)) \, dx\text{ since }\Phi \sim…
atomoutside
  • 493
1
vote
0 answers

Proving the strict stationary of $X(t)=e^{-t/2}W(e^t)$

Let $W(t)$ be a Wiener process with a parameter $\alpha=1$ and a process $X(t)=e^{-t/2}W(e^t)$. Show that X(t) is stationary in the strict sense. Resolution attempt: $$E(X(t))=E(e^{-t/2}W(e^t))=e^{-t/2}…
Pedro Gomes
  • 3,891
1
vote
0 answers

Locally bounded adapted process

The definition for Locally bounded is utilized in the following link. Which states that for the process $\Phi:[0,T]\times\Omega\to H$ ($H$ is a Hilbert space), $\Phi$ is locally bounded provided $$ \sup_{\Omega} \big\Vert…
raijin
  • 155
1
vote
2 answers

Relationship between moments

Suppose a random variable $\mathbf{X}$ with some statistical distribution. Let's say the $E[\cdot]$ is the expectation operator. Is it possible to establish a relation between $E[ \mathbf{X}^n]$ and $E[ \mathbf{X} ]^n$? I would like to see something…
Odds'n'Ends
  • 119
1
vote
1 answer

Continuous a.s. process

In Ross's Stochastic processes: A stochastic process $\{X(t), t \geq 0\}$ is said to be a Brownian motion process if $X(0) = 0$, $\{X(t), t \geq 0\}$ has stationary independent increments, and for every t > 0, $X(t)$ is normally…
Tim
  • 47,382
1
vote
0 answers

sum of moving average processes

Define the two moving average processes $$u(t)=a_0e(t)+a_1e(t-1)+\dots a_Ne(t-N)\qquad e(t)\sim WN(0,1)$$ $$v(t)=a_0\eta(t)+a_1\eta(t-1)+\dots a_N\eta(t-N)\qquad \eta(t)\sim WN(0,1)$$ where $e$ and $\eta$ are independet white noises. Is it true that…
Davide Maran
  • 1,149
1
vote
1 answer

Computing the Autocorrelation function of a WSS process with H(w)=iw

Problem: Consider a wide sense stationary stochastic process $X(t)$, with zero mean and auto correlation function $R_X(\tau)$. Consider its transformation by a linear time invariant derivative filter, which is the first derivative of $X(t)$, that…
Pedro Gomes
  • 3,891
1
vote
1 answer

Blumenthal's 0-1 law as implication of strong Markov property

There are several versions of the Blumenthal's 0-1 law and proofs of it. Many of them are only done for Brownian Motions. I'm doing a self-study in Stochastic processes and found a version as follows: if $A\in \mathcal{F}_{0^+}^X$ then…
Rocco van Vreumingen
  • 321
1
vote
2 answers

Independence of holding time and next state in continuous-time Markov chain

In a continuous-time Markov chain, I was wondering why the holding time and the next state are independent? Are the independence a conditional one given the current state? Quoted from Ross's Stochastic processes: The amount of time the process…
Tim
  • 47,382
1
vote
1 answer

Response function of derivative filter

Problem: Consider a wide sense stationary stochastic process $X(t)$, with zero mean and auto correlation function $R_X(\tau)$. Consider its transformation by a linear time invariant derivative filter, which is the first derivative of $X(t)$, that…
Pedro Gomes
  • 3,891
1
vote
1 answer

Are probabilities / expected times of transition between any two states within an irreducible class same?

In a discrete time Markov chain, consider an irreducible/communicating class, Are the probabilities of ever transition between any two states within the class the same? If the class is recurrent, the probabilities of each state ever transitioning…
Tim
  • 47,382
1
vote
0 answers

From Langevin to Fokker-Plank equation

I derived from the Langevin equation for the overdamped case of Brownian particle with mass m and charge q in 3D space with a constant homogeneous magnetic field $\vec{B}$ directed along z-axis and with 3-component Gaussian White noise the equation…
user314849
  • 95
1
vote
0 answers

Questions about stationary and limiting distributions of a discrete-time Markov chain

For a discrete-time Markov chain, Is it right that there are no more than one limiting distribution, i.e., limiting distribution is unique if any? If the chain has more than one recurrence irreducible classes, then is it right that the…
Tim
  • 47,382
1 2 3
…
46 47