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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.

16128 questions
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Harmonic oscillator with Rayleigh pdf noise

The problem of the harmonic oscillator with a random small perturbation is known. We can write the equation as: $$\ddot{x}(t)+\eta^2x(t)=0$$ where $\eta^2=\omega^2+\epsilon{W_t}$ with $\epsilon>0$ and $W_t$: white gaussian noise. My question is: if…
Riccardo.Alestra
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Expected value of time integral of a gaussian process

While working on a problem I've stumbled upon some expected values of time integrals of Gaussian stochastic processes. Some of them were addressed this question, but I have found also this one $$\left\langle\int…
Pincopallino
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Proving an equality for a Markov Chain

Let $X_n$ be an irreducible Markov chain taking values from the natural numbers (including $0$). Let $g,f$ be functions with $\mathbb{N}$ as domain (including $0$) such that $f = g + Pf$, where $P$ is the transition matrix for $X_n$. A note of…
user90593
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two definition of the Poisson process

I read the definition of Poisson process in Shereve's "Stochastic Analysis" which is constructed explicitly: (A) step 1:construct iid exponential r.v.$\tau_i$ with parameter $\lambda$ step 2:define $S_n=\sum_{k=1}^n \tau_k$. step 3:define…
Lookout
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Switching from one homogeneous Poisson process to another

Suppose that $\{N'(t) :t \geq 0\}$ and $\{N''(t) :t \geq 0 \}$ are the counting processes of two independent homogeneous Poisson processes on the line, each with with rate $\lambda$. Fix an integer $n\geq 1$ and let $S'_n$ be the time of the $n$-th…
James R.
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How can I know by inspection that a process is WSS?

I have some codes to generate three different Random Sequences: I am getting a 4x100 matrixes where 4 is the number of samples and 100 is the length of the process. I am getting these results: I know that in order a WSS if the mean is constant…
Peterson
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Existence of Truncated Brownian motion?

I am currently engaging a research that would really use your help. I am considering add a brownian-type shock to a "fraction" $\theta \in [0,1]$, for example $$d\theta_{t} = \sigma \theta_{t} dB_{t}$$ where $\sigma$ is a constant. However, the…
EconGuy
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Good resources on branching processes

I'm trying to understand branching processes. Do you know any good and written in a simple way resources / web pages / books. Free resources are welcome :).
Darqer
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What is the random process underlying this figure

I have the following random process and I would like to know what could be its underlying stochastic process: The closest I can get is by simulating the following: $ dX_t=\displaystyle\frac{dP_t}{k*Sin(t)} $ where $P_t$ is a Poisson process. In…
user13675
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Stochastic process vs Random process!

I am taking a course in stochastic process this time. As I read through a couple of books every one mentioned that stochastic process is also a random process. So, my confusion is why we call stochastic process just for a random process? For me…
user176667
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If $X_t, t \in T$ is separable random process, is event that it's sample path be continuous on $T$ measurable?

If $X_t, t \in T$ is separable random process (with values in metric space), an event that it's sample path be uniformly continuous on $T$ $$ \bigcap_{m \in \mathbb{N}} \bigcup_{n \in \mathbb{N}} \bigcap_{t_1 \in T} \bigcap_{t_2 \in K (t_1, 1 /…
Zoran Loncarevic
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Proof of finite expectation of renewal process (2)

I don't know if it is allowed here, to repost again his own question. I hope it is ok... I already asked this question here: Finite expectation of renewal process But I don't understand the last steps and I didn't get any answer in 4 months... And I…
mr_T
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Is there a conditional random variable?

Let $(\Omega,\Sigma,\mu)$ be a sample space. Let $F $ be a $\sigma$-subalgebra of $\Sigma$ and let $X$ be a real-valued random variable. So what does it mean $X/F$? How is this defined mathematically and books to read?
Juan Luis Fuentes Maguiña
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Change of measure for jump processes

I have some trouble understanding changes of measure for jump processes. I guess I'm missing some important bit of the theory. Consider a simple example. Let $N_t$ be a standard Poisson process with constant intensity $\lambda_t=1$ adapted to…
Julian Wergieluk
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Integrated gaussian process

I just want to know what kind of phenomenon a integrated gaussian process ($Y_{t}=\int_{0}^{t}X_{s}ds$ where X is a gaussian process) can modelize. Thanks.
kacou
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