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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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Hitting probability of simple Ito process

I have a noisy signal that I filter. I want to know what is the probability that this signal goes over some threshold $a > 0$ in an interval $t\in [0,T] $. In other words, I have some stochastic process $j(t) = \int_t^\infty f(\tau-t)…
Subap
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How do you define the filtration sigma algebras of a branching stochastic process?

Consider the following problem: at time $t=0$ a real valued stochastic process $X_{t}$ starts. At some random time $\tau>0$, $Y_{t}$ branches out of $X_{t}$ such that $Y_{0}=X_{\tau}$. After $\tau$ both processes continue to evolve to time $T$. In…
mn abdelghani
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Autocovariance function of stationary process

I read that for any stationary process the autocovariance function approaches 0 as the lag goes to infinity, but I can't find any proof of this concept. Is this really a general statement not depending on the kind of process? Thanks!
user381425
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What is the mathematical approach for determining the limit of predictability

Some physical phenomena, like turbulence, wind-driven ocean waves, and ship motion, can be demonstrated to be "random" in the long period. However, for short periods, the phenomena can be predicted accurately. My question is, for how long can a…
John
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Expected value of 2 random processes with Gaussian distribution and DSPP

I try to derive the expected value of $ E[S(t)X(t)] $ where S(t) is a Gaussian random process with mean M(t) and variance V(t), X(t) is a doubly stochastic Poisson process(DSPP) with intensity $$ \lambda (t) = KS(t) $$ where K is a constant. I…
silent_man
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If $S,T$ are stopping times, then is it necessary that $S-T$ is a stopping time?

If $S,T$ are stopping times in discrete-time, then I know $S+T$ is a stopping time. Is it necessary that $S-T$ is a stopping time? Intuitively, I can see that $S-T$ involves some events happening in the future, and hence not a stopping time. Can…
user430647
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Mean equations of species in a compartment-based diffusion model

Reading the Stochastic Modelling of Biological Processes Lecture Notes on modelling diffusion as a chain of chemical reactions (in Section 10.2, page 45), I'm struggling to flesh out the details left to the reader of multiplying the chemical master…
Ste Rose
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Expectation zero for stochastic process, with independent intervals

Let $\{X_k\}_{k\in[0,T]}$ be a stochastic process adapted to the filtered sigma algebra $\sigma_k$. I wonder, does the following equal zero? $\Bbb{E}[X_t-X_s|\sigma_s]$. I can show that it is true easily if $X$ is a Brownian motion process, but I…
Jason
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How to bound the total number of particles at time $t$ on a linear system. A question concerning Liggetts book Interacting Particle Systems

In Liggett's book, interacting particle systems, on pages 425 426 one reads and just before we see The proof of $E[\eta_t(u)]\leq (e^{Bt}\eta)(u)$ is left to the reader. Here is my attempt: $$E[\eta_t(u)] = \sum_{k=0}^\infty E[\eta^k_t(u)]…
Conrado Costa
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One question about proof of martingale representation theorem

Does any one know which book I can find the proof of martingale representation theorem in detail? I.E. Any $F_B$ adapted local martingale M is continuous and can be written as a integration of Brownian. There is a proof on my notes. It says there…
XXX11235
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Approximation of stochastic differential equations

Consider the two following real Stochastic Differential Equations (SDE) starting from the same initial condition: $$dx_t = f(x_t)dt + \sigma dB_t$$ $$dy_t = f(y_t)g_{\epsilon}(y_t)dt + \sigma dB_t$$ where $f$ and $g_{\epsilon}$ are such that there…
mellow
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I have some questions about structure function in Stochastic processes and random field.

First of all,thank you for your help! I have some questions about structure function in Stochastic processes and random field. Based on the definition of structure function in Stochastic processes and random field, the formula can be written…
张兴民
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How to obtain this exponential martingale? A question concerning a paper from Stroock and Varadhan

In page 364 of the article Diffusion processes with continuous coefficients I (Stroock Varadhan - 1969), one finds in lemma 3.5: The question is: how do we prove that $$ Y^s_\theta(t) = \exp \big\{ \langle \theta, \eta(t)\rangle - \frac{1}{2}…
Conrado Costa
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A question about the answer of another question

In the following link, Distribution of Sum of Brownian Motion and Integrated BM, it is said that 1- $W(t) + \int_0^T W(t) dt$ does not exist, why? 2- What topic should I read or learn to understand whatever "Saz" has commented under the question?…
Tayebe
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Family of probability measures

I need a characterization of a probability space wherein the probability measure is changing. I am not a mathematician, and do not know about stochastic processes, but I've been working through these notes:…
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