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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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A question on the expectation of Counting Process

Let $N(t)$ denote a counting process, $X_1$, $X_2$, ... denote the inter-arrival time, and $S_1$, $S_2$, ... denote the arrival timestamp. So $S_1=X_1$, $S_2=X_1+X_2$, ... Let $T$ be a constant, so $N(T)$ denote the number of arrivals in time…
Bloodmoon
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Is it an increasing process?

On a probability space $(\Omega,\mathscr{F},\mathbb{P})$ with filtration generated by Brownian motion, there is a progressivley process $(A_t)_{t\in[0,T]}$. If for any stopping times $0\leq \sigma\leq \tau\leq T$, $A_{\sigma}\leq A_{\tau}$, then the…
shixuejun
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Cellular automata (Random walk)

Here is the context of my question below. I cite from "Some Rigorous Results for the Greenberg-Hastings Model" by Richard Durrett- Consider the following cellular automaton known as the Greenberg-Hastings Model: The state space is…
mathfemi
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Unconditional variance of approximately simple oscillator

I am considering the Ornstein-Uhlenbeck like system $$ dx_t = -\sinh(x_t) dt + \cosh(x_t) \sqrt{1 + 2\sinh(x_t)} dW_t $$ and wish to compute the unconditional mean $E[ x_t sinh(x_t)]$ amongst other things. Any suggestions?
Quant
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Prove a particular random process is weakly stationary

Let $\left(X_i\right)_{i\ge1}$ be independent identically distributed random variables such that $\mathrm{P}\left( X_i = -1 \right)=\mathrm{P}\left( X_i = 1 \right)=\frac{1}{2}$. Let $X(t)=X_{\left \lfloor t \right \rfloor}$ be a stochastic process…
peter
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What is a right continuous(or left continuous ) stochastic process?

I understand the concept of left and right continuity in a real line , but how is it defined for a stochastic process? Do we fix $\omega$ and check the continuity of the path as time evolves, or is it something else?
user3503589
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Stochastic processes with full memory

Markov processes are stochastic processes with no memory. How are called the stochastic processes with full memory? Can't found anything on google.
gab06
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A Poisson process question

I saw an old post here, claiming that for a Poisson Process $X(t)$: $P[X(t) - X(s) = 1 \mid X(t) = 4]=\frac{4(t-s) s^3}{t^4}$ Am I missing something essential about stochastic processes, probability or the Poisson process? If I would have tried to…
Ana M
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How to prove that a jump diffusion has infinite total variation?

We have a jump diffusion: $X_t=bt + \sigma W_t + Y_t$ where b is the drift parameter, $\sigma$ the diffusion parameter, $W_t$ a Wiener process and $Y_t$ a CPP (compound Poisson process). We know that $W_t$ has infinite total variation (since its…
BGa
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Long memory of stochastic differential equation

It is well known that the solution to an ordinary stochastic differential equation has the Markov property so that if one tries to model some kind of long memory process one has to instead consider SDE's driven by e.g. fractional brownian…
htd
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Random process $X(t) = 10 \cos(Wt + A)$.

I am doing some exercises based on random process, but I can't find a way out on this: Let $X(t) = 10 \cos(Wt + A)$, where W is a Gaussian aleatory variable with parameters $N(10,2)$ and $A$ is uniform under $(0, 2\pi)$. Assuming that $W$ and $A$…
Bruno A
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Integral of Ito integral

I'm trying to calculate the distribution of $\int_{t_{n}}^{t_{n+1}}\sigma_{1}(\tau)\int_{t_{n}}^{\tau}\sigma_{2}(t)dW_{t}d\tau$ where $W_{t}$ is a brownian motion, that is $W_{t}|W_{t_{n}}\sim\mathcal{N}(W_{t_{n}},t-t_{n})\quad\forall t \ge t_{n}$…
user87889
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Renewal Process Stochastic

Suppose a machine operates with a certain component which comes in two different colors, purple and blue. When the component fails it is replaced immediately. The Probability a purple component is chosen during replacement is $\frac 34$, otherwise a…
John Lucas
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Branching processes extinction (homework)

This is my stochastic process course homework. I can solve (a)(b), which are easy to prove. But I have no idea about (c). Could you give me some idea?
user144374
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conditional probability poisson process

I am stuck on how to find the conditional probability of a poisson process. I know generally, if you have a poisson process with intensity parameter $\lambda$, then the conditional probability of having $m$ events in the first $t$ hours given $n$…
homegrown
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