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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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Simple Renewal Process Question

The question reads: Suppose the lifetime of a component $T_i$ in hour is uniformly distributed on $[100, 200]$. Components are replaced as soon as one fails and assume that this process has been going on long enough to reach equilibrium. Suppose…
jrad
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probability (waiting time = infinity) for a poisson process

I am new here so if I violate any rule, please inform. Consider the stochastic process given by $\{ N(t) : t \geq 0 \}$ which is time homogeneous poisson process with arrival rate $\lambda$. Let $W_n$ be the waiting time, i.e, $W_n = \inf…
Random
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Exit time of interval by diffusion using Girsanov's theorem

Suppose we have a process $X$ with $dX_t=\sigma dB_t + \mu dt$, for constants $\sigma$ and $\mu$, started at $x\in (a,b)$, for some constants $a$ and $b$, where $B$ is a Brownian motion. We'd like to determine the probability that $X$ exits $[a,b]$…
Ben Derrett
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Limit probability for some Hitting time of a Feller Process

I wanted to know if it was true that if we are given a one-dimensional Feller process taking values in $\mathbb{R}$ and a hitting time $\tau_A=\inf\{t>0 s.t. X_t\in A\}$ with $A$ a open set (this to avoid measurability complexities but I would be…
TheBridge
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Optimal Stopping Example

I've got a very basic question about optimal stopping. It may have just been me, but I feel like my professor didn't do a great job of explaining the topic too well. I was hoping one of you would be able to step me through this simple…
jrad
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Finding Characteristic Function

Can anyone help me with this problem? The random variable $X_n$ takes the values $\frac{k}{n}$, $k=1,2,\ldots,n$, each with probability $\frac{1}{n}$. Find its characteristic function and the limit as $n\to\infty$. Identify the random variable of…
kira
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convergence in law of an exponential brownian motion

I have a queston about the convergence in law of the following stochastic processe: $$\left\{I_t=\left(\int_0^te^{B_s}ds\right)^{1/\sqrt{t}}\right\}_{t\geq 0}$$ with $\{B_t\}_{t\geq 0}$ is a standrad brownian motion. Prove that $I_t\rightarrow…
Higgs88
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Poisson Process Arrival Probability

Just a quick question regarding two Poisson Processes: Let $X_t$ and $Y_t$ be two independent Poisson Processes with rate parameters $\lambda_1$ and $\lambda_2$, respectively, measuring the number of customers arriving in stores 1 and 2,…
jrad
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Brownian motion & Wiener Process

First of all, I keep on finding different answers to this question: Is Brownian motion (BM) process the same as Wiener process?or the Standard BM is Wiener? My second question is about the integral of a BM. If a BM process is the input of an…
Tayebe
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Correlation function between two Ornstein-Uhlenbeck processes

Is it possible to obtain an analytical expression for the quantity $$ C(t,s) =\langle X_t Y_{s} \rangle, $$ which I would call correlation function (but maybe I'm wrong), for two zero-mean Ornstein-Uhlenbeck processes $X_t$ and $Y_t$, i.e.…
Pincopallino
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Oblique bracket, stochastic integral

Let $X_t=\int_0^tsW_s^2dW_s$. How to set $<\int_0^tW_scos(s)dX_s>$? is it: $<\int_0^tW_scos(s)sW_s^2dW_s>=\int_0^ts^2W_s^6cos^2(s)ds$
wiwnes691
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Doob's maximal inequality

Let $X$ be a cadlag $L^{p}$ martingale ($p>1$). Let $q$ be the Hölder conjugate of $p$. Let $F$ be a finite subset of $[0,t]$. The following claim appears in a proof of Doob's maximal inequality that I am reading: $$ \mathbb{E}\left(\max_{s\in…
user303911
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Weak stationarity of a process modulus

Is it true that if a continuous-time stochastic process $X_t$ is weakly stationary then $|X_t|$ is also weakly stationary?
user5835
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Construct a martingale with two given distributions.

This is a follow up of another post: Construct a martingale with a given distribution? Given two distributions $f_1(\cdot)$ and $f_2(\cdot)$ on $\mathbb R$, under what condition can we construct a martingale $X_t$, such that $X_1$ has distribution…
Jay.H
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Some questions about continuous local martingales

Let $M=(M_t)$ be a continuous local martingale, i.e. it exists a sequences of stopping times $(\tau_n)$ converging to $+\infty$ $P-$a.s. such that $M^{\tau_n}=(M_{\tau_n\wedge t})$ is a martingale. Now I was able to show that that every continuous…
user20869
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