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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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Space of equivalence classes given by the autocovariance functions

The autocovariance function of stochastic process is pretty powerful tool to determine some property of the process -- mean square continuity, Riemann integrability and partially weakly stationarity and existence of the spectral density. So I am…
w8M
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Intensity of random Poisson walk

Let $M$ be a Poisson cloud with intensity $dxdy$ over $\mathbb{R}^2$. We denote $M_{\theta}$ and $M_R$ as the Poisson clouds on $R/2π \mathbb{Z}$ and $\mathbb{R}^+$, respectively, obtained from M by considering the angles and distances from the…
user1240705
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PSD of a realisation vs PSD of WSS Ergodic process

This is rather a straightforward question. Suppose a stochastic process is wide sense stationary (WSS) & it is ergodic as well. Is the power spectral density (PSD) say of a realisation, $\hat{S}(\omega)$, of this process equivalent to PSD of the…
Ricky
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Why is Bilateral Gamma process a pure jump process?

Recently (~10 years ago), Kuchler&Tappe have set up a new stochastic process called Bilateral Gamma process. This process is defined through its increments: $$\forall t\geq s, X_t-X_s\sim \Gamma_{BG}(\alpha_+(t-s), \lambda_+, \alpha_-(t-s),…
NancyBoy
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Ornstein-Uhlenbeck Operator as an Infinitesimal Generator of a Stochastic Process

This question continues the discussion from an earlier post on this website found here : Why the operator is termed as Ornstein–Uhlenbeck operator? I am interested in the relationship between the Ornstein-Uhlenbeck operator $L f(x)=\Delta f(x) -…
Timothy
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Basic Poisson Process

This is Exercise 2.1 of Varadhan's Stochastic Processes. Let $\tau_i$ be a sequence of independent identically distributed random variables with a common exponential distribution $e^{-\tau}\textrm{d}\tau$. Define $N(t) = 0$ if $0\leq t < \tau_1 …
M49
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First passage time of a discrete-time continous variable stochastic process.

There is a stochastic process $X(t)$, it is a discrete-time continuous-variable process. I can only know the distribution of $X(t)$ at discrete times $t = \tau, 2\tau, ...$. For such a process, given a constant threshold w, I concern the first…
YAN BINGXIN
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Proof of assertion in the book Weak convergence and Empirical processes

In page 290 of book Weak Convergence and Empirical Processes by Aad van der vaart, Jon Wellner, the last two lines say that the second assertion of theorem 3.2.5 can be proved by minor simplification of the preceding argument, I want someone to…
XL Deng
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Mean reversion with different reversion parameters when current value is above or below the mean?

Suppose we are modeling a process of P as: dP = h P (M - P) dt + s P dz where h is the mean reversion parameter and M is the long term mean level. In certain cases, you may revert to the mean more quickly when P is above the mean vs. when P is below…
JoeBass
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Distribution of jump-drift process

suppose I have a jump-drift process $y_t$ such that $y_t$ drifts at a rate $-\beta y_t dt$ and at some poisson rate $\lambda$ $y_t$ is drawn from a normal distribution with mean zero and variance $\sigma^2$. What is the distribution of…
David
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Probability of a wiener process greater than a time varing constant

Let $Xt$ be a Wiener process, and $f$ is a continuous funtion with $f(0)<0$. Is there any way to calculate $Prob(Xt-f(t)>0, \forall t \in (0,T))$?
di bao
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Waiting time distribution for combined stochastic processes

I have two particle detectors (random neutron source), for each I know the arrival waiting time distribution $PDF_1(t)$ and $PDF_2(t)$. They are non-exponential, since the detectors have dead-time, and their dead-times are different. Nevertheless,…
Vova N
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2d random walk average hitting time for 4 corners

I have found out that 1 and 2 dimensional random walks are recurrent. But considering the scenario of an infinite 2d space where the "targets" lie on the point (±2,±2). Is it possible to reach one of these 4 points within a limited time? If yes,…
leo2348
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Definition of Stochastic Process

My question is about the definition of a stochastic process. From the definition of the majority of textbooks, we know a stochastic process is defined as a collection of random variables defined on a common probability space $(\Omega, \mathcal{F},…
Alison
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Can extinction probabilities be negative?

Bacteria reproduce by cell division. In a unit of time, a bacterium will either die (with, $p=\frac{1}{4}$), stay the same (with, $p=\frac{1}{4}$), or split into 2 parts (with, $p=\frac{1}{2}$). The population starts with 100 bacteria at time $n =…
Niamh Lavelle
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