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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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An exchangeable version of the Indian Buffet Process

Indian Buffet Process: In the mathematical theory of probability, the Indian buffet process (IBP) is a stochastic process defining a probability distribution over sparse binary matrices with a finite number of rows and an infinite number of columns.…
Brian Uldrich
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"Minimal" definition of Wiener process

Almost everywhere (for example on Wiki) the Wiener process is defined as a process $W(t)$ such that: $W(0)=0 $; It has independent increments; The increments in an interval $\Delta t$ are gaussian with expected value 0 and variance $D\Delta t$; The…
Antonio19932806
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Viewing a stochastic process as a probability measure

Bernt Øksendal writes in his book Stochastic Differential Equations (page 11) that a stochastic process is a probability measure $P$ on the measurable space $((\mathbb{R}^n)^T, \mathcal{B})$. The sample space $(\mathbb{R}^n)^T$ is the set of all…
harisf
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Please tell me this stochastic process's name

Firstly, I'm not good at English, sorry. Please tell me this stochastic process's name $x_t=x_{t−1}−λ\operatorname{sgn}(x_{t−1})+ε$, where $ε$ is Gaussian noise and $λ>0.$
tn1021
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Show that a Wiener Process $X(t)$ is a normal process?

Show that a Wiener Process $X(t)$ is a normal process? Consider an arbitrary linear combination: $$\sum \limits_{i=1}^{n} a_iX(t_i) = a_1 X(t_1) + a_2 X(t_2) + \cdots + a_n X(t_n)\tag{1}$$ where $0 < t_1 < \cdots < t_n$ and $a_i$ are real constants,…
pico
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Is Gaussian process at random times also a Gaussian process?

I have a question I am not sure whether my answer is correct or not: I have a gaussian process $X_t$ (for $t\geq0$) and a random function $s(t):[0,\infty)\rightarrow[0,\infty)$. Does $X_{s(t)}$ (for $t\geq0$) also a Gaussian process? My answer is…
Moshefr
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Showing a conditional Poisson Process does not have independent increments

Suppose $N\left(\cdot\right)$ is a Poisson Process with rate $1$ and $Z$ is a positive non-constant random variable, define $N_{Z}\left(t\right)=N\left(Zt\right)$. I know that conditional on $Z$ this is a Poisson Process with rate $Z=z$ but without…
Serpahimz
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Pure birth process

Let $\lambda_n$ denote the arrival rate for a pure birth process of size $n$. Let $P_n(t)$ denote the probability of population size $n$ at time $t$. A stochastic process is dishonest if $\sum\limits_{j=0}^{\infty}P_j(t)<1$ for any $t$. If…
Mark
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What is $\mathcal F_{t^+}=\bigcap_{u>t}\mathcal F_u$ ? Is $\mathcal F_{t^+}$ anticipative?

Let $(\mathcal F_t)_{t\geq 0}$. I know that $$\mathcal F_{t^+}=\bigcap_{u>t}\mathcal F_u,$$ is a filtration. I know that if the filtratino is right continuous, then $\mathcal F_{t^+}=\mathcal F_t$. But suppose it's not, what is exactely $\mathcal…
Walace
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test for fractional brownian motion

Given a time series (real data), how can I check if this time series is a fractional brownian motion? I mean, I would start to check for stationary increments. Is it enough to do exploratory plots? Is there any test around, e.h. H0: this stochastic…
steffi
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$\Pr\{Z_S=\epsilon\}$ where S is a stopping time.

I have a process $Z_t$ that satisfies $\mathrm{d} Z_t = \dfrac{a}{Z_t}\mathrm{d}t +\mathrm{d}W_t$, Then I am given that $S=\min\{s>0: Z_s=\epsilon \text{ or } Z_s=\alpha\}$, then I need to find what is $\Pr\{Z_S=\epsilon\}$. Any help will be…
ActuariallyImpaired
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Derivation of Fokker Planck equation from master equation

I'm trying to derive a Fokker Planck equation for a stochastic process. Consider a population of size n, in which births and deaths are random events which occur at rates $\lambda$ and $\mu$ respectively. Let $P(n,t)$ denote the probability of the…
RBekker
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Brownian motion

Verify that $E( X(t) X(s) | X(0)=0 ) = min (t, s)$, where $X(t)$ is standard Brownian motion. I don't know where to start. Thanks!
user9636
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Show the memoryless property is equivalent to other expressions.

A random variable $X$ has memoryless property if $P( X \le s + t | X \gt s) = P(X \le t)$, $s, t \gt 0$. Show that the property above is equivalent to $P(X \gt s+ t | X \gt s) = P(X \gt t)$ and to $P(X \gt s +t) = P(X \gt s)P(X \gt t)$. I really…
Swayy
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Understanding two definitions of diffusion processes

From Wikipedia: A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker-Planck equation. I was wondering if "for which the Kolmogorov forward equation is the Fokker-Planck…
Tim
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