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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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What is the probability that $S \leq T$?

Suppose $S$ and $T$ are independent exponential random variables of parameters $\alpha$ and $\beta$ respectively. What is the probability that $S \leq T$ Could anyone explain to me how to calculate this? $$ \mathbb{P}(S \leq T). $$ But we only…
Nescrio
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Generlizing a result of Poisson point process

This is a follow up of What does it mean for a Poisson point process $\Phi$'s points in $A$, conditioned on $\Phi(A)=k$ to be uniform? My question: Are there similar results for the more general renewal process? To be specific, let's assume arrival…
Jay.H
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Is $\{T_a < T_b\} \perp \{T_b \leq T\}$ true ? $T_a =\inf\{t\geq 0 \mid B_t = a\}$

I'm trying to check wether two sets related to the Brownian motion are or not independent. Let $(B_t)$ be a Brownian motion and $a,b \in \mathbb{R}$, $a \neq b$. Consider the first time when the Brownian motion gets to a certain point: $$T_x =…
user90803
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A markov process which is not strong Markov (follow up)

This is a follow up discussion of https://mathoverflow.net/q/43833 The examples there are interesting, but it seems that they all rely on an "ambiguous" transition function, i.e., there is some state $x$, $p(X_t\in A|X_0=x)$ may have different…
Jay.H
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Is this a proof of $E\int^b_a f dZ = 0$?

I'm trying to prove that the expected value of the stochastic integral (integral w.r.t to the centered, right-continuous with orthogonal increments) is zero. I've come up with a very simple argument that I feel might be wrong. If I denote the…
Dahn
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Why is $Y e^{i\omega t}$ called an "elementary process"

In my seminar, we encountered the process: $X_t = Y e^{i\omega t}$, $t\in \mathbb R$, $Y$ centered complex random variable with $E|Y|^2 = \sigma < \infty$, $\omega \in \mathbb R$ is a constant and the supporting materials called this process an…
Dahn
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Generating functions and tumour cells

I have got G(s) = p+ $\ rs^2$ a p.g.f for a family size. Let K be the total number of tumour cells produced from a single original tumour cell Let R(s) = P[K=0] + sP[K=1]+.... be the p.g.f of this number Let the number of immediate descendants of…
Rosie
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2D simple symmetric random walks

I have a theorem which says that 2D symmetric random walks are recurrent. I understand this, the way my lecturer shows it is as follows; $\ p_{(0,0),(0,0)}^{(2n)} = (p_{0,0}^{(2n)})^2 = ((2nCn)(1/4)^n)^2 $ He then uses stirlings formula to say…
Rosie
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With what a and b this IS a Wiener Process

$aX_{t/9}-bX_{t/4}$, knowing that $X_t$ is a Wiener Process. This was already twice in the exams I failed, very likely it will be in todays exam in 2 hours. Can somebody help me with this one?
Andrius Naruševičius
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How to count the mean of the Poisson process

I am given Poisson process $X_t$ which has intensity $\lambda$. Is there a way to count the mean for it? I am terribly confused. I am able to count the mean of a variable which has a Poisson distribution (and that is $\lambda$), but as I understand…
Andrius Naruševičius
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How do you check whether the process is a Poisson process?

I am given $X_t^2$ to be checked, where $X_t$ is Poisson process. How do I check? What are the properties that need to be checked?
Andrius Naruševičius
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Stochastic process absorption probability

Let the transition matrix for a stochastic process be $$P=\begin{bmatrix} 1 & 0.7 & 0 & 0 & 0 \\ 0 & 0 & 0.5 & 0 & 0 \\ 0 & 0.3 & 0 & 0.65 & 0 \\ 0 & 0 & 0.5 & 0 & 0 \\ 0 & 0 & 0 & 0.35 & 1 \end{bmatrix}$$ and let $\mathbf{x}_0$ be the initial…
MathMag
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Integration with respect to continuous, Local martingale - A definition issue

Let $M$ be a local martingale, This means that there is a sequence of Stopping times $S_n$ $\lim_n S_n = \infty$ almost surely and $M^n_t = M_{t \wedge S^n}$ is a martingale. Now we would like to define the integral with respect to a local…
Conrado Costa
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Weak convergence for composition of cadlag stochastic processes

Let $(X^n_t)_{t \geq 0}$ be a sequence of cadlag stochastic processes, that is $X^n$ is a random element in the Skorokhod space $D([0, \infty), \mathbb{R}$) for each $n \in \mathbb{N}$. Also for each $n$ let $T^n_t$ be a cadlag stochastic process in…
yada
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Brownian motion Convergence

If $X$ is a standard 1d brownian motion and $M_t$ $= \mbox{max}\{X_S: 0 \le s \le t\} $, what can we say about $M_t/t$ as $t \rightarrow \infty$? Mainly, what can we say about the behavior of this martingale? My attempt: P(Msubt > a) = 2P(B(t) >a),…
mary
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