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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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Finding Simple Branching Process Recursive Generating Function

Say we have the following PMF for a simple branching process and want to find the eventual extinction probability, $$ P(Z_{1,1} = 0) = 0.25$$ $$ P(Z_{1,1} = 1) = 0.25$$ $$ P(Z_{1,1} = 2) = 0.50$$ Then, we can easily…
user130912
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Process with Markov property but not strong Markov property

I'm trying to find a simple example of a stochastic process with the Markov property, but not the strong Markov property, to give me an intuitive understanding of the distinction between them. All the processes I can think of off the top of my head…
Ben Derrett
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How to calculate the variance and the expected value of the MLE of stochasts $(X_1,\ldots,X_n)$ with density $f_\mu(x)=e^{x-\mu}1_{[0,\infty)}(x)$

So far i've got that the MLE is $\mu'= \min\{X_1,\ldots,X_n\}$ now i'm supposed to construct $$F_\mu'(x)= P(\mu'\le x)$$ The problem is that i don't understand how to construct this function for a stochast which is in the form of $\mu'$. I know…
Kees Til
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Continuous excessive (supermartingale) function

Consider a discrete time Feller Markov process $X$ on $\mathbb R$ with a kernel $K(x,dy) = \xi(x,y)dy$ and the transition operator $$ \mathcal Pf(x) = \int\limits_{\mathbb R}f(y)\xi(x,y)\,dy. $$ Here $\xi$ is a continuous and strictly positive…
SBF
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What is the hitting time distribution for white noise?

What is the distribution of the hitting time for a stochastic process $(W_t)_{t\in [0,T]}$, where $W_t$ are i.i.d. Gaussian random variables? How about in cases, in which $W_t$ are i.i.d. with a common distribution other than Gaussian?
user43130
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integration form of random variables

I'm reading Bernt Oksendal's "Stochasticc Differential Equations" and got confused at the form of integration on random variables. In chapter 2, page 9 (sixth edition) it (consider a given probability space $(\Omega, \mathscr{F}, P)$) defines the…
athos
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Definition of $L^p$-spaces of random variables

I'm reading Bernt Oksendal's "Stochasticc Differential Equations" and got confused at the definition of $L^p$-space of random variables. In chapter 2, page 9 (sixth edition) it says (consider a given probability space $(\Omega, \mathscr{F}, P)$): If…
athos
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On the characteristic of a semimartingale

Suppose $\{U_j\}$ are random variables. Consider the process $$ Y_t = \sum_{k=0}^{\sigma_t}U_k, $$ where $\sigma_t$ is some suitable increasing process taking values in $\mathbb{N}$. The jump times of $Y_t$ are $$ \tau_k = \inf\{t:\sigma_t\geq…
AlmostSureUser
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Does the linear combination of the elements of a basis with standard normal random coefficients always give a white noise?

Let us consider an orthonormal basis, e.g. $\{\psi_n(t)\}_{n\in\mathbb Z}$ of the space of real-valued functions that are square integrable $L^2(\mathbb R)$. Does the linear combination of the elements in $\{\psi_n(t)\}_{n\in\mathbb Z}$ with…
Mark
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Expectation and power spectrum density of a stochastic system

I am working on a problem and hoping to receive either confirmation that my work is correct or guidance on where I am going wrong. The problem is shown below; Here is my work; a) $$ E\{ X(t) \} = \eta_{_X} (t) = 2 E\{ Z(t) \} + 3 \lambda = 2…
AdamsK
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Filtrations and right continuity

Here is a question that has been posted on another forum (in french) that I couldn't answer and about which I'm really curious. So here it is, let's be given as a state space $\Omega=\{f\in C([0,1],R) s.t. f(0)=0\}$ associated with the borelian…
TheBridge
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Prove $\varphi(t_1,\,t_2,\,\lambda)=Ee^{i\lambda\left(\xi_{t_2}-\xi_{t_1}\right)}$ is continuous in $\left(t_1,\,t_2\right)\in \left[0,\,1\right]^2$.

Assume $\xi_t$ is a stochastically continuous random process on $\left.[0,\,\infty\right.)$. Prove $$\varphi(t_1,\,t_2,\,\lambda)=Ee^{i\lambda\left(\xi_{t_2}-\xi_{t_1}\right)}$$ is continuous in $\left(t_1,\,t_2\right)\in \left[0,\,1\right]^2$ where…
Knt
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Can Kolmogorov backward and forward equations hold at the same time?

I'm bit confused at Kolmogorov forward and backward equations: can they hold at the same time? I understand that Kolmogorov forward equation (KFE) describes the evolution of the pdf of $X(t)$ given a starting pdf (the initial condition) $$…
athos
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Hitting time inequality

In today's lecture, instructor said that $\mathbb{P}\{T_1>t\}\le ae^{-bt}\,\forall t>0$ for some positive $a,b$, as if it is a "trivial" fact. Although I know $\mathbb{E}[T_1]<\infty$, but I couldn't prove it after some try. $T_1=\min\{t:…
user998352
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Proof that a $\mathbb{F}$-Wiener-martingale is an $\mathbb{F}$ martingale

I am confused with the following proof, Claim: Let $W$ be a Wiener process adapted to a filtration $\mathbb{F}=\{\mathcal{F}_t\}_{t \geq 0}$, and suppose that $W_t-W_s$ is independent of $\mathcal{F}_s$ for all $s
tealing123
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