# Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

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### Probability of getting given pass generated by Gaussian process

Suppose I have a Gaussian Process $f \sim GP(\mu, k)$ with given mean function $\mu(x)$ and covariance function $k(x, x')$. I also have a trajectory $\textbf{p} = (p_1, \dots, p_n) \in \mathbb{R}^n$. ...
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### Ito-Formula for a Poisson-Process

I am new to Stochastic Theory and trying to understand (Prop 20.13) of this Article https://personal.ntu.edu.sg/nprivault/MA5182/stochastic-calculus-jump-processes.pdf (The Ito-Formula for a Poisson-...
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### Solution manual for "Continuous Time Markov Processes" by Liggett

Is there a solution manual for "Continuous Time Markov Processes: An Introduction" by Thomas M. Liggett? Thanks.
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### Behaviour of a spectral measure - question regarding certain terms used to describe its behaviour

Good morning, I'd like to ask a question based on this paper, which is about Gaussian Stationary Processes. One of the main concepts is the concept of a spectral measure. After Theorem 4.1 of this ...
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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 ...
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### Solve a quite easy Stochastic Differential Equation

Given $du=\mu dt +\sigma dB$ where B is an one-dimension standard Wiener process, then what's $u(t)$? Is there a general solution when $t\to 0$?
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### Calculating the expected hitting time of a specific birth and death chain

I am working with a specific birth and death chain, defined as follows. Consider a set of states $X = \{0,1,2,...,n\}$, where $x^* \in (0,n)$ is a recurrent state. Transition probabilities are defined ...
The infinitesimal generator of a standard Brownian motion (as Markovian process) in $\mathbb R$ can be computed with Af(x) := \lim_{t \downarrow 0} \frac{\mathbb{E}^x(f(X_t))-f(x)}{t} = \lim_{t \...