# Questions tagged [stochastic-processes]

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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### Interpretation of Lévy process with signed Lévy measures

Suppose that I have a non-decreasing, pure jump Lévy process of finite variation $X$ with Lévy measure $\pi$. The Lévy measure is then supported on $(0,+\infty)$. Suppose that the Lévy measure is a ...
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### Questions about Lamperti's criteria for stochastic process recurrence

I'm working through Lamperti's 1960 paper "Criteria for the recurrence or transience of stochastic process. I" (J. Math. Anal. Appl. 1(3–4), 314–330. DOI: 10.1016/0022-247x(60)90005-6) as ...
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### Book recommendation in functional analysis and probability

I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend? I'm looking for a book that has ...
1 vote
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### Boundary-condition-changing Operators for Free Boson BCFT with Dirichlet Boundary Conditions (or more general BCFTs)?

(NOTE: This is a crosspost from this Physics.SE post) Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. ...
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### Stochastic processes with variable dimensionalty

I am curious about stochastic processes where the dimensionality of the state space can grow over time which could have application to describe real world processes such evolution or technological ...
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### Pathwise Hölder continuity of Ito diffusions - is this result written anywhere?

Let $X$ be the solution to the multidimensional SDE $$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$ with $W$ a Brownian motion, and $\mu, \sigma$ Lipschitz continuous with $\sigma$ nowhere zero. I'm ...
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### Construction of a "Dirac" jump process

We work on a probability space $(\Omega,\mathscr{F},\mathbb{P})$ endowed with a filtration $\mathbb{F}$, and consider the positive line $[0,\infty)$. I am wondering if one can make sense of the ...
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### Is the average of a $\alpha$-Hölder process Hölder continuous of every order less than $\alpha$?

Let $X_t$ be a stochastic process on $[0, 1]$ that is almost surely Hölder continuous of order $\alpha > 0$, and almost surely uniformly bounded by some deterministic constant. It is not hard to ...
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### Integral functional minimal value problems

\begin{align} & F_n(\theta)=\int_0^T f_{n}(t,\theta(t)) \, dt \\[6pt] & f_n(t,\xi)=\int_\Omega\mathcal{L}(X(t) + Z_n(t,\omega),Y(t),\xi (\omega)) + R_n(\xi(\omega)) Pd(\omega) \end{align} ...
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### Simulation of Markov processes with exponential timestepping

Let $(Y_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. Numerical simulation of $(Y_t)_{t\ge0}$ can be done in the following way: Choose an initial ...
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### Simple version of fourth moment theorem of Nualart and Peccati

I'm trying to understand this theorem, which gives several conditions for a sequence of random variables in the $n$-th Weiner chaos to converge to a normal law. I am finding even the statements of the ...
1 vote
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### Uniform distribution as argument for copula likelihood

I am reading a well-known paper about copulas by Chen and Fan (2006). Specifically, Proposition 4.2 (see attached), in which all the arguments are uniform $U_{t-1}, U_t$. However, when the copula is ...
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### Constructing Wiener process on a given probability space

This is just a short question, and may be to basic, but: is there a way to construct a sequece of independent wiener processes on a given probability spaces?
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### MDP Average Reward independent of Initial State

Consider a Markov Decision Process where the state space $S$ and the action space $A$ are continuous and compact. In state $s$, if action $a$ is chosen and the next state becomes $s'$, the ...
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