All Questions
Tagged with stochastic-processes measure-theory
57 questions with no upvoted or accepted answers
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Constructing uncountably many independent random variables with same distribution from Brownian motion?
It is well known one cannot construct uncountable many independent random variables on $([0, 1], \mathcal{B}[0, 1], \lambda)$. ($\lambda$ Lebesgue measure.)
Also, one can clearly construct infinitely ...
user152718
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“Chapman-Kolmogorov”-convolution vs. smoothness
Let $K:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a so-called "integral-kernel": we certainly require $K(x,.)$ and $K(.,y)$ to be Lebesgue measurable for almost all $x,y \in \mathbb{R}^n$. An ...
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Derivatives in unusual support domains
Originally posted on Math.StackExchange, here, but I was advised to post it on MathOverflow as it is a research question. Now two final, great answers have been posted, see on Math.StackExchange.
I ...
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How can we show that the total variation distance of $X_s$ and $Y_s$ is bounded by the distance of $(X_t)_{t\ge s}$ and $(Y_t)_{t\ge s}$?
Let $(X_t)_{t\ge0}$ and $(Y_t)_{t\ge0}$ be real-valued time-homogeneous Markov processes with a common transition semigroup $(\kappa_t)_{t\ge0}$. Let $\mathcal L(Z)$ denote the distribution of a ...
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If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?
Let
$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces
$(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$
$(N_t)_{t\ge0}$ be a $\mathbb ...
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What is the sigma field of the derivative of a process?
When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...
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(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes
Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$,
\begin{equation}
\int_0^te^{-\lambda ...
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