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Stopping time sigma-fields

Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence). Is it true that the associated sigma-fields satisfy $F_{S_n}\...
nullptr's user avatar
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1 vote
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Does weak convergence of filtrations preserve progressive measurability?

Suppose on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ I have a sequence of filtrations $\mathbb{F}^n =(\mathcal{F}^n_t)_{t \geq 0}$ generated by Brownian motions $W^n$ for each $n$, ...
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If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
0xbadf00d's user avatar
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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 ...
0xbadf00d's user avatar
  • 167
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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,...
Filtrask's user avatar