0
votes
0answers
10 views

Functional representation of adapted jointly measurable stochastic processes

It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO. Let $X_t : \Omega \to E, \ t \geq 0$ be ...
3
votes
1answer
174 views

Unusual augmentation of a filtration

consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...
0
votes
1answer
85 views

Measurable functions lifted onto a space of point measures are measurable

I've been reading [1] and attempting to prove statements given without proof. In the paper the authors construct a measurable space of measures over a base space, and as an aside show an elegant way ...
3
votes
2answers
87 views

Is this generating family of a measurable space of point measures a pi-system?

I'm learning some probability and measure theory and working my way through the first few paragraphs of [1]. My question is perhaps too basic for Math Overflow, but I hope it is welcome here. Point ...
4
votes
0answers
155 views

Progressively measurable vs adapted

I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...
3
votes
1answer
115 views

Wiener measure of hitting sets A,B but not C (or easier hitting A but not C)

I am trying to formulate the measure of event $E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$, where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise ...
1
vote
1answer
201 views

question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see: http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...
2
votes
0answers
76 views

Existence of a conditional distribution

Let $X$ and $Y$ be standard Borel spaces and let $J$ be an analytic subset of $X\times \mathcal P(Y)$ where $\mathcal P(\Omega)$ is a set of probability measures on a Borel space $\Omega$ endowed ...
5
votes
0answers
275 views

Skorohod theorem (weak convergence) on a discrete setting

I have a question about the application of Skorohod representation theorem. The questions arises in this paper about robust hedging in mathematical finance. It is about the very last equation on page ...
2
votes
1answer
143 views

A question about stochastic kernels and invariant measures

Suppose that $E$ is a metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In addition, suppose that $p:E\times ...
3
votes
1answer
119 views

Generalization of Lévy's continuity theorem for nuclear spaces

I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions ...
1
vote
1answer
191 views

dual space of the subspace of the space of probability measures [closed]

I have a question which maybe so naive but I want to know the result about it. Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Then by some materiau such as ...
2
votes
1answer
156 views

Empirical estimator fot the total variation distance on a finite space

I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$): ...
4
votes
1answer
209 views

Coupling of non-probability/sub-probability measures

A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such ...
10
votes
2answers
693 views

Why do we want maps to be measurable (in countably-additive setting)

When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have ...
4
votes
0answers
65 views

Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...
4
votes
1answer
99 views

A terminal coalgebra of a certain functor on Mes

Let $\mathfrak C = \mathsf{Mes}$ be the category of meausurable spaces and measurable maps. For any object $X\in \mathfrak C_0$ we assign a measurable space $\mathcal P(X)$ whose elements $\mu$ are ...
2
votes
1answer
178 views

Coupling of vectors

Let $X = (X_1,X_2)$ and $\hat X = (\hat X_1,\hat X_2)$ be two random variables where $X_i,\hat X_i$ are taking values over the Polish space $E_i$ endowed with their Borel $\sigma$-algebras, where ...
3
votes
1answer
452 views

Topological conditions of Kolmogorov Extension Theorem

KET is often used to construct stochastic processes in continuous time when the state space is $\Bbb R^d$. As far as I am familiar with its proof, it uses standard monotonic class-like arguments ...
12
votes
1answer
756 views

Bochner integral of stochastic process = path by path Lebesgue integral?

After some helpful comments, I realized that I had to repost this question in a more systematic way. On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
11
votes
0answers
653 views

surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
2
votes
2answers
293 views

Family of Brownian Motions

I am trying to show the following statement Let $D\subset \mathbb{R}^2$ be an open and bounded subset. $\Pi=(P^x : x \in D )$ a Family of standard Brownian Motions started at $x \in D$. Then $\Pi$ ...
3
votes
1answer
587 views

Difference in probability distributions from two different kernels

Let $(E,\mathscr E)$ be a measurable space and $P,\tilde P$ be two stochastic kernels on that space. I wonder how the induced measures $\mathsf P_x$ and $\tilde{\mathsf P}_x$ differ on the space of ...
3
votes
2answers
279 views

Choice of predictable (or jointly measurable) eigenvalues and eigenvectors of nuclear-operator-valued stochastic process

Let $q^{ij}$, $i,j\in\mathbb{N}$, be predictable real-valued stochastic processes. Let $(e^i)$, $i\in\mathbb{N}$ be an ONB of a separable Hilbert space $H$. Assume that $Q=\sum_{i,j=1}^\infty ...
4
votes
3answers
690 views

Kolmogorov probability axioms without non-negativity condition

What is a minimal consistent modification of probability axioms to include negative values? Is it enough to use a minimal modification of axioms obtained by formal exclusion of non-negativity ...
2
votes
1answer
600 views

Regular Conditional Probability given a natural filtration of a stochastic process

OK, this is kind of re-posting, but I think I can clarify the question more, so it's worth a shot. Consider a real valued process $(X_t)_{t \leq T}$, cadlag on a probability space $(\Omega, ...
10
votes
3answers
1k views

Infinitesimal generators of stochastic processes

What's the $L^1$ analogue of Stone's theorem saying that any strongly continuous 1-parameter unitary groups has a unique self-adjoint generator? More precisely: let $X$ be a measure space ...
1
vote
2answers
595 views

Change of time or change of measure

Consider simple diffusion $dX_t = \sigma dw_t$ and a parameter $a>0$ and $X_0=x$. Let us denote $Y_t = X_{at}$ - thus we made a change of time. Let us denote an original measure as $P$. How to find ...