All Questions
Tagged with stochastic-processes measure-theory
150 questions
5
votes
1
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774
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Question/References on the Skorokhod M1 topology
Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
CommunityBot
- 1
3
votes
0
answers
145
views
What is an example of a non-tight probability measure?
Billingsley (Convergence of Probability Measures, 1968) and van der Vaart and Wellner (Weak Convergence and Empirical Processes, 2023) discuss the concept of tight probability measures and use the ...
- 277
1
vote
1
answer
185
views
Sum of $X_k$ with $\mathbb{P}(X_k=\pm 1) = 1/2\pm 1/(2\sqrt{k})$
Let $\{X_k\}$ be a sequence of mutually independent random variables with
\begin{align}
\mathbb{P}(X_k = 1) & = \frac{1}{2} + \frac{1}{2\sqrt{k}},
\\
\mathbb{P}(X_k = -1) & = \frac{1}{2} - \...
- 269
2
votes
0
answers
80
views
Stability of Hölder constants of frozen Itô stochastic integrals
$
\newcommand{\RR}{\mathbb{R}}
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- 835
3
votes
2
answers
922
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On representing a continuous time Markov chain by a stochastic integral of a Poisson random measure
Let $Q=(q_{ij})$ be the transition rate matrix of a continuous time Markov chain $\{ X_t \}$ with countable state space $M$. Let $q_i = -q_{ii}=\sum_{j \neq i}q_{ij}$, and let $\Gamma_{ij}$ be defined ...
1
vote
1
answer
276
views
Is it true that $F(X_0, \cdot) = X_0 + \int_0^T \sigma(s, X_0) \, \mathrm d B_s$ a.s.?
$
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- 6,155
2
votes
0
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66
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Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?
$
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- 835
0
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0
answers
80
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Measurable Extension
Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions ...
- 136
25
votes
2
answers
14k
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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 ...
CommunityBot
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0
votes
0
answers
54
views
Reference request: "doubly empirical" measure associated to a random measure
I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In ...
- 869
1
vote
1
answer
107
views
Interchange the deterministic and stochastic integrals
We fix $T >0$ and let $\mathbb T$ be the interval $[0, T]$. Let $(X_t, t \in \mathbb T)$ be a continuous adapted process on some filtered probability space $(\Omega, \mathcal A, (\mathcal F_t)_{t \...
- 128k
3
votes
3
answers
491
views
Progressive measurability intuition from Bichteler's *Stochastic integration with jumps* book
In the Stochastic Integration with Jumps Bichteler gives a very intuitive definition of progressive measurability I've never seen before:
Although I like this intuition very much, I cannot find a ...
- 111
0
votes
1
answer
450
views
A complex question related to a certain convergence of Lévy measures
Consider the sequence of stochastic processes $(X_n, n \geq 1)$, where $X_n = (X_{t;n})_{t\in \mathbb Z}$ and:
\begin{equation}\label{I}\tag{SP}
X_{t;n} = \sum_{j=0}^\infty \theta_{jn} \varepsilon_{t-...
- 724
2
votes
1
answer
133
views
Can convergence in distribution necessarily be realised by almost-sure convergence?
Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each ...
- 5,474
0
votes
0
answers
161
views
Markov process with time varying transition kernels
I cross post this question from StackExchange as it may be more appropriate.
I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...
2
votes
1
answer
159
views
Measurability of two hitting times at the stopped $\sigma$-algebra
Let $\mathcal{F}=(\mathcal{F}_t)_{t\ge 0}$ be the complete filtration generated by the Brownian motion $B $, and let $a<0<b$. Define the stopping times
$\tau_a=\inf\{t\ge 0\mid B_t=a\}$ and $\...
0
votes
0
answers
95
views
Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$
I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion.
I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
- 11
0
votes
1
answer
329
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 ...
- 2,821
4
votes
2
answers
205
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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 ...
- 2,821
1
vote
1
answer
168
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Resources to understand Lebesgue measure of Brownian motion's path [closed]
[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47]
Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
- 5,474
2
votes
0
answers
155
views
Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
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-1
votes
1
answer
129
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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 ...
- 1,989
1
vote
0
answers
115
views
Concatenation of Markov processes and independence
In chapter 14 of Sharpe's General Theory of Markov Processes the concatenation of Markov processes $X^1$ and $X^2$ is described. I've posed the relevant part at the bottom of this post.
It is rather ...
- 167
1
vote
1
answer
96
views
Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials
This question is cross-posted from https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials
I am trying to study the asymptotic behavior ...
- 3,937
2
votes
1
answer
469
views
Textbook definition for "path measure" or "probability measure over paths"
I need a formal definition for the path measure for stochastic differential equations.
Which textbook or paper should I consult?
CommunityBot
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2
votes
1
answer
181
views
If $X$ is a Markov process, can we find a mild assumption ensuring that $\frac1t\operatorname E_x\left[\int_0^tc(X_s)\:{\rm d}s\right]\to c(x)$?
Let
$(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$ for all $x\in E$
$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$
$(\kappa_t)_{t\ge0}$ ...
CommunityBot
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0
votes
0
answers
78
views
Different measurability of Hilbert-space valued random variable
My question is motivated by this link.
Let $(\Omega,\mathcal{F})$ and $(Y,\mathcal{B})$ be measurable spaces, a measurable map $T:\Omega\to Y$ is called a $Y$-valued random variable.
Now let $H$ be a ...
- 503
1
vote
0
answers
79
views
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 $n$, ...
- 35.5k
1
vote
1
answer
725
views
Is the integral of an adapted, measurable process adapted?
Let $X_s(\omega)$ be measurable and adapted.
Under what conditions will the process
$$
F_{t}(\omega) = \int_0^t X_s(\omega) \, ds
$$ also be adapted?
To me it seems that adaptedness and ...
- 5,474
1
vote
0
answers
175
views
Interpretation of the Lévy measure of an infinitely divisible random vector
We know that a random vector $X$ is infinitely divisible (ID) if for all $n \in \mathbb N$, there exist $X_1^n,..., X_{n}^n$ i.i.d. random vectors such that:
\begin{equation}
X = X_1^n + ...+ X_n^...
- 13
0
votes
2
answers
182
views
Distribution of local martingale is absolutly continuous to that of the Brownian motion?
Let $B(t, \omega)$ be a Brownian motion defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, adapted to a filtration $\{\mathcal{F}_t\}$. Let $\phi(t, \omega)$ be a $\{\mathcal{F}_t\}$-...
- 724
1
vote
0
answers
44
views
Measurability in a product space of a set defined only along its fibers
Consider the probability space $([0,1],\mathcal{B}([0,1]),\lambda)$, where $\mathcal{B}([0,1])$ denotes the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ is the Lebesgue measure in $[0,1]$. Then, ...
3
votes
3
answers
1k
views
Continuity of Brownian motion constructed from Kolmogorov extension theorem?
I'm trying to construct Brownian motion using the Kolmogorov extension theorem.
I am happy with the construction of a process with the required FDDs as (the canonical process associated with) a ...
- 620
2
votes
3
answers
458
views
More natural example of measurable but not progressive process
All examples of measurable but not progressive processes I have ever seen seemed to be based on the huge difference between $\mathcal{F}$ and $\mathcal{F}_\infty$. Here is what I mean.
Consider ...
- 620
1
vote
1
answer
422
views
Motivation for Ionescu-Tulcea extension theorem (as opposed to Kolmogorov's)
I recently asked a question on the differences between Ionescu-Tulcea and Kolmogorov extension theorems (ITET and KET for short). A lot of my confusion has been cleared there and what I understood ...
- 620
0
votes
1
answer
77
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Meyer's example of a separable process with no path regularity
This question is a cross-post from math.stackexchange.com. I am reposting it here since I didn't receive an answer there. The original post can be found by this link.
In the following excerpt from ...
- 128k
6
votes
2
answers
756
views
Kolmogorov vs Ionescu-Tulcea extension theorem (again)
Disclaimer. This post is not a duplicate, I have carefully (best I could) read all posts on the subject both here and on math.se and my particular questions have not been asked there.
I've recently ...
- 1,989
0
votes
1
answer
96
views
What is the significance of Blumenthal and Getoor's result on the boundedness of paths of a standard Markov process?
In the book Markov processes and Potential Theory of Blumenthal and Getoor we can find the following result:
I don't understand the significance of this result. If I don't misinterpret the assertion, ...
CommunityBot
- 1
1
vote
0
answers
166
views
Wiener Integral and its distribution
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space.
Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field.
Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...
2
votes
1
answer
201
views
Joint irreducibility and aperiodicity of two independent Markov chains
Let $(X_i)_i, (Y_i)_i$ be two independent Markov chains on Polish state spaces $\mathcal{X}, \mathcal{Y}$ and with kernels $P, Q$. Suppose that they are both $\psi$-irreducible and aperiodic and have ...
- 128k
1
vote
0
answers
103
views
Continuity of Wiener measure on open balls
Let $\mu$ be the Wiener measure on $C_0 [0, T]$, the space of continuous functions starting at $0$, under the sup norm.
Question: Is it true that the function $r \mapsto \mu(B_r(x))$ is continuous in $...
- 6,155
1
vote
2
answers
235
views
Connection between invariant measure and positive recurrence for continuum state space markov chain
Let $\{ X_n(\omega,x)\}_{n \ge 0}$ be a Markov chain with and underlying probability space $(\Omega,\Sigma,\mathbb{P})$ and state space $X= \mathbb{S}^1$. Suppose this markov chain admits unique ...
- 14.2k
0
votes
1
answer
262
views
Construction of a Markov process with prescribed local behavior and state-dependent jump distribution
Let
$(E,\mathcal E)$ be a measurable space
$\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\...
- 167
0
votes
0
answers
72
views
If $\kappa$ is a Markov kernel with density $p$, does it generally hold $p(x,z)=\int p(x,y)p(y,z)\:{\rm d}y$?
Let $(E,\mathcal E)$ be a measurable space and $\kappa$ be a Markov kernel on $(E,\mathcal E)$. Assume that $$\kappa(x,B)=\int_Bp(x,y)\:\lambda({\rm d}y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$...
- 167
0
votes
0
answers
42
views
If $X$ is a right-continuous process, is $t\mapsto\operatorname E\left[X_\tau\mid\tau=t\right]$ right-continuous as well?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(X_t)_{t\in[0,\:\infty]}$ be a real-valued process on $(\Omega,\mathcal A,\operatorname P)$;
$\tau$ be an $[0,\infty]$-valued random ...
- 167
1
vote
1
answer
337
views
How can we determine the generator of this Markov process (at least formally)?
Let
$(\Omega,\mathcal A)$ be a measurable space;
$(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$;
$(Y_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued time-homogeneous Markov process on $(\...
- 6,045
5
votes
1
answer
175
views
For stochastic process $X_t$ with marginals $\mu_t$, is it true that the sample-path continuity of $X_t$ implies $\mu_t$ is weakly continuous in $t$?
I need to prove or disprove that for a stochastic process $(X_t)_{t \in [0,1]}$ with marginals $(\mu_t)_{t \in [0,1]}$ on $\mathbb{R}$, if the sample paths of $(X_t)_{t \in [0,1]}$ are continuous, ...
- 69
1
vote
0
answers
47
views
How do we need to argue in this step of the Itō-Lévy-Khintchine decomposition?
Let
$E$ be a $\mathbb R$-Banach space;
$(\Omega,\mathcal A,\operatorname P)$ be a probability space;
$(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$;
$(X_t)_{t\ge0}$ be an $E$-...
- 167
7
votes
3
answers
830
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 $\...
- 651
1
vote
0
answers
328
views
Preservation of variance for log-normal variables under change of measure
Aim: to show that changing a probability measure via the application of a Radon-Nikodym derivative preserves variance of a log-normally distributed random variable (for the case when variance is non-...
- 141