2
votes
1answer
16 views

Set of simple predictable processes is a vector space

I have a question, which is probably very easy for you to answer. How can I show that the set of simple predictable processes a vector space is? It's clear that I only have to show that the sum of ...
3
votes
1answer
119 views

Proving the reflection principle of Brownian motion

The reflection principle of Brownian motion states that Brownian motion reflected at some stopping time $\tau$ is still a Brownian motion. The proof found in Mörters & Peres (as well as in ...
3
votes
0answers
63 views

Measurability of one set of measures

Let $X,Y$ be a standard Borel spaces (a Borel subset of a complete separable metric space), and let $\mathcal B(X),\mathcal P(X)$ denote collection of Borel sets and Borel probability measures on $X$ ...
0
votes
0answers
41 views

The relationship of $\sigma(f(X))$ and $X$

If X is a random variable and f is a measurable function, 1) Is f(X) measurable with $\sigma(X)$ ? 2) Is X measurable with $\sigma(f(X))$ ? Please give proof & example or counter example. Ok ...
3
votes
1answer
62 views

Filtrations and Sigma-Algebras

I have been practising a question set by my lecturer and try to verify the answer, unfortunately I am unable to understand the following question and answer. $\textbf{Question:}$ Let ...
8
votes
4answers
198 views

Does the operator $T(f)(t) := f(t) - f(0)$ preserve measurability?

Denote by $\mathcal{B}$ the Borel field on $\mathbb{R}$, denote by $\mathbf{C}_{\left[0,\infty\right)}$ the set of continuous, real-valued functions over the domain $\left[0,\infty\right)$ and denote ...
0
votes
0answers
37 views

A filtration with usual condition if the process is Càdlàg

$\{ \mathcal F_t \}$ is a natural filtration associated to a process $\{X_t\}_{t \ge1}$. Show $\{ \mathcal F_t \}$ is a filtration with usual conditions if $X_t$ is Càdlàg. Here a function is Càdlàg ...
2
votes
1answer
28 views

An elementary question on stochastic processes

First of all some definitions: Let $(X_t)_{t\in T}$ be a family of random variables on the probability space $(\Omega,\mathcal{A},P)$. We call $(X_t)_{t\in T}$ a stochastic process. Set ...
0
votes
0answers
31 views

Question about the proof of Theorem 1.31 in Protter “Stochastic integration and differential equation”

I studying the following proof of theorem 1.31 in Protter "Stochastic integration and differential equation". We are given a probability space $(\Omega,\mathcal{F},P)$ satisfying the usual conditions. ...
1
vote
0answers
48 views

Question on stochastic process

let $(\Omega, \mathcal{F},\pi)$ be a probability space with $\sigma$-algebra $\mathcal{F}$ and measure $\pi$. Let $$X:[0,+\infty)\times \Omega\rightarrow \mathbb{R}$$ a family of random variables ...
1
vote
1answer
54 views

Adaptedness of random variables

Suppose we have an RCLL adapted process $(X_t)$. Moreover we are given a stopping time $T$. We define $\mathcal{F}_T=\{A\in\mathcal{F}\mid A\cap\{T\le t\}\in \mathcal{F}_t, \text{ for all }t\ge0\}$. ...
3
votes
0answers
71 views

Showing Measurability of empirical process (with respect to ball measurability)

I'm currently working on a problem in a certain proof which i do not fully comprehend, so i'm asking here to hopefully get some help for understanding :-) The situation of the problem is the ...
1
vote
1answer
55 views

Integration of progressively measurable process

Let $X=\{X_{t},\cal{F}_{t}; 0\leq t<\infty\}$ be a progressively measurable process and $f(t,x):[0,\infty)\times \mathbb{R}^{d}\rightarrow \mathbb{R}$ be a bounded, $\cal{B}([0,\infty))\otimes ...
3
votes
1answer
49 views

is continuity preserved under Expectation?

Let's say I have a random function $X(t)$ that is continuous in $t$, almost surely. Is it true that $$\mathbb E(X(t_1)) = \mathbb E\left(\lim_{t\to t_1} X(t)\right)?$$ This seems incorrect to me ...
1
vote
0answers
42 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
2
votes
1answer
52 views

Does uniqueness always hold for the Kolmogorov's Extension Theorem?

Kolmogorov's Extension Theorem (KET) implies the existence and uniqueness of a product measure given its finite-dimensional distributions (FDDs), provided that the latter are consistent. KET puts some ...
6
votes
1answer
58 views

Measurability of the pushforward operator on measures

Let $X$, $Y$ and $Y'$ be (standard) Borel spaces. We let $\mathcal B(X)$ be the Borel $\sigma$-algebra of $X$ and $\mathcal P(X)$ to be the space of all Borel probability distributions on $X$ endowed ...
3
votes
0answers
32 views

Right-continuity of filtrations on product spaces

Let $(\Omega^1, \mathcal{F}^1)$ and $(\Omega^2,\mathcal{F}^2)$ be two measurable space and let $(\mathcal{F}^2_s)_{s \geq 0}$ be a filtration on $(\Omega^2,\mathcal{F}^2)$. Moreover, let $t\geq 0$ be ...
2
votes
1answer
53 views

transition kernel

I've got some trouble with transition kernels. We look at Markov process with statspace $(S,\mathcal{S})$ and initial distribution $\mu^0$. We have a transition kernel $P:S\times ...
2
votes
2answers
73 views

Relative sizes of Skorokhod and product topologies on space of sample paths

Let $E$ denote a compact metric space. Let $T$ denote the non-negative reals. Let $E^T$ denote the class of all functions from $T$ to $E$. Let $C$ denote the subset of $E^T$ consisting of càdlàg ...
2
votes
1answer
108 views

Need help with Sigma-algebra

I am confused on how to determine a Sigma-algebra. The following partitions of a set are given: $$ A1 = \{1,3\} $$ $$ A2 = \{2,4,6,8\} $$ $$ A3 = \{5,7,9\}$$ And Omega is $$ \Omega = ...
0
votes
1answer
45 views

A minor clarification on completion of $\sigma$-algebras

This is from Karatzas + Shreve Definition: The stochastic process $X$ is adapted to filtration $\{\mathcal{F}_t\}$ if, for each $t\geq 0$, $X_t$ is an $\mathcal{F}_t$-measurable random variable. ...
1
vote
1answer
89 views

measurability question with regard to a stochastic process

Here are two related exercise from Karatzas and Shreve Let $X$ be a process, every sample path of which is right continuous with left limits. Let $A$ be the event that $X$ is continuous on $[0,t_0)$. ...
3
votes
2answers
96 views

Can we define probability of an event involving an infinite number of random variables?

Consider a collection $(X_a)_{a\in[0,1]}$ of i.i.d. random variables following the uniform distribution on [0,1]. That is, for every real number $a \in [0,1]$ we have a random variable $X_a$. Can we ...
3
votes
2answers
80 views

Ergodic for the mean, but not ergodic stochastic process?

Is there an example of a strictly stationary (zero mean, finite variance) stochastic process $(X_t\mid t\in \mathbb{N})$ that satisfies the conclusion of the ergodic theorem, i.e., the sample mean ...
1
vote
1answer
42 views

Moving boundaries for Ornstein-Uhlenbeck processes

Let $\tau(X_t)$ be the first-passing time to the moving boundary $a(t)$ for an Ornstein-Uhlenbeck process $X_t$. I wonder how general an $a$ can be allowed in order to guarantee that $\tau$ becomes ...
1
vote
2answers
108 views

Is random variable $X_i$ measurable on ${\mathcal F_{i+1}}$ or ${\mathcal F_{i-1}}$?

Let say I have a filtration $\mathcal F_i$ with $\mathcal F_1$ contained in $\mathcal F_2$, $\mathcal F_2$ contained in $\mathcal F_3$ and so on...$\mathcal F_n$. $X_i$ is a stochastic process, $X_i$ ...
2
votes
2answers
39 views

Finding conditional variance

I know the marginal Variance of $\operatorname{Var}(Y) = E(Y^2)- (E(Y))^2$ and conditional variance of $\operatorname{Var}(Y|X)$ is $E((Y-E(Y|X))^2\mid X=x)$. I am trying to expand out the last ...
1
vote
1answer
79 views

What is the intuition behind Adapted Process

I am reading up on stochastic process and in particular adapted process. I know that if $X_t$ is $F_t$ measurable for each t, then it is an adapted process. But I do not understand the intuition ...
0
votes
2answers
207 views

Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions

Consider $\mathbb R^{[0,1]}$ the space of all functions from $[0,1]$ to $\mathbb R$ and the cylindrical sigma algebra $\mathcal B$ on it. The question is: how to prove that $C[0,1]\notin \mathcal ...
3
votes
2answers
115 views

Counter-examples of right-continuous filtrations

A filtration $(\mathcal{F}_t)$ is said to be right continuous if $\mathcal{F}_t = \bigcap\limits_{h > 0} \mathcal{F}_{t + h}$. (A filtration $( \mathcal{F}_t)$ is a collection such that each ...
0
votes
0answers
37 views

Two Markov chains: Optimal choice of initial distributions

Consider two Markov chains on the state space $X = [1;n] = \{1,2,\dots,n\}$ given by stochastic matrices $P$ and $Q$. Let $\alpha$ be the initial distribution for the first Markov Chain and $\beta$ is ...
2
votes
2answers
95 views

Progressive measurability of a specific set related to Brownian motion

Let $\{W_t: t \in R_+\} $ be a standard Brownian motion process on a given probability space. I am interested in assessing the progressive measurability of the following set: $Z(\omega) := \{t: ...
3
votes
1answer
117 views

Probabilities of uncountable intersection of events

In order to determine a probability for some event $A\in\Omega$, I ended up with $$ \mathbb{P}\left(X_t>f(t),\quad \forall [0,T]\right)≤ \mathbb{P}(A)≤\mathbb{P}\left(X_t≥f(t),\quad \forall ...
3
votes
4answers
98 views

On which measure space is $S_n = X_1 + \dots + X_n$ considered?

A common setting in law of large number theories is letting $X_1, X_2, \dots$ be independent indentical random variables on probability space $(\Omega, \mathcal{B}, P)$. Let $S_n = X_1 + \dots + X_n$. ...
3
votes
0answers
42 views

Decisive equivalence of collections of probability measures

Working on the optimal decision theory in stochastic setting, I've found out that the following notion of equivalence is very useful. Let $(X,\mathscr A)$ be a measurable space, and let $\mathrm ...
1
vote
1answer
70 views

Why universally and not just Borel policies

In a famous book Stochastic Optimal Control: The Discrete-Time Case by Bertsekas and Shreve they use universally measurable policies that come up with some handy features: e.g. they show that every ...
1
vote
1answer
50 views

Process with Feller stochastic kernel?

Let $m: \mathcal{B}(\mathbb{R}^m) \rightarrow [0,1]$ be a probability measure. Let $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ be measurable, and continuous in the first argument. ...
1
vote
0answers
46 views

Distributions representable as Ito diffusions

This is inspired by the following question. Let $X_t$ be an Ito diffusion on the interval $t\in [0,1]$: $$ \mathrm dX_t = a(X_t)\mathrm dt+ b(X_t)\mathrm dW_t $$ where say $a,b$ are Lipschitz ...
2
votes
1answer
78 views

Measure of $\{t:B_t\in E\}$ for some null set $E$.

I am wondering if the following result can be found in any textbook or if you have a proof of it. When $E$ is a null set and $B_t$ is the Brownian motion, we have almost surely : ...
2
votes
1answer
101 views

approximating essential supremum

Let $(\Omega,\mathbb{F},P)$ be a filtred probability space. For $t\in [0,T]$, we are given sets $U_t$ of non negative stochastic processes $X=\{X_s;0\le s\le T\}$. We know that for $s\le t$ we have ...
3
votes
0answers
95 views

Measurability of number of upcrossing $U_I(\alpha,\beta; X)$ in continuous time

These definitions come from Karatzas and Shreve, Brownian Motion and Stochastic Calculus. We may take for granted that $U_F(\alpha,\beta; X(\omega))$, the number of upcrossings over $[\alpha,\beta]$ ...
2
votes
1answer
74 views

Lebesgue–Stieltjes integral from 0 to $\infty$ on $\mathbb{R}^+$

In the Stochastic analysis course we encountered the following integral $\int_0^\infty H^2_sd[M,M]_s$, where $H_s$ is a predictable process, $M_s$ is a uniformly integrable martingale in $L^2$, ...
2
votes
1answer
81 views

A theorem about the Poisson Point process.

In the proof of the Levy-Khintchine theorem, I saw a theorem about the Poisson point process. The theorem states that if $\Pi$ is a poission point process on $S$ with intensity measure $\mu.$ Let ...
1
vote
1answer
510 views

Stochastic process, Gaussian, with zero mean is a Wiener process

Let $(\Omega, \mathcal F , \mathbb P)$ be a probability space and let $\mathcal F = \{\mathcal F_t\}_{t\ge} $ a filtration. Let $W=\{W_t;t ≥ 0\}$ be a stochastic process adapted to $\mathcal F$. ...
4
votes
2answers
92 views

Optimal probability measure

Let $A$ be a finite set and let $\Bbb P$ be a probability measure on $A^{\Bbb N_0}$. Further, let $x_i:A^{\Bbb N_0}\to A$ be projection maps, so that $(x_i)_{i=0}^\infty$ can be treated as a ...
4
votes
1answer
496 views

Interchange supremum and expectation

Let $B_n:=\{f\in L^\infty_+\mid f\le n \}$, where we consider $L^\infty$ with the weak$^*$ topology. I have the following sets $$D(z):=\{h\in L^0_+(\mathcal{F}_T)\mid h\le Z_T \mbox{ for a }Z\in ...
1
vote
0answers
128 views

Change of probability measure and a continuous-time Markov chain

Let $(\Omega,\mathcal{F},\mathbb{P},\mathbb{F})$ be a complete filtered probability space, with $W$ a Wiener process and $\alpha$ a continuous-time Markov chain (taking values in $\{1,...,M\}$). We ...
1
vote
0answers
98 views

Cylindrical sigma algebra answers countable questions only.

I got a missing link in some in the following (standard) textbook question: Show that the cylindrical sigma algebra $\mathcal{F}_T$ on $\mathbb{R}^T$ (equals $\bigotimes_{t\in ...
0
votes
1answer
67 views

About equivalent characterization of ergodicity

Can anyone give me some hint on the following problem? Many thanks! Given a probability space $(X, \Sigma, \mathbb{P})$ and a $\mathbb{P}$-preserving map $\tau: X\to X$, show that the following three ...