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I understand the following: Consider a probability space $(\Omega, \mathcal{A},P)$ and a Brownian motion $B=\{B_t, t\in [0,1]\}$ on this space and denote $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
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Let $B$ be a standard Brownian motion on a probability Space $(\Omega, \mathcal{F}, P)$ and let $\mathbb F:=(\mathcal{F}_t)_{t\in [0,T]}$ denote the natural filtration, i.e. $\mathcal{F}_t = ... 0answers 35 views ### The projective limit of probability spaces and the Kolmogorov-Daniell theorem Does the "projective limit" concept exist for probability spaces? The only result that I know of seems to be the Kolmogorov-Daniell theorem, but this is just a particular case where the spaces ... 1answer 48 views ### An example of stochastic process I use the following definition for a stochastic process. Let$(\Omega, \mathcal F, P)$be a probability space,$(E, \mathcal E)$be a measurable space, and$T$be a non-empty set. A collection ... 1answer 84 views ### Every Lipschitz function is the primitive of a measurable function I was doing exercise 5 of this exercise sheet and I don't know how to conclude. I need to prove that if$f \colon [0,1]\to \mathbb{R}$is Lipshitz,$X$is a uniform$(0,1)$random variable and ... 1answer 24 views ### Ito integrals and the Euler scheme I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ... 0answers 22 views ### Ultrametric space of stochastic filtration Let$\Omega$be an arbitrary set and$(\mathscr F_t)_{t\in \Bbb R_+}$be a non-decresing sequence of$\sigma$-algebras on$\Omega$such that any subset of$\Omega$is contained is some of them, that ... 0answers 30 views ### Transition kernel that is not Markov Let$(X,\mathcal{F})$and$(Y,\mathcal{G})$be two measurable space. A transition kernel$K$is a function$K : X \times \mathcal{G} \to \overline{\mathbb{R}}_+$suche that$K(\cdot,B)$is measurable ... 1answer 36 views ### Measurability and knowledge there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process$(X_n)_n$, then for example a stopping time ( other examples would be martingales, ... 1answer 20 views ### Extensions of the Ito integral This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate? 1answer 33 views ### Wasserstein metric: conditions for the existence of minimizer and duality Let$(X,d)$be a metric space and let$\mathcal P(X)$be the set of all Borel probability measures on$(X,d)$. The Wasserstein distance on$\mathcal P(X)$is given by $$W_d(\mu,\bar\mu):=\inf_{M\in ... 1answer 72 views ### How to make sense out of this: Ergodic theorem for Markov chains We had the ergodic theorem for Markov chains, stating that: For a state space S \subset \mathbb{N} and all functions f \in L^1 (meaning that \sum_{s \in S} |f(s)|\pi(s) < \infty) and an ... 1answer 38 views ### law of iterated logarithm Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ... 0answers 18 views ### Filtrations and sigma algebras [duplicate] I have a doubt concerning the basilar aspects of the filtrations in the stochastic theory. A filtration is an increasing sequence of \sigma-algebras on a measurable space. That is, given a ... 1answer 61 views ### Why this function is continuous? Let (\Omega,\Sigma,\mu) be a sample space and let L^2= \lbrace f:\Omega \rightarrow R / \int f^2d\mu <\infty \rbrace be a Hilbert space. Let L_n=L^2\times L^2 \times .... \times L^2 (n ... 2answers 97 views ### Problem with infinite product measures Given some measurable space \left(X,\mathcal{F}\right) and two probability measures \mu and \nu on this space one can define ... 1answer 22 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 ... 1answer 302 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 ... 0answers 70 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 ... 0answers 43 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 ... 1answer 74 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 ... 4answers 201 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 ... 0answers 63 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 ... 1answer 38 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 ... 0answers 38 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. ... 0answers 69 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}$$a family of random variables ... 1answer 57 views ### Adaptedness of random variables Suppose we have an RCLL adapted process (X_t). Moreover we are given a stopping time T. We define \mathcal{F}_T=\{A\in\mathcal{F}\mid A\cap\{T\le t\}\in \mathcal{F}_t, \text{ for all }t\ge0\}. ... 0answers 75 views ### Showing Measurability of empirical process (with respect to ball measurability) I'm currently working on a problem in a certain proof which i do not fully comprehend, so i'm asking here to hopefully get some help for understanding :-) The situation of the problem is the ... 1answer 64 views ### Integration of progressively measurable process Let X=\{X_{t},\cal{F}_{t}; 0\leq t<\infty\} be a progressively measurable process and f(t,x):[0,\infty)\times \mathbb{R}^{d}\rightarrow \mathbb{R} be a bounded, \cal{B}([0,\infty))\otimes ... 1answer 56 views ### is continuity preserved under Expectation? Let's say I have a random function X(t) that is continuous in t, almost surely. Is it true that$$\mathbb E(X(t_1)) = \mathbb E\left(\lim_{t\to t_1} X(t)\right)?$$This seems incorrect to me ... 0answers 56 views ### Infinite discounted sum of weakly dependent Normal random variables Say I have the expected value of a sum of weakly dependent Normal random variables of the form \mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right], where 0<a<1. I was wondering under what ... 1answer 56 views ### Does uniqueness always hold for the Kolmogorov's Extension Theorem? Kolmogorov's Extension Theorem (KET) implies the existence and uniqueness of a product measure given its finite-dimensional distributions (FDDs), provided that the latter are consistent. KET puts some ... 1answer 79 views ### Measurability of the pushforward operator on measures Let X, Y and Y' be (standard) Borel spaces. We let \mathcal B(X) be the Borel \sigma-algebra of X and \mathcal P(X) to be the space of all Borel probability distributions on X endowed ... 0answers 34 views ### Right-continuity of filtrations on product spaces Let (\Omega^1, \mathcal{F}^1) and (\Omega^2,\mathcal{F}^2) be two measurable space and let (\mathcal{F}^2_s)_{s \geq 0} be a filtration on (\Omega^2,\mathcal{F}^2). Moreover, let t\geq 0 be ... 1answer 73 views ### transition kernel I've got some trouble with transition kernels. We look at Markov process with statspace (S,\mathcal{S}) and initial distribution \mu^0. We have a transition kernel P:S\times ... 2answers 94 views ### Relative sizes of Skorokhod and product topologies on space of sample paths Let E denote a compact metric space. Let T denote the non-negative reals. Let E^T denote the class of all functions from T to E. Let C denote the subset of E^T consisting of càdlàg ... 1answer 122 views ### Need help with Sigma-algebra I am confused on how to determine a Sigma-algebra. The following partitions of a set are given:$$ A1 = \{1,3\}  A2 = \{2,4,6,8\}  A3 = \{5,7,9\}$$And Omega is$$ \Omega = ... 1answer 48 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. ... 1answer 107 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$[0,t_0)$. ... 2answers 108 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 ... 2answers 92 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 ... 1answer 50 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 ... 2answers 112 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,$X_i$... 2answers 44 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 ... 1answer 154 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 ... 2answers 334 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 ...
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 ...