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Questions tagged [wiener-measure]

The Wiener measure is the probability law on the space of continuous functions $g$ with $g(0)=0$, induced by the Wiener process.

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Differentiating a stochastic integral

How do i differentiate the following stochatic integral? $$\frac {d}{dW_t} \int_{0}^t \frac{1}{1-u} dW_u$$ My guess is $$\frac {d}{dW_t} \int_{0}^t \frac{1}{1-u} dW_u = \left.\frac {1}{1-u} \...
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Some Questions on Construction of Wiener Measure/Wiener Process

To pose my question, I first have to describe the construction which I use (due to Polletta): Assume that $\Omega = \Pi_{t \in [0, \infty)} \dot{\mathbb{R}}$ with topology of uniform convergence and ...
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Construct Correlated Wiener Processes

Construction Hello.. I am trying to better understand how one can correlate independent Wiener processes given a correlation matrix. Please see the attached notes. This method uses the Cholesky ...
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Absolutely continuous with respect to Wiener measure

Consider the Wiener measure $\mu$ on $\Omega:=\{x:[0,1]\to\mathbb{R}:x\in\mathcal{C}([0,1]),x(0)=0\}$ and define for $\theta\in\mathbb{R}$, \begin{align} \frac{d\mu_\theta}{d\mu}(x)=e^{\theta x(1)-\...
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How to show that the Wiener measure is singular with respect to a normal law [closed]

We have a Gaussian process $X$, $X_t:=B_t - tB_1$, where $B$ is a $BM$, $t\in[0,1]$. Let $\nu$ be the law of $X$ and $\mu$ the Wiener measure. How can I show that $\mu$ is singular with respect to $...
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Wiener space, probability measure, local martingale

I hope you can help me the following problem. Assume a Wiener space, that means a probability space $(\Omega,\mathbb F,\mathbb P)$, where $\Omega = C([0,\infty))$, $X$ is the coordinate process, $\...
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Has the Hamiltonian Path Integral Been Made Rigorous?

It is well known that the Lagrangian formulation of the path integral has been made rigorous, via the Wiener measure and/or the Trottier product formula. I haven't seen mathematicians discuss the ...
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Question on Levinson's proposed discrete form of Wiener filter: the stationarity assumption

The whole foundation of Levinson's discrete version of Wiener filter is based on the assumption of stationarity of a time series, and aims to predict a value based on the past observed values. Now, if ...
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Why does $C(s,t)=\min(s,t)$ mean it is not the wide - sense stationary?

I saw some information about Wiener process on the internet,and I can't really understand what it means. It says: The covariance of Wiener process is $C(s,t)=\min(s,t)$ when $0\lt s \lt t$, so the ...
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What does it mean when standard deviation is higher than the variance?

I am currently studying Wiener's process from Hull, and it says that the path of the process is jagged because when $\Delta t$ is small, the standard deviation i.e. $\sqrt{\Delta t}$ is bigger than $\...
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Approximation of Poisson by Wiener

Recently I learned that it is a widespread idea in applied math to approximate high rate Poisson processes by a Wiener process. I.e. take $N$ to be a homogeneous Poisson with rate $\lambda$, then for ...
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integral of stochastic function over ds

I have a question about the "hand-wavy" definition of these terms, or just whatever helps to form my intuition. When I encounter terms like $\int_0^t f(s) dW_s$ for $f(s)$ deterministic function, I ...
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What is a unit Wiener process?

Currently i am reading some papers where the term "mutually indepedent unit Wiener processes" is used. Does that mean that a Wiener process must have zero mean and variance 1? The papers i read: http:/...
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Markov property of Wiener process via characteristic function

I'm reading a book 'Introduction to theory of Random process' written by N. V. Krylov. The author prove the following theorem via characteristic functions (p.53) Theorem. Let $(w_t,\mathcal{F}_t)$ ...
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Embedding $L^p \subset L^q$ compact? And relation to abstract Wiener spaces

I am currently reading Hui Hsiung Kuo's book "Gaussian Measures in Banach Spaces" and there is an exercise (Exercise 21, p. 86) in which you are asked to show that for $1 \leq p < 2$, $(i, L^{2}[0,...
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Sum of Wiener processes is a Wiener Process

Let $w_0,w_1,...$ be independent Wiener processes on $[0,1]$ and define $$W(t)=w_0(1)+w_1(1)+...+w_{\left \lfloor{t}\right \rfloor-1}(1)+w_{\left \lfloor{t}\right \rfloor}(t-\left \lfloor{t}\right \...
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Correlation between Wiener and Orstein Ulenbeck processes

can anyone help me with finding correlation between: $x(t)= W(t)$ and $y(t) = \int_o^t exp((s-t)a)dW(s)$ Thanks in advance!
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Ito Isometry against non-Brownian SDE

Suppose $X_t$ is a Semi-martingale and $H_t$ is $X_t$-predictable. I know that if $X_t=W_t$ is a Wiener process then $$ \mathbb{E}[H\cdot W_T^2] = \mathbb{E}[\int_0^TH_t^2dt], $$ where $H\cdot W_T$ ...
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Question about Wiener measure

In this post user SM2 provides a way of constructing the Wiener measure $\nu$ on the measurable space $(C([0,\infty),\mathcal{B}(C([0,\infty))))$ where $\mathcal{B}(C([0,\infty)))$ is the Borel $\...
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Why is this local martingale a martingale?

Let $a(t, \omega)$ be a product measurable ($t$ refering to time and $\omega$ to an elementary event of some probability space) and adapted (to some filtration satisfying the "usual" conditions) ...
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References about distances between singular probability measures

I would be interested in references on the topic of distances between probability measures that are singular with one another and not reduced to trivial ones. For example from here we know that total ...
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Hitting time or normalized Brownian motion (divided by square root of t)

What I mean by a normalized Brownian motion is the following: $$ U(t) = \frac{W(t)}{\sqrt{t}}, $$ where W(t) is a standard Brownian motion (with $\mu = 0$ and $\sigma = 1$). Is there a name for such a ...
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Can Wiener process be axiomized without normal increments

A common characaterization of Wiener's process is the following which I took directly from Wikipedia: $W_0 = 0$ a.s. $W$ has independent increments: $W_{t+u} - W_t$ is independent of $σ(W_s : s ≤ t)$...
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Transformed Wiener Process

How is it that if $W$ is a Wiener process and if we define $\widetilde{W}=W + \frac{\mu - r}{\sigma}t$, then $-\frac{\widetilde{W}}{\sqrt{T}}$ is standard normal?
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Wiener's construction of the Wiener Measure

I am writing an essay about Norbert Wiener and I already have sufficient info about him in general and his history, but now I would like to know how he constructed the Wiener measure. I found some ...
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Why is Wiener measure on $C[0,1]$ strictly positive?

The question is as stated. I have thought about this for a while and can't really get anywhere. Here strictly positive means non-zero on non-empty open sets (in this case with a finite interval we are ...
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Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. Link here If this is not acceptable - administrators ...
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Pathwise measurability of Ito integral under supremum norm

I'm doing my first research project on Stochastic Analysis and in order to prove something which is crucial, I need to prove the following claim: LEMMA: Denote by $(C_{0}[0,\,T],\,||\centerdot||_{\...
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Donsker's theorem and multidimensional CLT

I want to prove that the linear interpolation $X_t^n(\omega):=\frac{1}{\sqrt{n}}\sum_{k=1}^{[nt]}{Y_k}(\omega)+\frac{1}{\sqrt{n}}Y_{[nt]+1}(\omega)(nt-[nt])$ of $\sum_{k=1}^{n}{Y_k}(\omega)$ for r.v. $...
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Wiener measure on continuous function space

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. I have following problem: Given is the map $W:\Omega\rightarrow C[0,1]$ (it is not given but I think it is implicit a Wiener process). ...
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Wiener measure of smooth function in space of continuous function.

How do we show that the Wiener measure of class of smooth functions in $C[0, \infty)$ is 1?
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How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by $\...