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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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Finding Cameron-Martin space of process

The question Find Cameron-Martin space of known process was not answered, but I found it rather interesting. It is well-known that Cameron-Martin space of Wiener measure is space $W_0^{2,1}$ (see ...
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Question related to canonical construction of Brownian Motion

I have confusion when I went through the canonical construction of 1-dimensional Brownian Motion. Here we take $\Omega:=C(\mathbb{R}_+,\mathbb{R})$, and we equip $\Omega$ with the smallest sigma ...
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Confused with the construction of cylindrical set measures - abstract Wiener space

I was reading the Wikipedia page on Abstract Wiener spaces to get some intuition on the Cameron-Martin spaces. I am really confused with their definition of the "measure" defined on ...
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Measurability of the Wiener measure with respect to the starting point

Let $M$ be a Riemannian manifold, and let $C = \{ c : [0,1] \to M \mid c \text{ is continuous}\}$. Endow $C$ with the Wiener measure $\mathbb P_x$ concentrated on the curves $c \in C$ with $c(0) = x \...
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Construction of the Classical Wiener Measure

I'm reading about the construction of the classical Wiener space in Schilling-Partzsch, Chapter 4. The gist of the construction is as follows. Consider the space $$ C_0 = \{ f: [0, \infty) \to \mathbb{...
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For a Wiener process, when can one exchange “for all $t$” and “almost surely”?

Certain local properties of the Wiener process $W_t$ are quick to prove at $t = 0$, for instance: almost surely $W_t$ is monotonous on no interval beginning at $t = 0$; almost surely $W_t$ is not ...
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Find Cameron-Martin space of known process

Find Cameron-Martin space of process $W_t-4tW_1$, where $W_t$ is Wiener process. As I know the definition of Cameron-Martin space is: $$H(\gamma)=\left\{ h \in X :|h|_{H(\gamma)}:= \sup_{\substack{f\...
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Is there a Feynman–Kac formula for the heat equation with a quartic term accounting for radiative heat loss?

Reading the Wikipedia page on the heat equation, I've found the following passage: An additional term may be introduced into the equation to account for radiative loss of heat. According to the ...
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Feynman–Kac formula: conditional expectation vs. Wiener integral

The Feynman–Kac formula for the solution $u(t,x)$ of the one-dimensional heat equation \begin{align*} \partial_t u &= \frac{1}{2}\Delta_x u,\\ u(0,x) &= f(x) \end{align*} is given by \...
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Connections of Gaussian measures, abstract Wiener space, etc. to other areas of mathematics and physics

Recently I've been trying to learn more about probability theory, stochastic process, and stochastic analysis and came upon the following set of lecture notes: Lunardi–Miranda–Pallara, Infinite ...
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Functional root of Wiener process

Does there exist a stochastic process $X$ such that when two such trajectories are sampled, their composition is Wiener-distributed? It would be natural to call such a process a functional square root ...
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Conditional expectation and indicator function

Is there a simple way of computing the following \begin{equation}\tag{1} \mathrm{E}(S_T\mathbf{1}_{S_T>X}|S_t=s) \end{equation} (here $S_t$ follows a Geometric Brownian Motion) with $t<T$? My ...
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Transition Probability of Orstein-Uhlenbeck Process using Girsanov's Theorem.

How I can find the weak solution and transition distribution of Orstein-Uhlenbeck Process using Girsanov's Theorem. More specifically, suppose that $$ dx_t = -\lambda \:x_t\:dt + \sigma\:d\beta_t, $$ ...
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Integral not well-defined for non-Cameron-Martin shift

Background: Let $E = C[0,1]$ be the space of continuous functions on $[0,1]$ and let $\mu$ be the classical Wiener measure on $E$, i.e. the distribution of a Brownian motion and let $H$ be the Cameron-...
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What is the filtered probability space used to study linear SDEs with constant coeficients?

Context: I am currently working with Kalman-like filters. As a result I deal with linear Stochastic Differential Equations (SDEs) with constant coefficients such as: $$ dx(t) = Ax(t)dt + Bdw(t) $$ ...
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On the calculation of mean square limit for the Ito integral

Most handouts and books on Ito calculus show a simple way to solve the integrals shown below without making use of Ito's lemma \begin{equation} \int_{t_0}^t\mathrm{d}W(s)\\ \int_{t_0}^t W(s)\mathrm{d}...
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$\int_0^1 \mathbb{1}_{\{f(t) >0\}}dt$ on $C[0,1]$ is continuous except on a set of Wiener measure 0.

The question is the following This question considers the occupation time of Brownian motion on the positive half time $\int_0^1 \mathbb{1}_{\{B_t >0\}}dt$. Show that the functional $\psi$ on $C[0,...
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What is Wiener's conditional measure and how can I operate with it?

For any integer $n$, any choice of $0 < t_1 < \cdots < t_n \leq 1$, and any Lebesgue measurable set, $E \in \mathbb{R}^n$ define the “cylinder” $$I = I(n;t_1;\cdots;t_n;E) := \{ \beta(·) \in ...
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Deriving trapezoid rule from conditional expectation of Brownian motion

I have read here and in P. Diaconis' paper Bayesian Numerical Analysis that, in particular, $$\mathbb{E}\left(\int_0^1 B_t dt | B_{t_0}, B_{t_1}, \dotsc, B_{t_{n-1}}, B_{t_n}\right)$$ yields the ...
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How to solve this equation $\mathbb{P}\{ W_1 > 2 | W_4 = 4\}$?

Assume that $W_t$ is a standard Wiener (Brownian motion) process. Calculate: $$\mathbb{P} \{ W_1 > 2 | W_4 = 4 \}$$ This my way of solving this problem but I am not sure is it right: src) \begin{...
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Obtaining Classical Wiener Space from abstract Wiener measure

The question I'm working on understanding the Abstract Wiener Space construction and wanted to rederive the defining property of the classical counterpart, $$\require{cancel} \xcancel{\xi_{t+s} - \...
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The convergence of the disintegrations of a sequence of measures

Let $C = \{c : [0,1] \to \mathbb R ^n \mid c \text{ is continuous and } c(0)=0 \}$ be endowed with the Wiener measure $P$. Consider an exhaustion $\mathbb R^n = \bigcup _{k \ge 0} U_k$ where each $U_k ...
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Weak convergence of law of scaled biased random walk

Let $(X_n:n\in\mathbb{N})$ be a sequence of independent, identically distributed random variables of finite mean $m$ and finite variance $\sigma^2$. Set $S_0=0$ and $S_n=X_1+\dots+X_n$ for $n\in\...
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How to determine the reflecting horizontal hidden barrier for Ito diffusion

The first article at the link below talks about that the Bi-Directional Grid Constrained (BGC) stochastic processes for a random variable X over time t is one in which the further it departs from the ...
Mailfilter's user avatar
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Existence of Gaussian process

Let $\Omega=C([0,1],\mathbb{R})$, $(\Pi_t)_{t\in [0,1]}$ the canonical process with $\Pi_t(\omega)=\omega_t$, $\mathcal{F}=\sigma(\Pi)$ and $\mathbb{F}$ the filtration generated by $\Pi$. Let $F:[0,1] ...
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Brownian Motions, Integrals, and Orthonormal Functions

I am out of my element with a topic I am working on. I think I have dwindled down the part I am stuck on to the following. If $\phi_n(t)$ is a sequence of orthonormal (eigenfunctions) functions and $...
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How to calculate the distribution of $2W(1.5) - W(2)$ where $W(t)$ is the Wiener process?

Suppose $W(t)$ is a Wiener process (which satisfies the following properties): $W(0)=0$ $W(T) - W(t) \sim \mathcal{N}(0, T-t)$ for any $t<T$ For any $0<t_{1}<...t_{j}<....<t_{n}$, ...
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Is Wiener process NOT stochastic at $t=0$?

I am learning about stochastic processes, in particular Wiener and related processes. Wiener process $W(t)$ has the property that $W(0) = 0$. Does this mean that in fact for $t=0$ Wiener process is ...
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Construction of Wiener measure(or Brownian motion)

In this definition of Wiener's measure, we define the measure of a standard Brownian motion by extending the f.d.d. distribution on set of all continuous functions. However, we also know that the set ...
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Calculus: taking a derivative of a cdf

Let $W_t$ be a standard Wiener process and $\tau_x (x > 0)$ be the first passage time to level $x (\tau_x = min \{t; W(t) = x\})$ . What are the probability density function and the expected value ...
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Topology on $C(\mathbf{R}_+,\mathbf{R})$ to get Wiener measure

Below is an extract from Le Gall's Brownian Motion, Martingales, and Stochastic Calculus, p27. I am having trouble seeing why "$\mathscr{C}$ coincides with the Borel $\sigma$-field on $C(\mathbf{...
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Explicit distribution of the d-dimensional Ornstein-Uhlenbeck process

I am trying to compute the distribution of the d-dimensional Ornstein-Uhlenbeck process $dX_t^{\epsilon}=-QX_t^{\epsilon}dt + \epsilon dB_t$, $X_0^{\epsilon}=x \in \mathbb{R}$. I know that, by Ito's ...
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Construction and properties of an Abstract Wiener Process on $C(\mathbb{R}^n,\mathbb{R}^n)$

I would like to consider a generalization of the classical Wiener process, also known as Brownian Motion. While the Brownian Motion is a process $W \colon [0,\infty) \times \Omega → \mathbb{R}$, I ...
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Is Brownian bridge adapted to a filtration

For t∈[0,1], we define X(t)=B(t)−tB(1), where {B(t):t≥0} is a standard Brownian motion. My question is, is X(t) adapted to the filtration generated by the standard brownian motion B(t) over the ...
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Covariance of Wiener Process with nonzero $x(0)$

Let $x(t)$ be a Wiener process and let $\sigma_1^2 = var(x(1))$, $\sigma_0^2 = var(x(0))$. If $x(0) = 0$, then we know that $C(t,s) = \sigma_1^2 min(t,s)$ (please see Mean and covariance of Wiener ...
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What are the differences between a Wiener measure and integral and the Lebesgue measure and integral?

I'm a physicist working with the Feynman-Kac integral. It is an integral over the Wiener measure. I'm interested in knowing if I can treat is as a Lebesgue integral in the following manner: When using ...
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If $(i, H, B)$ is an abstract Wiener space, then $B^*$ is dense in $H$?

Let $(i, H, B)$ be an abstract Wiener space - that is, $H$ is a Hilbert space, $||\cdot ||$ is a measurable norm on $H$, $B$ is the completion of $H$ with respect to $|| \cdot||$, $i$ is the inclusion ...
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Is there a "canonical" (probability) measure on continuous functions?

I'm looking for a "canonical" probability measure $\nu$ on the space $C^0([0,1];\mathbb{R})$, endowed with the Borel $\sigma$-algebra. I know I can take any measurable function $f:C^0([0,1];\...
Riccardo Ceccon's user avatar
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101 views

Gluing two Brownian paths together

I need help with the following problem. Let $B_1$ and $B_2$ be two independent random elements distributed as the Wiener measure. Define $$ B(t) = \begin{cases}B_1(t), \text{ if }t<1/2; \\ B_2(t),...
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Simulating brownian motion

I have this BM(Wiener process) which i need to simulate: Where sigma1 = 1 I know that the standard deviation of B1(t) is sqrt(t), so if we wouldnt have the constant sigma1, the simulation we would ...
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Finding the variance of a Wiener process

I am studying for an exam next week and got stuck on this problem. $W = {W(t)|0 \leq t \leq \infty}$ is a Wiener process. Set for $0\leq t \leq 1$ $W^o(t) = W(t) - t\cdot W(1)$ Find the probability $P(...
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The finite-dimensional distributions for a Wiener process are given by this formula?

A stochastic process $X = \{X_t\}$ on is Wiener Process if the following properties hold $X_0 = 0$ $X$ has independent increments: for any $n\in\mathbb{N}$ and any $0 < t_0<\ldots < t_m$ we ...
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Uniqueness of Brownian Motion from properties

I am trying to understand how the following properties for a stochastic process $B_t$ is a Gaussian process $B_t$ has independent increments $t\to B_t(\omega)$ is continuous for almost all $\omega$ ...
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463 views

Finding the expectation of a Wiener Process

My question is on how to find $\mathbb E[W_t^n]$ where $n= 0,1,2,...$ and $W_t$ is a standard normal Wiener process. Would we need to use a moment generating function. Thanks.
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Can a Wiener process be obtained as the limit of a "memoryless collision time" model?

Let $(N_t)_{t \geq 0}$ be a Poisson process of intensity $1$, and for each $\lambda>0$ and $t \geq 0$ let $$ W^{(\lambda)}_t = \sqrt{\lambda} \int_0^t (-1)^{N_{\lambda s}} \, ds = \frac{1}{\sqrt{\...
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Any Hilbert-Schmidt Operator Generates a Measurable Semi-Norm

I am reading Hui-Hsiung Kuo's "Gaussian measures in Banach spaces". In the page 59, there is the following exercise: Let $D$ be a Hilbert-Schmidt operator and define $||x|| := |Dx|$, for all $x \in H$...
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Justification for solution of Stochastic Differential Equation

If I have a Stochastic Differential Equation such as: $dX_t=(W_{t}^{3}+t^2)dt-2dW_t $ with $X_0$=$1$ and $dX_t=(W_t+t)dt+W_{t}^{2}dW_t $ with $X_0$=$1$ I would not be looking for the solution. ...
Jacob Mitch's user avatar
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Taking the expectation of Brownian motion cubed

I'm just having a bit of trouble figuring out what the what the expectation of Brownian motion cubed is. I know $E(W_t)$ = 0 $E(W_{t}^{2})$ = $t$ but what would $E(W_{t}^{3})$ (The expectation of ...
Jacob Mitch's user avatar
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What is wrong, or what is missing from the following statement?

What is missing from the following statement? If $W$ is a Wiener process, then it has independent increments, which means for arbitrary $$0=t_0\leq t_1\leq ...\leq t_n\leq s\leq t,$$ $$ \left\{ W_{t_{...
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Orthogonality of homogeneous chaoses.

Some notation: Let $B(t)$ be a brownian motion, let $\mathcal F^B$ the sigma algebra generated by $\{B(t):a\leq t\leq b\}$. Let $L_B^2(\Omega)$ be the Hilbert space of square-integrable functions ...
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