# 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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### 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 ...
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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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### 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. ...
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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 ...
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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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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 ...