Questions tagged [wiener-measure]

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 process). I want to prove that if $x(0)$ is not zero…

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)-\theta^2/2}. \end{align} How do I compute the…

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?

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:…