# Expectation of Wiener Measure

Fix $$T>0$$. Write $$C$$ for the space of all continuous functions $$f:[0,T] \to \mathbb{R}$$. For a standard Brownian motion $$B = \{B_t\}_{t \in [0,T]}$$ defined on probability space $$(\Omega,F,P)$$, let $$W$$ be the classical Wiener measure. That is, we treat $$B$$ as a random variable $$B: \Omega \to C$$ and $$B(\omega)\in C$$ is the continuous sample path, so $$W$$ is the distribution law of $$B$$: if $$E \subseteq C$$ is a measurable set, then $$W(E) = P(\{\omega \in \Omega | B(\omega) \in E\})$$.

Now fix $$t \in [0, T]$$. I am asked to compute this integral:

$$\begin{equation*} \int_{C}\phi(t)dW(\phi) \end{equation*}$$

How do I even start? I have no idea how to approach such a problem. In a special case which I integrate on a finite set $$E=\{\phi_1,\cdots,\phi_n\} \subseteq C$$, I can write:

$$\begin{equation*} \int_{E}\phi(t)dW(\phi) = \sum_{i=1}^n\phi_i(t)W(\{\phi_i\}) = \sum_{i=1}^n\phi_i(t) P\{B(\omega) = \phi_i\} \end{equation*}$$

Even in such a special case I don't know how to calculate $$P\{B(\omega) = \phi_i\}$$ .

• What is $\phi$? Is it a set of continuous functions? If so, what is $\phi(t)$? Commented Mar 11, 2022 at 16:35
• $W$ is a probability measure on $C$. $\phi$ is an element of $C$ so $\phi(t) \in \mathbb{R}$
– 温泽海
Commented Mar 11, 2022 at 16:39
• for a fixed $t$ you have $W|_t$ is just a normal distrtibution with parameters $\mu = 0$, $\sigma^2 = t$, so that integral $\int_C \phi(t)W(\mathrm d \phi) = 0$. You can understand it also as follows: for every $\phi$ you are as likely to get $-\phi$
– SBF
Commented Mar 11, 2022 at 16:49

This all may become simpler to grasp if you add some formalism. You have a measure $$W$$ on $$C$$ and you are asked to compute an integral $$\int_C e_t(\phi)W(\mathrm d\phi)$$ where $$e_t:C \to \Bbb R$$ is the evalution map that assigns to each function $$\phi\in C$$ its value $$\phi(t) \in \Bbb R$$. So you can treat $$(C,\mathcal C, W)$$ as a probability space and hence $$e_t$$ would just be a random variable, whose expectation you are asked to find. You very well know the distribution of this random variable: $$W(\phi:e_t(\phi) \in A) = P(\omega: B(t, \omega) \in A) = \mathcal N(0, t)(A).$$ The rest should be easy.