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Have a problem regarding the expected value of the Wiener process inside a function, namely:

Compute $E[cos(W_t)]$.

To extend my question, what is the general method of computing these E´s when it is wrapped up inside some function? For this I have a hunch of having to use some Taylor series for the cosine but how do I know? When do I need some special method apart from just using Ito´s?

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2 Answers 2

up vote 3 down vote accepted

In this particular case, the simplest way to compute the expected value is to write $\cos(x) = \Re(e^{ix})$ and use the formula for the characteristic function of a Gaussian variable: if $Z \sim \mathcal{N}(\mu,\sigma^2)$, $E[e^{iuZ}] = e^{iu\mu - \frac{1}{2}u^2 \sigma^2 }$ (simply write the expected value as an integral $\int_{\mathbb{R}} e^{iuz} \frac{1}{\sqrt{2\pi \sigma^2}} e^{\frac{(z-\mu)^2}{2\sigma^2}} dz$, regroup the exponentials and "complete the square").

So, since $W_t \sim \mathcal{N}(0,t)$, we get $$ E[\cos(W_t)] = E[\Re(e^{iW_t})] = \Re(E[e^{iW_t}]) = \Re(e^{-t/2}) = e^{-t/2}. $$

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How do you mean I should use the char. function? Would this way work? $E[cos(W_t)]=\sum (-1)^k\frac{1}{(2k)!}E[W^{2k}]$? @YBL –  user2069136 Apr 20 at 11:05
    
Edited the answer. Using the Taylor expansion can work but it is much more complicated. The exponential of a Gaussian variable is really easy to work with and appears a lot: exponential martingales, geometric brownian motion (Black-Scholes process), Girsanov theorem etc... –  YBL Apr 20 at 22:39
    
If the OP is not comfortable with using $\cos x = \Re \{ e^{i x} \} $, let $\cos x = \frac{e^{i x} + e^{-i x}}{2}$ and proceed from there. –  William S. Wong Apr 21 at 15:36
    
Thanks y´all! Anyone knows any good reading about expectation values of brownian motions and normal distributions? @YBL –  user2069136 Apr 22 at 11:07

Yes, I was thinking Taylor series approximation. Another possibility is to use bootstrapping or Jacknife, which is a linear approximation of bootstrapping.

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