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Assume that $X(t)$ is a zero-mean, unit variance Gaussian process, how to find the probability distribution of the integral \begin{equation} Y = \int_0^T \exp{(iuX(t))}dt \end{equation} where $u$ and $T$ are postive constants and $i=\sqrt{-1}$.

By searching the Internet, I find there are literatures on the geometric brownian motion, i.e., $\exp{(uX(t))}$. But the one involving complex number $i$ cannot be traced.

Is there an answer to this problem? Thank you.

ecook
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    The hypothesis that X is a zero-mean, unit variance Gaussian process is not sufficient to determine the distribution of Y. Different dependence structures of X yield different distributions of Y. – Did Jan 07 '15 at 17:43
  • As @Did said, this question is not currently well posed. Even if you mean that $X(t)$ is Brownian motion, this question is non-trivial. The distribution of $\int_0^T \text{exp}(u X(t))dt$ ($u$ real) has been studied by Matsumoto and Yor: http://arxiv.org/pdf/math/0511517.pdf I do not recall seeing anything for the complex case in any of their papers, but that is where I would start looking. – Chris Janjigian Jan 07 '15 at 18:23

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