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It is known that the expectation of an Ito integral is zero. Is there some property like this for the variance of an Ito integral? In particular, I am dealing with the following integral:

$Var[\int_{0}^{t}e^{-\gamma(t-s)}dW_{s}]$

Where $\gamma$ is constant. How should I go about solving this?

Thanks!

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For any (deterministic) function $f$ the random variable $\int_0^t f(s)dW_s$ has normal distribution with mean 0 and variance $\int_0^t f^2(s)ds.$ The variance formula is because of the Ito isometry. If $f$ is an square-integrable adapted process, the formula for the variance is the same but the distribution of the integral is not normal anymore.

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