Expectation and Variance of stochastic equation

Question

My questions is related to this question: Stochastic Differential equation, expectation and variance

I.e how do you calculate the variance and expectation of $U_t = e^{-\gamma t}U_0 + \int_0^t e^{\gamma (s-t)}\sigma dX_s$?

Answer 1

If $X$ is a Brownian motion then, as the function inside the stochastic integral is bounded, the stochastic integral is centered and, if $U_0$ is deterministic: $$ E(U_t) = e^{-\gamma t} U_0 $$

Then, using the Ito isometry: $$ E\left[U_t-E(U_t)\right]^2 = E\left( \int_0^t e^{\gamma(s-t) } \sigma dX_s \right)^2 = \int _0^t e^{2\gamma(s-t) } \sigma^2ds $$