Show normality of a process.

Question

I am given $(X_t)_{t \geq 0} = e^{-t}B_{e^{2t}}$ where $(B_{t})_{t \geq 0}$ is a standard Brownian Motion. It is also defined $M_t = X_t - X_0 + \int_0^tX_sds$.
I want to show that $M_t$ is a Gaussian random variable. I started computing the char. function: $$E[e^{iX_tu - iX_0u + iu\int_0^tX_sds}]$$ I know that the process $(X_t)_{t \geq 0}$ is stationary, would this be useful for computing the above expectation? How should I proceed? Any hint on how to compute that integral?

Answer 1

All linear combinations of $B_t$'s are normally distributed, so the same is true of linear combinations of $X_t$'s. Use Riemann sums to see that $M_t$ is almost sure limit of sequence of normal random variables which implies that $M_t$ is also normal.