I would like to compute $G$ defined as follows $$G(t):= \exp(-\int _0^t h_s~ ds )$$ with $h$ being a geometric Brownian Motion.
For that I would need first to compute $$\int_0^t e^{(\mu-\sigma^2/2)s+ \sigma W_s }~ ds$$ and know its law.
All I could figure out (that was not much) is that one can obtain all the moments by deriving the Laplace Transform and making the Laplace coefficient tends to zero witch is pretty straightforward
Could someone give me a help with that ? Any advice is appreciated.
Note that this question is not a duplicate since my question is to compute the exponential of an integral of a geometric Brownian motion and to compute the integral of the GBM itself.There is no expected value involveld
Many thanks