I have looked everywhere for a satisfactory answer to this, including Shreve's textbooks, but I can't find one.
If I want to integrate a some deterministic function f(t) with respect to brownian motion, i.e. $$ \int_0^tf(s)\mathrm dB_s $$ what is the solution?
I understand how to decompose this into a Ito integral. I also understand that the integral must have a normal distribution, and it is relatively simple to calculate the mean $= 0$ and the variance: $$ \mathsf{Var} = \mathsf E (\int_0^tf(s)\mathrm dB_s)^2 = \mathsf E\int_0^tf^2(s)\mathrm ds. $$ so does this mean I can write the answer as $$ \int_0^tf(s)\mathrm dB_s = \sqrt{\mathsf{Var}}B_t? $$ Because this would have a normal distribution with the correct mean and variance. It also makes sense to write the solution as N(0, variance).
Is either of these a correct solution to the integral of a generic deterministic function with respect to B.M.?
any help would be greatly appreciated.
Thanks, Paul