Let's first see if we agree on an example: take S(t) = W(t), a BM and q(.) = 1, and make the upper integration limit c.ThenG(t,X) = e^(-r t) int(X,c) W(u) dW(u) = -e^(-r t)[ (1/2) W(X)^2 - (1/2) X + "constant"]Do you agree that this is the result of the integration? [The "constant" is actually a random variable with zero expectation, but more importantly, independent of X]Now take S(t) = W(t) and q(S) = 1/S, so G(t,X) = e^(-r t) int(X,c) dW(u) = -e^(-r t) [ W(X) + "constant"].Agree with that one?So, really, your X is a temporal variable, probably better to write X = u. When we try to develop the derivative wrt u, the constants drop out. The problem is dW(u)/du doesn't really exist, although sometimes it is formally manipulated. So, I will guess your derivative doesn't exist in the case where S(t) is a diffusion process. If S(t) is not a diffusion process, but simply an ordinary function of time, so that the integral is not a stochastic integral, then the ordinary calculus rule works.
Last edited by Alan
on May 11th, 2012, 10:00 pm, edited 1 time in total.