Exercise 4.3 Let o(t) be a given deterministic function of time and define the process X by X, =. o(s)dW. (4.39) Use the technique described in Example 4.17 in order to show that the characteristic function of X, (for a fixed t) is given by Ele",]= exp " 1.o' ( s ) ds , uER , ( 4.40 ) thus showing that X is normally distibuted with zero mean and a variance given by Var [ X, ] = 1. o' (s)ds.Taking expected values will make the stochastic integral vanish. After moving the expectation within Example 4.17 Compute Ele"", ]. the integral sign in the ds-integral and defining m by m, = E[Z ] we obtain the equation Solution: Define Z by Z(t) = e". . The Ito formula gives us mt = 1- mads . dZ(t) =- a ew,dt + deadW,, 2 This is an integral equation, but if we take the t- derivative we obtain the ODE so we see that Z satisfies the stochastic differential m, equation (SDE) m, , dZ, mo =1 La' Z, dt + aZ,dW,, Zo 2 =1. Solving this standard equation gives us the answer In integral form this reads E[ew, ] = E[Z, ] = m, = e?'. Zt = 1+ o Zads ta Z,dW . We end with a useful lemma