2 Let W be a Brownian motion. Define X by dXt =(Xt)dt+dWt where , , , and X0 are all constants. (a) Write eT XT as the sum of a constant, a Riemann integral with respect to dt, and an Ito integral with respect to dWt, such that both integrands may depend on t, but not on X. Hint: As a first step, calculate d(etXt) = . . . (b) Find explicit formulas for the mean and variance of XT . In part (b) you may use the following fact (without providing a proof): If t is a nonrandom piecewise continuous function of t, then essentially because the sum of independent normals is normal; more specifically because ZTZT tdWt has distribution: Normal mean 0, variance t2dt 00 N1 X tn Wtn N1 Normalmean 0, variance X t2 t has distribution: forallpositiveintegerN,wheret:=T/N andtn :=ntandWtn :=Wtn+1 Wtn