Please answer the question below:

Part B. (Difficult) (e) We consider the function h(x) = 1(1,20) (2) = 0 if a >1, if a $ 1. We would like to determine E[h(X7) | F.] = g(t, X.). Use Feynman-Kac formula to write the PDE (with its terminal condition) that g has to solve. (f) We would like to solve the PDE obtained in (b). Let f(t, ) be a function such that g(t, x) = f(T - t, In(x)) f (t, x) = g(T - t, er). Use the PDE in (e) to determine the PDE (with initial condition) that the function f has to solve. (g) We know that the solution to the heat equation t - = 0 with intial condition uo is given by u(t, I) = 1 2nt uo(y) dy. Use this result to determine the function f and, then, the function g. (h) Notice that the result gives us an estimate of the probability P(Xr > 1 | F ), i.e. the best approximation of P(Xr > 1) given the observation until time t. Determine P(Xr > 1 | Fo) from (g) and verify that it corresponds to the result obtained by direct computation