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Define the stochastic process Xt by the dynamics dX, = u(t, X,)dt +o(t, X,)dW,, X, =x. where We is a a standard Brownian motion. a) Consider the following boundary value problem in the domain [0, 7] x R: OF Ox ~+ u(t,x) OF + 10' (t,x)- O'F at 2 ax + k(t, x) = 0, F(T,x) = D(x), where u, o, k and & are assumed to be known functions. Use the Feynman-Kac stochastic representation formula to show that this problem has the stochastic representation formula F (t, x) = ED(X, )] + [, E[k(s, X, )]ds. b) Using the result in (a), find the solution of the following boundary value problem OF 1 32F at 2 dx2 + x = 0 F(T, x) = In(x2)