Assume that the given boundary value problems can be solved using the Feynman-Kac stochastic representation formula. a) Consider the following boundary value problem in the domain [0, ] : + 1 2 2 2 2 + = 0 (, ) = ln( 2). Find the solution of this problem. [6] b) Define the stochastic process by the dynamics ( , , , ) ( ) . t t t t t t t x dX X dt X dW X = = + where is a a standard Brownian motion. Consider the following boundary value problem in the domain [0, ] : + (, ) + 1 2 2(, ) 2 2 + (, ) = 0, (, ) = (), where , , and are assumed to be known functions. Show that this problem has the stochastic representation formula (, ) = [( )] + [(, )] . [4