Define the stochastic process Xt by the dynamics dXtXt=μ(t,Xt)dt+σ(t,Xt)dWt,=x. where Wt is a a standard Brownian motion. a) Consider the following boundary value problem in the domain [0,T]×R : ∂t∂F+μ(t,x)∂x∂F+21σ2(t,x)∂x2∂2F+k(t,x)F(T,x)=0,=Φ(x), where μ,σ,k and Φ 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)=E[Φ(XT)]+∫tTE[k(s,Xs)]ds. b) Using the result in (a), find the solution of the following boundary value problem ∂t∂F+21x2∂x2∂2F+x=0F(Tˉ,x)=ln(x2)!
