PDE Approach to Exponential Utility Problem Sample Clauses

PDE Approach to Exponential Utility Problem. Here, we would like to solve the exponential utility problem via stochastic control approach. Starting from initial value x, the value process can be described as J(t, x) = ess sup E U x + ∫ πudSu Ft ∫ which seeks the maximal utility over all πt in the time horizon [t, T ]. Consider any possible trading strategy π, assume the above supremum is obtained by some strategy πˆ. For 0 ≤ s ≤ t ≤ T , it follows that ∫ E[J(t, Xπ)|Fs] = E[E[U (Xπ + πˆudSu)|Ft]|Fs] ∫ = E[E[U (Xπ + ≤ E[E[U (Xπ + = J(s, Xπ), πudSu + ∫ πˆudSu + ∫ πˆudSu)|Ft]|Fs] πˆudSu)|Ft]|Fs] which shows that he value process J(t, Xπ) is a martingale and is a supermartingale under πˆ. Hence, by the martingale optimality principle (lemma 2.2.2.1.), the equality holds for the optimal trading strategy πˆ. By the Ito˜ formula, we can differentiate the value function J(t, Xπ) as dJ(t, Xπ) = (J + µJ π S + 1 σ2J (π S )2)dt + σJ