Dynamic Hedging with Stochastic Differential Utility

process having the stochastic differential representationdW θt = π 0 dS t + dX θ t . (8)2. Preferences of the agent over wealth at time T are given by von Newman-Morgenstern utility function U : R → R, which is monotonic, twicecontinuously differentiable, strictly concave, with U 0 and U 00 each satisfyinga (linear) growth condition. This leaves the problemmax E £ ¡ ¢¤t U WθT . (9)θ∈ΘOf course the optimal position now is the one that maximizes statement9. Then we can state the proposition:Proposition 2 The optimal futures position strategy, by maximizing the terminalutility, is θ TV , whereθ TVt·(J ww + J wx )= −(J ww +2J wx + J xx ) (υ tυ 0 t) −1 υ t σ 0 tπ t +J ¸w + J x(J ww + J wx ) m t(10)Proof. Theproofisthesameastheonepresentedinthelastsectionexcept that π t+s = π, c t+s =0, ∀ s, t ≥ 0, andwedefine the value functionJ : R 2 → R byJ (z t )=maxθ∈Θ E £ U ¡ W θtT −t¢¤.Since we are maximizing as expected utility at a future date T , the HJB14