Dynamic Hedging with Stochastic Differential Utility

process having the stochastic differential representation
dW θ
t = 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, twice
continuously differentiable, strictly concave, with U 0 and U 00 each satisfying
a (linear) growth condition. This leaves the problem
max E £ ¡ ¢¤
t U W
θ
T . (9)
θ∈Θ
Of course the optimal position now is the one that maximizes statement
9. Then we can state the proposition:
Proposition 2 The optimal futures position strategy, by maximizing the terminal
utility, is θ TV , where
θ TV
t
·
(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. Theproofisthesameastheonepresentedinthelastsection
except that π t+s = π, c t+s =0, ∀ s, t ≥ 0, andwedefine the value function
J : R 2 → R by
J (z t )=max
θ∈Θ E £ U ¡ W θt
T −t¢¤
.
Since we are maximizing as expected utility at a future date T , the HJB
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