Dynamic Hedging with Stochastic Differential Utility

indicating that the ’increment’ θ 0 sdF s to the margin account at time
s is re-invested at the rate r, implying a corresponding increment of
e r(t−s) θ 0 sdF s to the margin account by time t. Itsequivalentstochastic
differential equation applying Ito’s Lemma is
dX θ t = ¡ rX θ t + θ 0 tm t
¢
dt + θ
0
tυ t dB t . (4)
7. Let π t ∈ R M be a bounded measurable function standing for the agent
spot commitment. For simplicity, we also assume that the agent does
not invest in any risky asset. Hence, the total wealth of the agent at
time t, given a futures position strategy θ, isthenWt θ ,whereW θ is the
Itoprocesshavingthestochasticdifferential representation
dW θ
t = π 0 tdS t + dX θ t − c t dt, (5)
where c t ∈ V is the consumption rate at time t.
8. Preferences of the agent over wealth at time t are given by the stochastic
differential utility 6 U : V → R, whose ”aggregator”, (f,k), is defined
as f : R D × R → R, and the variance multiplier, k, ask : R → R.
We define f to be regular, meaning that f is continuous, Lipschitz in
utility, and satisfies a growth condition in consumption 7 . In addition,
we assume that f is increasing and concave in consumption 8 .Consider
6 More details about SDU are given in the appendix B
7 More details about the meaning of these concepts, see DJ and Duffie andEpstein
(1992, p. 366).
8 Then, we can apply freely Propositions 3 (monotonicity) and 5 (concavity) of Duffie
and Epstein (1992).
6