Dynamic Hedging with Stochastic Differential Utility

happens to:
E t
Z
s≥t+ε
1
2 k (J) J z 0 1
s
ΣJ zs ds − E t
Zs≥t 2 k (J) J z 0 s
ΣJ zs ds =
= −E t
Z t+ε
t
1
2 k (J) J 0 z s
ΣJ zs ds
Divide by ε and take the limit when ε ↓ 0:
R t+ε 1
E t t
− lim
k (J) J 2 z 0 s
ΣJ zs ds
ε↓0 ε
= − 1 2 k (J) J 0 z t
ΣJ zt
For another derivation attack, we refer to Duffie and Epstein (1992). For
aheuristicproof:
J(z t )=εu(c t )+e −δ² h −1 (T ² h (J)) .
Consequently:
·
0 = lim
ε↓0
"
= lim
ε↓0
¸
u(c t )+ e−δ² h −1 (T ² h (J)) − J
=
²
u(c t )+ −δe−δ² h −1 [T ² h (J)] + e −δ² A d h(J)
h 0 (J)
1
= u(c t ) − δJ(z t )+ A dh(J)
h 0 (J) .
#
=
The proof follows now by the same lines as in Proposition 14.
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