Dynamic Hedging with Stochastic Differential Utility

happens to:E tZs≥t+ε12 k (J) J z 0 1sΣJ zs ds − E tZs≥t 2 k (J) J z 0 sΣJ zs ds == −E tZ t+εt12 k (J) J 0 z sΣJ zs dsDivide by ε and take the limit when ε ↓ 0:R t+ε 1E t t− limk (J) J 2 z 0 sΣJ zs dsε↓0 ε= − 1 2 k (J) J 0 z tΣJ ztFor another derivation attack, we refer to Duffie and Epstein (1992). Foraheuristicproof:J(z t )=εu(c t )+e −δ² h −1 (T ² h (J)) .Consequently:·0 = limε↓0"= limε↓0¸u(c t )+ e−δ² h −1 (T ² h (J)) − J=²u(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 14.40