Dynamic Hedging with Stochastic Differential Utility

Proposition 10 Assume u ∈ L and δ > 0. Then the HJB equation is:u(c t ) − δJ + A d J =0,as:Proof. (You may apply the theorem as well) Define the value functionConsequently:J(z t )=E t·ZJ(z t+ε )=E t+ε·Zs≥ts≥t+εTake the conditional expectations at t:T ε J(z t )=E t·Z¸e −δ(s−t) u (c s ) ds ,t≥ 0.¸e −δ(s−t−ε) u (c s ) ds ,t+ ε ≥ 0.s≥t+εSubtract the first equation from the last one:T ε J (z t ) − J (z t )=E t·Zs≥t+ε¸e −δ(s−t−ε) u (c s ) ds .Ze −δ(s−t−ε) u (c s ) ds −s≥t¸e −δ(s−t) u (c s ) ds .Divide by ε and take the limit when ε ↓ 0:LHSlimε↓0 εT ε J (z t ) − J (z t )= lim= A d J.ε↓0 ε38