Dynamic Hedging with Stochastic Differential Utility

Proposition 13 Suppose we want to define the following value functionZ£ ¡ ¢¤J (z t )=E t U Wθt 1 TT +2 E t k (J (z s )) Jz 0 sΣJ zs ds,s≥twith boundary condition J(z T )=U (w).Then, the relevant HJB equation isA d J + 1 2 k(J)J 0 z tΣJ zt =0.Proof. See earlier proofs.We could alternatively specify the following:Proposition 14 Suppose we want to define the following value functionJ (z t )=h ¡ £ ¡ ¡ ¢¢¤¢ −1 E t h U WθtT ,J(z T )=U (w).Then the relevant HJB equation is:A d J + 1 2 k(J)J 0 z tΣJ zt =0.Proof. Observe that:£ ¡ ¡ ¢¢¤h (J (z t )) = E t h U WθtT42