Dynamic Hedging with Stochastic Differential Utility

stated. When well defined, the utility process, J, for a given consumptionprocess, c, is the unique Ito process J having a stochastic differential representationof the formdJ(z t )=·−f(c t ,J(z t )) − 1 2 k(J)J 0 z tΣJ zt¸dt + J 0 z tΛ (z t ) dB t ,where the subscript of J indicates derivative with respect to the argument.In this framework the pair (f,k) is called ”aggregator”, that determinesthe consumption process, c, such that the utility process, J, is the uniquesolution toJ(z t )=E t·Zs≥t½f [c s ,J(z s )] + 1 ¾ ¸2 k(J)J z 0 sΣJ zs ds ,t≥ 0.We think of J(z t ) as the continuation utility of c at time t, conditional oncurrent information; and k(J) as the variance multiplier, applying a penalty(or reward) as a multiple of the utility ”volatility” J 0 zΣJ z . In a discrete timesetting, we could say that at time t, the intertemporal utility J (·,t+1)fortheperiodaheadandbeyondisarandomvariable.Thusfirst the agentcomputes the certainty equivalent, m (∼ J (·,t+1)|= t ), of the conditionaldistribution ∼ J (·,t+1)|= t of J (·,t+ 1), given information = t at time t.Then (s)he combines the latter with c t via the aggregator. The function fencodes the intertemporal substitutability of consumption and other aspectsof ”certainty preferences”, also generating a collateral risk attitude underuncertainty. The certainty equivalent function, m, encodes the risk aversionin the sense described in Epstein and Zin (1989).36