An Exact Connection between two Solvable SDEs and a Non Linear Utility Stochastic PDEs
Résumé
The paper proposes a new approach to consistent stochastic utilities, also called forward dynamic utility, recently introduced by M. Musiela and T. Zariphopoulou. These utilities satisfy a property of consistency with a given incomplete financial market which gives them properties similar to the function values of classical portfolio optimization. First, we derive a non linear stochastic PDEs that satisfy consistent stochastic utilities processes of Itô type and their dual convex conjugates. Then, under some assumptions of regularity and monotony on the stochastic flow associated with the optimal wealth as function of the initial capital, and on the optimal state price dual process, we characterize all consistent utilities for a given increasing optimal wealth process from the composition of the dual optimal process and the inverse of the optimal wealth. This allows us to reduce the resolution of fully nonlinear second order utility SPDE to the existence of monotone solutions of two stochastic differential equations. We also, express the volatility of consistent utilities as an operator of the first and the second order derivatives of the utility in terms of the optimal primal and dual policies.
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