An Exact Connection between two Solvable SDEs and a Nonlinear Utility Stochastic PDE

Motivated by the work of Musiela and Zariphopoulou \cite{zar-03}, we study the Itô random fields which are utility functions $U(t,x)$ for any $(\omega,t)$. The main tool is the marginal utility $U_x(t,x)$ and its inverse expressed as the opposite of the derivative of the Fenchel conjuguate $\tU(t,y)$. Under regularity assumptions, we associate a $SDE(\mu, \sigma)$ and its adjoint SPDE$(\mu, \sigma)$ in divergence form whose $U_x(t,x)$ and its inverse $-\tU_y(t,y)$ are monotonic solutions. More generally, special attention is paid to rigorous justification of the dynamics of inverse flow of SDE. So that, we are able to extend to the solution of similar SPDEs the decomposition based on the solutions of two SDEs and their inverses. The second part is concerned with forward utilities, consistent with a given incomplete financial market, that can be observed but given exogenously to the investor. As in \cite{zar-03}, market dynamics are considered in an equilibrium state, so that the investor becomes indifferent to any action she can take in such a market. After having made explicit the constraints induced on the local characteristics of consistent utility and its conjugate, we focus on the marginal utility SPDE by showing that it belongs to the previous family of SPDEs. The associated two SDE's are related to the optimal wealth and the optimal state price density, given a pathwise explicit representation of the marginal utility. This new approach addresses several issues with a new perspective: dynamic programming principle, risk tolerance properties, inverse problems. Some examples and applications are given in the last section.