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Motivated by the work of Musiela and Zariphopoulou \cite{zar-03}, we studythe It\^o 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 inverseexpressed as the opposite of the derivative of the Fenchel conjuguate$\tU(t,y)$. Under regularity assumptions, we associate a $SDE(\mu, \sigma)$ andits adjoint SPDE$(\mu, \sigma)$ in divergence form whose $U_x(t,x)$ and itsinverse $-\tU_y(t,y)$ are monotonic solutions. More generally, specialattention is paid to rigorous justification of the dynamics of inverse flow ofSDE. So that, we are able to extend to the solution of similar SPDEs thedecomposition based on the solutions of two SDEs and their inverses. The secondpart is concerned with forward utilities, consistent with a given incompletefinancial market, that can be observed but given exogenously to the investor.As in \cite{zar-03}, market dynamics are considered in an equilibrium state, sothat the investor becomes indifferent to any action she can take in such amarket. After having made explicit the constraints induced on the localcharacteristics of consistent utility and its conjugate, we focus on themarginal utility SPDE by showing that it belongs to the previous family ofSPDEs. The associated two SDE's are related to the optimal wealth and theoptimal state price density, given a pathwise explicit representation of themarginal utility. This new approach addresses several issues with a newperspective: dynamic programming principle, risk tolerance properties, inverseproblems. Some examples and applications are given in the last section.

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