  Motivated by the work of Musiela and Zariphopoulou \cite{zar-03}, we study
the 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 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.
