An exact connection between two solvable SDEs and a nonlinear utility stochastic PDE (Q2873147)

The authors study a model for which the economic agent adjusts its performances in an uncertain universe based on the information that is revealed over time and represented by a filtration \(F=\{F_t,\; t\geq 0\}.\) The agent starts with today's specification of its utility \(U(0,x)=U(x)\), and then builds the process \(U(t,x)\) for \(t\geq 0\) in relation to the information given. This is the concept of forward dynamic utility.\newline The considerations are based on positive progressive utility, which is a collection of Itō's semimartingales with respect to a \(d\)-dimensional Brownian motion \(W\) depending on a spatial parameter \(x\in \mathbb{R}^+\) also called the wealth. The main tool is the marginal utility \(U_x(t,x)\) and its inverse expressed as the opposite of the derivative of the Fenchel conjugate \(\tilde{U}(t,y)\). Under regularity assumptions, the authors associate an \(\mathrm{SDE}(\mu,\sigma)\) and its adjoint \(\mathrm{SPDE} (\mu,\sigma)\) in divergence, from whose \(U_x(t,x)\) and its inverse \(\tilde{U}_y(t,y)\) are monotonic solutions.\newline 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.\newline Market dynamics are considered in an equilibrium state, so that the investor becomes indifferent to any action which can be taken. The authors present explicit constraints induced on the local characteristics of the consistent utility and its conjugate. Then, they focus on the utility SPDE. The two associated SDEs are related to the optimal wealth and the optimal state price density, given a pathwise explicit representation of the marginal utility.\newline The proposed approach addresses several issues with a new perspective: the dynamic programming principle, risk tolerance properties, and inverse problems. Some examples and applications are given in the last section.