BSDEs driven by $|z|^2/y$ and applications to PDEs and decision theory

TL;DR

This paper establishes existence and uniqueness results for a class of singular backward stochastic differential equations (BSDEs) with applications to PDEs with singular terms and to decision theory problems like portfolio optimization and stochastic utility.

Contribution

It introduces new existence and uniqueness results for BSDEs with singular terms and applies these to PDEs with singularities and complex decision theory models.

Findings

Proved existence and uniqueness for BSDEs with |z|^2/y terms.

Applied results to PDEs with singular gradient terms.

Solved portfolio optimization problems with utility functions.

Abstract

Existence and uniqueness is established for a large class of backward stochastic differential equations which contain singular terms of the form $\pm|z|^2/y$. The results are applied to investigate singular partial differential equations (PDEs) and to decision theory problems that cannot be studied using classical regular BSDEs. The application to PDEs concerns the existence of viscosity solutions to PDEs containing a singular term of the form $\pm|\nabla v|^2/v$ with rather weak assumptions on the regularity of the coefficients. Such PDEs with singularity in the value process appear in several applications in physics and economics. Regarding the application to decision theory, on the one hand, we use singular BSDEs to solve portfolio optimization problems with logarithm and power utility and non-trivial terminal endowment. Moreover, we derive existence and uniqueness of the general version of the non-Markovian Kreps-Porteus stochastic differential utility defined by Duffie and Lions [17] and constructed, in the Markovian case using PDE arguments by Duffie and Lions [19].