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

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].