  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].
