  In this paper, we derive sufficient and necessary maximum principles for a
stochastic optimal control problem where the system state is given by a
controlled stochastic differential equation with default. We prove existence of
a unique solution to the controlled default stochastic differential equation.
Furthermore, we prove existence and uniqueness of solution to the adjoint
backward stochastic differential equation which appears in connection to the
maximum principles. Finally, we apply the maximum principles to solve a utility
maximisation problem with logarithmic utility functions.
