  A large collection of financial contracts offering guaranteed minimum
benefits are often posed as control problems, in which at any point in the
solution domain, a control is able to take any one of an uncountable number of
values from the admissible set. Often, such contracts specify that the holder
exert control at a finite number of deterministic times. The existence of an
optimal bang-bang control, an optimal control taking on only a finite subset of
values from the admissible set, is a common assumption in the literature. In
this case, the numerical complexity of searching for an optimal control is
considerably reduced. However, no rigorous treatment as to when an optimal
bang-bang control exists is present in the literature. We provide the reader
with a bang-bang principle from which the existence of such a control can be
established for contracts satisfying some simple conditions. The bang-bang
principle relies on the convexity and monotonicity of the solution and is
developed using basic results in convex analysis and parabolic partial
differential equations. We show that a guaranteed lifelong withdrawal benefit
(GLWB) contract admits an optimal bang-bang control. In particular, we find
that the holder of a GLWB can maximize a writer's losses by only ever
performing nonwithdrawal, withdrawal at exactly the contract rate, or full
surrender. We demonstrate that the related guaranteed minimum withdrawal
benefit contract is not convexity preserving, and hence does not satisfy the
bang-bang principle other than in certain degenerate cases.
