A priori estimates and exact solvability for non-coercive stochastic control equations

We establish, for the first time, explicit a priori and regularity estimates for solutions of the Dirichlet problem for Hamilton-Jacobi-Bellman operators from stochastic control, whose principal half-eigenvalues have opposite signs. In addition, if the negative eigenvalue is not too negative, the problem can have exactly two, one or zero solutions, depending on the valuation function. This is a novel exact multiplicity result for fully nonlinear equations, which also yields a generalization of the Ambrosetti-Prodi theorem to such equations.