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

TL;DR

This paper derives explicit a priori estimates and investigates the solution multiplicity for a class of nonlinear stochastic control equations with non-coercive Hamilton-Jacobi-Bellman operators, extending classical theorems.

Contribution

It provides the first explicit a priori estimates and a novel multiplicity result for solutions of fully nonlinear stochastic control equations with non-coercive operators.

Findings

Explicit a priori and regularity estimates established.

Exact solution multiplicity depends on eigenvalues and valuation function.

Generalization of the Ambrosetti-Prodi theorem to nonlinear stochastic control equations.

Abstract

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.