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

Maria Luísa Pasinato, Boyan Sirakov

TL;DR

This work advances the theory of noncoercive, fully nonlinear Hamilton-Jacobi-Bellman equations by providing explicit ABP-type a priori and regularity estimates when the principal half-eigenvalues $\\lambda_1^+(F,\\Omega)$ and $\lambda_1^-(F,\\Omega)$ have opposite signs. By introducing the asymptotic operator $F_\infty$ and analyzing the Dirichlet problem under a spectral gap, the authors establish a nonlinear Ambrosetti-Prodi phenomenon: for the decomposition $F[u]=h+t\phi$ there exists a threshold $t^*(h)$ with no solution for $t<t^*(h)$, exactly one (or a line segment) at $t=t^*(h)$, and exactly two ordered solutions for $t>t^*(h)$. They prove sharp a priori bounds in terms of the half-eigenvalues and right-hand sides, show the two solution branches are continuous in $t$, and describe the asymptotic limits (scaled profiles solving $F_\infty[w]=\phi$) that govern the AP-type bifurcation. The results substantially extend Ambrosetti-Prodi type multiplicity to fully nonlinear HJB operators and provide a precise framework for understanding existence, nonexistence, and multiplicity in this noncoercive setting.

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.

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

TL;DR

This work advances the theory of noncoercive, fully nonlinear Hamilton-Jacobi-Bellman equations by providing explicit ABP-type a priori and regularity estimates when the principal half-eigenvalues and have opposite signs. By introducing the asymptotic operator and analyzing the Dirichlet problem under a spectral gap, the authors establish a nonlinear Ambrosetti-Prodi phenomenon: for the decomposition there exists a threshold with no solution for , exactly one (or a line segment) at , and exactly two ordered solutions for . They prove sharp a priori bounds in terms of the half-eigenvalues and right-hand sides, show the two solution branches are continuous in , and describe the asymptotic limits (scaled profiles solving ) that govern the AP-type bifurcation. The results substantially extend Ambrosetti-Prodi type multiplicity to fully nonlinear HJB operators and provide a precise framework for understanding existence, nonexistence, and multiplicity in this noncoercive setting.

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.
Paper Structure (5 sections, 30 theorems, 110 equations)

This paper contains 5 sections, 30 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Main results and Examples
  3. Preliminaries
  4. Proofs of the main results
  5. Appendix

Key Result

Theorem 1.1

Let $g\in L^p(\Omega)$ for some $p>n$. Assume there exist $0<\lambda\leq \Lambda$, for which $A_{\alpha} \in C(\mathcal{S}^N)$ satisfy $\lambda I \leq A_{\alpha} \leq \Lambda I$, and $||b_{\alpha}||_{L^{\infty}(\Omega)}, ||c_{\alpha}||_{L^{\infty}(\Omega)}, \leq \Lambda$, for all $\alpha\in{\cal A} I. There is $C_0=C_0(n,\lambda,\Lambda, \Omega)>0$ such that the following bounds are valid for ($\

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1: dongkrylov
  • Remark 3.1
  • Theorem 3.2: QUAAS2008105
  • Remark 3.2
  • Corollary 3.3: QUAAS2008105
  • ...and 53 more