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Mean viability theorems and second-order Hamilton-Jacobi equations

Christian Keller

Abstract

We introduce the notion of mean viability for controlled stochastic differential equations and establish counterparts of Nagumo's classical viability theorems (necessary and sufficient conditions for mean viability). As an application, we provide a purely probabilistic proof of a comparison principle and of existence for contingent and viscosity solutions of second-order fully nonlinear path-dependent Hamilton-Jacobi-Bellman equations. We do not use compactness and optimal stopping arguments, which are usually employed in the literature on viscosity solutions for second-order path-dependent PDEs.

Mean viability theorems and second-order Hamilton-Jacobi equations

Abstract

We introduce the notion of mean viability for controlled stochastic differential equations and establish counterparts of Nagumo's classical viability theorems (necessary and sufficient conditions for mean viability). As an application, we provide a purely probabilistic proof of a comparison principle and of existence for contingent and viscosity solutions of second-order fully nonlinear path-dependent Hamilton-Jacobi-Bellman equations. We do not use compactness and optimal stopping arguments, which are usually employed in the literature on viscosity solutions for second-order path-dependent PDEs.
Paper Structure (28 sections, 21 theorems, 132 equations)

This paper contains 28 sections, 21 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Background and motivation.
  3. Stochastic and mean viability: A short overview and our contributions
  4. Contributions to partial differential equations
  5. Organization of the rest of the paper
  6. Notation
  7. Data and standing assumptions
  8. Mean viability
  9. The notions
  10. Mean tangency
  11. Necessary and sufficient conditions for mean viability
  12. Approximate solutions
  13. Quasi-contingent solutions of HJB equations
  14. Quasi-contingent supersolutions
  15. Quasi-contingent subsolutions
  16. ...and 13 more sections

Key Result

Theorem 4.6

Suppose that $(\widehat{v},K)$ is mean viable or approximately mean viable for E:mv:CSDE. Then, for each $(t,\mathbf{x},y)\in [0,T)\times\Omega\times{\mathbb{R}}$ with $v(t,\mathbf{x})\le y$, we have

Theorems & Definitions (66)

  • Definition 4.1
  • Definition 4.2
  • Remark 4.3
  • Remark 4.4
  • Definition 4.5
  • Theorem 4.6
  • proof
  • Remark 4.7
  • Theorem 4.8
  • Remark 4.9
  • ...and 56 more