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A general and sharp regularity condition for integro-differential equations with non-dominated measures

Alexandros Saplaouras

Abstract

The aim of this work is to present the regularity condition (also known in the literature as structure condition) an integro-differential operator may satisfy in order for the domination principle to hold for (sub-,super-) solutions of polynomial growth. More precisely, the framework presented in Hollender [13], in which power functions are used in order to determine the integrability conditions, is weakened by substituting the power functions with Young functions. The use of Young functions allows for sharp integrability conditions, which are crucial when one deals with limit theorems. As an immediate application, it is considered the case of parabolic Hamilton-Jacobi-Bellman (HJB) operators, for which the regularity condition is satisfied and, consequently, the comparison principle as well. The parabolic HJB operator presented in this work can be associated to second-order (decoupled) forward-backward stochastic differential equations with jumps.

A general and sharp regularity condition for integro-differential equations with non-dominated measures

Abstract

The aim of this work is to present the regularity condition (also known in the literature as structure condition) an integro-differential operator may satisfy in order for the domination principle to hold for (sub-,super-) solutions of polynomial growth. More precisely, the framework presented in Hollender [13], in which power functions are used in order to determine the integrability conditions, is weakened by substituting the power functions with Young functions. The use of Young functions allows for sharp integrability conditions, which are crucial when one deals with limit theorems. As an immediate application, it is considered the case of parabolic Hamilton-Jacobi-Bellman (HJB) operators, for which the regularity condition is satisfied and, consequently, the comparison principle as well. The parabolic HJB operator presented in this work can be associated to second-order (decoupled) forward-backward stochastic differential equations with jumps.
Paper Structure (16 sections, 20 theorems, 204 equations)

This paper contains 16 sections, 20 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Domination Principle for Hamilton--Jacobi--Bellman Equations
  3. Admissibility and regularity condition
  4. A prominent example: Hamilton--Jacobi--Bellman Equations
  5. Remarks on the assumptions and on possible relaxations
  6. Young functions
  7. Proofs of the results presented in Section \ref{['sec:HJB_Equations']}.
  8. Proof of Theorem \ref{['theorem:Domination_Principle']}
  9. Proof of Lemma \ref{['lemma:admissibility_HJB']}
  10. Auxiliary lemmata for the proof of Proposition \ref{['HJB:Regularity_Condition']}
  11. Preparatory remarks and lemmata
  12. Auxiliary lemmata for the main body of the proof of Proposition \ref{['HJB:Regularity_Condition']}
  13. Proofs of the results presented in Section \ref{['sec:YoungFunctions']}.
  14. Proof of Proposition \ref{['prop:UI_Young']}
  15. Proof of Lemma \ref{['lemma:UI_Young_improvement']}
  16. ...and 1 more sections

Key Result

Theorem 2.5

Suppose that $p> 0$, $T>0$ and $k\in\mathbb{N}$. Moreover, suppose that $u_i\in\textup{USC}_p([0,T]\times \mathbb{R}^d)$ are viscosity solutions in $(0,T)\times \mathbb{R}^d$ of for $i\in\{1,\ldots,k\}$ and operators $(G_i)_{i\in\{1,\ldots,k\}}$ which satisfy assumption:operator_F and the regularity condition described in def:regularity_condition for $\beta_1,\ldots,\beta_k>0$ and a twice continu

Theorems & Definitions (53)

  • Definition 2.1: Viscosity Solutions
  • Remark 2.3
  • Definition 2.4: Regularity Condition
  • Theorem 2.5: Domination Principle
  • proof
  • Corollary 2.6: Comparison Principle
  • proof
  • Remark 2.7
  • Definition 2.8: Hamilton--Jacobi--Bellman Equations
  • Remark 2.9
  • ...and 43 more