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Viscosity solutions of the integro-differential equation for the Cramér--Lundberg model with annuity payments and investments

Platon Promyslov

Abstract

This note is an addendum to the work initiated by Promyslov on the integro-differential equation arising in the ruin problem for annuity payment models. First, the existence of viscosity solutions is proved. Then the regularity of these solutions is established, showing that they are indeed classical solutions.

Viscosity solutions of the integro-differential equation for the Cramér--Lundberg model with annuity payments and investments

Abstract

This note is an addendum to the work initiated by Promyslov on the integro-differential equation arising in the ruin problem for annuity payment models. First, the existence of viscosity solutions is proved. Then the regularity of these solutions is established, showing that they are indeed classical solutions.

Paper Structure

This paper contains 6 sections, 10 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Model description and problem formulation
  3. Definition of a viscosity solution
  4. Existence of a viscosity solution
  5. Comparison principle and uniqueness
  6. Smoothness of the solution and transition to the classical solution

Key Result

lemma 1

The following limit holds: $\lim_{u \to 0+} \Phi(u) = 0$. Setting $\Phi(0)=0$ yields a function that is right-continuous at zero. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (23)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • definition 1
  • remark 1
  • remark 2
  • definition 2
  • theorem 1
  • proof
  • ...and 13 more