Canonical insurance models: stochastic equations and comparison theorems

TL;DR

This paper introduces a unified, model-agnostic stochastic framework for insurance models based on Thiele's differential equation, enabling direct comparison across diverse models with different probability measures.

Contribution

It develops a canonical, path-wise construction of insurance models that supports any intertemporal dependence structure, simplifying comparison theorems.

Findings

Supports any canonical insurance model regardless of dependence structure

Provides comparison theorems that handle non-equivalence of probability measures

Offers a model-lean, unified approach to stochastic insurance modeling

Abstract

Thiele's differential equation explains the change in prospective reserve and plays a fundamental role in safe-side calculations and other types of actuarial model comparisons. This paper presents a `model lean' version of Thiele's equation with the novel feature that it supports any canonical insurance model, irrespective of the model's intertemporal dependence structure. The basis for this is a canonical and path-wise model construction that simultaneously handles discrete and absolutely continuous modeling regimes. Comparison theorems for differing canonical insurance models follow directly from the resulting stochastic backward equations. The elegance with which these comparison theorems handle non-equivalence of probability measures is one of their major advantages over previous results.