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Stochastic mortality models: An infinite dimensional approach

Stefan Tappe, Stefan Weber

TL;DR

This work develops an infinite-dimensional stochastic framework for forward mortality, introducing forward mortality improvements as a flexible mechanism to capture cohort effects within a two-parameter mortality surface framework. By embedding forward rates and improvements into a Hilbert-space valued SPDE and proving HJM-type drift consistency conditions, the authors unify forward mortality dynamics with Musiela parametrization, and illustrate the construction with a Lévy-driven Gompertz-Makeham example. The results provide a rigorous basis for market-consistent mortality modeling, enabling robust risk pricing and longevity risk management using two-dimensional surfaces and stochastic shocks. Overall, the paper advances stochastic mortality modeling by delivering a principled, mathematically rigorous forward-rate/spatial framework with explicit consistency conditions and a concrete Lévy-driven illustration.

Abstract

Demographic projections of future mortality rates involve a high level of uncertainty and require stochastic mortality models. The current paper investigates forward mortality models driven by a (possibly infinite dimensional) Wiener process and a compensated Poisson random measure. A major innovation of the paper is the introduction of a family of processes called forward mortality improvements which provide a flexible tool for a simple construction of stochastic forward mortality models. In practice, the notion of mortality improvements are a convenient device for the quantification of changes in mortality rates over time that enables, for example, the detection of cohort effects. We show that the forward mortality rates satisfy Heath-Jarrow-Morton-type consistency conditions which translate to the forward mortality improvements. While the consistency conditions of the forward mortality rates are analogous to the classical conditions in the context of bond markets, the conditions of the forward mortality improvements possess a different structure: forward mortality models include a cohort parameter besides the time horizon; these two dimensions are coupled in the dynamics of consistent models of forwards mortality improvements. In order to obtain a unified framework, we transform the systems of Itô-processes which describe the forward mortality rates and improvements: in contrast to term-structure models, the corresponding stochastic partial differential equations (SPDEs) describe the random dynamics of two-dimensional surfaces rather than curves.

Stochastic mortality models: An infinite dimensional approach

TL;DR

This work develops an infinite-dimensional stochastic framework for forward mortality, introducing forward mortality improvements as a flexible mechanism to capture cohort effects within a two-parameter mortality surface framework. By embedding forward rates and improvements into a Hilbert-space valued SPDE and proving HJM-type drift consistency conditions, the authors unify forward mortality dynamics with Musiela parametrization, and illustrate the construction with a Lévy-driven Gompertz-Makeham example. The results provide a rigorous basis for market-consistent mortality modeling, enabling robust risk pricing and longevity risk management using two-dimensional surfaces and stochastic shocks. Overall, the paper advances stochastic mortality modeling by delivering a principled, mathematically rigorous forward-rate/spatial framework with explicit consistency conditions and a concrete Lévy-driven illustration.

Abstract

Demographic projections of future mortality rates involve a high level of uncertainty and require stochastic mortality models. The current paper investigates forward mortality models driven by a (possibly infinite dimensional) Wiener process and a compensated Poisson random measure. A major innovation of the paper is the introduction of a family of processes called forward mortality improvements which provide a flexible tool for a simple construction of stochastic forward mortality models. In practice, the notion of mortality improvements are a convenient device for the quantification of changes in mortality rates over time that enables, for example, the detection of cohort effects. We show that the forward mortality rates satisfy Heath-Jarrow-Morton-type consistency conditions which translate to the forward mortality improvements. While the consistency conditions of the forward mortality rates are analogous to the classical conditions in the context of bond markets, the conditions of the forward mortality improvements possess a different structure: forward mortality models include a cohort parameter besides the time horizon; these two dimensions are coupled in the dynamics of consistent models of forwards mortality improvements. In order to obtain a unified framework, we transform the systems of Itô-processes which describe the forward mortality rates and improvements: in contrast to term-structure models, the corresponding stochastic partial differential equations (SPDEs) describe the random dynamics of two-dimensional surfaces rather than curves.

Paper Structure

This paper contains 20 sections, 14 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution and Outline:
  3. Definitions and Elementary Properties
  4. A Conditional Law of Large Numbers
  5. Compensators
  6. Infinite-Dimensional Formulation
  7. Consistent HJM type dynamics of the forward mortality rates
  8. Consistent HJM type dynamics of the forward mortality improvements
  9. Consistent Musiela type dynamics of the forward mortality rates
  10. Consistent Musiela type dynamics of the forward mortality improvements
  11. Example: A Lévy process driven Gompertz-Makeham model
  12. Lévy process driven mortality models
  13. The Gompertz-Makeham model
  14. Appendix to Section \ref{['sec:lln']}: A Conditional Law of Large Numbers
  15. Appendix to Section \ref{['sec:comp']}: Compensators
  16. ...and 5 more sections

Key Result

Theorem 3.4

Letting $x\in\mathbb{R}$, we denote by $(N^{ n} (x))_{n\in \mathbb{N}}$ the survival indicators associated to a family of individuals born at date $-x$ with $\mathbb{F}$-DSCI death times. Then for all $t \geq -x \vee 0$: Setting $\mathcal{G}_t: = \sigma\{ N^n(x)_s: s\leq t, n\in \mathbb{N} \} \vee \mathcal{F}_t$, $t\in\mathbb{R}_+$, $\mathbb{G} = (\mathcal{G}_t)_{t \in \mathbb{R}_+}$ is the fu

Figures (1)

  • Figure 1: The domain $\Xi$ and the action of the shift semigroup $(S_t)_{t \geq 0}$.

Theorems & Definitions (60)

  • Remark 2.1
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 3.2
  • Definition 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 50 more