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On a retarded stochastic system with discrete diffusion modeling life tables

Tomás Caraballo, Francisco Morillas, José Valero

TL;DR

This work proposes a method for modeling and forecasting mortality rates by incorporating both the historical evolution of the mortality phenomenon and its random behavior and applies it to forecast mortality rates in Spain, showing that it yields better results than classical methods.

Abstract

This work proposes a method for modeling and forecasting mortality rates. It constitutes an improvement over previous studies by incorporating both the historical evolution of the mortality phenomenon and its random behavior. In the first part, we introduce the model and analyze mathematical properties such as the existence of solutions and their asymptotic behavior. In the second part, we apply this model to forecast mortality rates in Spain, showing that it yields better results than classical methods.

On a retarded stochastic system with discrete diffusion modeling life tables

TL;DR

This work proposes a method for modeling and forecasting mortality rates by incorporating both the historical evolution of the mortality phenomenon and its random behavior and applies it to forecast mortality rates in Spain, showing that it yields better results than classical methods.

Abstract

This work proposes a method for modeling and forecasting mortality rates. It constitutes an improvement over previous studies by incorporating both the historical evolution of the mortality phenomenon and its random behavior. In the first part, we introduce the model and analyze mathematical properties such as the existence of solutions and their asymptotic behavior. In the second part, we apply this model to forecast mortality rates in Spain, showing that it yields better results than classical methods.

Paper Structure

This paper contains 14 sections, 5 theorems, 72 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Properties of solutions
  3. The linear case
  4. The non-linear case
  5. Asymptotic behaviour
  6. Application to Life Tables
  7. Life Tables: mortality modeling through a stochastic delay approach
  8. Dynamical kernel graduation: combining stochastic behavior and historical data
  9. Numerical simulation
  10. Data and parameters
  11. Indicators
  12. Software
  13. Numerical results
  14. Conclusions

Key Result

Lemma 1

Assume that $g_{r}(t)\geq0,$ for all $r$ and $t,$ and that $\overline{\alpha}_{i}\leq1$, for all $i$. Then for any $\phi\in C([-h,0],\mathbb{R}_{+}^{m})$ there exists a unique globally defined solution $u\left( \text{\textperiodcentered}\right)$ such that $u\left( t\right) \in\mathbb{R}_{+}^{m}$ a

Figures (8)

  • Figure 1: Mean trajectory: $b=0.1$
  • Figure 2: Confidence Intervals for several confidence levels ($b=0.1$).
  • Figure 3: Confidence Intervals for several confidence levels ($b=0.025$).
  • Figure 4: Ensemble of realizations: $b=0.1$
  • Figure 5: Mean trajectories
  • ...and 3 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Corollary 2
  • Lemma 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Theorem 7