# Stochastic Mortality Modelling for Dependent Coupled Lives

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## Abstract

**:**

## 1. Introduction

#### 1.1. Review of Existing Literature

#### 1.2. Broken-Heart Syndrome and Shared Frailty Dependence

#### 1.3. Novelty of the Approach

## 2. Data Set

## 3. Model Description

#### 3.1. Probabilistic Mechanism

**Theorem**

**1**

- 1.
- The joint probability density function $\rho ({t}_{x},{t}_{y})$ for the time of death of two coupled lives $({\tau}_{x},{\tau}_{y})$ is given by the reduced form expression$$\rho ({t}_{x},{t}_{y})=\left\{\begin{array}{cc}\mathbb{E}\left[{\lambda}_{p}\left({t}_{x}\right){e}^{-{\int}_{0}^{{t}_{x}}{\lambda}_{x}\left(u\right)+{\lambda}_{y}\left(u\right)du}\mathbb{E}\left[{\tilde{\lambda}}_{q}\left({t}_{y}\right){e}^{-{\int}_{{t}_{x}}^{{t}_{y}}{\tilde{\lambda}}_{q}\left(u\right)du}\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}{\mathcal{G}}_{T}\right]\right],\hfill & {t}_{x}<{t}_{y}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{(7a)}\hfill \\ \mathbb{E}\left[{\lambda}_{p}\left({t}_{y}\right){e}^{-{\int}_{0}^{{t}_{y}}{\lambda}_{x}\left(u\right)+{\lambda}_{y}\left(u\right)du}\mathbb{E}\left[{\tilde{\lambda}}_{q}\left({t}_{x}\right){e}^{-{\int}_{{t}_{y}}^{{t}_{x}}{\tilde{\lambda}}_{q}\left(u\right)du}\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}{\mathcal{G}}_{T}\right]\right],\hfill & {t}_{x}>{t}_{y}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{(7b)}\hfill \end{array}\right.$$
- 2.
- The marginal probability density function ${\rho}_{p}\left(t\right)$ for the time of the first occurring death ${\tau}_{p}$ is$${\rho}_{p}\left(t\right)=\mathbb{E}\left[{\lambda}_{p}\left(t\right){e}^{-{\int}_{0}^{t}{\lambda}_{x}\left(u\right)+{\lambda}_{y}\left(u\right)du}\right],$$

#### 3.2. Stochastic Mortality Model with Non-Mean-Reverting Cox–Ingersoll–Ross Mortality Processes

**Remark**

**1.**

**Definition**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Proposition**

**1.**

**Corollary**

**1.**

**Corollary**

**2.**

**Remark**

**4.**

**Corollary**

**3.**

**Remark**

**5.**

## 4. Indifference Price Calculation for a Joint-Life Insurance Product

**Remark**

**6.**

#### Numerical Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

**Proof.**

## Appendix B. Proof of Corollary 2

**Proof.**

## Appendix C. Derivation of the Hamilton–Jacobi–Bellman Equation for U(w,t)

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**Figure 2.**Male versus female year of death plot (

**a**), female year of death vs interarrival time of deaths (

**b**) and number of deaths per year of bereavement (

**c**).

**Figure 3.**Male vs female year of death plot (

**a**), female year of death vs interarrival time of deaths (

**b**) and number of deaths per year of bereavement (

**c**), for cohort of couples with male spouse born between 1899 and 1926.

**Figure 4.**Male vs female year of death plot (

**a**), female year of death vs interarrival time of deaths (

**b**) and number of deaths per year of bereavement (

**c**), for cohort of couples with male spouse born between 1927 and 1954.

Dependent Price | Independent Price | |
---|---|---|

$a=2.0$ | 0.8199 | 1.2005 |

$a=1.0$ | 0.6953 | 0.9291 |

$a=0.1$ | 0.5376 | 0.5764 |

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Henshaw, K.; Constantinescu, C.; Menoukeu Pamen, O.
Stochastic Mortality Modelling for Dependent Coupled Lives. *Risks* **2020**, *8*, 17.
https://doi.org/10.3390/risks8010017

**AMA Style**

Henshaw K, Constantinescu C, Menoukeu Pamen O.
Stochastic Mortality Modelling for Dependent Coupled Lives. *Risks*. 2020; 8(1):17.
https://doi.org/10.3390/risks8010017

**Chicago/Turabian Style**

Henshaw, Kira, Corina Constantinescu, and Olivier Menoukeu Pamen.
2020. "Stochastic Mortality Modelling for Dependent Coupled Lives" *Risks* 8, no. 1: 17.
https://doi.org/10.3390/risks8010017