Stochastic mortality models: an infinite-dimensional approach

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.

Forward Mortality Rates in Discrete Time II: Longevity Risk and Hedging Strategies

“…However, as discussed in Hunt and Blake (2015a), the projection of cohort parameters is difficult, and made more complicated by the nature of the partial information we have regarding them at any specific date. In part because of this, the forward mortality models proposed to date, such as those in the Heath-Jarrow-Morton framework in Barbarin (2008), Bauer et al (2008) and Tappe and Weber (2013), the semi-parametric factor model of Bauer (2011a,b, 2014), or the Olivier-Smith model developed in Olivier and Jeffrey (2004), Smith (2005), Cairns (2007) and Alai et al (2013), have not been able to incorporate cohort effects.…”

Dynamic term structure modelling with default and mortality risk: new results on existence and monotonicity

“…Thereafter we consider less specific models for portfolio credit risk in a top-down setting and related infinite-dimensional credit risk models. Finally, we give an account of stochastic mortality modelling following Tappe and Weber (2013). 5.1.…”

“…In this section, we shall briefly illustrate our developed methods concerning positivity and monotonicity from Section 4 in the context of stochastic mortality models. In the sequel, we follow the framework of Tappe and Weber (2013), to which we refer for further details.…”

“…Or it could take the role of the number of already occurred losses in the context of credit portfolio modelling and collateralized debt obligations (see Filipović, Overbeck, and Schmidt (2011) and references therein). In the context of life insurance, η denotes the age of the considered individual and models the effect that a higher age influences the survival probability as in Tappe and Weber (2013). Also models for market or liquidity impacts have a similar structure, compare the recent approach in Jarrow and Roch (2013).…”

Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates

“…A more recent development is the concept of forward mortality rates, which are frequently modeled by a Heath-Jarrow-Morton framework similarly to forward interest rates. References that use diffusion process are, e.g., [12][13][14][15][16][17][18][19][20][21][22][23][24], just to mention a few. Although negative mortality rates do not have a meaningful interpretation, many of those references use diffusion models, where negative values occur with a positive probability.…”