This study proposes a linear-rational joint survival mortality model based on the Wishart process. The Wishart process, which is a stochastic continuous matrix affine process, allows for a general dependency between the mortality intensities that are constructed to be positive. Using the linear-rational framework along with the Wishart process as state variable, we derive a closed-form expression for the joint survival annuity, as well as the guaranteed joint survival annuity option. Exploiting our parameterisation of the Wishart process, we explicit the distribution of the mortality intensities and their dependency. We provide the distribution (density and cumulative distribution) of the joint survival annuity. We also develop some polynomial expansions for the underlying state variable that lead to fast and accurate approximations for the guaranteed joint survival annuity option. These polynomial expansions also significantly simplify the implementation of the model. Overall, the linear-rational Wishart mortality model provides a flexible and unified framework for modelling and managing joint mortality risk.

翻译：本研究提出了一种基于Wishart过程的线性有理联合生存死亡率模型。Wishart过程作为一种随机连续矩阵仿射过程，能够为构建的正死亡率强度之间建立广义依赖关系。通过将线性有理框架与作为状态变量的Wishart过程相结合，我们推导出联合生存年金及其保证型期权的闭式解。利用我们对Wishart过程的参数化方法，我们明确了死亡率强度的分布及其依赖结构。我们给出了联合生存年金的分布（密度函数与累积分布函数）。同时，我们针对基础状态变量开发了多项式展开方法，从而实现了对保证型联合生存年金期权快速而精确的近似计算。这些多项式展开也显著简化了模型的实施过程。总体而言，线性有理Wishart死亡率模型为联合死亡率风险的建模与管理提供了一个灵活而统一的框架。