Application of Continuous Non-Gaussian Mortality Models with Markov Switchings to Forecast Mortality Rates  
[PERSON]  
1 Department of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University, [PERSON] 1/3, 01-938 Warsaw, Poland; [EMAIL_ADDRESS]  
Footnote 1: email: {[PERSON],  
by [27; 28; 29; 30; 31; 32]. The authors proposed extended Milevsky and Promislov models with the non-Gaussian linear scalar filters for modelling of empirical mortality coefficients (\(\mu_{x,t}\)). Compared to the Gaussian linear scalar filters model with switchings (GLSFs) and Lee-Carter with switchings (LCs), the model proposed above allows for a more precise estimate of \(\mu_{x,t}\) for some fixed age groups of X. The proposed models with switchings seem to represent the natural way the epidemic oscillates. Unfortunately, the switching times were assumed to be deterministic in all mentioned approaches ([33]).  
In this paper, we first propose a special case when the empirical data of \(\mu_{x,t}\) can be divided into three basic parts. The observed data are \"regular\" (lower dispersion) in their first and third part. The second part is \"nonregular\", chaotically distributed, and with higher dispersion. The mortality models corresponding to the first and third part of the observed data are assumed to be extended Milevsky and Promislow models with the non-Gaussian linear scalar filters. The Markov switched model is proposed for the second part of the empirical data, which is a consequence of the switching between the first and the third part of the observed data. We will use the first and second moments of mortality rates in an iterative procedure to estimate the model parameters and the switching points between three basic parts. To the author's knowledge, the proposed approach is a new one.  
The paper is organised as follows. In Section 2.1, basic notations and definitions of stochastic hybrid systems with Markov switchings are introduced. In Sections 2.2 and 2.3, two new basic models with Gaussian linear scalar filters and continuous non-Gaussian excitation are presented, respectively. The moment equations are derived, and the stationary solutions for part variables are found. It is shown that only some of the parameters of these models can be unambiguously estimated. In Section 2.4, parameter estimation is performed based on the adapted numerical algorithm of a nonlinear minimisation problem (random search). The hybrid models are obtained by: parameter estimation procedures and the determination of switching points based on the Chow test for both models. The following steps of the technical work in Sections 2.1-2.4 were performed:  
1. The family of extended Milevsky and Promislov mortality models with Gaussian linear scalar filters (GLSF) is studied. This family is described by stochastic processes representing a mortality rate \(\mu_{x,t}\) for a person aged X at time t. The solutions of the mentioned stochastic differential equations are considered with switches (Section 2.1).
2. Considering the \(\ln\)-function of \(\mu_{x,t}\), and applying the Ito formula, a new vector state with unknown parameters is introduced (Section 2.2).
3. Using moment equations for GLSF with Markov switches, the first- and the second-order moments of equations for a particular case of two subsystem models (stationary and nonstationary solutions) are obtained, and approximate solutions are analysed (Sections 2.2.1-2.2.3).
4. A similar analysis to step S3 for the non-Gaussian linear scalar filters (nGLSF) model with Markov switches is repeated (Section 2.3).
5. The estimation procedure of the parameters (introduced in step S2) is applied (Section 2.4).  
In Section 3, we have compared empirical mortality rates with theoretical ones obtained from the standard LC model and the models proposed in Section 2.1. Section 4 contains the discussion. Conclusions research are then given in Section 5.