Stochastic Mortality Projection Models

Updated 14 June 2026

Stochastic mortality projection models are statistical methods that incorporate randomness to forecast future age-specific death rates using state-space frameworks.
These models employ Bayesian inference, particle MCMC, and nonparametric techniques to obtain predictive distributions for life expectancies and actuarial liabilities.
Advanced features like stochastic volatility, regime-switching, and machine learning boost predictive accuracy and enhance scenario analysis.

Stochastic mortality projection models are a class of statistical models developed to forecast future mortality rates under uncertainty, explicitly accounting for both systematic (trend and volatility) and idiosyncratic risks in observed and future death rates. Distinguished from deterministic trend models, these approaches produce full predictive distributions for quantities of actuarial relevance—such as age-year-specific mortality rates, period and cohort life expectancies, and embedded liabilities—by modeling the stochastic evolution of key components (e.g., mortality indices, volatility processes, covariate effects). Modern research integrates hierarchical state-space structures, nonparametric Bayesian methods, robust uncertainty quantification, and the capacity to handle exogenous shocks or structural changes.

1. Foundations and State-Space Representations

A principal paradigm is the recasting of popular mortality models—such as the Lee–Carter (LC) family—into a general state-space framework. Consider the basic Lee–Carter structure for logged death rates: $y_{x,t} = \ln \hat m_{x,t} = \alpha_x + \beta_x \kappa_t + \varepsilon_{x,t}$ with $x$ (age), $t$ (year), and $\varepsilon_{x,t}$ representing homoscedastic or heteroscedastic measurement error. The stochastic evolution of the mortality period index $\kappa_t$ typically follows a random walk with drift: $\kappa_t = \kappa_{t-1} + \theta + \omega_t$ where $\omega_t$ may have static or time-varying variance. A notable extension is the introduction of stochastic volatility (SV) in the innovation variance of $\kappa_t$ , achieved by letting $\omega_t|\gamma_t \sim N(0,\exp\{\gamma_t\})$ and $\gamma_t$ itself evolve as an AR(1) process: $x$ 0 This leads to the "LCSV" (Lee–Carter stochastic-volatility) and "LCSV-H" (with heteroscedastic error) models, comprehensively capturing time-heterogeneity and age-specific variance in observed death trends (Fung et al., 2016).

Identification within bilinear models is critical. Rather than sum-to-one or sum-to-zero constraints on $x$ 1 and $x$ 2, an anchor-one-age rule ( $x$ 3, $x$ 4 fixed) is preferred for both likelihood maximization and MCMC inference, yielding trivial, robust identifiability in both frequentist and Bayesian estimation (Fung et al., 2016).

2. Bayesian Inference and Particle MCMC Schemes

Full joint Bayesian estimation is undertaken by assigning independent, weakly-informative priors to all parameters and latent states, encompassing scale, sensitivity, drift, volatility, and autoregressive parameters. For LCSV-type models, this produces a posterior over $x$ 5. A hybrid block-Gibbs sampler is constructed with:

Forward-filtering/backward-sampling for $x$ 6
Particle Independent Metropolis–Hastings (PIMH) for the stochastic volatility path $x$ 7
Closed-form Gibbs steps for static parameters (Gaussian for scale/loadings/drift, inverse-gamma for variances).

This Rao–Blackwellized Particle MCMC scheme converges to the true posterior and accommodates incorporation of both heteroscedasticity and time-varying volatility, improving fit and calibration over two-step SVD/ARIMA routines. For Danish male data, LCSV-H achieves best in-sample DIC and provides realistic predictive intervals in forecasting (Fung et al., 2016).

3. Nonparametric and Gaussian Process Extensions

Beyond parametric trend models, Gaussian Process (GP) regression provides a flexible, nonparametric family for modeling (log-)mortality rates or improvement factors across age and year. In the sub-population context, a GP prior is imposed on the deflator process $x$ 8 (relative log-mortality to a reference table), with covariance structure determined by length-scales in age and year, and hyperpriors for all parameters: $x$ 9 where $t$ 0 may be constant or encode core age/year effects, and $t$ 1 typically employs squared-exponential kernels (possibly separable). A negative binomial model accommodates overdispersion in sparse populations (Melo et al., 4 Jun 2025). Bayesian inference proceeds via full MCMC (e.g., in R Stan), yielding joint posteriors and out-of-sample predictive distributions. GP-based frameworks, when compared using scoring rules, outperform standard fixed-effect or AR(1) deflators and produce credible intervals for both rates and counts that respect data sparsity. Such approaches are particularly effective for actuarial projection in small pension funds relative to large industry, national, or cohort benchmarks (Melo et al., 4 Jun 2025); see also (Ludkovski et al., 2016) for GP smoothing and improvement factor modeling in national populations.

4. Flexible Dispersion and Distributional Modelling

Modeling of death counts' dispersion structure is addressed by replacing the Poisson or negative binomial likelihood with the Conway–Maxwell–Poisson (CMP) distribution, allowing flexible data-driven estimation of overdispersion or underdispersion: $t$ 2 with mean and variance functions

$t$ 3

Hierarchical Bayesian inference is then performed, with $t$ 4 as a strictly positive random parameter (Gamma-prior), enabling robust integration over process, distributional, and parameter uncertainty via MCMC. CMP-based models consistently outperform Poisson and NB in both in- and out-of-sample fit, especially in the presence of strong data dispersion (England & Wales males, 1963–2021) (Wong et al., 4 Jan 2026).

5. Structured Nonparametric and Machine Learning Models

Generalized Additive Models (GAMs) and machine learning tools extend stochastic mortality projection by fitting highly flexible semiparametric smoothers to age, period, and cohort components, typically through penalized B-splines (Eilers–Marx P-splines) for age-period-cohort (APC) surfaces. GAMs are fitted by penalized quasi-likelihood, with smoothing parameters selected by REML or GCV. Machine learning corrections (e.g., tree-based residual modeling) can augment classical trend models, capturing residual features missed by standard parametric components (Nalmpatian et al., 2023). Such models enable integration of time-varying exogenous covariates (e.g., COVID-19 indicators), multi-population pooling, and scenario-based projection under pandemic regimes. Forecast intervals and scenario uncertainty are quantified by Monte Carlo, drawing from both spline posterior and profile-likelihood error (Nalmpatian et al., 2023).

6. Contemporary Models: Delays, Jumps, High-Frequency, and Structural Enhancements

Recent developments introduce more complex stochastic architectures:

Retarded stochastic systems model age-year mortality surfaces using stochastic delay differential or difference equations, integrating discrete diffusions (across age), explicit memory kernels (long-term delay in temporal transmission), and bounded multiplicative stochasticity. Such models achieve superior predictive coverage and plausibility over classic Lee–Carter and Renshaw–Haberman in empirical applications, particularly by capturing path-dependence and attenuating unwarranted forecast variance (Caraballo et al., 26 May 2025).
Catastrophe and regime-switching models embed stochastic process models with hidden Markov structures, accommodating multi-year, age-specific mortality shocks (e.g., war, pandemic) by regime-switching in the residuals. Prediction intervals expand to reflect calibrated shock frequency/severity and provide solvency capital requirements that may diverge substantially from standard formulae (Robben et al., 2023).
Mixed-frequency state-space models leverage both annual and high-frequency (e.g., monthly) mortality data by embedding Lee–Carter structures in a joint state-space with SARIMA monthly latent processes. EM/Kalman procedures pool information across frequencies in real time, improving intra-year nowcasts and forecast intervals compared to annual-only or separately reconciled models (Li et al., 9 Jan 2026).

7. Applications, Comparative Validation, and Empirical Performance

Rigorous validation employs:

In-sample and out-of-sample scoring rules: DIC, log-score, residual-based $t$ 5 measures, RMSE, and proper scoring rules (CRPS, RPS).
Posterior predictive coverage and fan charts for both rates and improvement factors.
Comparative calibration (e.g., to Danish, English & Welsh, Brazilian, Dutch, or Spanish data), with models such as GP, LCSV, CMP, retarded NLSD, and regime-switching consistently exhibiting better fit, shorter- or better-calibrated forecast intervals, or both, relative to classical linear trend models (Fung et al., 2016, Melo et al., 4 Jun 2025, Wong et al., 4 Jan 2026, Caraballo et al., 26 May 2025, Robben et al., 2023).

Actuarial applications include liability valuation, SCR calculation (VaR under simulation), scenario-analysis under pandemic or climate-risk regimes, and assessment/forecast of mortality improvement rates for use in pricing and reserving frameworks.

Key references: (Fung et al., 2016) – stochastic-volatility Lee–Carter models in a Bayesian state-space framework; (Melo et al., 4 Jun 2025, Ludkovski et al., 2016) – Gaussian Process mortality and sub-population models; (Wong et al., 4 Jan 2026) – Bayesian Lee–Carter with Conway–Maxwell–Poisson count specification; (Nalmpatian et al., 2023) – GAM-APC and ML extensions; (Caraballo et al., 26 May 2025) – retarded discrete-diffusion SDE models; (Robben et al., 2023) – regime-switching multi-population with catastrophe risk; (Li et al., 9 Jan 2026) – mixed-frequency state-space modeling.