Maize yield prediction in the sub-Saharan region is imperative for mitigation of risks emanating from crop loss due to changes in climate. Temperature, rainfall amount, and reference evapotranspiration are major climatic factors affecting maize yield. They are not only interdependent but also have significantly changed due to climate change, which causes nonlinearity and nonstationarity in weather data. Hence, there exists a need for a stochastic process for predicting maize yield with higher precision. To solve the problem, this paper constructs a joint stochastic process that acquires joints effects of the three weather processes from joint a probability density function (pdf) constructed using copulas that maintain interdependence. Stochastic analyses are applied on the pdf and process to account for nonlinearity and nonstationarity, and also establish a corresponding stochastic differential equation (SDE) for maize yield. The trivariate stochastic process predicts maize yield with R 2 = 0.8389 and M A P E = 4.31 % under a deep learning framework. Its aggregated values predict maize yield with R 2 up to 0.9765 and M A P E = 1.94 % under common machine learning algorithms. Comparatively, the R 2 is 0.8829% and M A P E = 4.18 % , under the maize yield SDE. Thus, the joint stochastic process can be used to predict maize yield with higher precision.

Many actuarial science researchers on stochastic modeling and forecasting of systematic mortality risk use Cairns-Blake-Dowd (CBD) Model (2006) due to its ability to consider the cohort effects. A three-factor stochastic mortality model has three parameters that describe the mortality trends over time when dealing with future behaviors. This study aims to predict the trends of the model, $k_t^{\left(2\right)}$ by applying the Recurrent Neural Networks within a Short-Term Long Memory (an artificial LSTM architecture) compared to traditional statistical ARIMA (p,d,q) models. The novel deep learning (machine learning) technique helps integrate the CBD model to enhance its accuracy and predictive capacity for future systematic mortality risk in countries with limited data availability, such as Kenya. The results show that Long Short-Term Memory network architecture had higher levels of precision when predicting the future systematic mortality risks than traditional methods. Ultimately, the results can be implemented by Kenyan insurance firms when modeling and forecasting systematic mortality risk helpful in the pricing of Annuities and Assurances.

The classical mortality models such as the Cairns-Blake-Dowd (CBD), Lee-Carter (LC), Linear Regression (LR) models are used to model Systematic Mortality Risk (SMR) for many developed countries populations for actuarial product valuations. This research study aims at incorporating the Bühlmann credibility approach (BCA) to improve the SMR models to fit sub-Saharan African populations like Kenya. Since the Kenyan population does not exhibit the Gaussian properties used in modeling the classical error terms, we proposed using Normal Inverse Gaussian distribution to model these error terms instead of a Gaussian distribution. We model the error terms of the classical models (LC, CBD, and LR) as a Normal Inverse Gaussian (NIG) distribution through the Bühlmann credibility approach. This novel approach demonstrates an improved precision of the predicted SMR as shown by the values of MAPE and RMSE measures compared to those under classical mortality risk models. Ultimately, we have done actuarial valuations of annuities and assurances using our determined SMR, thus concluding that this BCA approach improves the accuracy of actuarial products sold in the Kenyan market.

This study considered the modeling of systematic mortality risk for populations with finite data using the Poisson incorporated Credibility regression model. For novelty, we have included the credibility regression approach to modelling mortality by assuming the number of annual deaths follow a Poisson distribution. Our model shows improvement in precision levels when estimating mortality risk compared to classical models used in European countries. We have illustrated that our model works optimally when using Kenyan mortality data, comparing male and female lives under the different strategies, thus making better predictions than the classical LeeCarter (LC) and CairnsBlakeDowd (CBD) models. The mean absolute forecast error (MAFE), mean absolute percentage forecast error (MAPFE), root mean square error (RMSE), and root mean square forecast error (RMSFE) under the incorporated credibility regression model are much lower than the values obtained without incorporation of the Buhlmann credibility approach. The findings of this research will help insurance companies, pension firms, and government agencies in sub-Saharan countries model and forecast systematic mortality risks accurately. Finally, the results are essential in actuarial modelling and pricing, thus making life assurance products affordable for most people in low-income African countries.

The explosion of time series count data with diverse characteristics and features in recent years has led to a proliferation of new analysis models and methods. Significant efforts have been devoted to achieving flexibility capable of handling complex dependence structures, capturing multiple distributional characteristics simultaneously, and addressing nonstationary patterns such as trends, seasonality, or change points. However, it remains a challenge when considering them in the context of long-range dependence. The Lévy-based modeling framework offers a promising tool to meet the requirements of modern data analysis. It enables the modeling of both short-range and long-range serial correlation structures by selecting the kernel set accordingly and accommodates various marginal distributions within the class of infinitely divisible laws. We propose an extension of the basic stationary framework to capture additional marginal properties, such as heavy-tailedness, in both short-term and long-term dependencies, as well as overdispersion and zero inflation in simultaneous modeling. Statistical inference is based on composite pairwise likelihood. The models flexibility is illustrated through applications to rainfall data in Guinea from 2008 to 2023, and the number of NSF funding awarded to academic institutions. The proposed model demonstrates remarkable flexibility and versatility, capable of simultaneously capturing overdispersion, zero inflation, and heavy-tailedness in count time series data.

This paper establishes Stochastic Partial Differential Equation (SPDE) as a framework to model the dynamics of wealth associated with unsecured loans. The approach discussed herein recognizes the significance of the correlation between market forces, loan-specific characteristics, and idiosyncratic borrower factors. We develop a model that incorporates these elements through a combination of diffusion terms and a Gaussian Kernel noise component. We detail the derivation of the SPDE, emphasizing the role of Gaussian noise to capture the proportional impact of fluctuating economic conditions on loan wealth. The incor-poration of Gaussian decay in our model allows us to address the memory and fading influence of past economic shocks, providing a realistic depiction of risk and return dynamics in a loan portfolio. Analytical solution to the SPDE is explored under Dirichlet’s natural constrains of the process. Through numerical simulations, we demonstrate the model's application in es-timating probability of default of borrowers given various credit scores under different economic conditions. The resultant model’s use in unsecured loans opens a new chapter of SPDE applications in finance.