Abstract This paper investigates the 𝕃 p {\mathbb{L}^{p}} -solutions ( p ∈ ( 1 , 2 ) {p\in(1,2)} ) for backward stochastic differential equations driven by Teugels martingales associated with a Lévy process (BSDELs). While existence and uniqueness were established by El Jamali (2023) under Lipschitz conditions, the present work focuses on establishing a comparison theorem specifically tailored for the 𝕃 p {\mathbb{L}^{p}} framework. Building upon this fundamental result, we prove the existence of 𝕃 p {\mathbb{L}^{p}} -solutions for BSDELs under significantly weakened assumptions on the coefficient. Specifically, we consider cases where the generator is either continuous with linear growth, or satisfies a monotonicity condition-being increasing (resp. decreasing)-combined with left (resp. right) continuity. These results extend the theoretical reach of BSDEL theory to a broader class of stochastic processes and coefficients.
Medium- and long-term insurance products must participate in equity market returns to remain competitive. To eliminate downside risk, guarantees should be included, which naturally gives rise to the problem of valuing contracts in a unified insurance-finance framework. We study a general setup that allows for joint modelling of financial markets, mortality, and policy holder behaviour. Within this framework, we propose a general affine approach exploiting the tractability of affine processes to obtain explicit valuation formulas for variable annuities and related contracts. The model permits flexible dependence between mortality and equity dynamics. Moreover, surrender intensities are modelled as functions of the driving affine process, thereby introducing endogenous market dependence into lapse behaviour - a feature rarely treated analytically in the literature. The resulting framework combines analytical tractability with sufficient flexibility to capture key features of long-term insurance products.
This paper investigates the L-p-solutions (p is an element of(1, 2)) for backward stochastic differential equations driven by Teugels martingales associated with a L & eacute;vy process (BSDELs). While existence and uniqueness were established by El Jamali (2023) under Lipschitz conditions, the present work focuses on establishing a comparison theorem specifically tailored for the L-p framework. Building upon this fundamental result, we prove the existence of L-p-solutions for BSDELs under significantly weakened assumptions on the coefficient. Specifically, we consider cases where the generator is either continuous with linear growth, or satisfies a monotonicity condition-being increasing (resp. decreasing)-combined with left (resp. right) continuity. These results extend the theoretical reach of BSDEL theory to a broader class of stochastic processes and coefficients.
The concept of tail variability in statistical analysis plays a central role in various fields, including economics, finance, and public policy. In this context, this paper aims to explore the asymptotic properties of Tail Extended Gini-type measures for a bivariate random vector ( X , Y ) {(X,Y)} , where X represents the examined loss variable and Y serves as a benchmark variable. Within this framework, we introduce a new measure of variability called the Joint Tail Extended Gini, which considers the tail characteristics of both X and Y. Specifically, we examine the asymptotic properties of the proposed measure, taking into consideration the degree of risk aversion of investors. Additionally, we generalize the Joint Tail-Gini functional in order to provide a more flexible risk measure. Subsequently, we present examples and demonstrate a practical application of our results. These findings have significant implications for understanding the extreme behaviors of investors and can be applied in various fields, including decision-making, to measure and analyze tail risk.
Real issues with the application of Minimum Capital Requirements (MCR) on the organisational level of an insurance holding has been the starting point of this paper. In contrast to the Solvency Capital Required (SCR), which is the day-to-day risk measure for insurers, the MCR comes into play in adverse circumstances. After the presentation of the concept of MCR, its formal properties are given. Applications highlight some of its shortcomings. Then a mathematical framework based on concepts of Aubin's viability theory sketches a unifying framework for both the SCR and MCR. Finally, a control theoretic framework is illustrated in a Cramer-Lundberg setting.
