Risk measures and stochastic differential equations Theory and applications

This thesis advances the theory and applications of risk measures and Backward Stochastic Differential Equations (BSDEs) across four chapters, providing both theoretical insights and practical tools.
Chapter 1 introduces dynamic star-shaped risk measures and their connection to BSDEs. It shows that star-shaped functionals can be represented as the pointwise minimum of convex functionals, leading to the construction of dynamic convex risk measures. The chapter also explores star-shaped supersolutions for BSDEs and their practical applications in capital allocation and portfolio optimization.
Chapter 2 develops new characterizations for law-invariant star-shaped functionals, emphasizing their connections with Value-at-Risk and Expected Shortfall. Kusuoka-type representations are derived, demonstrating adaptability to diverse settings, including finance and insurance, without requiring monotonicity or cash additivity.
Chapter 3 introduces Geometric Backward Stochastic Differential Equations (GBSDEs) and two-driver BSDEs for modeling dynamic return risk measures. It investigates BSDEs with logarithmic and quadratic growth rates, proving existence, uniqueness, and stability under general conditions. The proposed GBSDE framework ensures meaningful financial properties, such as multiplicative convexity and positive homogeneity, even with unbounded coefficients or terminal conditions.
Chapter 4 focuses on general dynamic Capital Allocation Rules (CARs) for cash-subadditive risk measures, addressing challenges like interest rate ambiguity. A one-to-one correspondence is established between cash-subadditive CARs and risk measures, without assuming Gâteaux differentiability. Additionally, CARs derived from BSDEs with monotonic drivers are shown to solve Backward Stochastic Volterra Integral Equations (BSVIEs).
This work provides a comprehensive framework for advancing risk measurement, dynamic capital allocation, and BSDE methodologies, with applications in finance, insurance, and related fields.