Hawkes and Affine Processes in Risk Modeling: Theory and Applications in Finance and Cybersecurity

The following Ph.D. thesis consists of three chapters and explores various applications of Hawkes processes and affine models within the field of risk modeling. Each chapter corresponds to a distinct paper. In the first project, we address an hedging problem, specifically we look for a semi-static variance-optimal strategy. We minimize the variance of the hedging error, combining static and dynamic positions in different market instruments. The problem is analyzed in an affine modeling framework, featuring stochastic volatility and self-exciting jumps in the log-price. The optimal strategy is characterized analytically through multi-dimensional complex integrals and computed numerically. We also perform a parameter sensitivity analysis and examine the impact of incorporating jumps on the hedging error. The second work focuses on a stochastic control problem applied to cyber-risk mitigation. A continuous-time stochastic model is developed, incorporating Hawkes processes to describe the arrival of cyberattacks targeting a specific entity. We formulate a control problem which is solved via dynamic programming, and we determine the optimal investment strategy. We then perform numerical experiments to highlight the role that attack modeling plays in determining the optimal response and resource allocation. The third chapter addresses a theoretical problem. Given an affine process under a certain probability measure, we characterize the family of all stable measure transformations that preserve the affine structure of the process. This theoretical insight is fundamental for applications such as pricing and risk management, ensuring that the affine properties are maintained under different probability measures.