Theory of Neural Networks with Applications in Finance, Insurance and Climate-Economy Modelling

This thesis sits at the intersection of three key research areas: (1) stochastic calculus, (2) artificial intelligence, with an emphasis on deep learning, and (3) financial and actuarial mathematics. In recent years, machine learning has increasingly influenced quantitative finance and actuarial science, with fascinating insights and applications. Topics such as the hedging of risky positions, synthetic scenario generation, and model calibration have all benefited from machine learning, making it possible to study previously intractable problems with remarkable precision and computational efficiency. The goal of this thesis is to advance the existing literature by introducing new results in the theory of neural networks from the point of view of financial and actuarial mathematics, and to exemplify these contributions through interesting case studies, which can be broadly grouped into the following three main thematic areas. The first theme, Deep Measure Projections, focuses on the optimal projection of a given measure onto a set of algorithmically generated measures, a concept that aligns well with the existing quantitative finance literature. Incomplete financial markets typically require the choice of an appropriate pricing measure for arbitrage-free pricing of financial derivatives. Similarly, variance reduction techniques for Monte Carlo (MC) methods seek efficient sampling measures for computing MC estimators. Chapter 1 addresses this by studying variance reduction through changes in sampling measures that are calculated via feedforward neural networks. The second theme, Deep Surrogate Models, examines the use of deep neural networks to approximate complex input-output maps, a method that is theoretically justified by universal approximation theorems. This approach is particularly useful when solving high-dimensional stochastic control problems, often encountered in quantitative finance and actuarial science, by least-squares Monte Carlo (LSMC) methods. Chapter 2 studies optimal insurance purchases when bequest motives are age-varying and life insurance and life annuities both carry loads, revealing up to two distinct periods of non-participation. Chapter 3 then extends the application of the LSMC method to complex stochastic climate-economy models, demonstrating how deep neural networks can improve the accuracy and efficiency of optimal policy derivation in uncertain, high-dimensional environments. The final theme, Algorithmic Strategies, involves the use of neural networks to approximate optimal decisions, such as those faced in dynamic trading, hedging, or reinsurance strategies, in feedback form. This approach, particularly deep hedging, has become a cornerstone method in quantitative finance and actuarial science for generating computationally feasible algorithms that identify optimal strategies. Chapter 4 introduces an application to algorithmic reinsurance policies that optimize the expected utility of terminal wealth perturbed by a modified Gerber--Shiu penalty function. Finally, Chapter 5 establishes universal approximation theorems for algorithmically generated stochastic (integral) processes, demonstrating that a stochastic calculus can be developed using algorithmic strategies.