This thesis explores the confluence of financial mathematical models and advanced machine-learning techniques, specifically focusing on pricing, hedging and volatility modelling. By transcending the limitations of traditional models, it leverages cutting-edge advancements in deep learning to handle complex financial calculations, providing valuable insights in these areas.

One significant contribution is the development of neural stochastic differential equations, a novel approach that enhances pricing and hedging by capturing realistic market behaviour while providing reliable bounds. Notably, the approach offers flexible calibration under risk-neutral and real-world measures, enabling more accurate simulations of diverse market scenarios.

Next, to tackle the challenges posed by path-dependent partial differential equations arising from rough volatility models, this thesis introduces a numerical algorithm utilizing reservoir-type neural networks. The algorithm merges theoretical convergence properties with practical regression-based solutions, effectively overcoming some of the limitations of traditional numerical methods.

Furthermore, this study delves into variance reduction in stochastic volatility models, with a particular focus on the Heston model. It presents a comprehensive analysis of importance sampling, based on large and moderate deviation principles. The variance reduction of the resulting Monte Carlo estimators is demonstrated through theoretical guarantees and empirical studies.

Lastly, the thesis aims to address a significant gap in a rough volatility model used in equity markets, namely the Rough Bergomi model. It proposes a generalized model that departs from the log-normality constraint, incorporating self-similarity and stationary increments. Through this refinement, it seeks to provide a more accurate reflection of VIX market data.

In summary, we hope this research casts a new light on the symbiosis between financial mathematics and modern data science techniques. Its goal is to propel a paradigm shift in our understanding of modelling financial markets, magnifying the transformative potential of deep learning techniques on traditional mathematical models currently used in the financial industry.