<p><br></p><p dir="ltr">The complexity of modern financial markets poses substantial challenges for analyzing time-series data, as traditional diffusion models often fail to capture the intricate dynamics of real-world market behavior. This dissertation develops data-driven, non-Markovian methodologies to overcome these limitations, enhancing the analysis of dependencies in financial and insurance data. The research employs advanced stochastic models and machine learning techniques to address critical phenomena in finance and insurance. Notable findings include the following: a Hawkes process framework is introduced to model cascading health transitions, capturing how past events amplify the likelihood of future occurrences (Chapter 3); a discrete-time Hawkes process is used to quantify time-varying lead-lag relationships between intraday and overnight returns, uncovering predictive dynamics in asset price movements (Chapter 4); and attention-based models are applied to high-dimensional, spatiotemporal limit order book (LOB) data, enabling robust analysis and forecasting of its complex structure and behavior (Chapter 5). These findings highlight the limitations of traditional Markovian models, particularly in representing memory-dependent systems, high-frequency data, and multi-state processes. By advancing non-Markovian methods, this dissertation provides practical tools for analyzing momentum effects, cascading health transitions, and intricate market microstructures. These contributions establish a robust analytical foundation for understanding memory-dependent dynamics in finance and insurance, addressing key limitations of Markovian assumptions and opening new avenues for research and application.</p>