Volterra Processes and Path-Signatures in Finance

15h

Memory effects play a crucial role in shaping financial phenomena, from asset price and order flow dynamics to portfolio optimization and derivative pricing. Many financial processes exhibit path-dependent behaviors such as long/short-memory effects and lead-lag relationships, which cannot be adequately captured by traditional Markovian models. Understanding and incorporating these dependencies is essential for accurate trading strategies, risk management and investment decisions.

Recent advancements in stochastic modeling, including Volterra processes and path signatures, offer rigorous frameworks to model memory in quantitative finance. Volterra processes are (in their simplest versions) the continuous-time analogs of weighted moving averages and encompass fractional Brownian motion and Hawkes processes, while path signatures, sequences of iterated integrals, have proven to be powerful tools for feature extraction in time-series analysis, particularly in machine learning applications. Signatures are central objects in rough path theory. In the context of mathematical finance, signatures are a powerful tool both for modeling phenomenon with memory and for providing numerical methods in the absence of the Markov property. Despite their promise, these methods still present significant theoretical and practical challenges.

This course will develop memory-aware models and numerical methods suitable for processes with memory, including models and methods based on Volterra processes and path signatures. Applications include volatility modeling, derivative pricing, portfolio optimization under market frictions and risk management.