Research

Synopsis: Broadly speaking, in the last decades we have witnessed these two parallel research-lines in mathematical finance: on one side, numerous applications of techniques coming from machine learning theory to classical problems of quantitative finance, ranging from portfolio optimization to derivative pricing and calibration, from wealth management to time-series prediction; on the other side, "infinite-dimensional-modelling", namely the re-formulation of classical entities and objects from financial mathematics in terms of infinite-dimensional systems. Some classical instances of this are given by interest rate theory, implied volatility surfaces, risk neutral densities; more recent examples are infinite-dimensional models of liquidity and dynamics in limit order books, stochastic partial differential equations for portfolio choice and energy markets.

At present, my research focus is to fuse together these two research currents, namely applying techniques coming from deep learning to infinite-dimensional problem pertaining to finance and trading, whence the term infinite-dimensional deep learning, and rigorously examining their theoretical justification in terms of their universal approximation properties as well as their interpretability.   

Overall, this is a wide and ongoing project, which nicely combines tools from abstract subjects (e.g. functional analysis, PDE theory, infinite-dimensional stochastic analysis, probability) with topics coming from more applied disciplines (like econometrics, functional data analysis, statistics and computer science) and which allows me to use my background in pure mathematics in collaboration with experts from other fields.

Research areas

• Theoretical Deep Learning geared towards applications in finance.

• Geometric PDEs 

• SPDEs motivated by fluid dynamics problems.

Current and future directions 

• Limit Order Books and Operator Learning: generative models and optimal order execution.

• Long time behaviour of generative models: model collapse.

• Graph neural networks for collective socio-economic phenomena: mitigation of the spread of fake news through social media.