OT techniques in data driven methodology: theory and practice from mathematical finance and statistics
Abstract
Wasserstein distances, or Optimal Transport methods more generally, offer a powerful non-parametric toolbox to conceptualise and quantify model uncertainty in diverse applications. Importantly, they work across the spectrum: from small uncertainty around a selected model (e.g., the empirical measure) to large uncertainty of considering all models consistent with the data. I will showcase this using examples from mathematical finance (pricing and hedging of options, optimal investment) and statistics (non-parametric estimators, regularised regression methods). I will illustrate the large uncertainty regime using Martingale OT problems. For the small uncertainty regime I will consider a generic stochastic optimization problem and its distributionally robust version using Wasserstein balls. I will derive explicit formulae for the first order correction to both the value function and the optimizer. Throughout, I will present both theoretical result, as well as comments on the available numerical methods.
The talk will be borrow from many joint works, including with Daniel Bartl, Samuel Drapeau, Stephan Eckstein, Gaoyue Guo, Tongseok Lim and Johannes Wiesel.
