Bicausal optimal transport for SDEs with irregular coefficients

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Abstract

Many natural phenomena that exhibit randomness can be modelled by stochastic differential equations (SDEs), often having less regularity than the classical case of SDEs with Lipschitz coefficients. In such settings, we are interested in quantifying model uncertainty and the impact of model choice on the value of stochastic optimisation problems. To this end, we seek an appropriate notion of distance on the space of models. In particular, we study the adapted Wasserstein distance between the laws of SDEs. This is a special case of a bicausal optimal transport problem, in which the classical optimal transport problem is constrained to respect the flow of information inherent in stochastic processes.

Under minimal regularity assumptions on the coefficients, we show that the value of the bicausal optimal transport problem between the laws of one-dimensional SDEs is attained by the synchronous coupling. This is the coupling induced by taking a common Brownian motion as the driving noise for each SDE. Our proof is based on a discretisation method, exploiting monotonicity properties of the resulting discrete-time processes. A key tool in our work is a transformation-based semi-implicit Euler—Maruyama scheme for SDEs whose drift coefficient may have discontinuities and exponential growth. We prove the first strong existence and uniqueness result for such SDEs, and we obtain strong convergence rates for the implicit scheme. Moreover, our results provide a method for efficient computation of the adapted Wasserstein distance.

This talk is based on joint work with Michaela Szölgyenyi (University of Klagenfurt), and with Julio Backhoff-Veraguas (University of Vienna) and Sigrid Källblad (KTH Stockholm).

About Ben

Ben is a Postdoctoral Assistant in the Department of Statistics at the University of Klagenfurt, where he is working in the Stochastic Processes group of Michaela Szölgyenyi. Prior to this, he spent three years working in the Stochastic Analysis and Mathematical Finance group of Mathias Beiglböck and Walter Schachermayer at the University of Vienna. Originally from the UK, Ben completed his first degree at the University of Bristol and his postgraduate studies at the University of Bath, obtaining his PhD in 2020. Ben was a member of the Doctoral Training Centre SAMBa at the University of Bath, where his PhD thesis was supervised by Alexander Cox on the topic Stochastic Control Problems for Multidimensional Martingales.

Currently, Ben is working on problems in optimal transport with stochastic constraints, the analysis and numerical analysis of SDEs with irregular coefficients, and the problem of fitting martingales to prescribed marginal data. More generally, his research interests span the areas of applied probability, stochastic analysis, and mathematical finance.