The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions

A novel discretization is presented for decoupled forward–backward stochastic differential equations (FBSDE) with differentiable coefficients, simultaneously solving the BSDE and its Malliavin sensitivity problem. The control process is estimated by the corresponding linear BSDE driving the trajectories of the Malliavin derivatives of the...

Stability of backward stochastic differential equations: the general Lipschitz case

In this paper, we obtain stability results for backward stochastic differential equations with jumps (BSDEs) in a very general framework. More specifically, we consider a convergent sequence of standard data, each associated to their own filtration, and we prove that the associated sequence of (unique) solutions is also convergent. The...

A Novel Method for Solving High-Dimensional Backward Stochastic Differential Equations Using Malliavin Calculus and Deep Learning

Backward stochastic differential equations (BSDE) are known to be a powerful tool in mathematical modeling due to their inherent connection with second-order parabolic partial differential equations (PDE) established by the non-linear Feynman-Kac relations. The fundamental power of BSDEs lies in the fact that with them one does not merely obtain...

Fourier Methods for Multidimensional Problems and Backward SDEs in Finance and Economics

In this thesis we deal with processes with uncertainties, such as financial asset prices and the global temperature. We model their evolutions by so-called stochastic processes. Many of these stochastic processes are based on the Wiener process, whose increments are normally distributed. Other models may contain jump components, to model, for...

A numerical method for backward stochastic differential equations with applications in finance.

This thesis starts by discussing the foundations of mathematical finance and some theoretical results on backward stochastic differential equations. We discuss some examples of these equations in mathematical finance (primarily option pricing) and develop a numerical method that can approximate solutions to these equations. Subsequently, we...