A non linear approximation method for solving high dimensional partial differential equations: Application in finance

José Infante Acevedo (1) , Tony Lelièvre (1, 2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS

We study an algorithm which has been proposed by Chinesta et al. to solve high-dimensional partial differential equations. The idea is to represent the solution as a sum of tensor products and to compute iteratively the terms of this sum. This algorithm is related to the so-called greedy algorithm introduced by Temlyakov. In this paper, we investigate the application of the greedy algorithm in finance and more precisely to the option pricing problem. We approximate the solution to the Black-Scholes equation and we propose a variance reduction method. In numerical experiments, we obtain results for up to 10 underlyings. Besides, the proposed variance reduction method permits an important reduction of the variance in comparison with a classical Monte Carlo method.

Mots clés: Variance reduction; Black-Scholes partial differential equation; Greedy algorithms

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

José Infante Acevedo, Tony Lelièvre. A non linear approximation method for solving high dimensional partial differential equations: Application in finance. Mathematics and Computers in Simulation, 2018, 143, pp.14-34. ⟨10.1016/j.matcom.2016.07.013⟩. ⟨hal-00861892v2⟩