2021/10/11 by Bálint Négyesi, Kristoffer Andersson, Negyesi, Balint +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2110.05421
openalex publication_date 2021/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A novel discretization is presented for 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 solution pair, which implies the need to provide accurate Γ estimates. The approximation is based on a merged formulation given by the Feynman-Kac formulae and the Malliavin chain rule. The continuous time dynamics is discretized with a theta-scheme. In order to allow for an efficient numerical solution of the arising semi-discrete conditional expectations in possibly high-dimensions, it is fundamental that the chosen approach admits to differentiable estimates. Two fully-implementable schemes are considered: the BCOS method as a reference in the one-dimensional framework and neural network Monte Carlo regressions in case of high-dimensional problems, similarly to the recently emerging class of Deep BSDE methods [Han et al. (2018), Huré et al. (2020)]. An error analysis is carried out to show L2 convergence of order 1/2, under standard Lipschitz assumptions and additive noise in the forward diffusion. Numerical experiments are provided for a range of different semi- and quasi-linear equations up to 50 dimensions, demonstrating that the proposed scheme yields a significant improvement in the control estimations.