A backward differential deep learningbased algorithm for solving highdimensional nonlinear backward stochastic differential equations
Abstract
In this work, we propose a novel backward differential deep learningbased algorithm for solving highdimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained not only on the inputs and labels but also the differentials of the corresponding labels. This is motivated by the fact that differential deep learning can provide an efficient approximation of the labels and their derivatives with respect to inputs. The BSDEs are reformulated as differential deep learning problems by using Malliavin calculus. The Malliavin derivatives of solution to a BSDE satisfy themselves another BSDE, resulting thus in a system of BSDEs. Such formulation requires the estimation of the solution, its gradient, and the Hessian matrix, represented by the triple of processes $\left(Y, Z, \Gamma\right).$ All the integrals within this system are discretized by using the EulerMaruyama method. Subsequently, DNNs are employed to approximate the triple of these unknown processes. The DNN parameters are backwardly optimized at each time step by minimizing a differential learning type loss function, which is defined as a weighted sum of the dynamics of the discretized BSDE system, with the first term providing the dynamics of the process $Y$ and the other the process $Z$. An error analysis is carried out to show the convergence of the proposed algorithm. Various numerical experiments up to $50$ dimensions are provided to demonstrate the high efficiency. Both theoretically and numerically, it is demonstrated that our proposed scheme is more efficient compared to other contemporary deep learningbased methodologies, especially in the computation of the process $\Gamma$.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.08456
 arXiv:
 arXiv:2404.08456
 Bibcode:
 2024arXiv240408456K
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Machine Learning;
 Quantitative Finance  Computational Finance;
 65C30;
 68T07;
 60H07;
 91G20
 EPrint:
 40 pages, 5 figures, 5 tables