Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations
This work addresses computational challenges in solving high-dimensional BSDEs, which are important in fields like finance for pricing problems, but it appears incremental as it builds on existing deep learning and optimization methods.
The authors tackled the problem of solving high-dimensional nonlinear backward stochastic differential equations (BSDEs) by proposing a deep learning-based scheme that reformulates it as a global optimization with local loss functions, achieving accurate approximations for high-dimensional cases including financial pricing problems.
In this work, we propose a new deep learning-based scheme for solving high dimensional nonlinear backward stochastic differential equations (BSDEs). The idea is to reformulate the problem as a global optimization, where the local loss functions are included. Essentially, we approximate the unknown solution of a BSDE using a deep neural network and its gradient with automatic differentiation. The approximations are performed by globally minimizing the quadratic local loss function defined at each time step, which always includes the terminal condition. This kind of loss functions are obtained by iterating the Euler discretization of the time integrals with the terminal condition. Our formulation can prompt the stochastic gradient descent algorithm not only to take the accuracy at each time layer into account, but also converge to a good local minima. In order to demonstrate performances of our algorithm, several high-dimensional nonlinear BSDEs including pricing problems in finance are provided.