The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations

Shuaiqiang Liu,Lech A Grzelak,Cornelis W Oosterlee

doi:10.3390/risks10030047

Shuaiqiang Liu, Lech A Grzelak + Show 1 more

Open Access

https://doi.org/10.3390/risks10030047

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Abstract

We propose an accurate data-driven numerical scheme to solve stochastic differential equations (SDEs), by taking large time steps. The SDE discretization is built up by means of the polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Basic error analysis indicates that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression–decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. As a proof of concept, 1D numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.

Highlights

The highly successful deep learning paradigm (LeCun et al 2015) receives a lot of attention in science and engineering
As the first method component, we evaluate the quality of the artificial neural network (ANN) which defines the collocation points, for the GBM dynamics
We developed a data-driven numerical solver for stochastic differential equations, by which large time step simulations can be carried out accurately in the sense of strong convergence

Summary

Introduction

The highly successful deep learning paradigm (LeCun et al 2015) receives a lot of attention in science and engineering. The aim with machine learning is to either speed up the solution process or to solve high-dimensional problems that are not handled by the traditional numerical methods. We develop a highly accurate numerical discretization scheme for scalar stochastic differential equations (SDEs), which is based on taking possibly large discrete time steps. We “learn” to take large time steps, with the help of the stochastic collocation Monte. There are quite a few applications that could benefit from an accurate and efficient numerical method on the basis of a large time step discretization (Li et al 2021), for example, in finance, the valuation of path-dependent financial derivatives or financial risk management where counterparty credit risk plays a role

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