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

Shuaiqiang Liu, Lech Grzelak and Cornelis Oosterlee
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Shuaiqiang Liu: Applied Mathematics (DIAM), Delft University of Technology, 2628 CD Delft, The Netherlands

Risks, 2022, vol. 10, issue 3, 1-27

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.

Keywords: artificial neural network; stochastic differential equations; large time step simulation; stochastic collocation Monte Carlo sampler; numerical scheme; path-dependent options (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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Related works:
Working Paper: The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations (2021)
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