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.
Original language | English |
---|---|
Article number | 47 |
Journal | Risks |
Volume | 10 |
Issue number | 3 |
DOIs | |
Publication status | Published - Mar 2022 |
Externally published | Yes |
Bibliographical note
Funding Information:Acknowledgments: S.L. would like to thank the China Scholarship Council (CSC) for the financial support.
Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.
Funding
Acknowledgments: S.L. would like to thank the China Scholarship Council (CSC) for the financial support.
Keywords
- Artificial neural network
- Large time step simulation
- Numerical scheme
- Path-dependent options
- Stochastic collocation Monte Carlo sampler
- Stochastic differential equations