A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equations. (English) Zbl 1517.65009
Summary: Recent work on path-dependent partial differential equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using regression. However, a limitation of this approach is to require the selection of a basis in a function space. In this paper, we overcome this limitation by the use of deep learning methods, and we show that this setting allows for the derivation of error bounds on the approximation of conditional expectations. Numerical examples based on a two-person zero-sum game, as well as on Asian and barrier option pricing, are presented. In comparison with other deep learning approaches, our algorithm appears to be more accurate, especially in large dimensions.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 68T07 | Artificial neural networks and deep learning |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 91G60 | Numerical methods (including Monte Carlo methods) |
Keywords:
path-dependent partial differential equations (PPDEs); deep neural networks; numerical methods for PPDEsReferences:
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