Syoiti Ninomiya

Syoiti Ninomiya, Yuming Ma

A new deep-learning neural network architecture based on high-order weak approximation algorithms for stochastic differential equations (SDEs) is proposed. The architecture enables the efficient learning of martingales by deep learning models. The behaviour of deep neural networks based on this architecture, when applied to the problem of pricing financial derivatives, is also examined. The core of this new architecture lies in the high-order weak approximation algorithms of the explicit Runge--Kutta type, wherein the approximation is realised solely through iterative compositions and linear combinations of vector fields of the target SDEs.

Syoiti Ninomiya, Nicolas Victoir

The authors present a new simple algorithm to approximate weakly stochastic differential equations in the spirit of [1] and [2]. They apply it to the problem of pricing Asian options under the Heston stochastic volatility model, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast. [1] Shigeo Kusuoka, ``Approximation of Expectation of Diffusion Process and Mathematical Finance,'' Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp. 147--165. [2] Terry Lyons and Nicolas Victoir, ``Cubature on Wiener Space,'' Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460 (2004), pp. 169--198.