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

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.

On Pricing of Discrete Asian and Lookback Options under the Heston Model

“…The method uses a Stochastic Collocation [10] (SC) based approach to approximate the target distribution. Then, artificial neural networks (ANNs) "learn" the proxy of the desired distribution, allowing for fast recovery [16,19].…”

“…Inspired by the works in [16,19], the method relies on the technique of Stochastic Collocation (SC) [10], which is employed to accurately approximate the targeted distribution by means of piecewise polynomials. Artificial neural networks (ANNs) are used for fast recovery of the coefficients which uniquely determine the piecewise approximation.…”

“…The method proposed in this article constitutes an extension of [19], where the problem of sampling from time-integrated stochastic bridges was addressed. The model relies on the Seven-League scheme [16], where artificial neural networks are employed to "learn" the distribution of the random variable of interest utilizing stochastic collocation points [10]. The method results in a robust procedure for Monte Carlo pricing.…”

Nysalt: Nyström-Type Inference-Based Schemes Adaptive to Large Time-Stepping

“…Consequently, the inferred integrators are robust and generalizable beyond the training set (see Section 3). A few work of similar spirit can be found in [17,48,80], where neural network based approximations lead to large time-stepping integrators. However, the neural networks are computationally expensive to train and their parameters are often sensitive to training data, making it difficult to systematically investigate its properties such as the maximal admissible time step size of stability.…”

“…When the system is known, MDNet based on graph neural network in [80] enables the simulation of microcanonical (i.e. Hamiltonian) molecular dynamics with large steps; a stochastic collocation method in [48] and a parametric inference in [45] have lead to large time-stepping integrators for SDEs. This study extends the parametric inference approach in [45] to Hamiltonian systems.…”