Deep learning-based algorithms for high-dimensional nonlinear backward stochastic differential equations

My name is Lorenc Kapllani and I am a PhD student in the Applied and Computational Mathematics group at the University of Wuppertal. My research focuses on the development of efficient deep learning-based algorithms for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs) and quantifying their uncertainty.

The primary motivation for studying BSDEs lies in their importance as essential tools for modeling problems in various scientific domains, including finance, economics, and physics. This importance stems from their connection to partial differential equations (PDEs) through the well-known (nonlinear) Feynman-Kac formula. In most practical scenarios, BSDEs cannot be solved explicitly due to their nonlinearity. Moreover, some of the most important equations are naturally formulated in high dimensions. For example, the Black-Scholes option pricing equation shows the dimensionality of the BSDE with the number of underlying financial assets considered. Most existing numerical techniques are ill-suited for high-dimensional BSDEs due to the well-known challenge known as the “curse of dimensionality”. The computational cost of solving high-dimensional BSDEs grows exponentially with increasing dimensionality.

The first deep learning-based scheme tailored for this purpose, called deep BSDE, was introduced in [1]. The authors conducted numerical experiments on various examples, demonstrating the effectiveness of their proposed algorithm in high-dimensional settings. This method has paved the way for solving such problems in hundreds of dimensions within a reasonable computational time. Following the DBSDE scheme, many other works have been proposed [2, 3, 4]. In [3] it is shown that the deep BSDE scheme has some drawbacks. It may converge to an approximation far from the actual solution or even diverge when dealing with complex solution structures and long terminal times. Moreover, it is capable of obtaining much better approximations of the BSDE at the initial time than at other times. To address these issues, we introduced an improved scheme [5] that overcomes the drawbacks of the deep BSDE scheme.

It is well known that models using deep learning in decision making are prone to noise and model errors. Therefore, it is critical to evaluate the reliability of the model prior to practical use. An example of such decisions is the pricing and hedging of various contracts in finance. Companies can suffer significant financial losses as a result of poor decisions. Therefore, it is highly desirable to understand the uncertainties in the aforementioned deep learning-based systems and to develop methods to quantify them. In [6], we investigate uncertainty quantification (UQ) for the systems in [1, 5]. Typically, dealing with uncertainty involves computing the standard deviation (STD) of approximate solutions from multiple algorithm runs on different datasets, which is computationally expensive, especially for high-dimensional problems. Therefore, we have developed an efficient UQ model that estimates the STD and mean of the approximate solution using a single algorithm run. Our numerical experiments not only demonstrate the ability of the UQ model to reliably estimate the mean and STD of the approximate solution for the considered deep learning-based BSDE schemes, but also highlight several practical implications of using such a model.

When solving BSDEs, one must simultaneously approximate the solution and its gradient. In general, the latter is more difficult to approximate. The same problem is observed in current deep learning based algorithms. Therefore, I am currently developing a new algorithm based on differential deep learning to provide highly accurate approximations of the gradient of the solution.

[1] W. E, J. Han, and A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5 (2017), pp. 349-380.
[2] M. Raissi, Forward-backward stochastic neural networks: Deep learning of high-dimensional partial differential equations, arXiv preprint 1804.07010, 2018.
[3] C. Hure, H. Pham, and X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comp., 89 (2020), pp. 1547-1579.
[4] L. Kapllani, The effect of the number of neural networks on deep learning schemes for solving high dimensional nonlinear backward stochastic differential equations, in M. Ehrhardt, M. Günther (eds) Progress in Industrial Mathematics at ECMI 2021. ECMI 2021. Mathematics in Industry, vol. 39, Springer, Cham, 2022.
[5] L. Kapllani and L. Teng, Deep learning algorithms for solving high-dimensional nonlinear backward stochastic differential equations, Discrete Contin. Dyn. Syst. – B, (2023).
[6] L. Kapllani, L. Teng, and M. Rottmann, Uncertainty quantification for deep learning-based schemes for solving high-dimensional backward stochastic differential equations, arXiv preprint 2310.03393, 2023.

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