Core Concepts

A novel backward differential deep learning-based algorithm is proposed for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained not only on the inputs and labels but also the differentials of the corresponding labels.

Abstract

The authors propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear BSDEs. The key idea is to formulate the BSDE problem as a differential deep learning problem by using Malliavin calculus. This allows the estimation of the solution, its gradient, and the Hessian matrix, represented by the triple of processes (Y, Z, Γ).
The algorithm works as follows:
The BSDEs are discretized using the Euler-Maruyama method.
DNNs are employed to approximate the unknown processes (Y, Z, Γ).
The DNN parameters are backwardly optimized at each time step by minimizing a differential learning type loss function, which is defined as a weighted sum of the dynamics of the discretized BSDE system.
This approach ensures high accuracy not only for the process Y but also for the processes Z and Γ, which are important for financial applications. Compared to other deep learning-based schemes, the proposed algorithm is more efficient in computing the process Γ.
The authors provide an error analysis to show the convergence of the proposed algorithm. Numerical experiments up to 50 dimensions demonstrate the high efficiency of the method.

Stats

The authors state that the proposed scheme is more efficient compared to other contemporary deep learning-based methodologies, especially in the computation of the process Γ.

Quotes

"High-accurate gradient approximations are of great significance, especially in financial applications, where the process Z represents the hedging strategy for an option contract."
"Except the works in [38, 48], other deep learning-based schemes does not discuss in detail the approximations for Z in high-dimensional spaces, as it is generally more challenging than approximating Y for BSDEs."

Key Insights Distilled From

by Lorenc Kapll... at **arxiv.org** 04-15-2024

Deeper Inquiries

The proposed differential deep learning approach can be extended to other deep learning-based schemes for solving BSDEs that are formulated forward in time as a global optimization problem by modifying the loss function and training strategy. In the DLBDP scheme, the DNNs are trained backwardly at each time step by minimizing a differential learning type loss function. To adapt this approach to schemes formulated forward in time, the loss function can be redefined to incorporate the dynamics of the processes in a global optimization framework. By formulating the loss function to consider the forward evolution of the processes and optimizing the DNN parameters accordingly, the DLBDP scheme can be extended to handle BSDEs formulated forward in time.

While the differential deep learning formulation offers advantages in terms of accuracy and convergence for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), there are potential limitations or drawbacks compared to the traditional supervised deep learning approach. One limitation is the increased complexity in training the DNNs to approximate not only the labels but also their derivatives with respect to the inputs. This additional requirement may lead to higher computational costs and training time. Moreover, the need for explicit information about the dynamics of the processes, such as the process Z and the Hessian matrix Γ, can introduce challenges in modeling and optimization, especially in high-dimensional spaces. Additionally, the reliance on Malliavin calculus for formulating the BSDE as a differential learning problem may require a deeper understanding of advanced mathematical concepts, potentially limiting the accessibility of the approach to practitioners without a strong mathematical background.

The proposed algorithm can be adapted to handle other types of high-dimensional stochastic differential equations beyond BSDEs, such as forward-backward stochastic differential equations (FBSDEs) or stochastic control problems, by modifying the formulation of the loss function and the training strategy. For FBSDEs, the DNNs can be trained to approximate the solutions and their Malliavin derivatives in a similar manner to the BSDEs, with adjustments made to account for the forward evolution of the processes. In the case of stochastic control problems, the algorithm can be extended to incorporate the control variables and optimize the DNN parameters to approximate the optimal control strategies. By adapting the differential deep learning approach to these different types of stochastic differential equations, it can provide efficient and accurate solutions for a broader range of high-dimensional problems in various fields.

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