Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations (1709.05963v1)

Abstract: High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using ${\rm T{\small ENSOR}F{\small LOW}}$ in ${\rm P{\small YTHON}}$ illustrate the efficiency and the accuracy of the method in the cases of a $100$-dimensional Black-Scholes-Barenblatt equation, a $100$-dimensional Hamilton-Jacobi-BeLLMan equation, and a nonlinear expectation of a $ 100 $-dimensional $ G $-Brownian motion.

• The paper introduces a novel method merging fully nonlinear PDEs with 2BSDEs to address high-dimensional challenges in financial models.
• It employs deep neural networks with temporal discretization to overcome the curse of dimensionality in complex systems.
• SGD-driven optimization and extensive numerical experiments validate its efficiency in approximating 100-D Black-Scholes-Barenblatt and HJB equations.

Overview of Machine Learning Approximation Algorithms for High-Dimensional Nonlinear PDEs

The paper provides a comprehensive method to solve high-dimensional fully nonlinear partial differential equations (PDEs) and second-order backward stochastic differential equations (2BSDEs) through machine learning techniques. The approach is significant in addressing the computational challenges posed by the "curse of dimensionality" in PDEs that arise in financial modeling, such as derivative pricing and portfolio optimization.

Key Contributions

1. Theoretical Framework and Problem Setting: The problem focuses on high-dimensional PDEs - particularly those seen in the financial industry where the dimensionality corresponds to the number of assets in a portfolio. The paper highlights the difficulties of solving these equations due to their nonlinear nature and reliance on factors such as default risks and transaction costs.
2. Merging PDEs with 2BSDEs: A novel method is proposed that capitalizes on the connection between fully nonlinear second-order PDEs and 2BSDEs. This connection is pivotal for developing algorithms that can approximate the solution of such complex equations.
3. Temporal and Spatial Approximations: The methodology involves temporally discretizing the 2BSDEs into simpler processes and employing deep neural networks for spatial approximations. This blend of techniques marks a shift from traditional numerical methods to learning-based approaches that are more scalable in higher dimensions.
4. Stochastic Gradient Descent Optimization: The optimization process is driven by stochastic gradient descent (SGD) types of algorithms, allowing the solution search to navigate the high-dimensional space effectively. This aspect is critical in adjusting the parameters of the neural networks to converge to an efficient representation of the solution.
5. Numerical Experiments and Results: The paper demonstrates the effectiveness of its proposed method through experiments with 100-dimensional Black-Scholes-Barenblatt equations and Hamilton-Jacobi-BeLLMan equations. The results obtained via TensorFlow implementations validate the scheme's capacity to handle intricate financial modeling tasks.

Numerical Results and Implications

The numerical simulations showcase significant advancements in computing solutions to high-dimensional financial models. The results underscore the neural network's ability to learn and generalize, enabling robust solution approximations for nonlinear PDEs under high-dimensional constraints.

Potential and Speculative Future Developments

The paper's methodology offers a pathway for scalable and efficient solutions in AI applications for complex mathematical problems. Future research could explore:

• Enhancements in deep learning architectures to further streamline computational costs and speed.
• Generalization of the proposed framework to other domains such as physics and engineering where high-dimensional PDEs are prevalent.
• Integration with advanced probabilistic and generative models to improve approximation accuracies for stochastic processes.

In conclusion, this paper fosters a link between machine learning techniques and classical numerical methods, showing potential for revolutionizing how complex high-dimensional systems are addressed, particularly in the financial sector. The blend of PDE theory with emerging AI tools presents a promising avenue for tackling a broader range of challenges in applied mathematics.