Convergence Analysis of Iterative Deep Learning Algorithms for Fully Nonlinear BSPDEs in Non-Markovian Utility...

Area of Science:

• Quantitative Finance
• Computational Finance
• Numerical Analysis

Background:

• Utility maximization in non-Markovian settings involves solving fully nonlinear backward stochastic partial differential equations (BSPDEs), specifically stochastic Hamilton-Jacobi-Bellman (HJB) equations.
• Traditional methods for solving these complex equations can be computationally intensive and may struggle with non-Markovian dynamics.

Purpose of the Study:

• To propose novel iterative deep learning algorithms for solving stochastic Hamilton-Jacobi-Bellman (HJB) equations in non-Markovian settings.
• To analyze the convergence properties of these proposed deep learning algorithms.
• To provide error estimates for time discretization, policy iteration, and the deep learning schemes.

Main Methods:

• Development of iterative deep learning algorithms tailored for fully nonlinear BSPDEs.
• Derivation of theoretical error estimates for time discretization schemes applied to BSPDEs.
• Derivation of theoretical error estimates for policy iteration schemes on discretized BSPDEs.
• Derivation of theoretical error estimates for the iterative deep learning schemes on discretized BSPDEs.

Main Results:

• The proposed iterative deep learning algorithms demonstrate convergence for solving the targeted BSPDEs.
• The algorithms exhibit accuracy, validated through numerical examples.
• Error estimates were successfully derived for the various discretization and iteration schemes.

Conclusions:

• Iterative deep learning provides an effective approach for solving complex HJB equations in non-Markovian financial models.
• The developed algorithms show promising convergence and accuracy, applicable to models like the Heston and rough volatility models.
• This work contributes a robust computational framework for advanced problems in quantitative finance.