- The paper introduces a model-based deep learning method using two neural networks to solve finite-horizon continuous-time stochastic control problems with jumps.
- The approach iteratively trains two neural networks to approximate the policy and value function, effectively handling high-dimensional problems with jumps.
- The algorithms were validated on high-dimensional problems including LQR with jumps and an optimal consumption-investment problem, demonstrating practical applicability.
Deep Learning for Continuous-time Stochastic Control with Jumps
This paper presents a model-based deep learning approach to solve finite-horizon continuous-time stochastic control problems involving jump processes. It addresses dynamic decision-making scenarios under uncertainty, modeled as stochastic control problems, where the system evolves in continuous time with random jumps. The authors propose a novel method that combines neural network representation with the continuous-time dynamic programming principle, aiming to capture complex stochastic dynamics efficiently.
Methodology
The approach involves training two neural networks iteratively: one to approximate the optimal policy and the other to estimate the value function, leveraging variants of the Hamilton--Jacobi--Bellman (HJB) equation as training objectives. The primary advantage of this method is its ability to handle high-dimensional state spaces and integrate the stochastic dynamics without relying on discretization methods.
Two different algorithms, GPI-PINN 1 and GPI-PINN 2, are introduced:
- GPI-PINN 1: Utilizes a Physics-Informed Neural Network (PINN) approach, minimizing residuals of the HJB equation to approximate the value function. It performs well in scenarios without jumps but faces challenges in efficiency when jump components are present.
- GPI-PINN 2: Designed specifically to address high-dimensional problems with jumps, using a technique to bypass the computation of gradients and Hessians, thus enabling faster convergence. It circumvents the need for scaling intensive computations, such as expectation evaluation over jump distributions, by employing an expectation-free Hamiltonian.
Numerical Experiments
The algorithms are validated across various high-dimensional stochastic control problems:
- Linear-Quadratic Regulator (LQR) with Jumps: The paper demonstrates the efficacy of GPI-PINN 2 for solving the LQR problem involving controlled jump intensities. Despite increased problem complexity, GPI-PINN 2 efficiently approximated the value function and optimal policy without prohibitive computation times.
- Optimal Consumption-Investment Problem: The approach was extended to a practical financial application involving stochastic volatility and jump intensity processes. The convergence behavior of the algorithms was assessed through numerical simulation, where GPI-PINN 2, notably, handled the incorporation of jumps into the dynamics expertly.
Implications and Future Directions
This dual neural network strategy presents a significant advancement in solving high-dimensional stochastic control problems efficiently. Its practical applicability spans finance, operations research, and engineering disciplines, where complex dynamics are involved. While promising, the approach depends on known system dynamics—an assumption that can limit application in certain real-world scenarios where dynamics must be inferred rather than prescribed.
The future work may explore improving the stability of the training process across all dimensions of control problems, especially those involving intricate stochastic processes. Additionally, further research could extend these techniques to partially observed systems, where dynamics are not fully observable, thus broadening the scope of applicability in real-world systems.
The methodological innovation introduced in this paper aligns with a broader trend in computational research—leveraging deep learning frameworks to transcend traditional numerical limits in solving intricate dynamical systems. Indeed, as the landscape of AI and machine learning progresses, tools like the ones presented here will play a pivotal role in pushing the envelope of automation and intelligent decision-making.