Deep Learning-Based Algorithms for Stochastic Control of Jump Diffusion in Finance

Abstract

PDEs, like HJB-equations, can be solved using grid-based methods. These methods are inefficient for solving high-dimensional HJB-equation, because they suffer from the Curse of Dimensionality. Neural networks may overcome this problem. In this research, we solve high dimensional Partial Integro Differential Equations (PIDE) using neural networks. PIDE are PDEs that are associated with a jump-diffusion process. In this work, we only use finite activity jump processes. This means that the jump has a compensation component that to make it a martingale. We show two methods to solve PIDEs: a forward method (H-dBSDE, dBSDE-Jump) and a backward method (DBDP-MC). Both methodologies use neural networks to regress the solution and its derivative. The DBDP-MC is extended to jumps by calculating the compensation of the jumps with an offline Monte Carlo simulation. We tested this methodology on Bermudan basket options with 50 dimensions. The method was able to price them correctly. The dBSDE was extended by adding a new set of neural networks. These networks are learned with a different extra loss function. We argue that we can learn the two losses in a hierarchical way, leading to the Hierarchical dBSDE (H-dBSDE) method. Other work was done by minimizing the two loss functions simultaneously by using the sum of them. Easier problems like pricing European option can be solved correctly by the dBSDE-Jump method. However, we show that this can lead to wrong terminal fits, which makes it difficult to solve complex problems efficiently.