Shaolin Ji

Shaolin Ji, Shige Peng, Ying Peng et al.

In this paper,we mainly focus on the numerical solution of high-dimensional stochastic optimal control problem driven by fully-coupled forward-backward stochastic differential equations (FBSDEs in short) through deep learning. We first transform the problem into a stochastic Stackelberg differential game problem (leader-follower problem), then a bi-level optimization method is developed where the leader's cost functional and the follower's cost functional are optimized alternatively via deep neural networks. As for the numerical results, we compute two examples of the investment-consumption problem solved through stochastic recursive utility models, and the results of both examples demonstrate the effectiveness of our proposed algorithm.

Qiang Han, Shaolin Ji, Yunzhang Li

A deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations is proposed. Building upon the established DBDP method (Huré et al., 2020), our algorithm introduces a reformulation of the local loss functions that are sequentially optimized via backward induction at each time step. The core of this approach involves reformulating simulated backward stochastic difference equations into their conditional expectation representations, thereby recasting a projection-based stochastic optimization problem as a deterministic function-approximation task. By explicitly incorporating conditional expectations, the DBR scheme facilitates a denoising mechanism prior to loss evaluation. This architecture substantially mitigates numerical variance, resulting in enhanced training stability and superior generalization performance. Numerical results demonstrate that the DBR scheme consistently outperforms the DBDP1 method, maintaining accuracy up to d=200 for bounded solutions (see Table 1). Notably, for complex unbounded PDEs where the DBDP1 method fails beyond d=10, the DBR scheme remains robust up to $d=20$ with relative errors under 9.7% (see Table 6}). Theoretically, we derive rigorous upper error bounds and establish half-order convergence for the proposed scheme. Extensions to variational inequalities are also provided.

Shaolin Ji, Shige Peng, Ying Peng et al.

In this paper, we mainly focus on solving high-dimensional stochastic Hamiltonian systems with boundary condition, which is essentially a Forward Backward Stochastic Differential Equation (FBSDE in short), and propose a novel method from the view of the stochastic control. In order to obtain the approximated solution of the Hamiltonian system, we first introduce a corresponding stochastic optimal control problem such that the extended Hamiltonian system of the control problem is exactly what we need to solve, then we develop two different algorithms suitable for different cases of the control problem and approximate the stochastic control via deep neural networks. From the numerical results, comparing with the Deep FBSDE method developed previously from the view of solving FBSDEs, the novel algorithms converge faster, which means that they require fewer training steps, and demonstrate more stable convergences for different Hamiltonian systems.

Shaolin Ji, Shige Peng, Ying Peng et al.

In this paper, we aim to solve the high dimensional stochastic optimal control problem from the view of the stochastic maximum principle via deep learning. By introducing the extended Hamiltonian system which is essentially an FBSDE with a maximum condition, we reformulate the original control problem as a new one. Three algorithms are proposed to solve the new control problem. Numerical results for different examples demonstrate the effectiveness of our proposed algorithms, especially in high dimensional cases. And an important application of this method is to calculate the sub-linear expectations, which correspond to a kind of fully nonlinear PDEs.

Shaolin Ji, Shige Peng, Ying Peng et al.

Recently, the deep learning method has been used for solving forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). It has good accuracy and performance for high-dimensional problems. In this paper, we mainly solve fully coupled FBSDEs through deep learning and provide three algorithms. Several numerical results show remarkable performance especially for high-dimensional cases.