This paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation. We solve this highly nonlinear partial differential equation (PDE) with a second order backward stochastic differential equation (2BSDE) formulation. The convex structure of the problem allows us to describe a dual problem that can either verify the original primal approach or bypass some of the complexity. The second algorithm utilises the full power of the duality method to solve non-Markovian problems, which are often beyond the scope of stochastic control solvers in the existing literature. We solve an adjoint BSDE that satisfies the dual optimality conditions. We apply these algorithms to problems with power, log and non-HARA utilities in the Black-Scholes, the Heston stochastic volatility, and path dependent volatility models. Numerical experiments show highly accurate results with low computational cost, supporting our proposed algorithms.

本文提出了两种解决深度学习随机控制问题的算法，重点是效用最大化问题。第一个算法通过 Hamilton Jacobi Bellman (HJB) 方程解决马尔可夫问题。我们使用二阶后向随机微分方程 (2BSDE) 公式求解这个高度非线性的偏微分方程 (PDE)。问题的凸结构允许我们描述一个对偶问题，它可以验证原始原始方法或绕过一些复杂性。第二种算法利用对偶方法的全部力量来解决非马尔可夫问题，这些问题通常超出了现有文献中随机控制求解器的范围。我们解决了满足双重最优性条件的伴随 BSDE。我们将这些算法应用于权力问题，Black-Scholes、Heston 随机波动率和路径相关波动率模型中的 log 和 non-HARA 实用程序。数值实验显示出高精度的结果，计算成本低，支持我们提出的算法。