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.
翻译:本文提出了用深层学习解决随机控制问题的两种算法, 重点是实用最大化问题。 第一个算法通过汉密尔顿· 雅各比· 贝尔曼(HJB) 等式解决Markovian 问题。 我们用第二顺序后向偏向分异方程(2BSDE) 的配方解决这种高度非线性局部方程(PDE) 。 问题的共通结构使我们能够描述一个可以验证原始原始原始原始方法或绕过某些复杂方法的双重问题。 第二个算法利用双重方法的全能解决非马尔科维安问题, 这些问题往往超出现有文献中随机控制解答器的范围。 我们用一种符合双重最佳性条件的联结的 BSDE 。 我们将这些算法应用于与电源、 日志和非 HARA 工具在黑雪花中存在的问题、 Heston 随机性波动和路径依赖的波动模型。 数字实验以低计算成本显示非常准确的结果, 支持我们提议的算法。