论文标题
深度学习有限的效用最大化
Deep Learning for Constrained Utility Maximisation
论文作者
论文摘要
本文提出了两种算法，用于解决深度学习的随机控制问题，重点是实用性最大化问题。第一种算法通过汉密尔顿雅各比·贝尔曼（HJB）方程解决了马尔可夫问题。我们用二阶向后随机微分方程（2BSDE）公式解决了这种高度非线性的部分微分方程（PDE）。问题的凸结构使我们能够描述一个双重问题，该问题可以验证原始原始方法或绕过一些复杂性。第二种算法利用二元方法的全部力量来解决非马克维亚问题，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佳条件的伴随BSDE。我们将这些算法应用于黑色choles中的功率，日志和非HARA实用程序的问题，Heston随机波动率以及依赖路径的波动率模型。数值实验显示出高度准确的结果，计算成本低，从而支持我们提出的算法。
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.
