Deep Learning for Constrained Utility Maximisation
This addresses constrained utility maximization in finance, offering efficient solutions for complex stochastic control problems, though it appears incremental by extending existing duality and deep learning approaches.
The paper tackles stochastic control problems for utility maximization by proposing two deep learning algorithms: one for Markovian problems using a 2BSDE formulation of the HJB equation, and another for non-Markovian problems via duality methods. Numerical experiments on various models show highly accurate results with low computational cost.
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.