Delta family approach for the stochastic control problems of utility maximization

TL;DR

This paper introduces a novel delta family approach for solving high-dimensional stochastic control problems in utility maximization, enabling explicit recursive computation of the value function with applications to complex financial models.

Contribution

The paper presents a new series representation-based method for stochastic control, extending to high dimensions and mixed control problems, with practical applications to advanced volatility models.

Findings

Explicit series representation of the value function.

Effective recursive time-stepping scheme for high-dimensional problems.

Successful application to complex stochastic volatility models.

Abstract

In this paper, we propose a new approach for stochastic control problems arising from utility maximization. The main idea is to directly start from the dynamical programming equation and compute the conditional expectation using a novel representation of the conditional density function through the Dirac Delta function and the corresponding series representation. We obtain an explicit series representation of the value function, whose coefficients are expressed through integration of the value function at a later time point against a chosen basis function. Thus we are able to set up a recursive integration time-stepping scheme to compute the optimal value function given the known terminal condition, e.g. utility function. Due to tensor decomposition property of the Dirac Delta function in high dimensions, it is straightforward to extend our approach to solving high-dimensional stochastic control problems. The backward recursive nature of the method also allows for solving stochastic control and stopping problems, i.e. mixed control problems. We illustrate the method through solving some two-dimensional stochastic control (and stopping) problems, including the case under the classical and rough Heston stochastic volatility models, and stochastic local volatility models such as the stochastic alpha beta rho (SABR) model.