# Delta family approach for the stochastic control problems of utility maximization

*Jingtang Ma*,
*Zhengyang Lu* and
*Zhenyu Cui*

Papers from arXiv.org

**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.

**Date:** 2022-02

**New Economics Papers:** this item is included in nep-ore and nep-upt

**References:** View references in EconPapers View complete reference list from CitEc

**Citations:** View citations in EconPapers (1) Track citations by RSS feed

**Downloads:** (external link)

http://arxiv.org/pdf/2202.12745 Latest version (application/pdf)

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:arx:papers:2202.12745

Access Statistics for this paper

More papers in Papers from arXiv.org

Bibliographic data for series maintained by arXiv administrators ().