First Order BSPDEs in higher dimension for optimal control problems

TL;DR

This paper investigates first order backward stochastic partial differential equations in higher dimensions, providing a framework for solving complex stochastic optimal control problems in financial modeling without predefined dynamics.

Contribution

It introduces a novel approach to multidimensional BSPDEs that generalizes Hamilton-Jacobi-Bellman equations for control problems with unspecified dynamics.

Findings

Develops a theory for higher-dimensional first order BSPDEs with boundary conditions.

Applies the framework to financial modeling scenarios.

Provides methods to construct value functions without known process structures.

Abstract

The paper studies the First Order BSPDEs (Backward Stochastic Partial Differential Equations) suggested earlier for a case of multidimensional state domain with a boundary. These equations represent analogs of Hamilton-Jacobi-Bellman equations and allow to construct the value function for stochastic optimal control problems with unspecified dynamics where the underlying processes do not necessarily satisfy stochastic differential equations of a known kind with a given structure. The problems considered arise in financial modelling.