Dual method for continuous-time Markowitz's Problems with nonlinear wealth equations

TL;DR

This paper develops a dual method approach to solve continuous-time Markowitz portfolio problems with nonlinear wealth dynamics and bankruptcy constraints, deriving conditions for optimal terminal wealth and characterizing the optimal strategies.

Contribution

It introduces a novel dual method framework for nonlinear wealth equations in continuous-time portfolio optimization, including necessary and sufficient optimality conditions.

Findings

Optimal terminal wealth satisfies a specific condition derived via terminal perturbation.

Optimal wealth and portfolio are characterized as solutions to a constrained forward-backward stochastic differential equation.

The approach handles nonlinear wealth dynamics and bankruptcy prohibition effectively.

Abstract

Continuous-time mean-variance portfolio selection model with nonlinear wealth equations and bankruptcy prohibition is investigated by the dual method. A necessary and sufficient condition which the optimal terminal wealth satisfies is obtained through a terminal perturbation technique. It is also shown that the optimal wealth and portfolio is the solution of a forward-backward stochastic differential equation with constraints.