The convergence rate from discrete to continuous optimal investment stopping problem

TL;DR

This paper establishes the rate at which the solution to a discrete optimal investment stopping problem converges to its continuous counterpart, using quadratic reflected BSDEs and advanced stochastic calculus techniques.

Contribution

It characterizes the continuous problem via quadratic reflected BSDEs with unbounded terminal conditions and derives the convergence rate from discrete to continuous solutions under a Markovian framework.

Findings

Proves uniform convergence of discrete to continuous quadratic reflected BSDEs.

Provides explicit convergence rate estimates.

Develops auxiliary SDE techniques for solution estimates.

Abstract

We study the optimal investment stopping problem in both continuous and discrete case, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth. Based on the work [9] with an additional stochastic payoff function, we characterize the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equation (BSDE for short) with unbounded terminal condition. In regard to discrete problem, we get the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provide some useful prior estimates about the solutions with the help of auxiliary forward-backward SDE system and Malliavin calculus. Finally, we obtain the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.