The High Contact Principle with Reward Functions Involving Initial Points

Dongmei Guo,Chunli Chu,Yi Hu,Jiankang Zhang

doi:10.1155/2014/801791

Abstract

This paper examines a class of general optimal stopping problems in which reward functions depend on initial points. Two points of view on the initial point are introduced: one is to view it as a constant, and the other is to view it as a constant process starting from the point. Based on the two different views, two versions of the generalized high contact principle are derived. Finally, we apply the generalized high contact principle to one example.

Highlights

The “high contact principle,” first introduced by Samuelson [1] and McKean [2] and developed in greater depth by Øksendal [3] and Brekke and Øksendal [4], is a very useful tool to verify if a given function h(x) is a solution to the following optimal stopping problem: u (x) = sup τ E [g (Xτx)] (1)where g is a bounded continuously differentiable function in Rk, Xtx is an Ito process satisfying the following stochastic differential equation: dXt = b (Xt) dt + σ (Xt) dBt, X0 = x, (2)b : Rk → Rk and σ : Rk → Rk×m are Lipschitz continuous functions with at most linear growth, and Bt is an mdimensional standard Brownian motion.Generally, it is very hard to solve optimal stopping problem (1) directly
This paper examines a class of general optimal stopping problems in which reward functions depend on initial points
The “high contact principle,” first introduced by Samuelson [1] and McKean [2] and developed in greater depth by Øksendal [3] and Brekke and Øksendal [4], is a very useful tool to verify if a given function h(x) is a solution to the following optimal stopping problem: u (x) where g is a bounded continuously differentiable function in Rk, Xtx is an Ito process satisfying the following stochastic differential equation: dXt = b (Xt) dt + σ (Xt) dBt, X0 = x, (2)

Summary

Introduction

The “high contact principle,” first introduced by Samuelson [1] and McKean [2] and developed in greater depth by Øksendal [3] and Brekke and Øksendal [4], is a very useful tool to verify if a given function h(x) is a solution to the following optimal stopping problem: u (x). The aim of this paper is to derive a generalized high contact principle which can be used to check if a given function h is a solution to (3). We use two approaches for deriving two versions of the generalized high contact principle: one is more practical from applied perspective and the other is more intuitive from theoretical perspective.

The Reward Functions Not Involving Initial Points

The Reward Functions Involving Initial Points

The Main Results

One Application

Some Remaining Problems