一类随机优化问题的最大值原理及其在风险动态度量中的应用

The maximum principle for one kind of Stochastic

Optimization Problem and Application in Dynamic

Measure of Risk*

Shaolin Ji Zhen Wu

School of Mathematics and System Science

Shandong University

Jinan 250100P.R.China

jsl@ wuzhen@

Abstract

In this paper we get the maximum principle for one kind of stochastic optimization

problem motivated by the dynamic measure of risk.The dynamic measure of risk to the

investor in …nancial market can be studied in our framework where the wealth equation may

have nonlinear coe¢cient.

Key words.backward stochastic di¤erential equation,perturbation method,Ekeland varia-tional principle,dynamic measure of risk.

AMS 1991subject Classi…cation:Primary 90A09,93E20;secondary 60H10

*Research supported by Chinese National Natural Science Foundation(No.10371067and No.10201018).

1Introduction

In the …nancial market,if one investor has not enough initial wealth x ,he cannot perfectly hedge the liability C at future time T .One interesting problem is how to quantify such a risk.

There are many excellent works concerning this subject (see Föllmer &Leukert[1][2],Cvitanic &Karatzas [3]).In our paper,we will get one kind of maximum principle for the stochastic optimization problem in next section and apply the result to study the problem introduced by Cvitanic &Karatzas [3].They quanti…ed the risk by

inf ( )2A (x )

E 0 C X x; (T )S 0(T ) +where ( )is the portfolio strategy,S 0( )is the price of the risk-free instrument and X ( )represents the investor’s wealth.Föllmer &Leukert [1][2],Edirisinghe,Naik &Uppal [4]proposed the related criteria.This measure associates with a stochastic optimal control problem in which the control (portfolio strategy)and state (wealth)constraints are imposed .Cvitanic &Karatzas [3]employed the tools of convex duality and proved the existence of the optimal strategy.

This is a dynamic optimization problem and it isn’t easy to extend the result in [7]to nonlinear wealth equation case even without convex assumption.

It is well-known that nonlinear backward stochastic di¤erential equation (BSDE for short)has been independently introduced by Pardoux and Peng [5]and by Du¢e and Epstein [6].BSDE theory is proved to be a useful tool for mathematical …nance (see [7],El.Karoui,Peng and Quenez).In Quenez’s Ph.D thesis and El.Karoui,Peng and Quenez [8]a terminal perturbation method is suggested:the terminal condition of BSDE is regarded as "control variable"in a recursive utility optimization problem.This method can easily deal with some state constraints for the dynamic optimization problem.Iintroducing the lagrange multiplier and using convex analysis method,they get the maximum principle when the coe¢cients are assumed to be convex.

Motivated by their method,we transfer the dynamic measure of risk proposed by Cvitanic &Karatzas [3]to a static optimization problem which subjects to an initial constraint.Without the convexity assumption,we use the terminal perturbation method and the Ekeland variational principle to deal with the corresponding initial constraint and obtain the necessary condition which the optimal objective satis…es i.e.maximum principle.Our method is di¤erent with that in [8]and the result is extended to the case with the assumption of smooth coe¢cient instead of convex one.For the linear case,we obtain the same result as that in [3].For the nonlinear case,we give a necessary condition that the terminal wealth of an investor must satisfy.

This paper is organized as follows:in the second section,we deal with one kind of stochastic control problem motivated by the above dynamic measure of risk.This problem is described by BSDE and control variable is the terminal condition of ing the terminal perturbation method and Ekeland variational principle,we obtain the maximum principle.In section 3,at …rst,the dynamic measure of risk problem [3]where the portfolio strategy is the control is transformed to stochastic optimal control problem studied in section 2.Then,we apply the maximum principle obtained in section 2to study the dynamic measure of risk in which the wealth equation is linear and obtain the same result as that in [3].In section 4,we consider the dynamic measure of risk problem when the wealth equation is nonlinear without convexity assumption.In this case we cannot use the convex duality method introduced in [3]to deal with the dynamic measures of risk.Applying the result of section 2we can obtain the similar results as that in section 3.Our method can be applied more widely in the …nancial market.