Wealth Optimization Models with Stochastic Volatility and Continuous Dividends

[a] School of Science XI’AN University of Science and Technology， XI’AN， China.

*Corresponding author.

Supported by NSFC（71103143）， SSTF（2009KRM99）， NSFS（2011JQ1016）， NPFC（20110491672， 2012T50809）， SSFE（12JK0858）， XUST（201240）.

Received 1 January 2013； accepted 14 February 2013

Abstract

This paper study the problem of wealth optimization.It is established that the behavior model of the stock pricing process is jump-diffusion driven by a count process and stochastic volatility. Supposing that risk assets pay continuous dividend regarded as the function of time. It is proved that the existence of an optimal portfolio and unique equivalent martingale measure by stochastic analysis methods. The unique equivalent martingale measure ，the optimal wealth process， the value function and the optimal portfolio are deduced.

Key words： Jump-Diffusion process； Stochastic volatility； Dividends； Incomplete financial market； Wealth optimization

INTRODUCTION

The wealth optimization problem and the portfolioselection theory are always the kernel problems on financial mathematics. The domestic and foreign scholars have done a great dral of researches on the wealth optimization problem and obtained many results which is instructive to financial practice. When markets are complete， the existence of optimal strategies can be found Merton （1）， Jeanblanc and Pontier （2）， Follmer and Leukert （3）， Pham （4）， Nakano （5） discussed continuous and jump-diffusion modes.

In this paper， We define the wealth optimization problem：

where Xx，π（t）is the wealth process and A is the set of a dmissible portfolios. When the wealth is equal to x at the time t. we consider an economic agent whose behavior facing the risk is determined by a utility function （6）.Utility function is non decreasing， strictly concave， obviously U’（·）admits an inverse I（□）.He invests his wealth in the two assets and wants to maximize the expected utility of wealth at time T.Our work extends those studies and analyses the wealth optimization problem when markets is incomplete and driven by discontinuous prices.We consider that price of underlying asset price obeys jump-diffusion process， jump process generalized conforms to the actual situation of stock price movement. This paper discusses jump-diffusion asset price model being driven by a count proces that more general than Poisson process. Supposing that risk assets pay continuous dividend regarded as the function of time. It is proved that the existence of an optimal portfolio and unique equivalent martingale measure by stochastic analysis methods. The unique equivalent martingale measure ，the optimal wealth process， the value function and the optimal portfolio are deduced.

REFERENCES

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