Stochastic Pantograph Equations Driven By Fractional Brownian Motion And Its Stochastic Maximum Principle

The paper mainly introduces the stochastic pantograph equations driven by fractional Brownian motion and its stochastic maximum principle.The stochastic process can describe many problems happened in our life,and we pursuit the optimal solutions of the problems.So the optimal problem becomes an important branch of mathematics.The stochastic optimal control problem is one of the basic problems in stochastic control science.It has a wide range of applications in the industrial field,economic field,biomedical fields and so on.With the rapid development of economy and technology,the practical problems in various fields have strict demands on time,especially in the stock trading and the satellite launch which have a significant impact on the time delay.So the stochastic differential equation with time delay is widely used.Stochastic pantograph equations is a special kind of stochastic differential equations with time delay.In the process of the study,many areas has gradually revealed its fractal characteristics,especially in the financial field,a number of studies have confirmed that the financial markets have fractal characteristics.Compared to the standard Brown movement,fractional Brown motion can describe the character better.So in the paper we study the stochastic pantograph equations driven by fractional Brownian motion.In this paper,we use the connection between the brownian motion and Gaussian motions to transfer some results.And for the optimal control problem of the stochastic pantograph equations driven by fractional Brownian motion,we use the Pontryagin maximum principle to establish the Pontryagin maximum principle of the stochastic pantograph equations driven by fractional Brownian motion and deduce the necessary conditions for the stochastic optimal control.