Stochastic Differential Systems Driven by Fractional Brownian Motion

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0204613Hu It is well-known that the fractional Brownian motions are not semimartingales. The powerful stochastic calculus for semimartingales are not applicable to them. Motivated by the urgent need from applications the principal investigator and his collaborators have developed a new stochastic calculus of Ito type based on the Wick product. He proposes to continue this research topic and to study the stochastic differential systems driven by fractional Brownian motions. First he shall study the existence, uniqueness and approximation of global solutions to stochastic differential systems driven by fractional Brownian motions. Many researchers have attempted to obtain result on these aspects with little success. The principal investigator has discovered a relationship between stochastic differential systems driven by fractional Brownian motions and quasilinear hyperbolic equations (of infinitely many variables). It is well-known that the latter equations are also difficult to solve. However, there are a number of results which are useful. This connection will lead to a better understanding of stochastic differential systems and the PI plans to explore this relation. Secondly, in application of the stochastic systems driven by fractional Brownian motions, one also needs to identify the coefficients and the Hurst parameter. The PI proposes to study one such identification problem and apply it to the investigation of the stochastic volatility model in financial market. To obtain the maximum benefit from a physical or social system, one needs to understand the system in the most precise way possible. This requires building a mathematical model for the dynamic evolution of the system. When the system is under the influence of some uncertain factors, the system should be modeled by a random process. Up to now one of the random processes which has received the most attention and has been studied the most is stochastic differential equations based on the so-called Brownian motion. Brownian motion has some nice properties such as Markovian: Its future state depends only on the present state and does not depend on the past. This simplicity makes the mathematics for it easy and very profound results have been achieved. In fact there have been enormous work on it over the past century. However, this elegant property also limits the applicability of such a random process, since it cannot be used to describe those systems whose future depend not only the present but also on past history! Fractional Brownian motions are random processes having this long range dependence and may be used to describe such systems. This proposal aims to construct mathematical tools for the fractional Brownian motions which have already found applications in hydrology, climatology, network traffic analysis, and finance. This research will have impact on these areas as well as in life science.