The study of stochastic differential equations with jumps

点击翻译按钮获取中文摘要

The study of the Malliavin calculus on the Wiener space was initiated by Malliavin in l980's and appears now completed form by works of many researchers. The result is applied to the stochastic differential equation based on the Wiener process and many interesting results are obtained for the smoothness of the law of the solution. However, for the study of the stochastic differential equations with jumps, the Malliavin calculus can not be applied. We need the analysis of the Poisson space (Poisson random measure) which describes random jumps. In this research program, we developed the Malliavin calculus to the product of the Wiener space and the Poisson space and then applied it to the smoothness of the law of the solution of a stochastic differential equation with jumps. In the course of the research we corporated with Yasushi Ishikawa in Ehime University and we wrote a joint paper on this subject.Furthermore, we studied the structure of martingales on the filtered probability space generated by a Levy process, and we applied it to a problem in mathematical finance. If a stochastic process describing the movement of a stock (price process) has jumps the market is not complete. Then the risk neutral probabilities (equivalent martingale measures) is not uniquely determined. There are infinitely many equivalent martingale measures. Further, contingent claims such as options are not always attainable. In this research he showed that if a process is a martingale for any equivalent martingale measure, then the process can be represented by a stochastic integral based on the discounted price process. Using the result, he determined the upper and the lower prices of a contingent claim.