The study of stochastic differential equations with jumps

Wiener空间上的Malliavin演算的研究始于20世纪80年代，经过许多研究者的工作，现已趋于完善。将所得结果应用于基于Wiener过程的随机微分方程，得到了解的光滑性规律的许多有趣结果。然而，对于带跳的随机微分方程的研究，Malliavin积分不能被应用。我们需要分析描述随机跳跃的泊松空间（泊松随机测度）。在本研究计划中，我们将Malliavin演算发展到Wiener空间和Poisson空间的乘积，然后将其应用于带跳随机微分方程解的光滑性。在研究过程中，我们与爱媛大学的Yasushi石川合作，共同撰写了一篇关于这个问题的论文。此外，我们还研究了由Levy过程生成的过滤概率空间上的鞅的结构，并将其应用于金融数学中的一个问题。如果一个描述股票运动的随机过程（价格过程）有跳跃，那么市场是不完整的。则风险中性概率（等价鞅测度）不是唯一确定的。存在无穷多个等价鞅测度。此外，期权等或有债权并非总是可以实现。在这项研究中，他表明，如果一个过程是一个鞅的任何等价鞅措施，那么这个过程可以表示为一个随机积分的基础上折扣价格过程。利用这个结果，他确定了未定权益的最高价格和最低价格。

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.