The study of stochastic differential equations with jumps

维纳空间上的Malliavin演算的研究是由Malliavin于20世纪80年代提出的，目前已由许多研究者的工作完成。将所得结果应用于基于Wiener过程的随机微分方程，得到了许多关于解律的光滑性的有趣结果。然而，对于具有跳变的随机微分方程的研究，Malliavin微积分是不能应用的。我们需要分析描述随机跳跃的泊松空间（泊松随机测度）。在本研究项目中，我们将Malliavin演算推广到Wiener空间与Poisson空间的乘积中，并将其应用于具有跳变的随机微分方程解律的光滑性。在研究过程中，我们与爱媛大学的Yasushi Ishikawa合作，并就这一主题共同撰写了一篇论文。在此基础上，研究了由Levy过程生成的过滤概率空间上鞅的结构，并将其应用于数学金融中的一个问题。如果描述股票运动的随机过程（价格过程）有跳跃，市场就不完整。那么风险中性概率（等效鞅度量）就不是唯一确定的。有无穷多个等价鞅测度。此外，期权等或有权利要求并不总是可以实现的。在这项研究中，他证明了如果一个过程是任意等价鞅测度的鞅，那么这个过程可以用基于折现价格过程的随机积分来表示。利用这一结果，他确定了一项或有债权的最高和最低价格。

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.