Stochastic Model of Floating Interest rate and positive research

本研究的目的是对最近开始的随机偏微分方程模型进行理论和实证研究。我们打算只考虑政府债券或伦敦银行间拆放款利率，它们应该是无风险的。然而，最近人们开始对所谓的与违约债券有关的信用衍生品感兴趣。所以我们稍微改变了一下计划，想到了一个包含违约概率的模型。首先，在一定的边界条件和系数条件下，证明了与利率相关的随机偏微分方程解的存在唯一性。同时，我们给出了一个条件，使解保持在正范围内。接下来，我们对违约债券的价格进行了理论研究。在这里，我们给出了一个反例公式，人们普遍相信风险率过程和条件违约概率之间的关系。并给出了连续时间滤波模型中危险率过程的计算公式。在数学金融中，价格的重要实用公式是根据期望给出的。在估计统计参数时，精确而迅速地计算这些期望是很重要的。本文介绍了一种新的扩散模型数值计算方法，该方法在理论上是有效的。在计算对冲策略时，我们需要更复杂的预期，但对它们的研究被推迟到未来。随机过程模型的统计考虑将变得越来越重要。我们的计划包括为它建立一个理论。在这方面，我们只得到了关于次椭圆扩散过程的加性泛函概率律收敛性的渐近展开式。

The purpose of this research was to do theoretical and positive research on stochastic partial differential equation models begun recently. We were planning to think of only government bond or LIBOR which are supposed to be riskless. However, quite recently people got interested in so-called credit derivatives concerning defaultable bonds. So we slightly changed our plan and thought of model containing default probability.First we showed the existence and uniqueness of solution to stochastic partial differential equations related to interest rate under certain conditions for boundary conditions and coefficients. Also, we gave a condition so that the solution remains in positive range.Next we did theoretical research on prices of defaultable bonds. Here we gave a counter-example for a formula which people widely believed on the relationship between hazard rate processes and conditional default probabilities. Also, we gave a formula on hazard rate process in continuous-time filtering models.In mathematical finance, practically important formula for prices are given in terms of expectations. In estimates of statistical parameters, it is important to compute such expectations precisely and rapidly. We introduced a new numerical computation method in diffusion models, which are rather restrictive but widely used, and we showed that it is quite effective theoretically. In computing hedging strategies, we need more complicated expectations, but the research of them are postponed to the future.The statistical consideration for stochastic process models will be getting important more and more. Our plan contained the construction of a theory for it. In this respect we only got an asymptotic expansion formula related to convergence of probability law of additive functionals for hypo-elliptic diffusion processes.