Collaborative Research: Small time behavior of multiscale diffusions motivated by stochastic volatility models

成熟的大偏差理论描述了扩散过程在短时间内从一点移动到状态空间的另一部分的小概率。在金融数学的背景下，这是特别有趣的，因为人们对股票价格在短时间内的行为感兴趣。例如，这将与短期期权价格或在短时间内达到违约水平的可能性有关。在过去的几年里，人们对多因素随机波动率模型进行了深入的研究，发现它们对于描述观察到的隐含波动率的微笑/倾斜非常有用。在短期期限内，出现在大偏差估计中的利率函数通常不会以封闭形式给出，并且将取决于模型的一些细节，例如波动率的波动率。另一方面，多尺度随机波动率模型及其渐近性对于获得期权价格和套期保值策略的精确逼近是非常有用的，这些模型可以用少量易于校准的数据组参数有效地计算出来。在波动率时间尺度分离的情况下，研究速率函数的渐近展开式是很自然的。特别是，与随机波动因子的平均回归时间相比，成熟度小但大的制度变得非常有趣和具有挑战性。该奖项支持的研究是关于推导并充分证明这种渐近展开式的。该问题将被表述为扩散过程的某些泛函的指数矩的对数渐近，这反过来又与Hamilton-Jacobi-Bellman方程的均匀化/平均理论相联系。该研究的协作性质结合了多尺度随机挥发性建模领域的专业知识，大偏差理论领域的专业知识，以及HJB方程的粘度解和均质化/平均。该奖项将支持一项多学科研究工作。它涉及金融数学领域中非常实际的建模问题，以及在扩散过程和非线性偏微分方程均匀化的大偏差理论中的理论收敛结果。期望在小违约概率下得到的结果将对理解信贷市场的违约机制有很大帮助。因此，这项研究是非常及时的。UCSB的高级PI积极参与组织会议，研讨会和特别会议，促进学术研究人员和实践者之间的互动。这项研究将在UCSB新成立的金融数学与统计研究中心（CRFMS）的活动中得到充分展示。该合作项目还将作为UCSB和KU两所机构的研究生和博士后的培训工具。

The well-established theory of large deviations describes the small probabilities that a diffusion process moves from one point to another part of the state space in a short time. This is particularly interesting in the context of financial mathematics in which one is interested in the behavior of the stock price in a short time. For instance, this will be relevant to option prices at short maturities, or probabilities to reach a default level in a short time. Multi-factor stochastic volatility models have been studied intensively in the past years, and they have been found very useful to describe the observed smile/skew of implied volatilities. At short maturities the rate functions appearing in the large deviation estimates will usually not be given in closed form, and will depend on some details of the models like volatility of volatility. On the other hand it has been shown that multiscale stochastic volatility models and their asymptotics are very useful to obtain accurate approximations of option prices and hedging strategies which can be efficiently computed with only a few group parameters easy to calibrate to data. It is then natural to study asymptotic expansions of the rate function in the regime of separation of volatility time scales. In particular, the regime in which the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor turns out to be very interesting and challenging. The research supported by this award is about deriving and fully justifying such asymptotic expansion. The problem will be formulated as logarithmic asymptotic for exponential moments of certain functionals of diffusion processes, which in turn is connected with homogenization/averaging theory for Hamilton-Jacobi-Bellman equations. The collaborative nature of the research combines expertise in the area of multiscale stochastic volatility modeling, in the areas of large deviation theory, and viscosity solution and homogenization/averaging for HJB equations. This award will support a multidisciplinary research effort. It concerns very practical modeling issues in the area of financial mathematics as well as theoretical convergence results in the theories of large deviations for diffusion processes and homogenization of nonlinear partial differential equations. It is expected that the results obtained on small default probabilities will be very helpful in understanding default mechanisms in credit markets. This research therefore is very timely. The senior PI at UCSB is actively involved in organizing conferences, workshops and special sessions which promote interaction between academic researchers and practitioners. The research will be given full exposure in the activities of the newly created Center for Research in Financial Mathematics and Statistics (CRFMS) at UCSB. The collaborative project will also serve as a training tool for graduate students and postdoctoral fellows at both institutions, UCSB and KU.