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状态空间的一个点移动到另一个部分的小概率。这在金融数学的背景下特别有趣，因为人们对股票价格在短时间内的行为感兴趣。例如，这将与短期到期的期权价格或在短时间内达到违约水平的概率有关。多因素随机波动率模型在过去的几年中得到了广泛的研究，并且发现它们非常有用地描述了所观察到的隐含波动率的微笑/偏斜。在短期内，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，如波动率的波动率。另一方面，它已被证明，多尺度随机波动率模型及其渐近性是非常有用的，以获得准确的近似的期权价格和对冲策略，可以有效地计算，只有少数几个组参数容易校准的数据。然后，很自然地研究在波动率时间尺度分离的制度下的利率函数的渐近展开。特别是，在该制度中，成熟度是小的，但大的随机波动率因子的均值回复时间相比，原来是非常有趣和具有挑战性的。该奖项支持的研究是关于推导和充分证明这种渐近展开。这个问题将被制定为对数渐近的某些泛函的扩散过程，这反过来又与均匀化/平均理论的Hamilton-Jacobi-Bellman方程的指数矩。 该研究的协作性质结合了多尺度随机波动率建模领域的专业知识，在大偏差理论领域，以及HJB方程的粘性解和均匀化/平均化。该奖项将支持多学科的研究工作。它涉及非常实际的建模问题，在金融数学领域，以及理论上的收敛结果的理论大偏差的扩散过程和均匀化的非线性偏微分方程。我们期望在小违约概率下得到的结果对理解信贷市场中的违约机制有很大帮助。因此，这项研究非常及时。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从业人员之间的互动。该研究将在UCSB新成立的金融数学与统计研究中心（CRFMS）的活动中得到充分的曝光。该合作项目还将作为UCSB和KU这两个机构的研究生和博士后研究员的培训工具。

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.