Well-Posedness and Stability Analysis of Two Classes of Generalized Stochastic Volatility Models
Well-Posedness and Stability Analysis of Two Classes of Generalized Stochastic Volatility Models
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DOI：
10.1137/20m1336199
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发表时间：
2020-10
期刊：
影响因子：
--
通讯作者：
Ning Ning-Ning;Jing Wu
中科院分区：
文献类型：
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作者：
Ning Ning-Ning;Jing Wu
在本文中，为了应对快速发展的定量金融模型导致缺乏足够的理论支持，我们研究了两类广义随机波动率模型，建立了它们的强解的适定性，并进行了小扰动的稳定性分析。在第一类中，一个多维路径依赖过程由另一个多维路径依赖过程驱动。第二类是具有H\"旧连续系数的广义一维随机波动率模型。这两类模型的最大区别在于，过程及其相关驱动过程都有自己的次微分算子，其一个特例是多边障碍的一般反射算子。因此，所研究的模型完全涵盖了各种新探索的适定性未知的随机波动率模型变体，自然可以为多维、路径依赖性和多边障碍反射。
In this paper, to cope with the shortage of sufficient theoretical support resulted from the fast-growing quantitative financial modeling, we investigate two classes of generalized stochastic volatility models, establish their well-posedness of strong solutions, and conduct the stability analysis with respect to small perturbations. In the first class, a multidimensional path-dependent process is driven by another multidimensional path-dependent process. The second class is a generalized one-dimensional stochastic volatility model with H\"older continuous coefficients. What greatly differentiates those two classes of models is that both the process and its correlated driving process have their own subdifferential operators, whose one special case is the general reflection operators for multi-sided barriers. Hence, the models investigated fully cover various newly explored variants of stochastic volatility models whose well-posedness is unknown, and naturally serve as the rigorous mathematical foundation for new stochastic volatility model development in terms of multi-dimension, path-dependence, and multi-sided barrier reflection.