By Gyöngy's theorem, a local and stochastic volatility model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility (LV) function is equal to the ratio of the Dupire LV function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a stochastic differential equation (SDE) nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented in Guyon and Henry‐Labordère [Risk Magazine, 25, 92–97], provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. When the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level, we prove existence of a solution to the associated Fokker–Planck equation under the condition that the range of the squared stochastic volatility factor is not too large. We then deduce existence to the calibrated model by extending the results in Figalli [Journal of Functional Analysis, 254(1), 109–153].

中文翻译：

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校准制度转换局部波动率模型的存在

根据Gyöngy定理，如果本地波动率（LV）函数等于Dupire LV函数与均方根均方根之比，则以具有正到期日和行使价的所有欧洲看涨期权的市场价格校准局部随机波动率模型给定现货价值的随机波动率因子。在McKean的意义上，这导致了非线性的随机微分方程（SDE）。基于条件期望的核近似的粒子法，如Guyon和Henry‐Labordère所述[ Risk Magazine，25[92-97]，即使当随机波动因子的范围很大时，即使可能出现一些校准错误，也可以提供有效的校准程序。但是到目前为止，就McKean而言，尚无全局存在的结果可用于SDE非线性。当随机波动率因子是一个跳变过程，且该跳变过程具有有限的多个值并且其跳变强度取决于现货水平时，我们证明存在相关的Fokker-Planck方程的解是在平方随机波动率因子的范围不是太大了。然后，我们通过扩展Figalli中的结果来推断校准模型的存在性[ Journal of Functional Analysis，254（1），109-153]。