一般混合扩散SDE及其与波动率与资产去相关的不确定波动率期权模型的关系
The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation
摘要：
在本文中，给定一个不断发展的混合概率密度，我们定义了一个候选扩散过程，其边际法遵循相同的演变。作为一个特例，我们导出了一个随机微分方程（ε），它有一个唯一的强解，且其密度演化为高斯密度的混合。我们提出了一个有趣的结果之间的比较所获得的过程和它的平方扩散系数的瞬时和终端相关性。作为对金融数学的应用，我们构造了边缘密度为对数正态密度混合的扩散过程。我们解释了这样的过程可以用来模拟市场微笑现象。我们证明了对数正态混合动力学是一个合适的不确定波动率模型的一维扩散版本，并适当地重新解释了早期的相关结果。我们探讨了数值扩散和不确定波动版本的未来微笑结构之间的关系。
英文摘要：
In the present paper, given an evolving mixture of probability densities, we define a candidate diffusion process whose marginal law follows the same evolution. We derive as a particular case a stochastic differential equation (SDE) admitting a unique strong solution and whose density evolves as a mixture of Gaussian densities. We present an interesting result on the comparison between the instantaneous and the terminal correlation between the obtained process and its squared diffusion coefficient. As an application to mathematical finance, we construct diffusion processes whose marginal densities are mixtures of lognormal densities. We explain how such processes can be used to model the market smile phenomenon. We show that the lognormal mixture dynamics is the one-dimensional diffusion version of a suitable uncertain volatility model, and suitably reinterpret the earlier correlation result. We explore numerically the relationship between the future smile structures of both the diffusion and the uncertain volatility versions.
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