arXiv 2208.099862022-08math.PRmath.APq-fin.MF

非正则 McKean–Vlasov 方程与局部随机波动率模型中的校准问题

Non--regular McKean--Vlasov equations and calibration problem in local stochastic volatility models

Mao Fabrice Djete

AI总结：

本文在系数关于测度变量仅具极小连续性的条件下研究非正则 McKean–Vlasov 方程，证明了解的存在性与混沌传播，并由此推出校准的局部随机波动率模型的存在性。

摘要：

为了处理金融中校准的局部随机波动率模型的存在性问题，我们研究了一类 McKean–Vlasov 方程，其中对系数施加了极小的连续性假设。具体而言，漂移系数，尤其是波动率系数，在 Wasserstein 拓扑下关于测度变量不一定连续。本文给出了该类 McKean–Vlasov 方程的一个存在性结果，并证明了其可由 N–粒子系统逼近，即混沌传播性质。作为直接结果，我们能够对适当选择的随机波动率参数推导出校准的局部随机波动率模型的存在性。相应的混沌传播结果也得到了证明。

英文摘要：

In order to deal with the question of the existence of a calibrated local stochastic volatility model in finance, we investigate a class of McKean--Vlasov equations where a minimal continuity assumption is imposed on the coefficients. Namely, the drift coefficient and, in particular, the volatility coefficient are not necessarily continuous in the measure variable for the Wasserstein topology. In this paper, we provide an existence result and show an approximation by $N$--particle system or propagation of chaos for this type of McKean--Vlasov equations. As a direct result, we are able to deduce the existence of a calibrated local stochastic volatility model for an appropriate choice of stochastic volatility parameters. The associated propagation of chaos result is also proved.

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