INVERTING THE MARKOVIAN PROJECTION, WITH AN APPLICATION TO LOCAL STOCHASTIC VOLATILITY MODELS

• 成果类型: Article
• 署名作者: Lacker, Daniel; Shkolnikov, Mykhaylo; Zhang, Jiacheng
• 署名单位: Columbia University; Princeton University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1420
• 发表日期: 2020
• 页码: 2189-2211
• 关键词: convergence; mimicking; time

摘要

We study two-dimensional stochastic differential equations (SDEs) of McKean-Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such SDEs arise when one tries to invert the Markovian projection developed in (Probab. Theory Related Fields 71 (1986) 501-516), typically to produce an Ito process with the fixed-time marginal distributions of a given one-dimensional diffusion but richer dynamical features. We prove the strong existence of stationary solutions for these SDEs as well as their strong uniqueness in an important special case. Variants of the SDEs discussed in this paper enjoy frequent application in the calibration of local stochastic volatility models in finance, despite the very limited theoretical understanding.