Annals of Probability

Inverting the Markovian projection, with an application to local stochastic volatility models

Daniel Lacker, Mykhaylo Shkolnikov, and Jiacheng Zhang

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Abstract

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 Itô 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.

Article information

Source
Ann. Probab., Volume 48, Number 5 (2020), 2189-2211.

Dates
Received: May 2019
Revised: December 2019
First available in Project Euclid: 23 September 2020

Permanent link to this document
https://projecteuclid.org/euclid.aop/1600826469

Digital Object Identifier
doi:10.1214/19-AOP1420

Mathematical Reviews number (MathSciNet)
MR4152640

Zentralblatt MATH identifier
07276922

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 35Q84: Fokker-Planck equations
Secondary: 35J60: Nonlinear elliptic equations

Keywords
Fokker–Planck equations local stochastic volatility Markovian projection McKean–Vlasov equations mimicking nonlinear elliptic equations pathwise uniqueness regularity of invariant measures strong solutions

Citation

Lacker, Daniel; Shkolnikov, Mykhaylo; Zhang, Jiacheng. Inverting the Markovian projection, with an application to local stochastic volatility models. Ann. Probab. 48 (2020), no. 5, 2189--2211. doi:10.1214/19-AOP1420. https://projecteuclid.org/euclid.aop/1600826469


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