Open Access
September 2020 Inverting the Markovian projection, with an application to local stochastic volatility models
Daniel Lacker, Mykhaylo Shkolnikov, Jiacheng Zhang
Ann. Probab. 48(5): 2189-2211 (September 2020). DOI: 10.1214/19-AOP1420


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.


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Daniel Lacker. Mykhaylo Shkolnikov. Jiacheng Zhang. "Inverting the Markovian projection, with an application to local stochastic volatility models." Ann. Probab. 48 (5) 2189 - 2211, September 2020.


Received: 1 May 2019; Revised: 1 December 2019; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152640
Digital Object Identifier: 10.1214/19-AOP1420

Primary: 35Q84 , 60H10
Secondary: 35J60

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

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • September 2020
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