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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

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.

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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. https://doi.org/10.1214/19-AOP1420

Information

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

Subjects:
Primary: 35Q84, 60H10
Secondary: 35J60

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 5 • September 2020
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