Annals of Probability
- Ann. Probab.
- Volume 48, Number 5 (2020), 2189-2211.
Inverting the Markovian projection, with an application to local stochastic volatility models
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.
Ann. Probab., Volume 48, Number 5 (2020), 2189-2211.
Received: May 2019
Revised: December 2019
First available in Project Euclid: 23 September 2020
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Fokker–Planck equations local stochastic volatility Markovian projection McKean–Vlasov equations mimicking nonlinear elliptic equations pathwise uniqueness regularity of invariant measures strong solutions
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