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

Daniel Lacker, Mykhaylo Shkolnikov, Jiacheng Zhang

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)2189-2211
Number of pages23
JournalAnnals of Probability
Volume48
Issue number5
DOIs
StatePublished - Sep 2020

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

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

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