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

Daniel Lacker; Mykhaylo Shkolnikov; Jiacheng Zhang

doi:10.1214/19-AOP1420

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

Daniel Lacker, Mykhaylo Shkolnikov, Jiacheng Zhang

Research output: Contribution to journal › Article › peer-review

5 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 language	English (US)
Pages (from-to)	2189-2211
Number of pages	23
Journal	Annals of Probability
Volume	48
Issue number	5
DOIs	https://doi.org/10.1214/19-AOP1420
State	Published - 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

Access to Document

10.1214/19-AOP1420

Cite this

@article{5b6ee221ef1e43328eb1e3743267c1c4,

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

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 It{\^o} 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.",

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

author = "Daniel Lacker and Mykhaylo Shkolnikov and Jiacheng Zhang",

note = "Funding Information: M. Shkolnikov is supported in part by the NSF Grant DMS-1811723 and a Princeton SEAS innovation research grant. Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2020",

year = "2020",

month = sep,

doi = "10.1214/19-AOP1420",

language = "English (US)",

volume = "48",

pages = "2189--2211",

journal = "Annals of Probability",

issn = "0091-1798",

publisher = "Institute of Mathematical Statistics",

number = "5",

}

TY - JOUR

T1 - INVERTING THE MARKOVIAN PROJECTION, WITH AN APPLICATION TO LOCAL STOCHASTIC VOLATILITY MODELS

AU - Lacker, Daniel

AU - Shkolnikov, Mykhaylo

AU - Zhang, Jiacheng

N1 - Funding Information: M. Shkolnikov is supported in part by the NSF Grant DMS-1811723 and a Princeton SEAS innovation research grant. Publisher Copyright: © Institute of Mathematical Statistics, 2020

PY - 2020/9

Y1 - 2020/9

N2 - 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.

AB - 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.

KW - Fokker–Planck equations

KW - local stochastic volatility

KW - Markovian projection

KW - McKean– Vlasov equations

KW - mimicking

KW - nonlinear elliptic equations

KW - pathwise uniqueness

KW - regularity of invariant measures

KW - strong solutions

UR - http://www.scopus.com/inward/record.url?scp=85134615312&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85134615312&partnerID=8YFLogxK

U2 - 10.1214/19-AOP1420

DO - 10.1214/19-AOP1420

M3 - Article

AN - SCOPUS:85134615312

SN - 0091-1798

VL - 48

SP - 2189

EP - 2211

JO - Annals of Probability

JF - Annals of Probability

IS - 5

ER -

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

Abstract

All Science Journal Classification (ASJC) codes

Keywords

Access to Document

Other files and links

Fingerprint

Cite this