[Paper Review] Inverting the Markovian projection, with an application to local stochastic volatility models

Daniel Lacker, Mykhaylo Shkolnikov+1 more|arXiv (Cornell University)|May 15, 2019

Stochastic processes and financial applications15 references4 citations

TL;DR

This paper establishes the strong existence and uniqueness of stationary solutions for a class of two-dimensional McKean–Vlasov SDEs that invert the Markovian projection, enabling the construction of local stochastic volatility models with pre-specified marginal distributions and richer dynamics. The key contribution is proving that when the volatility multiplier is squared, the invariant distribution factorizes, ensuring independence of the components under stationarity.

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 by Gyöngy (1986), 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.

Motivation & Objective

• To address the lack of theoretical foundations for inverting the Markovian projection in local stochastic volatility model calibration.
• To establish the existence and uniqueness of stationary solutions for a class of two-dimensional McKean–Vlasov SDEs arising from such inversion.
• To characterize the invariant distribution of the resulting SDEs, particularly under the condition that the volatility multiplier is squared.
• To provide a rigorous mathematical framework for time-changed diffusion processes that preserve stationary product-form distributions.

Proposed method

• Formulates a two-dimensional SDE where the drift and diffusion coefficients of the first component depend on the conditional expectation of functions of the second component.
• Imposes regularity and ergodicity conditions on the coefficients (drift, diffusion, and multiplier functions) to ensure strong existence and uniqueness.
• Applies the Fokker-Planck equation and stationary measure analysis to characterize the invariant distribution of the joint process.
• Uses a time-changed diffusion interpretation to show that the invariant measure remains product-form when the multiplier function is squared.
• Applies results from ergodic theory and weak uniqueness (via Trevisan's theorem) to identify the unique stationary law.
• Demonstrates that under the condition $ h \equiv f^2 $, the invariant measure of the SDE is a product of the marginal invariant measures of the individual components.

Experimental results

Research questions

• RQ1 Under what conditions does a two-dimensional McKean–Vlasov SDE arising from inverting the Markovian projection admit a strong stationary solution?
• RQ2 When does the invariant distribution of such an SDE factorize into a product of marginal distributions?
• RQ3 What is the role of the squared multiplier function $ f^2 $ in ensuring independence of the components in stationarity?
• RQ4 How does a time-changed diffusion process preserve the same invariant measure as the original SDE?
• RQ5 Can the weak uniqueness of the stationary solution be established under minimal regularity assumptions on the coefficients?

Key findings

• Under Assumption A, the two-dimensional SDE (1.5) admits a unique strong stationary solution.
• When $ h \equiv f^2 $, the invariant distribution of the SDE is a product measure, implying independence of the components in stationarity.
• The stationary law of the first component matches the marginal law of the original one-dimensional diffusion, preserving the target distribution.
• The invariant density $ p(x,y) $ is explicitly shown to be $ m_1(x) m_2(y) $ when $ h \equiv f^2 $, where $ m_1 $ and $ m_2 $ are the stationary densities of the marginal SDEs.
• The time-changed process $ (X_{\tau_t}, Y_t) $ with $ \tau_t = \int_0^t f^2(Y_s) ds $ has the same invariant distribution as the original SDE, despite non-constant time change.
• The stationary solution is unique in law, and the components are independent under the invariant measure when $ h \equiv f^2 $.