Annals of Probability

Inverting the Markovian projection, with an application to local stochastic volatility models

Daniel Lacker, Mykhaylo Shkolnikov, and Jiacheng Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Ann. Probab., Volume 48, Number 5 (2020), 2189-2211.

Received: May 2019
Revised: December 2019
First available in Project Euclid: 23 September 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 35Q84: Fokker-Planck equations
Secondary: 35J60: Nonlinear elliptic equations

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.

Export citation


  • [1] Abergel, F. and Tachet, R. (2010). A nonlinear partial integro-differential equation from mathematical finance. Discrete Contin. Dyn. Syst. 27 907–917.
  • [2] Abergel, F., Tachet, R. and Zaatour, R. (2017). Nonparametric model calibration for derivatives. J. Math. Finance 7 571–596.
  • [3] Alfonsi, A., Labart, C. and Lelong, J. (2016). Stochastic local intensity loss models with interacting particle systems. Math. Finance 26 366–394.
  • [4] Bogachev, V. I., Krylov, N. and Röckner, M. (1996). Regularity of invariant measures: The case of non-constant diffusion part. J. Funct. Anal. 138 223–242.
  • [5] Bossy, M. and Jabir, J.-F. (2019). On the wellposedness of some McKean models with moderated or singular diffusion coefficient. In Frontiers in Stochastic Analysis—BSDEs, SPDEs and Their Applications. Springer Proc. Math. Stat. 289 43–87. Springer, Cham.
  • [6] Bossy, M., Jabir, J.-F. and Talay, D. (2011). On conditional McKean Lagrangian stochastic models. Probab. Theory Related Fields 151 319–351.
  • [7] Brunick, G. and Shreve, S. (2013). Mimicking an Itô process by a solution of a stochastic differential equation. Ann. Appl. Probab. 23 1584–1628.
  • [8] Cherny, A. S. (2002). On the uniqueness in law and the pathwise uniqueness for stochastic differential equations. Theory Probab. Appl. 46 406–419.
  • [9] Dermoune, A. (1999). Probabilistic interpretation of sticky particle model. Ann. Probab. 27 1357–1367.
  • [10] Donsker, M. D. and Varadhan, S. R. S. (1975). Asymptotic evaluation of certain Markov process expectations for large time. I. Comm. Pure Appl. Math. 28 1–47.
  • [11] Dupire, B. (1994). Pricing with a smile. Risk 7 18–20.
  • [12] Evans, L. C. (2010). Partial Differential Equations, 2nd ed. Graduate Studies in Mathematics 19. Amer. Math. Soc., Providence, RI.
  • [13] Gärtner, J. (1988). On the McKean–Vlasov limit for interacting diffusions. Math. Nachr. 137 197–248.
  • [14] Guyon, J. and Henry-Labordère, P. (2011). The smile calibration problem solved. Available at
  • [15] Guyon, J. and Henry-Labordère, P. (2012). Being particular about calibration. Risk 25 88.
  • [16] Guyon, J. and Henry-Labordère, P. (2014). Nonlinear Option Pricing. Chapman & Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton, FL.
  • [17] Gyöngy, I. (1986). Mimicking the one-dimensional marginal distributions of processes having an Itô differential. Probab. Theory Related Fields 71 501–516.
  • [18] Herrmann, S. and Tugaut, J. (2010). Non-uniqueness of stationary measures for self-stabilizing processes. Stochastic Process. Appl. 120 1215–1246.
  • [19] Hirsch, F., Profeta, C., Roynette, B. and Yor, M. (2011). Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series 3. Springer, Milan; Bocconi Univ. Press, Milan.
  • [20] Jourdain, B., Lelièvre, T. and Roux, R. (2010). Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process. ESAIM Math. Model. Numer. Anal. 44 831–865.
  • [21] Jourdain, B. and Zhou, A. (2016). Existence of a calibrated regime switching local volatility model and new fake Brownian motions. Preprint. Available at arXiv:1607.00077.
  • [22] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [23] Khasminskii, R. (2012). Stochastic Stability of Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 66. Springer, Heidelberg.
  • [24] Krylov, N. V. (1969). Itô’s stochastic integral equations. Teor. Veroyatn. Primen. 14 340–348.
  • [25] Krylov, N. V. (1984). On the relation between differential operators of second order and the solutions of stochastic differential equations. In Steklov Seminar 1985.
  • [26] Lax, P. D. (2002). Functional Analysis. Pure and Applied Mathematics (New York). Wiley, New York.
  • [27] Lelièvre, T., Rousset, M. and Stoltz, G. (2008). Long-time convergence of an adaptive biasing force method. Nonlinearity 21 1155–1181.
  • [28] Lipton, A. (2002). The vol smile problem. Risk 15 61–66.
  • [29] Lipton, A. and McGhee, W. (2002). Universal barriers. Risk 15 81–85.
  • [30] Piterbarg, V. (2006). Markovian projection method for volatility calibration. Available at
  • [31] Saporito, Y. F., Yang, X. and Zubelli, J. P. (2019). The calibration of stochastic local-volatility models: An inverse problem perspective. Comput. Math. Appl. 77 3054–3067.
  • [32] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Classics in Mathematics. Springer, Berlin.
  • [33] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Math. 1464 165–251. Springer, Berlin.
  • [34] Tian, Y., Zhu, Z., Lee, G., Klebaner, F. and Hamza, K. (2015). Calibrating and pricing with a stochastic-local volatility model. J. Deriv. 22 21–39.
  • [35] Trevisan, D. (2016). Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab. 21 Art. ID 22.
  • [36] Veretennikov, A. Yu. (1980). On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24 354–366.