Nonregular McKean–Vlasov Equations and Calibration Problem in Local Stochastic Volatility Models

References

• [1] Abergel F, Tachet R (2010) A nonlinear partial integro-differential equation from mathematical finance. Discrete Continuous Dynamical Systems 27(3):907–917.Crossref, Google Scholar
• [2] Adams RA, Fournier JJF (2003) Sobolev Spaces, Pure and Applied Mathematics (Elsevier Science, Amsterdam).Google Scholar
• [3] Aronson DG (1967) Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73(6):890–896.Crossref, Google Scholar
• [4] Bayer C, Belomestny D, Butkovsky O, Schoenmakers J (2022) A reproducing Kernel Hilbert Space approach to singular local stochastic volatility McKean-Vlasov models. Preprint, submitted March 2, https://arxiv.org/abs/2203.01160.Google Scholar
• [5] Bogachev VI, Krylov NV, Röckner M, Shaposhnikov SV (2015) Fokker–Planck–Kolmogorov Equations, Mathematical Surveys and Monographs (American Mathematical Society, Providence, RI).Crossref, Google Scholar
• [6] Bossy M, Jabir J-F (2017) On the wellposedness of some McKean models with moderated or singular diffusion coefficient. Springer Proc. Math. Statist. (Springer, Cham, Switzerland).Google Scholar
• [7] Demengel F, Demengel G (2012) Functional Spaces for the Theory of Elliptic Partial Differential Equations (Springer, London).Crossref, Google Scholar
• [8] Djete MF, Possamaï D, Tan X. McKean–Vlasov optimal control: Limit theory and equivalence between different formulations. Math. Oper. Res. 47(4):2891–2930.Link, Google Scholar
• [9] Dupire B (1994) Pricing with a smile. Risk 7(1):18–20.Google Scholar
• [10] Evans L (2010) Partial Differential Equations, Graduate Studies in Mathematics (American Mathematical Society, Providence, RI).Crossref, Google Scholar
• [11] Guyon J, Henry-Labordere P (2013) Nonlinear Option Pricing, Chapman and Hall/CRC Financial Mathematics Series (Taylor & Francis, Boca Raton, FL).Crossref, Google Scholar
• [12] Jourdain B, Méléard S (1998) Propagation of chaos and fluctuations for a moderate model with smooth initial data. Ann. L’Institut Henri Poincare (B) Probability Statist. 34(6):727–766.Crossref, Google Scholar
• [13] Jourdain B, Zhou A (2020) Existence of a calibrated regime switching local volatility model. Math. Finance 30(2):501–546.Google Scholar
• [14] Krylov NV (1980) Controlled Diffusion Processes (Springer, New York).Crossref, Google Scholar
• [15] Krylov NV (1996) Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics (American Mathematical Society, Providence, RI).Crossref, Google Scholar
• [16] Krylov NV (1999) An Analytic Approach to SPDEs. Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, vol. 64 (American Mathematical Society, Providence, RI).Google Scholar
• [17] Krylov NV (2007) Parabolic and elliptic equations with VMO coefficients. Comm. Partial Differential Equations 32(3):453–475.Crossref, Google Scholar
• [18] Kusuoka S (2017) Continuity and Gaussian two-sided bounds of the density functions of the solutions to path-dependent stochastic differential equations via perturbation. Stochastic Processing Appl. 127(2):359–384.Crossref, Google Scholar
• [19] Lacker D, Shkolnikov M, Zhang J (2020) Inverting the Markovian projection, with an application to local stochastic volatility models. Ann. Probability 48(5):2189–2211.Crossref, Google Scholar
• [20] Lipton A (2002) The vol smile problem. Risk Magazine 15:61–65.Google Scholar
• [21] Oelschläger K (1985) A law of large numbers for moderately interacting diffusion processes. Zeitschrift Wahrscheinlichkeitstheorie Verwandte Gebiete 69:1432–2064.Crossref, Google Scholar
• [22] Piterbarg V (2006) Markovian projection method for volatility calibration. Preprint, submitted June 6, https://dx.doi.org/10.2139/ssrn.906473.Google Scholar
• [23] Saporito YF, Yang X, Zubelli JP (2017) The calibration of stochastic local-volatility models: An inverse problem perspective. Comput. Math. Appl. 77:3054–3067.Crossref, Google Scholar
• [24] Stroock D, Varadhan S (1997) Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften, vol. 233 (Springer–Verlag, Berlin).Google Scholar
• [25] Tian Y, Zhu Z, Lee G, Klebaner F, Hamza K (2015) Calibrating and pricing with a stochastic-local volatility model. J. Derivatives 22(3):21–39.Crossref, Google Scholar
• [26] Villani C (2008) Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschafte, vol. 338 (Springer, Berlin).Google Scholar