Editing Fokker–Planck equation (section)

===Particular cases with known solution and inversion===
In [[mathematical finance]] for [[volatility smile]] modeling of options via [[local volatility]], one has the problem of deriving a diffusion coefficient <math>{\sigma}(\mathbf{X}_t,t)</math> consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of the Fokker–Planck equation: Given the density f(x,t) of the option underlying ''X'' deduced from the option market, one aims at finding the local volatility <math>{\sigma}(\mathbf{X}_t,t)</math> consistent with ''f''. This is an [[inverse problem]] that has been solved in general by Dupire (1994, 1997) with a non-parametric solution.<ref>[[Bruno Dupire]] (1994) Pricing with a Smile. ''Risk Magazine'', January, 18–20.</ref><ref>[[Bruno Dupire]] (1997) Pricing and Hedging with Smiles. Mathematics of Derivative Securities. Edited by M.A.H. Dempster and S.R. Pliska, Cambridge University Press, Cambridge, 103–111. {{ISBN|0-521-58424-8}}.</ref> Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility <math>{\sigma}(\mathbf{X}_t,t)</math> consistent with a solution of the Fokker–Planck equation given by a [[mixture model]].<ref>{{Cite journal| doi = 10.1142/S0219024902001511| year = 2002| last1 = Brigo | first1 = D.| last2 = Mercurio| first2 = Fabio| title = Lognormal-Mixture Dynamics and Calibration to Market Volatility Smiles| journal = International Journal of Theoretical and Applied Finance| volume = 5| issue = 4| pages = 427–446| citeseerx = 10.1.1.210.4165}}</ref><ref>{{Cite journal| doi = 10.1088/1469-7688/3/3/303| title = Alternative asset-price dynamics and volatility smile| year = 2003| last1 = Brigo | first1 = D.| last2 = Mercurio | first2 = F.| last3 = Sartorelli | first3 = G.| journal = Quantitative Finance| volume = 3| issue = 3| pages = 173–183| s2cid = 154069452}}</ref> More information is available also in Fengler (2008),<ref>Fengler, M. R. (2008). Semiparametric Modeling of Implied Volatility, 2005, Springer Verlag, {{ISBN|978-3-540-26234-3}}</ref> Gatheral (2008),<ref>[[Jim Gatheral]] (2008). The Volatility Surface. Wiley and Sons, {{ISBN|978-0-471-79251-2}}.</ref> and Musiela and Rutkowski (2008).<ref>Marek Musiela, Marek Rutkowski. ''Martingale Methods in Financial Modelling'', 2008, 2nd Edition, Springer-Verlag, {{ISBN|978-3-540-20966-9}}.</ref>