Editing Fokker–Planck equation (section)

==Solution==
Being a [[partial differential equation]], the Fokker–Planck equation can be solved analytically only in special cases. A formal analogy of the Fokker–Planck equation with the [[Schrödinger equation]] allows the use of advanced operator techniques known from quantum mechanics for its solution in a number of cases. Furthermore, in the case of overdamped dynamics when the Fokker–Planck equation contains second partial derivatives with respect to all spatial variables, the equation can be written in the form of a [[master equation]] that can easily be solved numerically.<ref>{{Cite journal| author= Holubec Viktor, Kroy Klaus, and Steffenoni Stefano |title=Physically consistent numerical solver for time-dependent Fokker–Planck equations |journal=Phys. Rev. E |volume=99 |issue= 4|pages=032117 |year=2019 |doi=10.1103/PhysRevE.99.032117|pmid=30999402 |arxiv=1804.01285 |bibcode=2019PhRvE..99c2117H |s2cid=119203025 }}</ref>
In many applications, one is only interested in the steady-state probability distribution <math> p_0(x)</math>, which can be found from <math display="inline">\frac{\partial p(x,t)}{\partial t} = 0</math>.
The computation of mean [[first passage time]]s and splitting probabilities can be reduced to the solution of an ordinary differential equation which is intimately related to the Fokker–Planck equation.

===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>