Fokker–Planck equation

A solution to the one-dimensional Fokker–Planck equation, with both the drift and the diffusion term. The initial condition is a Dirac delta function in x = 1, and the distribution drifts towards x = 0.

The Fokker–Planck equation describes the time evolution of the probability density function of the position of a particle, and can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck and is also known as the Kolmogorov forward equation. The first use of the Fokker–Planck equation was for the statistical description of Brownian motion of a particle in a fluid. The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov.

In one spatial dimension x, the Fokker–Planck equation for a process with drift D₁(x,t) and diffusion D₂(x,t) is

${\frac {\partial }{\partial t}}f(x,t)=-{\frac {\partial }{\partial x}}\left[D_{1}(x,t)f(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D_{2}(x,t)f(x,t)\right].$

More generally, the time-dependent probability distribution may depend on a set of $N$ macrovariables $x_{i}$. The general form of the Fokker–Planck equation is then

${\frac {\partial f}{\partial t}}=-\sum _{i=1}^{N}{\frac {\partial }{\partial x_{i}}}\left[D_{i}^{1}(x_{1},\ldots ,x_{N})f\right]+\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}\left[D_{ij}^{2}(x_{1},\ldots ,x_{N})f\right],$

where $D^{1}$ is the drift vector and $D^{2}$ the diffusion tensor; the latter results from the presence of the stochastic force.

Relationship with stochastic differential equations

The Fokker–Planck equation can be used for computing the probability densities of stochastic differential equations. Consider the Itō stochastic differential equation

$\mathrm {d} \mathbf {X} _{t}={\boldsymbol {\mu }}(\mathbf {X} _{t},t)\,\mathrm {d} t+{\boldsymbol {\sigma }}(\mathbf {X} _{t},t)\,\mathrm {d} \mathbf {W} _{t},$

where $\mathbf {X} _{t}\in \mathbb {R} ^{N}$ is the state and $\mathbf {W} _{t}\in \mathbb {R} ^{M}$ is a standard M-dimensional Wiener process. If the initial distribution is $\mathbf {X} _{0}\sim f(\mathbf {x} ,0)$, then the probability density $f(\mathbf {x} ,t)$ of the state $\mathbf {X} _{t}$ is given by the Fokker–Planck equation with the drift and diffusion terms

$D_{i}^{1}(\mathbf {x} ,t)=\mu _{i}(\mathbf {x} ,t)$

$D_{ij}^{2}(\mathbf {x} ,t)={\frac {1}{2}}\sum _{k}\sigma _{ik}(\mathbf {x} ,t)\sigma _{jk}(\mathbf {x} ,t).$

Similarly, a Fokker–Planck equation can be derived for Stratonovich stochastic differential equations. In this case, noise-induced drift terms appear if the noise strength is state-dependent.

Examples

A standard scalar Wiener process is generated by the stochastic differential equation

$\ \mathrm {d} X_{t}=\mathrm {d} W_{t}.$

Here the drift term is zero and the diffusion coefficient is 1/2; thus the corresponding Fokker–Planck equation is

${\frac {\partial f(x,t)}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}f(x,t)}{\partial x^{2}}},$

that is the simplest form of diffusion equation. If the initial condition is $f(x,0)=\delta (x)$, the solution is

$f(x,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {x^{2}}{2t}}}.$

Computational considerations

Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (the Monte Carlo method, canonical ensemble in molecular dynamics). However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider $f(\mathbf {v} ,t)d\mathbf {v}$, that is, the probability of the particle having a velocity in the interval $(\mathbf {v} ,\mathbf {v} +d\mathbf {v} )$ when it starts its motion with $\mathbf {v} _{0}$ at time 0.

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 Schrodinger equation allows the use of advanced operator techniques known from quantum mechanics for its solution in a number of cases. In many applications, one is only interested in the steady-state probability distribution $f_{0}(x)$, which can be found from ${\dot {f}}_{0}(x)=0$. The computation of mean first passage times 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 ${\sigma }(\mathbf {X} _{t},t)$ 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 ${\sigma }(\mathbf {X} _{t},t)$ consistent with f. This is an inverse problem that has been solved in general by Dupire (1994, 1997) with a non-parametric solution. Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility ${\sigma }(\mathbf {X} _{t},t)$ consistent with a solution of the Fokker-Plank equation given by a mixture model. More information is available also in Fengler (2008), Gatheral (2008) and Musiela and Rutkowski (2008).

Fokker-Planck equation and Path Integral

Every Fokker–Planck equation is equivalent to a path integral. The path integral formulation is an excellent starting point for the application of field theory methods. This is used, for instance, in critical dynamics. A derivation of the path integral is possible in the same way as in quantum mechanics, simply because the FP equation is formally equivalent to the Schrödinger equation. Here are the steps for a FP equation with one variable x. Write the FP equation in the form

${\frac {\partial }{\partial t}}f\left(x^{\prime },t\right)=\int _{-\infty }^{\infty }dx\left(\left[D_{1}\left(x,t\right){\frac {\partial }{\partial x}}+D_{2}\left(x,t\right){\frac {\partial ^{2}}{\partial x^{2}}}\right]\delta \left(x^{\prime }-x\right)\right)f\left(x,t\right).$

Integrate over a time interval $\varepsilon$,

$f\left(x^{\prime },t+\varepsilon \right)=\int _{-\infty }^{\infty }dx\left(\left(1+\varepsilon \left[D_{1}\left(x,t\right){\frac {\partial }{\partial x}}+D_{2}\left(x,t\right){\frac {\partial ^{2}}{\partial x^{2}}}\right]\right)\delta \left(x^{\prime }-x\right)\right)f\left(x,t\right)+O\left(\varepsilon ^{2}\right).$

Insert the Fourier integral

$\delta \left(x^{\prime }-x\right)=\int _{-i\infty }^{i\infty }{\frac {d{\tilde {x}}}{2\pi i}}e^{{\tilde {x}}\left(x-x^{\prime }\right)}$

for the $\delta$-function,

$f\left(x^{\prime },t+\varepsilon \right)=\int _{-\infty }^{\infty }dx\int _{-i\infty }^{i\infty }{\frac {d{\tilde {x}}}{2\pi i}}\left(1+\varepsilon \left[{\tilde {x}}D_{1}\left(x,t\right)+{\tilde {x}}^{2}D_{2}\left(x,t\right)\right]\right)e^{{\tilde {x}}\left(x-x^{\prime }\right)}f\left(x,t\right)+O\left(\varepsilon ^{2}\right)$

$=\int _{-\infty }^{\infty }dx\int _{-i\infty }^{i\infty }{\frac {d{\tilde {x}}}{2\pi i}}\exp \left(\varepsilon \left[-{\tilde {x}}{\frac {\left(x^{\prime }-x\right)}{\varepsilon }}+{\tilde {x}}D_{1}\left(x,t\right)+{\tilde {x}}^{2}D_{2}\left(x,t\right)\right]\right)f\left(x,t\right)+O\left(\varepsilon ^{2}\right).$

This equation expresses $f\left(x^{\prime },t+\varepsilon \right)$ as functional of $f\left(x,t\right)$. Iterating $\left(t^{\prime }-t\right)/\varepsilon$ times and performing the limit $\varepsilon \longrightarrow 0$ gives a path integral with Lagrangian

$L=\int dt\left[{\tilde {x}}D_{1}\left(x,t\right)+{\tilde {x}}^{2}D_{2}\left(x,t\right)-{\tilde {x}}{\frac {\partial x}{\partial t}}\right].$

The variables ${\tilde {x}}$ conjugate to $x$ are called "response variables". As in quantum mechanics it now would be possible to perform the ${\tilde {x}}$ path integrals. But in the FP case this normally is not a good idea - the perturbation theory is simpler with response variables.

See also

• Kolmogorov backward equation
• Boltzmann equation
• Navier–Stokes equations
• Vlasov equation
• Master equation
• Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations

Further reading

• Bruno Dupire (1994) Pricing with a Smile. Risk Magazine, January, 18-20.
• Dupire, B. (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.
• Damiano Brigo, Fabio Mercurio, Lognormal-mixture dynamics and calibration to market volatility smiles, International Journal of Theoretical and Applied Finance, 2002, Vol: 5, Pages: 427 - 446

• Brigo, D, Mercurio, F, Sartorelli, G, Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183, ISSN: 1469-7688
• Fengler, M. R. (2008). Semiparametric Modeling of Implied Volatility, 2005, Springer Verlag.

• Crispin W. Gardiner, "Handbook of Stochastic Methods", 3rd edition (paperback), Springer, ISBN 3-540-20882-8.

• Jim Gatheral (2008). The Volatility Surface. Wiley and Sons.
• Marek Musiela, Marek Rutkowski. Martingale Methods in Financial Modelling, 2008, 2nd Edition, Springer-Verlag.

• Hannes Risken, "The Fokker–Planck Equation: Methods of Solutions and Applications", 2nd edition, Springer Series in Synergetics, Springer, ISBN 3-540-61530-X.

External links

• Fokker–Planck equation on the Earliest known uses of some of the words of mathematics
• Backward Kolmogorov's equation Derivation of backward Kolmogorov's equation for various situations (accumulation of payoff, stopping time and so fourth).