Fokker Planck Equation
Langevin equation
In case of describing macro behavior of a particle as the equation of motion,it deals with stochastic by random except for the macro order parameter. The viewpoint is x, changes with the passage of time divide a drift force which change in response to x and a part of the random force w(t) that is written as following. This is the Langevin equation.
dx/dt = α(x) + w(t)
This equation of motion is known as a Brownian motion, it shows the position of x combined drift force α(x) with stochastic variables w(t) at the passage of time t hours. The path of x at the passage of time t describes stochastic motion. The path after the passage of time t shows as stochastic. The stochastic process, shown as the path to reach x at the passage of time t, is called as the Winner process.
Figure1. Sample paths of the Winner process
The distribution describes P(x,t), initial value x0 is on P(x0,0), probable value x and distribution P rely on P0(x0) after the passage of time t shown as figure.
Although initial condition x=x0 provided, the value of x is stochastic after the passage of time t due to the stochastic behavior of w(t). w(t) is called the Gaussian noise in this process.
The Financial technology progressed to apply for the new bonds development and the prediction of the market on the assumption that the price change after the passage of time t in the market relies on the Winner process. This assumption that the price changes in financial market relies on the Winner process, describes the “Efficient Market Hypothesis" in mathematics.
The well known ‘Black-Shoerls formula’ as option pricing model in financial technology introduced as a solution of this equation of motion, Langevin equation, by the solution to method of Ito type stochastic differential equation. The price change distribution after the passage of the time t is introduced the value of x as a solution of the equation of motion which relies on the Winner process.
Please refer to following figure. In case of drift motion, the value of x after the passage of time t is determined a value as dx/dt＝α(x), in difference with drift motion, the value of x after the passage of time t does not to corresponds to the value of x x０ at t=0.
In this way a term of the gaussian noise w(t) describes a stochastic motion, not deterministic motion.
Figure2. Stochastic distribution
The value of x at t=0 spreads after the passage of time t. The value of x is stochastic at time t due to the stochastic behavior.
The equation of the value x at time t of the stochastic distribution P(x,t) is provided as
∂P(x,t)/∂t = LP(x,t) ---------(a)
The standardization condition
∮-∞∞ P(x,t) dt = 1
equation(a) is written as
∂P/∂t + ∂J(x,t)/∂x = 0, --------(b)
J(x,t) = V・ P(x,t)
where V linear operator, J is the flux of diffusion V contained 2nd differential
V = α(x) - ∂β(x)/∂x
then equation(b) is∂P/∂t = { -∂α(x)/∂x + ∂2β(x)/∂x2}P
this is called Fokker-Planck equation
where put α(x)=0,β(x)=D(constant)∂P/∂t = D∂2P/∂x2
This equation called the diffusion equation
A solution is provided
P(x,t) = 1/(√(4πDt) exp(-(x-x0)2/ 4Dt)
This describes above figure2. The point x0 diffuses infinite points x at time t.
This shows modeling of uncertainty in financial market.