The Kolmogorov forward equation, also known as the Fokker-Planck equation, describes the time evolution of the probability density function
\( p(x,t) \) of a stochastic process. It is particularly useful for continuous-time and continuous-state-space Markov processes.
Mathematical Formulation
The Kolmogorov forward equation is given by:
$$ \frac{\partial p(x,t)}{\partial t} = -\frac{\partial}{\partial x} \left[ A(x,t) p(x,t) \right] + \frac{1}{2} \frac{\partial^2}{\partial x^2} \left[ B(x,t) p(x,t) \right] $$
where:
- \( p(x,t) \) is the probability density function of the state \( x \) at time \( t \),
- \( A(x,t) \) is the drift coefficient, representing the average rate of change of \( x \),
- \( B(x,t) \) is the diffusion coefficient, representing the variance rate of \( x \).
Derivation
Consider a stochastic process \( X(t) \) governed by the following Itô stochastic differential equation (SDE):
$$ dX(t) = A(X(t), t) dt + \sqrt{B(X(t), t)} dW(t) $$
where \( W(t) \) is a Wiener process (or standard Brownian motion).
To derive the Fokker-Planck equation, we start with the Chapman-Kolmogorov equation for the transition probability \( p(x,t | x_0, t_0) \):
$$ p(x, t + \Delta t | x_0, t_0) = \int_{-\infty}^{\infty} p(x, t + \Delta t | x’, t) p(x’, t | x_0, t_0) dx’ $$
Assuming \( \Delta t \) is infinitesimally small, we can expand \( p(x, t + \Delta t | x’, t) \) using a Taylor series:
$$ p(x, t + \Delta t | x’, t) \approx p(x, t | x’, t) + \Delta t \frac{\partial}{\partial t} p(x, t | x’, t) $$
Similarly, expand \( p(x’, t | x_0, t_0) \) around \( x \):
$$ p(x’, t | x_0, t_0) \approx p(x, t | x_0, t_0) + (x’ – x) \frac{\partial}{\partial x} p(x, t | x_0, t_0) + \frac{1}{2} (x’ – x)^2 \frac{\partial^2}{\partial x^2} p(x, t | x_0, t_0) $$
Substituting these expansions into the Chapman-Kolmogorov equation and integrating over \( x’ \), we obtain:
$$ p(x, t + \Delta t | x_0, t_0) – p(x, t | x_0, t_0) \approx \Delta t \frac{\partial}{\partial t} p(x, t | x_0, t_0) $$
$$ = \int_{-\infty}^{\infty} \left[ p(x, t | x’, t) + \Delta t \frac{\partial}{\partial t} p(x, t | x’, t) \right] \left[ p(x’, t | x_0, t_0) \right] dx’ $$
Considering the contributions from the drift and diffusion terms:
$$ \begin{align} \mathbb{E}[(x’ – x)] &= A(x, t) \Delta t, \ \mathbb{E}[(x’ – x)^2] &= B(x, t) \Delta t, \end{align} $$
and higher-order terms can be neglected as \( \Delta t \to 0 \). This yields:
$$ \frac{\partial}{\partial t} p(x, t) = – \frac{\partial}{\partial x} \left[ A(x, t) p(x, t) \right] + \frac{1}{2} \frac{\partial^2}{\partial x^2} \left[ B(x, t) p(x, t) \right] $$
Thus, we have derived the Kolmogorov forward equation.
Interpretation
- The term \( -\frac{\partial}{\partial x} \left[ A(x,t) p(x,t) \right] \) represents the advection (or drift) of the probability density due to the deterministic part of the process.
- The term \( \frac{1}{2} \frac{\partial^2}{\partial x^2} \left[ B(x,t) p(x,t) \right] \) represents the diffusion of the probability density due to the stochastic part of the process.
Example: Ornstein-Uhlenbeck Process
For an Ornstein-Uhlenbeck process, which is a type of Gaussian process, the SDE is given by:
$$ dX(t) = \theta (\mu – X(t)) dt + \sigma dW(t) $$
where \( \theta \), \( \mu \), and \( \sigma \) are constants. In this case:
- $$ ( A(x,t) = \theta (\mu – x) ) $$
- $$ ( B(x,t) = \sigma^2 ) $$
The Kolmogorov forward equation becomes:
$$ \frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x} \left[ \theta (\mu – x) p(x,t) \right] + \frac{\sigma^2}{2} \frac{\partial^2}{\partial x^2} p(x,t) $$
This equation describes how the probability density function of the Ornstein-Uhlenbeck process evolves over time.
