The Kolmogorov backward equation is a fundamental equation in the theory of stochastic processes, particularly for continuous-time Markov processes. It provides a way to describe the evolution of the expected value of a function of the process as a function of the initial state and time.
Mathematical Formulation
The Kolmogorov backward equation is given by:
$$ \frac{\partial u(x,t)}{\partial t} = A(x,t) \frac{\partial u(x,t)}{\partial x} + \frac{1}{2} B(x,t) \frac{\partial^2 u(x,t)}{\partial x^2} $$
where:
- \( u(x,t) = \mathbb{E}[f(X(T)) | X(t) = x] \) is the expected value of a function \( f \) of the process at time \( T \) given that the process is in state \( x \) at time \( t \),
- \( A(x,t) \) is the drift coefficient,
- \( B(x,t) \) is the diffusion coefficient.
Derivation
Consider a stochastic process \( X(t) \) governed by the 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.
Let \( u(x,t) \) be the expected value of a function \( f \) evaluated at \( T \), conditioned on \( X(t) = x \):
$$ u(x,t) = \mathbb{E}[f(X(T)) | X(t) = x] $$
By the definition of the conditional expectation and using the Itô lemma, we have:
$$ du = \frac{\partial u}{\partial t} dt + \frac{\partial u}{\partial x} dX + \frac{1}{2} \frac{\partial^2 u}{\partial x^2} (dX)^2 $$
Substituting the SDE into this equation:
$$ du = \frac{\partial u}{\partial t} dt + \frac{\partial u}{\partial x} \left( A(X(t), t) dt + \sqrt{B(X(t), t)} dW(t) \right) + \frac{1}{2} \frac{\partial^2 u}{\partial x^2} B(X(t), t) dt $$
Taking the expectation and noting that \( \mathbb{E}[dW(t)] = 0 \) and \( (dW(t))^2 = dt \), we get:
$$ \frac{\partial u}{\partial t} dt = \mathbb{E} \left[ \frac{\partial u}{\partial t} dt + A(X(t), t) \frac{\partial u}{\partial x} dt + \frac{1}{2} B(X(t), t) \frac{\partial^2 u}{\partial x^2} dt \right] $$
Cancelling the \( dt \) terms, we arrive at the Kolmogorov backward equation:
$$ \frac{\partial u(x,t)}{\partial t} = A(x,t) \frac{\partial u(x,t)}{\partial x} + \frac{1}{2} B(x,t) \frac{\partial^2 u(x,t)}{\partial x^2} $$
Interpretation
The Kolmogorov backward equation describes how the expected value of a function of the process evolves backward in time, from a future time \( T \) to the current time \( t \).
Example: Ornstein-Uhlenbeck Process
For an Ornstein-Uhlenbeck process, the SDE is:
$$ dX(t) = \theta (\mu – X(t)) dt + \sigma dW(t) $$
where \( \theta \), \( \mu \), and \( \sigma \) are constants.
In this case:
- $$ ( A(x,t) = \theta (\mu – x) ) $$
- $$ ( B(x,t) = \sigma^2 ) $$
The Kolmogorov backward equation becomes:
$$ \frac{\partial u(x,t)}{\partial t} = \theta (\mu – x) \frac{\partial u(x,t)}{\partial x} + \frac{\sigma^2}{2} \frac{\partial^2 u(x,t)}{\partial x^2} $$
This equation describes how the expected value of a function of the Ornstein-Uhlenbeck process evolves backward in time.
