Intuition of the Kolmogorov Equations

Question

So I understand the derivation of the Kolmogorov Forward and Backward Equations, but I don't quite understand the intuition. Here is from Stokey, 2008:

"The backward equation involves time $t$ and the initial condition $x$, with the current state $y$ held fixed. A similar PDE, the Kolmogorov forward equation (KFE), involves $t$ and $y$, with the initial state $x$ fixed. The forward equation is useful for characterizing the limiting distribution, if one exists."

In what situations do the two arrise? For instance, if I know the current state and am interested in a probability distribution over possible initial states, I use the backward equation. If I know the current state and am interested in the probability distribution over the state in the future, I use the forward equation. Is this correct?

On a more technical note, how does one define boundary conditions for these PDE's? Are the boundary conditions a result of the derivation? Perhaps Stokey is not the best reference on this topic...

Answer 1

I will try to answer to your last question. I did not read the paper but in models with higher dimensions, it is always difficult to find an analytical solution. If there exists an analytical solution (for a very basic model with a one-state variable), it is possible to derive the initial conditions for your control and state variable from your differential equations.

However, in systems where there does not exist an analytical solution, you are supposed to solve it numerically, in which case you must give a numerical value for one of your variables (or more until you find steady state values for all your variables.)

After, you can find the steady-state values of your variables in which case, your initial conditions should not be so far from the steady state level (otherwise, there would be some convergence problems in your model if you don't choose the appropriate initial values, close to steady state. It is another issue.)

Answer 2

Here's some information with regard to the intuition of each.

The Kolmogorov forward equation is often referred to as the Fokker-Planck equation. It is a partial differential equation (PDE) that describes the time evolution of the probability density function of a variable over a state. That is, suppose we have information about the state $x$ at a time $t$. The forward equation tells us the distribution of $x$ at a time $s>t$.

The Kolmogorov backward equation on the other hand is used to understand the probability of a state ending up in a set $B$ at some time $s$. Define a function of the state, $u_s(x)$. At the final time $s$, $u_s(x) = 1$ if $x \in B$. Otherwise, it is zero. Then, for any time $t < s$, $u_t(x)$ describes the probability that $X_s$ given $X_t = x_t$ will end up in the set $B$. The Kolmogorov backward equation described the evolution of the probability function. $u_s(x)$ then is the final condition of a this PDE.

I've taken this information mostly from Wikipedia:

• Kolmogorov equations
• Fokker–Planck equation
• Kolmogorov backward equations