## The forward equation

We now derive the forward Kolmogorov equation, which describes the behavior of q(x, t, z, s) as a function of x and t, treating z and s as parameters. This derivation is long and there are a few subtleties that we will need to explore. The easy way out, for me at least, would simply be to tell you the equation and cite some other places where the derivation could be found. However, I want you to understand how this tool arises.

Our starting point is the Chapman-Kolmogorov equation, which I write in a slightly different form than Eq. (7.56) (Figure 7.15)

q(x, t + dt,z, s) = q(y, t,z, s)q(x, t + dt,y, t)dy

That is: to be around the value x at time t + dt, the process starts at z at time s and moves from there to the value y at time t; from y at time t the process then has to move to the vicinity of x in the next dt.

Now we know that we are going to want the derivative of the transition density with respect to t, so let us subtract q(x, t, z, s) from both sides of Eq. (7.63)

q(x, t + dt, z, s) — q(x, t, z, s)= q(y, t, z, s)q(x, t + dt, y, t)dy — q(x, t, z, s)

Now here comes something subtle and non-intuitive (in the ''why did you do that?'' with answer ''because I learned to'' sense). Suppose that h(x) is a function for which we can find derivatives and which goes to 0 as |x| ! 1. We multiply both sides of Eq. (7.64) by h(x) and integrate over x:

Now we divide both sides of Eq. (7.65) by dt and let dt approach 0. The left hand side (LHS) becomes

and taking the limit inside the integral, we recognize the derivative of q with respect to t:

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