Forward Kolmogorov Equation = Fokker-Planck Equation

If one writes down any SDE for a spot process, it ought to be possible to work out the probability distribution for spot at a given time in the future. 

 

Here, we will show that the probability density function (PDF) is the solution of a PDE, and we will also show how to obtain this PDE. 

 

Example: SDE   -->    Kolmo 1

 

In order to find an equation for the PDF, we will take an arbitrary function B(S) of S and examine its expectation at a given time t. 

If we apply Ito's lemma to B, we obtain: 

Kolmo 2

 

Recall that dW is a random variable that is independent of the Filtration Ft, that is, independent of all information that can be known at time t. 

In particular, dw is independent of Kolmo 3.

In addition, since dW is normally distributed, we have E(dW) = 0 and by independence Kolmo 4

 

Taking the expectation on both sides, we obtain: 

Kolmo 5

 

Now let's introduce the probability density p(s,t) that at time t spot takes the value s. 

We can then write the expectations in the previous equation as integrals over the density: 

Kolmo 8

 

On the left hand side, we can take the total time derivative inside the integral by making it a partial derivative. 

On the right hand side, we can integrate by parts once for the first term and twice for the second. 

Kolmo 11

 

It follows that the following partial differential equation must be true: 

Kolmo 12

 

This is known as the forward Kolmogorov or Fokker-Planck equation. 

 

Why is this a Forward equation? 

 

Well, the boundary condition is at time t = 0     -->   Kolmo 13

So to find the PDF at future time t, one needs to solve this PDE forwards in time from time t = 0. 

This is in contrast to the BS pricing equation, which is a backwards PDE. When solving the BS equation, we start at the expiry time of the option and the boundary condition is the payout of the option at that time. Then we solve backward in time to find the value now at t = 0. 

 

 

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