20-Term Structure Page 629 Wednesday, February 4, 2004 1:33 PM
Term Structure Modeling and Valuation of Bonds and Bond Options 629∂p^12 ∂(^2) p
------ = ---σ---------
∂t (^2) ∂x^2
As a second example, consider the geometric Brownian motion whose
stochastic differential equation is
dXt = μXtdt+ σXtdBt , μ(Xt,t)= μXt , σ(Xt,t)= σXt
The associated Fokker-Planck equation is
2
∂p 1 2 ∂
2
(x p) ∂(xp)
------ = ---σ --------------------– μ---------------
∂t^2 ∂x^2 ∂x
The Fokker-Planck equation is a forward equation insofar it gives
the probability density at a future time tstarting at the present time s.
Another important PDE associated with Itô diffusions is the following
backward Kolmogorov equation:
∂ ∂^2 pxtys( ,, , ) ∂pxtys( ,, ,)
( ,, ,
1
- -----pxtys)= ---σ , ,
2
(xt)----------------------------------– μ(xt)-------------------------------
∂t (^2) ∂x^2 ∂x
The Kolmogorov backward equation gives the probability density that
we were at x,tgiven that we are now at y, s. Note that there is a fundamen-
tal difference between the backward and the forward Kolmogorov equa-
tions because the Itô processes are not reversible. In other words, the
probability density that we were at x,tgiven that we are now at y, sis not
the same as if we start the process at y, sand we look at density at x,t.
Thus far we have established an equivalence between stochastic dif-
ferential equations and associated partial differential equations in the
sense that they describe the same process. We have now to make an
additional step by establishing a connection between the expectations of
an Itô process and an associated PDE. The connection is provided by the
Feynman-Kac formula which is obtained from a generalization of the
backward Kolmogorov equation.
Consider the following PDE:
∂Fxt( , ) 1 ∂^2 Fxt( , ) ∂Fxt( , )
- --------------------= ---σ^2 (xt, )----------------------+ μ(xt, )--------------------
∂t (^2) ∂x^2 ∂x