Chapter 2: FAQs 119
What are the Forward and Backward
Equations?
Short Answer
Forward and backward equations usually refer to the
differential equations governing the transition probabil-
ity density function for a stochastic process. They are
diffusion equations and must therefore be solved in the
appropriate direction in time, hence the names.
Example
An exchange rate is currently 1.88. What is the prob-
ability that it will be over 2 by this time next year? If
you have a stochastic differential equation model for
this exchange rate then this question can be answered
using the equations for the transition probability density
function.
Long Answer
Let us suppose that we have a random variableyevolv-
ing according to a quite general, one-factor stochastic
differential equation
dy=A(y,t)dt+B(y,t)dX.
HereAandBare both arbitrary functions ofyandt.
Many common models can be written in this form,
including the lognormal asset random walk, and com-
mon spot interest rate models.
Thetransition probability density functionp(y,t;y′,t′)
is the function of four variables defined by
Prob(a<y<bat timet′|yat timet)
=
∫b
a
p(y,t;y′,t′)dy′.
