Estimation in Financial Models - RiskLab

Chapter 5
Some Diusion Models with
Explicit Solutions
In this chapter our representation follows [45], x4.2-4.4.
5.1 Linear stochastic dierential equations
We rst consider the class of linear stochastic dierential equations. To simplify
the exposition we concentrate on the scalar case and afterwards give
the extension to the multivariate case.
The general form of a scalar linear stochastic dierential equation is
dX t =(a 1 (t)X t + a 2 (t)) dt +(b 1 (t)X t + b 2 (t)) dW t ; (5.1)
where a 1 ;a 2 ;b 1 ;b 2 are "nice" 1 functions of time t or constants. As in the
case of ordinary dierential equations, the method of solution also involves
a fundamental solution of an associated homogeneous dierential equation.
Here the homogeneous linear equation belonging to the general equation (5.1)
is
dX t = a 1 (t)X t dt + b 1 (t)X t dW t : (5.2)
In order to nd its fundamental solution t;t0 , that means the solution of
(5.2) satisfying t0 ;t 0
= 1, we rst look at the special case b 1 (t) 0, that
means at the ordinary dierential equation
dX t
dt
= a 1 (t)X t : (5.3)
1 They are Lebesgue measurable and bounded on an interval 0 t T . Then a unique
(strong) solution X t on t 0 t T exists for each 0 t 0