Estimation in Financial Models - RiskLab

The second case to consider is the time-independent case. We want to reduce
the nonlinear autonomous stochastic dierential equation
to the linear stochastic dierential equation
dX t = a(X t )dt + b(X t )dW t (5.28)
dY t =(a 1 Y t + a 2 )dt +(b 1 Y t + b 2 )dW t ; (5.29)
by means of a suitable transformation Y t = U(X t ).
Equations (5.19) and (5.20) simplify to
a(x) dU
dx (x)+1 2 b2 (x) d2 U
dx 2 (x) =a 1U(x)+a 2 (5.30)
and
b(x) dU
dx (x) =b 1U(x)+b 2 : (5.31)
We assume b(x) 6 0 and b 1 6= 0 and conclude from (5.31)
with an arbitrary constant C and
U(x) =C exp(b 1 h(x)) , b 2
b 1
(5.32)
h(x) =
Z x
x 0
ds
b(s) : (5.33)
As in the previous case we want to nd conditions for a and b. Therefore
we substitute (5.32) in (5.30), dierentiate, multiply with
,1
dU
dx =
b(x)
exp(,b 1 h(x)) and obtain an expression with only a 1 on the right hand
b 1
side. Dierentiating again we get the following condition
b 1
dA
dx (x)+ d
dx
!
b(x)dA dx (x) =0 (5.34)
with
A(x) = a(x)
b(x) , 1 db
(x): (5.35)
2 dx
This condition is satised in the trivial case dA = 0 for any b dx 1, or in the case
0 1
d
@ d
dx b
dA
dx
A =0; (5.36)
dx
dA
dx
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