Estimation in Financial Models - RiskLab

where we assume the choice of b 1
b 1 = ,
d
dx
b
dA
dx
dA
dx
: (5.37)
If b 1 6= 0 then we may choose b 2 = 0 and use the transformation
U(x) =C exp(b 1 h(x)): (5.38)
If b 1 =0we use
U(x) =b 2 h(x)+C: (5.39)
Now we apply the derived theory to three groups of examples.
Example 1 The stochastic dierential equation
dX t = 1 2 g(X t)g 0 (X t )dt + g(X t )dW t ; (5.40)
where g is a given dierentiable function, is reducible with the general solution
X t = h ,1 (W t + h(X 0 )); (5.41)
with
h(x) =
Z x
x 0
ds
g(s) : (5.42)
In the following we show how to obtain the general solution (5.41). With
the notation of the theory above we see that A(x) = 0, and hence (5.34) is
satised for all b 1 . We choose b 1 =0,b 2 = 1 and obtain
U(x) =h(x)+C:
Inserting U in (5.30) gives a 1 (h(x)+C)+a 2 =0,and with a 1 =0,a 2 =0,
(5.29) reduces to dY t = dW t with the solution Y t = Y 0 + W t : With C =0we
have Y t = h(X t ), especially Y 0 = h(X 0 ), and hence obtain (5.41).
We remark that in the special case (5.40) we may nd a solution in another
more pleasantway. Equation (5.40) is equivalent to the Stratonovich stochastic
dierential equation (for the Stratonovich integral see e.g. [56], p.16, or
[45], x4.9)
dX t = b(X t ) dW t :
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