Estimation in Financial Models - RiskLab

where h is given by (5.42).
Again we want to show how (5.49) can be derived. We see A(x) = , and
hence b 1 can be chosen arbitrarily. We choose b 1 =0and b 2 = 1 and obtain
U(x) =h(x)+C:
Inserting U in (5.30) gives = h(x)a 1 + a 2 ; and hence a 1 =0,a 2 = . We
have dY t = dt + dW t with its solution Y t = t + W t + Y 0 . With C =0we
have Y t = h(x t ), especially Y 0 = h(X 0 ), and we obtain (5.49).
As in the previous example we remark that (5.48) is equivalent to the
Stratonovich stochastic dierential equation
dX t = b(X t )dt + b(X t ) dW t ;
which can be integrated directly in order to obtain the general solution (5.49).
The applications of example 1 we gave can all be modied to represent applications
to example 2. For instance, consider
dX t =
1
q q
2 X t + Xt 2 +1 dt + Xt 2 +1dW t ;
X t = sinh(t + W t + arcsinhX 0 ): (5.50)
Example 3 The stochastic dierential equation
dX t = g(X t )h(X t )+ 1
2 g(X t)g 0 (X t ) dt + g(X t )dW t ; (5.51)
where g is a given dierentiable function and h is given by (5.42), can be
reduced to the Langevin equation of the form
dY t = ,aY t dt + bdW t ;
with a = , and b = 1, and has the general solution
X t = h ,1
e t h(X 0 )+e t Z t
0
e ,s dW s
: (5.52)
Again let us show how we derive the solution (5.52). We see A(x) = h(x)
and by (5.37) we obtain b 1 =0.With b 2 =1we have
U(x) =h(x)+C:
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