Estimation in Financial Models - RiskLab

That means, instead of reducing (5.40) to a linear stochastic dierential
equation we may integrate the Stratonovich stochastic dierential equation
directly as well, obtaining (5.41) at once.
Applications of Example 1:
dX t = 1 2 a2 X t dt + aX t dW t ;
X t = X 0 exp(aW t ): (5.43)
a(a , 1)X
1,2=a
t
dX t = 1 2
X t = W t + X 1=a
0
dt + aX 1,1=a
t dW t ;
a
: (5.44)
q
dX t = 1dt +2
X t =
W t +
X t dW t ;
q
X 0
2
: (5.45)
dX t = 1 2 a2 mX 2m,1
t dt + aX m t dW t ; m 6= 1;
X t = X 1,m
0
, a(m , 1)W t
1=(1,m)
: (5.46)
dX t = 1 2 X tdt +
q
X 2 t +1dW t ;
X t = sinh(W t + arcsinhX 0 ): (5.47)
Example 2 The stochastic dierential equation
dX t = g(X t )+ 1
2 g(X t)g 0 (X t ) dt + g(X t )dW t ; (5.48)
with a given dierentiable function g, is reducible with the general solution
X t = h ,1 (t + W t + h(X 0 )); (5.49)
67