Estimation in Financial Models - RiskLab

The function U :[0;T] IR d 7! IR have continuous partial derivatives @U
@t , @U
@x k
and
@2 U
@x k @x i
for k; i = 1; 2;:::;d. Dene a scalar process fY t ; 0 t T g by
Y t = U(t; X t )=U(t; Xt 1 ;Xt 2 ;:::;Xt d ) w. p. 1. Then
2
3
dY t = 4 @U dX
@t + a k @U
t + 1 mX dX
Bt
i;j B k;j @ 2 U
5
t dt
@x k 2
@x i @x k
+
mX
dX
j=1 i=1
k=1
Bt
i;j @U
dWt j ;
@x i
j=1 i;k=1
w. p. 1 for 0 t T , where the partial derivatives are evaluated at (t; X t )
and where we denote a k t = a k (t; !), B i;j
t = B i;j (t; !).
For an extension of the It^o formula to a wider class of non-smooth functions
we refer to the recently published paper by Follmer, Protter and Shiryaev
[26].
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