Estimation in Financial Models - RiskLab

Appendix C
The It^o Formula
C.1 The one-dimensional case
The function U :[0;T] IR 7! IR have continuous partial derivatives @U , @U
@t @x
and @2 U
and X
@x 2 t satisfy the one-dimensional It^o stochastic dierential equation
dX t = a(t; !)dt + b(t; !)dW t ;
q
where jaj and b are in the space L 2 . Dene a process Y t by Y t = U(t; X t )
for 0 t T . Then
" @U
dY t =
@t (t; X @U
t)+a t
@x (t; X t)+ 1 #
@ 2 U
2 b2 t
@x (t; X t) dt
2
+ b t
@U
@x (t; X t) dW t ;
w. p. 1 for 0 t T , and with the notation a t = a(t; !);b t = b(t; !).
C.2 The multi-dimensional case
The process X t satisfy the d-dimensional It^o stochastic dierential equation
dX t = a(t; !)dt + B(t; !)dW t ;
where fW t ;t 0g is an m-dimensional Wiener process with independent
components, W t =(Wt 1
q ;Wt 2 ;:::;Wt
m ), and a :[0;T] 7! IR d , B :[0;T]
7! IR dm , satisfying ja k j and B k;j 2 L 2 for k =1;:::;d, j =1;:::;m.
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