8.10 It 's Formula for Semimartingales

Let X( t) be a semimartingale and f be a C ² function. Then f( X( t)) is a semimartingale, and It 's formula holds

The quadratic variation [ X, X] jumps at the points of jumps of X and its jumps ?[ X,X]( s) = ( ? X( s)) ². Thus the jump part of the integral ? ^t ₀ f ?( X( s ?)) d[ X,X]( s) is given by ? _{s ? t} f ?( X( s ?))( ? X( s)) ², leading to an equivalent form of the formula

where [ X,X] ^c is the continuous component of the finite variation function [ X, X]. Using the relationship between the square and the sharp brackets (8.50), we can write It 's formula with the sharp bracket process of X, provided the sharp bracket exists,

where X ^cm denotes the continuous martingale part of X.

Example 8.22

Let N( t) be a Poisson process. We calculate ? ^t ₀ N( s ?) dN ( s). The answer can be derived from the integration by parts formula (8.24), but now we use (8.59). Since ( N( t) ? t) ^cm = 0 (by Corollary 8.30)

Since N( s) = N