8.4 Integrals with respect to Semimartingales

In this section the stochastic integral ? ^T ₀ H( t) dS( t) is defined, where S( t) is a semimartingale. Due to representation S( t) = S(0) + M( t) + A( t) the integral with respect to S( t) is the sum of two integrals one with respect to a local martingale M( t) and the other with respect to a finite variation process A( t). The integral with respect to A( t) can be done path by path as the Stieltjes integral, since A( t), although random, is of finite variation.

The integral with respect to the martingale M( t) is new, it is the stochastic integral ? ^T ₀ H( t) dM( t). When M( t) is Brownian motion B( t), it is the It integral, defined in Chapter 4. But now martingales are allowed to have jumps and this makes the theory more complicated. The key property used in the definition of the It integral is that on finite intervals Brownian motion is a square integrable martingale. This property in its local form plays an important role in the general case. Conditions for the existence of the integral with respect to a martingale involves the martingale's quadratic variation, which was introduced in Section 7.6.