General Theory of Stochastic Integration

1. It is well-known and it is well emphasized that it is possible that a process is not integrable with respect to a local martingale but it is integrable if we consider the integrand as a semimartingale. It is less emphasized but perhaps it is the most surprising feature of the stochastic integration with respect to semimartingales that it is possible that a process is integrable with respect to a process with finite variation in semimartingale sense but it is not integrable in the ordinary sense. The counter example is the following: Let L be a compensated compound Poisson process with jump sizes 1,1/4,1/9.... Obviously L is a purely discontinuous local martingale and the trajectories of L have finite variation. One can “compress” L to a process on the interval [0,1]. Obviously the left-continuous, piecewise constant process H which jumps at the jump points of L and which at the n-th jump point is n is not integrable with respect to L in the ordinary pathwise sense. But

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so the integral as a stochastic integral with respect to a local martingale is well-defined. Of course if L is predictable, then it cannot be a local martingale. In this case L is a special semimartingale, so by the integration with respect to special semimartingales, that is by Theorem 4.49, a predictable process H is integrable with respect to L in semimartingale sense if and only if H is integrable in the ordinary sense.

2. One important characterization of the special semimartingales, which is not in the book, is that a semimartingale is a special semimartingale if and only if the square root of the quadratic variation of the semimartingale is locally integrable. Using this one can simplify the proof of Theorem 4.49 about the integration of special semimartingales. Download

3. On page 239. in the first line of the proof the indexes are mixed. On both sides the index is k and it runs from 1 to n.

 
ch4.txt · Utolsó módosítás: 2010/08/24 13:58 szerkesztette: medvegyev
 
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