About Stochastic Calculus in Presence of Jumps at Predictable Stopping Times
- 1 International Laboratory of Statistics of Random Processes and Quantitative Financial Analysis, Tomsk State University, Tomsk, Russia
Abstract
In this paper, some basic results of stochastic calculus are revised using the following observation: For any semimartingale, the series of jumps at predictable stopping times converges a.s. on any finite time interval, whereas the series of jumps at totally inaccessible stopping times diverges. This implies that when studying random measures generated by jumps of a given semimartingale, it is naturally to define separately a random measure μ generated by the jumps at totally inaccessible stopping times and an other random measure π generated by the jumps at predictable stopping times. Stochastic integrals f · （ μ － μ p ） are well defined for suitable functions f , where μ p is the predictable compensator of μ . Concerning the stochastic integral h · π , it is well defined without any compensating of the integer valued measure π .
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