Apart from correcting a number of printing mistakes, and some mathematical inaccuracies as well, this second edition contains some new material: indeed, during the fifteen years elapsed since the first edition came out, a large number of new results concerning limit theorems have of course been proved by many authors, and more generally mathematical life has been going on. This gave us the feeling that some of the material in the first edition was perhaps not as important as we thought at the time, while there were some neglected topics which have in fact proved to be very useful in various applications.So perhaps a totally new book would have been a good thing to write. Our natural laziness prevented us to do that, but we have felt compelled to fill in the most evident holes in this book. This has been done in the most painless way for us, and also for the reader acquainted with the first edition (at least we hope so ... ). That is all new material has been added at the end of preexisting chapters.There are essentially three "new" topics covered below. One concerns stochastic calculus per se with a view towards financial mathematics: Section 8 added to Chapter II, which contains a more thorough study of "stochastic exponentials" (or Doleans-Dade exponential) of semimartingales and its inverse, the "stochastic logarithm"; Sections 6 and 7 added to Chapter III, about stochastic integrals of vector-valued integrands with respect to a vector-valued semimartingale (in the finite-dimensional case only). The second topic concerns the so-called "predictable" uniform tightness condition for a sequence of semimartingales, a topic which has reached maturity by now: Section 6 of Chapter VI has been totally rewritten, and Section 6 has been added to Chapter IX and is concerned with the stability results for stochastic differential equations in the light of this uniform tightness condition. Finally, in order to deal with limit theorems for discretized semimartingales on a fixed interval we have added Section 7 to Chapter II (for the prerequisites on this topic) and Section 7 to Chapter IX, where the main and hopefully useful results are presented.
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