Could anybody give me some pointers on the differences between the following processes:Counting Process, Point Process, Jump Process, Pure Jump Process...Moreover, I am looking for the Ito formula for jumps that can handle dXt = mu(t) * dt + sigma(t)*dWt + dZtwhere Zt is more general than a jump process that is driven by a Poisson process. The literature I found all assumes Zt is driven by a Poisson process so the dZt can be written as Jt- * dNt, where Nt is a Poisson process and Jt- is the (random) jump size at time t-. I am looking for literature on more general forms of dZt, let's say some processes that are not driven by a Poisson process?Specificly, I am looking for the Ito formula to handle the general dZt term.For example, the current Ito formula with jumps can give me:f(Xt) = f(X0) + Integrate{ (... ) dt} + Integrate{(...) dWt} + summation{ f(Xs) - f(Xs-), s from 0 to t} ...But then how do I relate the "summation{ f(Xs) - f(Xs-), s from 0 to t}" to the dZt in the formula of dXt?Many literature gives result for dZt = Jt- * dNt, and then in fact I can write the summation as a Integrate{(...) dNt}, which is a lot easier to work with and in fact I can decompose the dNt into a martingale term plus an intensity term.But for general dZt, not driven by Poisson process, how can I change the summation into an integral and then decompose into a Martingale term and an intensity term?Could anybody give me some pointers on this?The Duffie-Pan-Singleton's paper and the popular Affine Jump Diffusion model all assume the dZt is driven by Poisson process, am I right?Thanks

Hi, Literature:There are a couple of books that conserns this subject, and which one you should read depends on how technical you want to get. You probably want to look into semimartingales, which can be regarded as "the set of nice integrators", i.e. processes that makes the integral "work". The advantage of this approach is that you develop a general integral, and don't have to look at the integral for all sort of different processes beeing the integrator. The classical work for people coming from a mathematical background would be Phillip Protters "Stochastic Integration and Differential Equations", on Springer Verlag. This book is quite technical. Here you will find Itos formula for semimartingales, which includes Levy processes, and hence also Poisson processes. An alternative would be Cont/Tankovs "Financial Modelling With Jump Processes", Chapman Hall. This is, as they claim, for experienced praticioners, or something. I find the book an easy read, but illustrative with a sort of practical argumentation and way of looking at things. However, i find a lot of typos which are a bit annoying. A third alternative is Applebaums "Levy Processes and Stochastic Calculus", which is also a good alternative which is not so demanding as Protters book. I couldn't tell you the difference between a counting and a point process, but I belive a counting process just counts the number of events - like a Poisson process that counts the times of jumps. A jump process is any process pluss a term involving a jump - typically a brownian motion with drift plus a compound poisson. A pure jump process contains only jumps - no drift or brownian component. Given a jump process one could identify the time of the jumps, then a associated random measure N and the summation{f(X(s-)) - f(X(s))} = summation{ delta(f(X(s))) } would equal integral{ delta(f(X(s))) N(t,dx) } -- in a wery heuristically fashion. Not trivial, although quite intuiutive, as you can imagine. Hope this is of any help -- S

I suppose you want the process to be self-consistent. Then the jumps follow Poisson process. Look at a certain jump size J. Let P(t) be the probability that a jump of size J did not happen during a time interval of length t. Now look at two consequtive intervals of length t and s. Selfconsistency gives P(t + s) = P(t)*P(s) and you concludethat P(t) = exp(-lambda*t), i.e. Poisson process. Basically each jump size J has its own frequency lambda(J). If lambda = integral lambda(J) dJ is finite then the it is a finite activity jump process and lambda(J)/lambda is the distribution of jump sizes given that a jump of any size occured. It could be useful for some formulas,eg Merton gaussian jumps or exponential jumps.If integral lambda(J) dJ is infinite then it is an infinte activity jump process, eg variance gamma and similar stuff.Both cases can be treated similarly via Fourier transform for european options.Unrelated:Of course you can make the diffusion vol and lambdas follow some jump+diffusion too ....

Does anyone use double exponential jumps?