Hey in this report (Approximation of Jump Diffusions in Finance and Economics by Bruti-Liberati and Platen) is described the Milstein formula (3.5) for simulation SDE with jump component. How it is calculated? In this formula we also have to compute value of a Wiener process in a jump time, how to do it? I think that simualtion of a Poisson process by increment will be insufficient in this situation. Or maybe the Euler scheme is preferable and the Milstein scheme is not used in this case?
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$\begingroup$ It would be much better if you included further details in your post so that one can still understand your question even if the link you have provided gets broken or the material is changed. $\endgroup$– AlperMar 2 at 22:37
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Those terms represent iterated integrals of the type $$ \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} dW(z) dJ(s) $$ and $$ \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} dJ(z) dW(s) $$ Which seem to be the third and fourth lines in that formula, respectively.
To quote that same report:
Furthermore, one needs to sample the Wiener process W at the jump times τi, for i ∈ {1, . . . , NT }
Which either means you have to sample $N$ for every $t_n$ and then sample $\Delta W$ an $N$ number of times for every $t_n$ or use jump adapted schemes that calculate when a jump occurs and sample the process there. These work on jump-diffusions (which is the type of SDE written on that link), so not on processes with infinite activity (like an alpha-stable process). For more info on these, you can see chapter 8 of the book at Numerical Solution of Stochastic Differential Equations with Jumps in Finance, which is by the same authors.
