Abstract
In this chapter, we extend the method of analysis introduced in Chapter 11 to a general framework. This method was essentially introduced by Norris to obtain an integration by parts (IBP) formula for jump-driven stochastic differential equations. We focus our study on the directional derivative of the jump measure which respect to the direction of the Girsanov transformation. We first generalize the method in order to consider random variables on Poisson spaces and then show in various examples how the right choice of direction of integration is an important element of this formula.
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Notes
- 1.
We hope that the context will not bring any confusion with the matrix notation or the dependence of constants with respect to certain parameters.
- 2.
Clearly, this definition is formal. Try to write the exact definition.
- 3.
Recall that as all matrix norms are equivalent, you can pick the one you like best.
- 4.
Note that the current set-up is not totally covered by the results of Part I of this book. If you prefer you may think instead of the above general setting that the Lévy measure corresponds to the tempered Lévy measure introduced in Sect. 11.4.
- 5.
It “almost” does have a density.
- 6.
We hope that this notation for the jump size random variables which has been used from the start will not confuse the reader with the derivative of the flow. The context should make clear which object we are referring to in each case.
- 7.
This sum convention is used from now on.
- 8.
It may be helpful to recall the discussion at the end of Chap. 10.
- 9.
We assume that the reader can extrapolate the conditions in Chap. 7 to the multi-dimensional case, otherwise they may also assume that the current chapter applies for the case \( d=m=u=1 \).
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Kohatsu-Higa, A., Takeuchi, A. (2019). Integration by Parts: Norris Method. In: Jump SDEs and the Study of Their Densities. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-32-9741-8_12
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DOI: https://doi.org/10.1007/978-981-32-9741-8_12
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