Abstract
In this chapter, we will generalize the previous construction of compound Poisson processes and allow the possibility of a infinite number of jumps on a fixed interval. The stochastic process constructed in this section will satisfy that the number of jumps whose absolute size is larger than any fixed positive value is finite in any fixed interval. Therefore the fact that there are infinite number of jumps is due to the fact that most of these jumps are small in size. The conditions imposed will also imply that the generated stochastic process has paths of bounded variation and therefore Stiltjes integration can be used to give a meaning to stochastic integrals. We also introduce the associated stochastic calculus.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Hint: Note that the process \( Z^+_t-Z^{(1,+)}_t \) does not include any jump of size bigger than one. Also, see Exercise 4.1.26 .
- 2.
Recall that \(\mathscr {N}^{(\varepsilon _{n})}=\mathscr {N}^{(\varepsilon _{n},+)}+\mathscr {N}^{(\varepsilon _{n},-)}\).
- 3.
Recall Theorem 1.1.6.
- 4.
One has to be careful with other texts as there are generalized versions of stable laws which include a parameter which measures symmetry.
- 5.
Recall Exercise 2.1.36.
- 6.
Recall Exercise 1.1.11.
- 7.
This is trivial.
- 8.
Hint: Decompose the expectation according to the number of jumps of \(Z_t\). Then use repeatedly the inequality \((x+y)^\beta \le x^\beta y^\beta \) for \(x, y >0\).
- 9.
Recall the notation issues discussed in footnote 22.
- 10.
We have taken the advantage of assuming that the reader knows about local martingales. Otherwise you may try to prove the same statement assuming that \(\mathbb {E}[\int _{\mathbb {R}\times [0,t]}|g(z,s)|\widehat{\mathscr {N}}(dz, ds)]<\infty \).
- 11.
This is just a consequence of the following property: Any càdlàg function has at most a countable number of discontinuities. Recall Exercise 3.4.9.
- 12.
We leave this as an exercise for the reader. One can do this using an argument similar to the one in the rest of the proof.
- 13.
That is, \(X_{t}(\omega )\) and \(\widetilde{X}_{t}(\omega )\) are càdlàg functions.
- 14.
For example, let \(f:\mathbb {R}\rightarrow \mathbb {R}\) such that f is Lipschitz with a Lipschitz constant smaller than 1. Then the equation \(f(x)=x\) has a unique solution. In our setting one has to choose a time small enough so that this idea can be applied.
- 15.
Of course, in the case that you may not need the càdlàg property then you can get by with this kind of solution.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Kohatsu-Higa, A., Takeuchi, A. (2019). Construction of Lévy Processes and Their Corresponding SDEs: The Finite Variation Case. In: Jump SDEs and the Study of Their Densities. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-32-9741-8_4
Download citation
DOI: https://doi.org/10.1007/978-981-32-9741-8_4
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-32-9740-1
Online ISBN: 978-981-32-9741-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)