Abstract
In this chapter, we consider a class of Lévy processes which are not of bounded variation as in the preceding chapter but instead they are processes with paths of infinite variation. From the pedagogical point of view, this chapter provides the construction of the Lévy process, leaving for the reader most of the developments related to the construction of the stochastic integral, the Itô formula and the associated stochastic differential equations. This is done in the exercises in order to let you test your understanding of the subject. This is done on two levels. You will find the ideas written in words in the proofs. If you do not understand them you may try a further description that may be given in Chap. 14. It is a good exercise to try to link the words and the equations so that you understand the underlying meaning. This is also a chapter that may be used for promoting discussion between students and the guiding lecturer.
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Notes
- 1.
In fact, one can prove that in any interval the number of jumps is infinite. This will left as an exercise for the reader.
- 2.
In fact, as an exercise prove that in this case, the integral \(\int _0^\infty (e^{\theta x}-1)f(x)dx\) is not finite.
- 3.
Although some moments will not be finite. Recall Exercise 4.1.27.
- 4.
Note that this construction is done using \(f-f_1\) in Sect. 4.1.
- 5.
In many situations we will assume that \( c=1 \) without further mentioning it.
- 6.
In fact, recall that if \(\varphi (\theta )\) is the characteristic function of some random variable X then \(\varphi (\theta )e^{-icIt\theta }\) is the characteristic function of \(X-ctI\) with \(I=\int _{|x|\le 1}xf(x)dx\). For more on this, see Exercise 5.1.3.
- 7.
Recall the arguments in Sect. 4.1.
- 8.
It is also a martingale sequence based on \(\varepsilon \). Prove this as an exercise.
- 9.
- 10.
Exercise: Describe a stronger norm where this sequence is a Cauchy sequence.
- 11.
Compensated means that the mean is being taken from the process so that the resulting process has mean zero. This will imply that the process becomes a martingale.
- 12.
Recall that the characteristic function of \(\bar{Z}_t^+\) can be computed as the limit of the characteristic functions of the sequence \(\bar{Z}_t^{(\varepsilon _n, +)}\).
- 13.
Some may object to the usage of \(Z^+\) as in the finite variation case this process is not compensated, while in the infinite variation case it becomes the compensated process.
- 14.
In fact, it may be easier to study the difference between the approximating processes.
- 15.
Hint: Recall the solution of Exercise 4.1.25. But also there is a very short way of proving this by computing \(\lim _{n\uparrow \infty }\mu ^{(\varepsilon _n, 1)}\).
- 16.
Therefore in general, variances of the process \(Z_t\) cannot be studied unless one adds extra conditions on f.
- 17.
Recall that in general, a compound Poisson process does not have a density.
- 18.
Clearly, one may modify the Lévy measure so that asymptotically it is equivalent to this example. We prefer this way of writing as it makes it clear the relation with the concept of a Lévy measure.
- 19.
- 20.
In fact, the limit is only determined by the behavior of f around zero. Therefore any change on the function f away from zero will not change the limit behavior.
- 21.
Prove this. Note that the first term converges in absolute variation, while the second only converges in the \(L^2(\varOmega )\) sense with the supremum norm in time. For this reason, for the first term one uses two derivatives, while for the second one only uses the first derivative.
- 22.
In fact, even the independent increment property will suffice.
- 23.
Otherwise as an exercise you can try to think each time how to prove the needed version of the Itô formula.
- 24.
- 25.
We hope that the abuse of notation using \(\lambda \) for the measure and the function that defines \(\lambda \) does not cause confusion.
- 26.
Note that we have slightly changed the notation so that it is easier to read. That is, \( Y\left( \int _0^t \lambda (s) ds \right) \equiv Y(u)|_{u=\int _0^t \lambda (s) ds} \).
- 27.
Hint: Prove that \(\int _0^t\mathbb {E}[|Ah(N_s)|]ds\le te^{\lambda t}\mathbb {E}[|Ah(N_t)|].\)
- 28.
Hint: Use the fact that \( \int _{u}^{t} h(N_{s-}+1)-f(N_{s-})dN_s=\sum _{k=1}^\infty (h(k)-h(k-1) \mathbf {1}_{\{u<T_k\le t\}} \).
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Kohatsu-Higa, A., Takeuchi, A. (2019). Construction of Lévy Processes and Their Corresponding SDEs: The Infinite Variation Case. In: Jump SDEs and the Study of Their Densities. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-32-9741-8_5
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