Abstract
In many applied problems, one needs to compute expectations of a function of a random variable which are obtained through a certain theoretical development. This is the case of \( \mathbb {E}[G(Z_t)] \), where Z is a Lévy process with Lévy measure \( \nu \) which may depend on various parameters. Similarly, G is a real-valued bounded measurable function which may also depend on some parameters and is not necessarily smooth. For many stability reasons one may be interested in having explicit expressions for the partial derivatives of the previous expectation with respect to the parameters in the model. These quantities are called “Greeks” in finance but they may have different names in other fields.
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Notes
- 1.
Still, as we will find out later the above methodology is far more general as in many cases explicit densities cannot be obtained.
- 2.
As is usually the case. The case \( \alpha =1 \) has to be dealt with separately due to the difference in representations.
- 3.
In the case that \( G\in C^1_b \) recall, for example, Exercise 5.1.13. Since \(\mathbb {E} \big [ E_+^p \big ] = \int _0^\infty \mathbb {P} [ E_+ > \lambda ^{1/p} ] \, d\lambda \), all we have to check is \(\mathbb {P} [ E_+ > \lambda ^{1/p} ]\) as \(\lambda \rightarrow \infty \), which can be seen in [36]. We can also obtain the pth moment on \(E_-\).
- 4.
The existence of such a function can be assured, if one assumes enough conditions on the function \( G_n \).
- 5.
If \(Z_{t}\) has a second-order moment for each \(t \in [0,T]\), the function \(G \in C_{b} (\mathbb {R})\) can be extended to the continuous function with linear growth order.
- 6.
We can discuss the case of [a, b), (a, b) and [a, b] similarly.
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Kohatsu-Higa, A., Takeuchi, A. (2019). Sensitivity Formulas. In: Jump SDEs and the Study of Their Densities. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-32-9741-8_11
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DOI: https://doi.org/10.1007/978-981-32-9741-8_11
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