Abstract
We learn early in any probability theory course that in order to compute any significant quantity we need to have information regarding the distribution function of random variables. This issue appears not only in applied problems where actual computation needs to be carried out but also in theoretical problems where qualitative information of the distribution function is needed.
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Notes
- 1.
Clearly there are many other ways to deal with the present problem. Notably, there are the analytical methods and most distinguishably methods developed in the partial differential equation literature.
- 2.
That is, \(\mathbb {E}[|X|^p]<\infty \) for all \(p>0\).
- 3.
Note that the indicator function used for the distribution function can also be written with \(G(y)=\mathbf {1}(y-x\le 0)\) and therefore \(\partial _x\mathbb {E}[G(X)]=-\mathbb {E}[G'(X)]\).
- 4.
More exactly, one proves the absolute continuity of the probability measure induced by X. For details, see Lemma 2.1.1 in [46].
- 5.
Changing the values of Y being used. In fact, in the last step one uses \(Y=H_{k-1}\).
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Kohatsu-Higa, A., Takeuchi, A. (2019). Overview. In: Jump SDEs and the Study of Their Densities. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-32-9741-8_8
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DOI: https://doi.org/10.1007/978-981-32-9741-8_8
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