Abstract
As explained in Sect. 8.2, the goal of this second part is to show how to obtain integration by parts (IBP) formulas for random variables which are obtained through systems based on an infinite sequence of independent random variables.
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Notes
- 1.
This restriction is tailor-made in order to fit \(\alpha ^{-1}\) stable distributions as our main example. In particular, one may simplify the set-up to \(\alpha =\frac{1}{2}\) and \(g(x)=\frac{1}{\sqrt{2\pi }}\exp (-\frac{x^2}{2})\) (the density of a standard normal random variable).
- 2.
In the continuous time setting this property corresponds to the adaptedness of the solution of Eq. (10.1).
- 3.
The concept that replaces this quantity in Malliavin Calculus is usually called the Malliavin (or stochastic) covariance matrix.
- 4.
This result requires much more knowledge than we give in this text and we therefore do not pursue it. This topic of research is commonly known as “regularization by noise”.
- 5.
See for example, [11] on the space D[0, 1] .
- 6.
If one prefers to put this in the framework of (10.1) then take \(n=1\), \(X_0=0\) and \(Z_0=\max _{0\le s\le 1}W_s=X_1\).
- 7.
This is a way of hiding the boundary terms that appeared in the previous exercise.
- 8.
Note that this property is related to the so-called adaptedness condition of X. In short, as \(X_j\) only depends on the values of \(Z_0,..., Z_{j-1}\), then \(\partial _kX_j=0\) if \(j\le k\).
- 9.
Exercise: Think of an example. Actually it is much harder to think of an example where this formula converges.
- 10.
Hint: After carrying out the appropriate change of variables, differentiate the formula with respect to \(\xi \) and evaluate it for \(\xi =0\).
- 11.
Also known as Gauss’s theorem or Ostrogradsky’s theorem.
- 12.
This example corresponds to \(\alpha ^{-1}\)-stable process. For example, \(\alpha =1\) corresponds to the Cauchy process for which \(g(x)=\frac{1}{\pi (1+x^2)}\), \(x\in \mathbb {R}\).
- 13.
How to resolve the issue that \(Z_k\) appears in the denominator will be further discussed in Example 10.3.4.
- 14.
In particular, our definition of “space transformation” includes the property that the transformation is one-to-one and that the inverse is differentiable.
- 15.
- 16.
\(\wedge \) means that the requirement is related to the representation of \(X_n\) using the new random variables U.
- 17.
This is a condition on \(\varDelta \).
- 18.
Note that the set-up is slightly different from the one given in Sect. 10.1 with respect to the form of the density of \(Z_k\).
- 19.
- 20.
That is, \(C_\zeta \rightarrow \infty \) as \(\zeta \rightarrow 0\). Clearly this can be done only in the case that \(\varepsilon _k=1\). But this can be done often if we choose \(\zeta \) as an appropriate function of \(\varDelta \) so that \(\mathbb {P}(\varepsilon _k=1)\) is close to one.
- 21.
See Exercise 10.4.7.
- 22.
Hint: Do not forget the boundary conditions for the integrals of \( h_k \) at \( \pm \infty \).
- 23.
Recall the result in Exercise 1.1.11.
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Kohatsu-Higa, A., Takeuchi, A. (2019). Basic Ideas for Integration by Parts Formulas . In: Jump SDEs and the Study of Their Densities. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-32-9741-8_10
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