Celebratio Mathematica

T. Funaki, Y. Hariya, and M. Yor: “Wiener integrals for centered Bessel and related processes, II,” ALEA, Lat. Am. J. Probab. Math. Stat. 1 (2006), pp. 225–240. Electronic only. Part I was published (with a slightly different title) in Markov Process. Relat. Fields 13:1 (2007). MR 2249656 Zbl 1112.60042 article

This is the second part of a series of papers on the construction of stochastic integrals of Wiener’s type for the centered \( \delta \)-dimensional Bessel processes (BES(\( \delta \))-processes in short) and their variants. The approach adopted in the present paper is via the Brascamp–Lieb inequality. This method works well for the BES(\( \delta \))-processes, BES(\( \delta \))-bridges with \( \delta \geq 3 \), the Brownian meander and their extensions described by a class of stochastic differential equations, but not for their powers. As we have seen in the first part, another approach via Hardy’s \( L^2 \) inequality is effective for BES(\( \delta \))-processes with \( \delta\geq 1 \) and their powers. The method used in this paper is powerful to establish a family of accurate bounds on the distributions of these Wiener integrals.

T. Funaki, Y. Hariya, F. Hirsch, and M. Yor: “On the construction of Wiener integrals with respect to certain pseudo-Bessel processes,” Stochastic Process. Appl. 116 : 12 (December 2006), pp. 1690–1711. MR 2307055 Zbl 1110.60055 article

In previous papers [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Processes Related Fields (2006) (in press); T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes, II, Alea (2006) (in press)], the authors have shown that it is possible to define the Wiener-type integrals \[ \int_0^1 h(s)\,d\overline{R}_s ,\] for every \( h\in L^2([0,1] \), \( ds) \) and \( (\overline{R}_s) \) any centered Bessel process with dimension \( d\gt 1 \). In this paper, various conditions are stated, showing that such a construction is possible for a large class of processes indexed by two square integrable Brownian functionals. In particular, some of the results previously obtained for the Bessel processes are thus recovered, and in fact shown to extend to certain processes of the form \[ \sqrt{t}f\left(\frac{R_t}{\sqrt{t}}\right) .\]

T. Funaki, Y. Hariya, F. Hirsch, and M. Yor: “On some Fourier aspects of the construction of certain Wiener integrals,” Stochastic Process. Appl. 117 : 1 (2007), pp. 1–22. MR 2287100 Zbl 1113.60055 article

The existence and best \( L^2 \)-bounds for the Wiener type integrals \[ \int_0^1 f(s)\,dX_s ,\] where \( X \) ranges through a wide class of Bessel-like centered processes, and \( f \) belongs to \( L^2([0,1]) \), are discussed in terms of Fourier transforms associated with some characteristics of \( X \), thus providing some unification of previous results on this topic obtained by the authors [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Processes and their Related Fields (2006), T. Funaki, Y. Hariya, F. Hirsch, M. Yor, On the construction of Wiener integrals with respect to certain pseudo-Bessel processes, Stochastic Processes and their Applications (2006)] as well as yielding new results.

T. Funaki, Y. Hariya, and M. Yor: “Wiener integrals for centered powers of Bessel processes, I,” Markov Process. Relat. Fields 13 : 1 (2007), pp. 21–56. Part II was published (with a slightly different title) in Lat. Am. J. Probab. Math. Stat. 1 (2006). MR 2321750 Zbl 1123.60035 article

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