Derivative of the Donsker delta functionals

Suryawan, Herry Pribawanto

Mathematica Bohemica, Tome 144 (2019) no. 2, pp. 161-176

Abstract (VO)

We prove that derivatives of any finite order of Donsker's delta functionals are well-defined elements in the space of Hida distributions. We also show the convergence to the derivative of Donsker's delta functionals of two different approximations. Finally, we present an existence result of finite product and infinite series of the derivative of the Donsker delta functionals.

Détail

DOI : 10.21136/MB.2018.0078-17

Classification : 28C20, 46F25, 60G20, 60H40
Keywords: Donsker delta functional; white noise analysis; distributional derivative

Bibliographie

[1] Aase, K., Ø{k}sendal, B., Ubø{e}, J.: Using the Donsker delta function to compute hedging strategies. Potential Anal. 14 (2001), 351-374. | DOI | MR | JFM

[2] Benth, F. E., Ng, S.-A.: Donsker's delta function and the covariance between generalized functionals. J. Lond. Math. Soc., II. Ser. 66 (2002), 1-13. | DOI | MR | JFM

[3] Bock, W., Silva, J. L. da, Suryawan, H. P.: Local times for multifractional Brownian motions in higher dimensions: a white noise approach. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19 (2016), Article ID 1650026, 16 pages. | DOI | MR | JFM

[4] Cesarano, C.: Integral representations and new generating functions of Chebyshev polynomials. Hacet. J. Math. Stat. 44 (2015), 535-546. | DOI | MR | JFM

[5] Draouil, O., Ø{k}sendal, B.: A Donsker delta functional approach to optimal insider control and applications to finance. Commun. Math. Stat. 3 (2015), 365-421 erratum ibid. 3 2015 535-540. | DOI | MR | JFM

[6] Gradshteyn, I. S., Ryzhik, I. M.: Table of Integrals, Series, and Products. Elsevier/\hskip0ptAcademic Press, Amsterdam (2015). | DOI | MR | JFM

[7] Grothaus, M., Riemann, F., Suryawan, H. P.: A white noise approach to the Feynman integrand for electrons in random media. J. Math. Phys. 55 (2014), Article ID 013507, 16 pages. | DOI | MR | JFM

[8] Hida, T., Kuo, H.-H., Potthoff, J., Streit, L.: White noise. An Infinite-Dimensional Calculus. Mathematics and Its Applications 253. Kluwer Academic Publishers, Dordrecht (1993). | DOI | MR | JFM

[9] Hu, Y., Watanabe, S.: Donsker's delta functions and approximation of heat kernels by the time discretization methods. J. Math. Kyoto Univ. 36 (1996), 499-518. | DOI | MR | JFM

[10] Hytönen, T., Neerven, J. van, Veraar, M., Weis, L.: Analysis in Banach Spaces. Vol. I. Martingales and Littlewood-Paley Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge 63. Springer, Cham (2016). | DOI | MR | JFM

[11] Kondratiev, Y. G., Leukert, P., Potthoff, J., Streit, L., Westerkamp, W.: Generalized functionals in Gaussian spaces: The characterization theorem revisited. J. Funct. Anal. 141 (1996), 301-318. | DOI | MR | JFM

[12] Kuo, H.-H.: Donsker's delta function as a generalized Brownian functional and its application. Theory and Application of Random Fields. Proc. IFIP-WG 7/1 Working Conf., Bangalore, 1982 Lect. Notes Control Inf. Sci. 49. Springer, Berlin (1983), 167-178. | DOI | MR | JFM

[13] Kuo, H.-H.: White Noise Distribution Theory. Probability and Stochastics Series. CRC Press, Boca Raton (1996). | MR | JFM

[14] Lascheck, A., Leukert, P., Streit, L., Westerkamp, W.: More about Donsker's delta function. Soochow J. Math. 20 (1994), 401-418. | MR | JFM

[15] Lee, Y.-J., Shih, H.-H.: Donsker's delta function of Lévy process. Acta Appl. Math. 63 (2000), 219-231. | DOI | MR | JFM

[16] Obata, N.: White Noise Calculus and Fock Space. Lecture Notes in Mathematics 1577. Springer, Berlin (1994). | DOI | MR | JFM

[17] Rosen, J.: Derivatives of self-intersection local times. 38th seminar on probability. Lecture Notes in Math. 1857 Springer, Berlin (2005), 263-281 M. Émery et al. | DOI | MR | JFM

[18] Stein, E. M., Shakarchi, R.: Complex Analysis. Princeton Lectures in Analysis 2. Princeton University Press, Princeton (2003). | MR | JFM

[19] Suryawan, H. P.: A white noise approach to the self-intersection local times of a Gaussian process. J. Indones. Math. Soc. 20 (2014), 111-124. | DOI | MR | JFM

[20] Watanabe, S.: Donsker's $\delta$-functions in the Malliavin calculus. Stochastic Analysis. Proc. Conf., Haifa, 1991 Academic Press, Boston (1991), 495-502 E. Mayer-Wolf et al. | DOI | MR | JFM

[21] Watanabe, S.: Some refinements of Donsker's delta functions. Stochastic Analysis on Infinite Dimensional Spaces. Proc. U.S.-Japan bilateral seminar, Baton Rouge, 1994 Pitman Res. Notes Math. Ser. 310. Longman Scientific Technical, Harlow (1994), 308-324 H. Kunita et al. | MR | JFM