INTEGRATING PATH-DEPENDENT FUNCTIONALS ON YEH-WIENER SPACE

TitleINTEGRATING PATH-DEPENDENT FUNCTIONALS ON YEH-WIENER SPACE
Publication TypeJournal Article
Year of Publication2015
AuthorsPIERCE, IAN, SKOUG, DAVID
Volume19
Issue4
Start Page643
Pagination15
Date Published2015
ISSN1083-2564
AMS28C20, 60J65
Abstract

Denote by ${ C_{a,b}(Q) }$ the generalized two-parameter Yeh-Wiener space with associated Gaussian measure. We investigate several scenarios in which integrals of functionals on this space can be reduced to integrals of related functionals over an appropriate single-parameter generalized Wiener space ${ C_{ \hat{a},\hat{b}} [0, T ]. }$This extends some interesting results of R. H. Cameron and D. A. Storvick.

URLhttp://www.acadsol.eu/en/articles/19/4/13.pdf
Refereed DesignationRefereed
Full Text

REFERENCES
[1] Earl Berkson and Thomas A. Gillespie, Absolutely continuous functions of two variables and
well-bounded operators, J. London Math. Soc. 30 (1984), 305–321.
[2] Robert H. Cameron and David A. Storvick, Two related integrals over spaces of continuous
functions, Pacific J. Math 55 (1974), 19–37.
[3] Joo Sup Chang, Chull Park, and David Skoug, Fundamental Theorem of Yeh-Wiener calculus,
Stochastic Anal. Appl. 9 (1991), 245–262.
[4] Kun Soo Chang and Il Yoo, A simple proof of converse measurability theorem for Yeh-Wiener
spaces, Bull. Korean Math. Soc. 23 (1986), 35–37.
[5] Seung Jun Chang, Hyun Soo Chung, and David Skoug, Some basic relationships among transforms,
convolution products, first variation, and inverse transforms, Cent. Eur. J. Math. 11 (2013), 538–551.
[6] Seung Jun Chang, Hyun Soo Chung, and David Skoug, Integral transforms of functionals in L2
(Ca,b[0, T ]), J. Fourier Anal. Appl. 14 (2009), 441–462.
[7] Seung Jun Chang and David Skoug, The effect of drift on conditional Fourier-Feynman transforms
and conditional convolution products, Int. J. Appl. Math. 2 (2000), 505–527.
[8] Seung Jun Chang and Dong Myung Chung, Conditional function space integrals with applications,
Rocky Mountain J. Math. 26 (1996), 37–62.
[9] Dong Hyun Cho, Analogues of conditional Wiener integrals with drift and initial distribution
in a function space, Abstr. Appl. Anal. (2014), Art. ID 916423.
[10] Dong Hyun Cho, Evaluation formulas for generalized conditional Wiener integrals with drift
on a function space, J. Funct. Spaces Appl. (2013), Art. ID 469840.
[11] Gerald W. Johnson and Michel L. Lapidus, The Feynman Integral and Feynman’s Operational
Calculus, Clarendon Press Oxford University Press, New York, NY, 2000.
[12] Bong Jin Kim, Byoung Soo Kim, and Yoo Il, A change of scale formula for a function space
integral on Ca,b[0, T ], Proc. Amer. Math. Soc. 87 (2013), 86–93.
[13] Byoung Soo Kim, Integral transforms of square integrable functionals on Yeh-Wiener space,
Kyungpook Math. J. 49 (2009), 155–166.
[14] S. R. Paranjape and Chull Park, Distribution of the supremum of the two-parameter YehWiener
process on the boundary, J. Appl. Probability 10 (1973), 875–880.
[15] Chull Park and David Skoug, Boundary-valued conditional Yeh-Wiener integrals and a KacFeynman
Wiener integral equation, J. Korean Math. Soc. 33 (1996), 763–775.
[16] Chull Park and David Skoug, Grid-valued conditional Yeh-Wiener integrals and a KacFeynman
Wiener integral equation, J. Integral Equations Appl. 8 (1996), 213–230.
[17] Chull Park and David Skoug, Distribution estimates of barrier-crossing probabilities of the
Yeh-Wiener process, Pacific J. Math. 78 (1978), 455–466.
[18] Ian Pierce and David Skoug, Comparing the distribution of various suprema on two-parameter
Wiener space, Proc. Amer. Math. Soc. 141 (2013), 2149–2152.
[19] Ian Pierce, On a family of generalized wiener spaces and applications, Ph.D. Thesis, University of Nebraska-Lincoln (2011).
[20] Ian Pierce and David Skoug, Integration formulas for functionals on the function space
Ca,b[0, T ] involving Paley-Wiener-Zygmund stochastic integrals, Panamer. Math. J. 18 (2008), 101–112.
[21] David Skoug, Converses to measurability theorems for Yeh-Wiener space, Proc. Amer. Math.
Soc. 57 (1976), 304–310.
[22] James Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker Inc., New York, NY, 1973.
[23] James Yeh, Orthogonal developments of functionals and related theorems in the Wiener space
of functions of two variables, Pacific J. Math. 13 (1963), 1427–1436.
[24] James Yeh, Cameron-Martin translation theorems in the Wiener space of functions of two
variables, Trans. Amer. Math. Soc. 107 (1963), 409–420.
[25] James Yeh, Wiener measure in a space of functions of two variables, Trans. Amer. Math. Soc.
95 (1960), 433–450.