[1] Abels, H. (2012) Pseudodifferential and singular integral operators.
An introduction with applications. De Gruyter Graduate Lectures. De
Gruyter, Berlin. x+222 pp. ISBN: 978-3-11-025030-5
[2] Adachi, T. (2014). Toward categorical risk measure theory. Theory and
Applications of Categories, 29(14):389–405.
[3] Adachi, T. and Ryu, Y.(2019). A Category of Probability Spaces. Journal
of Mathematical Sciences The University of Tokyo Vol.26, No.2, 2019,
p.201-221.
[4] Airault, H., Ren, J. and Zhang, X. (2000) Smoothness of local times
of semimartingales. C. R. Acad. Sci. Paris S´er. I Math. 330, no. 8, 719724.
[5] Ait Ouahra, M., Kissami, A. and Ouahhabi, H. (2014) On fractional derivatives of the local time of a symmetric stable process as a
doubly indexed process. Random Oper. Stoch. Equ. 22, no. 2, 99-107.
[6] Amaba, T. and Ryu, Y. (2018) Distributional Itˆo’s Formula and Regularization of Generalized Wiener Functionals. ALEA, Latin American
Journal of Probability and Mathematical Statistics, Vol.15, No.1, 2018,
p.703-753.
[7] Bender, C. (2003) An Itˆo formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stochastic Process. Appl. 104, no. 1, 81-106.
[8] Bongioanni, B. and Torrea, J. L. (2006) Sobolev spaces associated
to the harmonic oscillator. Proc. Indian Acad. Sci. Math. Sci. 116, no.
3, 337-360.
[9] Boufoussi, B. and Roynette, B. (1993) Le temps local brownien
1/2
appartient p.s. `a l’espace de Besov Bp.∞ . C. R. Acad. Sci. Paris S´er. I
Math. 316, no. 8, 843-848.
97
[10] Bouleau, N. and Yor, M. (1981) Sur la variation quadratique des
temps locaux de certaines semimartingales. C. R. Acad. Sci. Paris S´er.
I Math. 292, no. 9, 491-494.
¨ llmer, H., Protter, P. and Shiryayev, A. N. (1995) Quadratic
[11] Fo
covariation and an extension of Itˆo’s formula. Bernoulli 1, no. 1-2, 149169.
[12] Franz, U. (2003). What is stochastic independence? In Obata, N., Matsui, T., and Hora, A., editors, Quantum probability and White Noise
Analysis, Non-commutativity, Infinite-dimensionality, and Probability
at the Crossroads. QP-PQ, XVI, pages 254–274, Singapore. World Sci.
Publishing.
[13] Friedman, A. (1964) Partial differential equations of parabolic type.
Prentice-Hall, Inc., Englewood Cliffs, N.J. xiv+347 pp.
[14] Giry, M. (1982). A categorical approach to probability theory. In Banaschewski, B., editor, Categorical Aspects of Topology and Analysis,
volume 915 of Lecture Notes in Mathematics, pages 68–85. SpringerVerlag.
[15] Hasumi, M. (1961) Note on the n-dimensional tempered ultradistributions. Tˆohoku Math. J. (2) 13, 94-104.
[16] Ikeda, N. and Watanabe, S. (1989) Stochastic differential equations
and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd.,
Tokyo. xvi+555 pp. ISBN: 0-444-87378-3
[17] Krylov, N. V. (2008) Lectures on elliptic and parabolic equations in
Sobolev spaces. Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, xviii+357 pp. ISBN: 978-0-8218-4684-1
[18] Kubo, I. (1983) Itˆo formula for generalized Brownian functionals. Theory and application of random fields (Bangalore, 1982), 156-166, Lecture
Notes in Control and Inform. Sci., 49, Springer, Berlin.
[19] Kusuoka, S. and Stroock, D. (1985) Applications of the Malliavin
calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32, no. 1, 1-76.
[20] Lawvere, F. W. (1962). The category of probabilistic mappings. Preprint.
[21] Liang, Z. (2006) Fractional smoothness for the generalized local time
of the indefinite Skorohod integral. J. Funct. Anal. 239, no. 1, 247-267.
98
[22] Nualart, D. and Vives, J. (1993) Chaos expansions and local times.
Publ. Mat. 36 (1992), no. 2B, 827-836.
[23] Revuz, D. and Yor, M. (1999) Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293.
Springer-Verlag, Berlin. xiv+602 pp. ISBN: 3-540-64325-7
[24] Silva, J. S. E. (1958) Les fonctions analytiques comme ultradistributions dans le calcul op´erationnel. (French) Math. Ann. 136, 5896.
[25] Shuwen, L. and Cheng, O. (2016) Local times of stochastic differential equations driven by fractional Brownian motions. arXiv preprint,
arXiv:1602.07272
[26] Stein, E. M. (1956) Interpolation of linear operators. Trans. Amer.
Math. Soc. 83, 482-492.
[27] Thangavelu, S. Lectures on Hermite and Laguerre expansions. With
a preface by Robert S. Strichartz. Mathematical Notes, 42. Princeton
University Press, Princeton, NJ, 1993. xviii+195 pp. ISBN: 0-691-000484
[28] Trotter, H. F. (1958) A property of Brownian motion paths. Illinois
J. Math. 2, 425-433.
[29] Uemura, H. (2004) Tanaka formula for multidimensional Brownian
motions. J. Theoret. Probab. 17, no. 2, 347–366.
[30] Veretennikov, A. Ju. and Krylov, N. V. (1976) Explicit formulae for the solutions of stochastic equations. (Russian) Mat. Sb. (N.S.)
100(142), no. 2, 266-284, 336.
[31] Wang, A. T. (1977/78) Generalized Ito’s formula and additive functionals of Brownian motion. Z. Wahrscheinlichkeitstheorie und Verw.
Gebiete 41, no. 2, 153-159.
[32] Watanabe, S. (1987) Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15, no. 1, 1-39.
[33] Watanabe, S. (1991) Donsker’s δ-functions in the Malliavin calculus.
Stochastic analysis, 495-502, Academic Press, Boston, MA.
99
[34] Watanabe, S. (1993) Fractional order Sobolev spaces on Wiener space.
Probability theory and related fields 95, no. 2, 175–198.
[35] Watanabe, S. (1994) Some refinements of Donsker’s delta functions.
Stochastic analysis on infinite-dimensional spaces (Baton Rouge, LA,
1994), 308-324, Pitman Res. Notes Math. Ser., 310, Longman Sci. Tech.,
Harlow.
[36] Watanabe, S. (1994) Stochastic Levi sums. Comm. Pure Appl. Math.
47, no. 5, 767-786.
[37] Williams, D. (1991). Probability with Martingales. Cambridge University
Press.
[38] Yoshinaga, K. (1960) On spaces of distributions of exponential growth.
Bull. Kyushu Inst. Tech. Math. Nat. Sci. 6, 1-16.
100
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