A Fourier Analysis Based New Look at Integration

[1] A. Andresen, P. Imkeller, and N. Perkowski, Large deviations for Hilbert-space-valued Wiener processes: a sequence space approach, in: F. Viens et al (eds.), Malliavin Calculus and Stochastic Analysis, Springer, New York, 2013, pp. 115–138. Search in Google Scholar

[2] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., 343, Springer, Berlin, 2011. Search in Google Scholar

[3] P. Baldi and B. Roynette, Some exact equivalents for the Brownian motion in Hölder norm, Probab. Theory Related Fields 93 (1992), no. 4, 457–484. Search in Google Scholar

[4] G. Ben Arous, M. Gr€dinaru, and M. Ledoux, Hölder norms and the support theorem for diffusions, Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 3, 415–436. Search in Google Scholar

[5] G. Ben Arous and M. Ledoux, Grandes déviations de Freidlin–Wentzell en norme hölderienne, in: J. Azéma et al (eds.), Séminaire de Probabilités XXVIII, Lecture Notes in Math., 1583, Springer-Verlag, Berlin, 1994, pp. 293–299. Search in Google Scholar

[6] M. Eddahbi, M. N’zi, and Y. Ouknine, Grandes déviations des diffusions sur les es-paces de Besov–Orlicz et application, Stochastics Stochastics Rep. 65 (1999), no. 3–4, 299–315. Search in Google Scholar

[7] R. Catellier and M. Gubinelli, Averaging along irregular curves and regularisation of ODEs, Stochastic Process. Appl. 126 (2016), no. 8, 2323–2366. Search in Google Scholar

[8] Z. Ciesielski, On the isomorphisms of the space H_α and m, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 217–222. Search in Google Scholar

[9] Z. Ciesielski, G. Kerkyacharian, and B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math. 107 (1993), no. 2, 171–204. Search in Google Scholar

[10] P.K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Berlin, 2014. Search in Google Scholar

[11] P.K. Friz and N.B. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge Stud. Adv. Math., 120, Cambridge University Press, Cambridge, 2010. Search in Google Scholar

[12] N. Gantert, Einige große Abweichungen der Brownschen Bewegung, PhD thesis, Universität Bonn, 1991. Search in Google Scholar

[13] M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86–140. Search in Google Scholar

[14] M. Gubinelli, Ramification of rough paths, J. Differential Equations 248 (2010), no. 4, 693–721. Search in Google Scholar

[15] M. Gubinelli, P. Imkeller, and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi 3 (2015), e6, 75 pp. Search in Google Scholar

[16] M. Gubinelli, P. Imkeller, and N. Perkowski, A Fourier analytic approach to pathwise stochastic integration, Electron. J. Probab. 21 (2016), Paper No. 2, 37 pp. Search in Google Scholar

[17] M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504. Search in Google Scholar

[18] P. Imkeller and D.J. Prömel, Existence of Lévy’s area and pathwise integration, Commun. Stoch. Anal. 9 (2015), no. 1, 93–111. Search in Google Scholar

[19] A. Kamont, Isomorphism of some anisotropic Besov and sequence spaces, Studia Math. 110 (1994), no. 2, 169–189. Search in Google Scholar

[20] A. Lejay, Controlled differential equations as Young integrals: a simple approach, J. Differential Equations 249 (2010), no.8, 1777–1798. Search in Google Scholar

[21] T.J. Lyons, Uncertain volatility and the risk-free synthesis of derivatives, Appl. Math. Finance 2 (1995), no. 2, 117–133. Search in Google Scholar

[22] T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. Search in Google Scholar

[23] T.J. Lyons, M.J. Caruana, and T. Lévy, Differential Equations Driven by Rough Paths, Lecture Notes in Math., 1908, Springer, Berlin, 2007. Search in Google Scholar

[24] T.J. Lyons and Z.M. Qian, Calculus for multiplicative functionals, Itô’s formula and differential equations, in: N. Ikeda et al (eds.), Itô’s Stochastic Calculus and Probability Theory, Springer-Verlag, Tokyo, 1996, pp. 233–250. Search in Google Scholar

[25] T. Lyons and Z. Qian, Flow equations on spaces of rough paths, J. Funct. Anal. 149, (1997), no. 1, 135–159. Search in Google Scholar

[26] T. Lyons and Z. Qian, System Control and Rough Paths, Oxford Math. Monogr., Oxford Sci. Publ., Oxford University Press, Oxford, 2002. Search in Google Scholar

[27] T. Lyons and O. Zeitouni, Conditional exponential moments for iterated Wiener integrals, Ann. Probab. 27 (1999), no. 4, 1738–1749. Search in Google Scholar

[28] P. Mörters and Y. Peres, Brownian Motion, Camb. Ser. Stat. Probab. Math., 30, Cambridge University Press, Cambridge, 2010. Search in Google Scholar

[29] D. Nualart and S. Tindel, A construction of the rough path above fractional Brownian motion using Volterra’s representation, Ann. Probab. 39 (2011), no 3, 1061–1096. Search in Google Scholar

[30] D. Nualart and M. Zakai, On the relation between the Stratonovich and Ogawa integrals, Ann. Probab. 17 (1989), no. 4, 1536–1540. Search in Google Scholar

[31] S. Ogawa, Quelques propriétés de l’intégrale stochastique du type noncausal, Japan J. Appl. Math. 1 (1984), no. 2, 405–416. Search in Google Scholar

[32] S. Ogawa, The stochastic integral of noncausal type as an extension of the symmetric integrals, Japan J. Appl. Math. 2 (1985), no. 1, 229–240. Search in Google Scholar

[33] S. Ogawa, Noncausal stochastic calculus revisited – around the so-called Ogawa integral, in: N.M. Chuong et al (eds.), Advances in Deterministic and Stochastic Analysis, World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 297–320. Search in Google Scholar

[34] M. Rosenbaum, First order p-variations and Besov spaces, Statist. Probab. Lett. 79 (2009), no. 1, 55–62. Search in Google Scholar

[35] B. Roynette, Mouvement Brownien et espaces de Besov, Stochastics Stochastics Rep. 43 (1993), no. 3-4, 221–260. Search in Google Scholar

[36] J. Unterberger, A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering, Stochastic Process. Appl. 120 (2010), no. 8, 1444–1472. Search in Google Scholar

[37] J. Unterberger, Hölder continuous rough paths by Fourier normal ordering, Comm. Math. Phys. 298 (2010), no. 1, 1–36. Search in Google Scholar

[38] L.C. Young, General inequalities for Stieltjes integrals and the convergence of Fourier series, Math. Ann. 115 (1938), no. 1, 581–612. Search in Google Scholar

[39] A.K. Zvonkin, A transformation of the phase space of a diffusion process that will remove the drift, (in Russian), Mat. Sb. (N.S.) 93(135) (1974), 129–149, 152. Search in Google Scholar