Non-degeneracy of Wiener functionals arising from rough differential equations

Cass, Thomas; Friz, Peter; Victoir, Nicolas

doi:10.1090/S0002-9947-09-04677-7

Non-degeneracy of Wiener functionals arising from rough differential equations
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by Thomas Cass, Peter Friz and Nicolas Victoir PDF

Trans. Amer. Math. Soc. 361 (2009), 3359-3371 Request permission

Abstract:

Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Itô map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of the solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss the existence of a density for solutions to stochastic differential equations driven by a general class of non-degenerate Gaussian processes, including processes with sample path regularity worse than Brownian motion.

References

Fabrice Baudoin and Martin Hairer, A version of Hörmander’s theorem for the fractional Brownian motion, Probab. Theory Related Fields 139 (2007), no. 3-4, 373–395. MR 2322701, DOI 10.1007/s00440-006-0035-0
Erhan Bayraktar, Ulrich Horst, and Ronnie Sircar, A limit theorem for financial markets with inert investors, Math. Oper. Res. 31 (2006), no. 4, 789–810. MR 2281230, DOI 10.1287/moor.1060.0202
Denis R. Bell, The Malliavin calculus, Dover Publications, Inc., Mineola, NY, 2006. Reprint of the 1987 edition. MR 2250060
Jean-Michel Bismut, Large deviations and the Malliavin calculus, Progress in Mathematics, vol. 45, Birkhäuser Boston, Inc., Boston, MA, 1984. MR 755001
P. Friz, T. Lyons, and D. Stroock, Lévy’s area under conditioning, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 1, 89–101 (English, with English and French summaries). MR 2196973, DOI 10.1016/j.anihpb.2005.02.003
Peter Friz and Nicolas Victoir, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 4, 703–724 (English, with English and French summaries). MR 2144230, DOI 10.1016/j.anihpb.2004.05.003
Peter Friz and Nicolas Victoir, A note on the notion of geometric rough paths, Probab. Theory Related Fields 136 (2006), no. 3, 395–416. MR 2257130, DOI 10.1007/s00440-005-0487-7
Friz, P.; Victoir, N.: Differential Equations Driven by Gaussian Signals I. arXiv-preprint.
Friz, P.; Victoir, N.: Multidimensional Stochastic Processes as Rough Paths. Theory and Applications, Cambridge University Press (in preparation)
Paolo Guasoni, No arbitrage under transaction costs, with fractional Brownian motion and beyond, Math. Finance 16 (2006), no. 3, 569–582. MR 2239592, DOI 10.1111/j.1467-9965.2006.00283.x
Guasoni, P.; Rasonyi, M.; Schachermayer, W.: The Fundamental Theorem of Asset Pricing for Continuous Processes under Small Transaction Costs. Preprint (2007)
Martin Hairer, Ergodicity of stochastic differential equations driven by fractional Brownian motion, Ann. Probab. 33 (2005), no. 2, 703–758. MR 2123208, DOI 10.1214/009117904000000892
Shigeo Kusuoka, The nonlinear transformation of Gaussian measure on Banach space and absolute continuity. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 567–597. MR 687592
Shigeo Kusuoka, Dirichlet forms and diffusion processes on Banach spaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 1, 79–95. MR 657873
K. Kusuoka, On the regularity of solutions to SDE, Asymptotic problems in probability theory: Wiener functionals and asymptotics (Sanda/Kyoto, 1990) Pitman Res. Notes Math. Ser., vol. 284, Longman Sci. Tech., Harlow, 1993, pp. 90–103. MR 1354163
Shigeo Kusuoka and Daniel Stroock, Applications of the Malliavin calculus. I, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 271–306. MR 780762, DOI 10.1016/S0924-6509(08)70397-0
S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391–442. MR 914028
Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (1998), no. 2, 215–310. MR 1654527, DOI 10.4171/RMI/240
T. J. Lyons and Z. M. Qian, Calculus of variation for multiplicative functionals, New trends in stochastic analysis (Charingworth, 1994) World Sci. Publ., River Edge, NJ, 1997, pp. 348–374. MR 1654380
Terry Lyons and Zhongmin Qian, Flow of diffeomorphisms induced by a geometric multiplicative functional, Probab. Theory Related Fields 112 (1998), no. 1, 91–119. MR 1646428, DOI 10.1007/s004400050184
Terry Lyons and Zhongmin Qian, System control and rough paths, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR 2036784, DOI 10.1093/acprof:oso/9780198506485.001.0001
Paul Malliavin, Stochastic analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 313, Springer-Verlag, Berlin, 1997. MR 1450093, DOI 10.1007/978-3-642-15074-6
H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
David Nualart, The Malliavin calculus and related topics, 2nd ed., Probability and its Applications (New York), Springer-Verlag, Berlin, 2006. MR 2200233
Nualart, D.; Saussereau, B.: Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Preprint (2005)
Hu, Y.; Nualart, D.: Differential equations driven by Hölder continuous functions of order greater than 1/2; ArXiv (math.PR/0601628)
Hiroshi Sugita, Sobolev spaces of Wiener functionals and Malliavin’s calculus, J. Math. Kyoto Univ. 25 (1985), no. 1, 31–48. MR 777244, DOI 10.1215/kjm/1250521157
Ichiro Shigekawa, Stochastic analysis, Translations of Mathematical Monographs, vol. 224, American Mathematical Society, Providence, RI, 2004. Translated from the 1998 Japanese original by the author; Iwanami Series in Modern Mathematics. MR 2060917, DOI 10.1090/mmono/224
Nasser Towghi, Multidimensional extension of L. C. Young’s inequality, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), no. 2, Article 22, 13. MR 1906391
A. Süleyman Üstünel and Moshe Zakai, Transformation of measure on Wiener space, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1736980, DOI 10.1007/978-3-662-13225-8
L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251–282. MR 1555421, DOI 10.1007/BF02401743