Small ball properties and representation results. (English) Zbl 1353.60050
Summary: We show that small ball estimates together with Hölder continuity assumption allow to obtain new representation results in models with long memory. In order to apply these results, we establish small ball probability estimates for Gaussian processes whose incremental variance admits two-sided estimates and the incremental covariance preserves sign. As a result, we obtain small ball estimates for integral transforms of Wiener processes and of fractional Brownian motion with Volterra kernels.
MSC:
| 60H05 | Stochastic integrals |
| 60G15 | Gaussian processes |
| 60G22 | Fractional processes, including fractional Brownian motion |
Keywords:
integral representations; generalized Lebesgue-Stieltjes integral; small ball estimate; Gaussian processes; quasi-helix; fractional Brownian motionReferences:
| [1] | Aurzada, F.; Simon, T., Small ball probabilities for stable convolutions, ESAIM Probab. Stat., 11, 327-343 (2007) · Zbl 1184.60010 |
| [3] | Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A., Sub-fractional Brownian motion and its relation to occupation times, Statist. Probab. Lett., 69, 4, 405-419 (2004) · Zbl 1076.60027 |
| [4] | Cheridito, P.; Kawaguchi, H.; Maejima, M., Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab., 8, 3, 14 (2003), electronic · Zbl 1065.60033 |
| [5] | Dudley, R. M., Wiener functionals as Ito integrals, Ann. Probab., 5, 140-141 (1977) · Zbl 0359.60071 |
| [6] | Houdré, C.; Villa, J., An example of infinite dimensional quasi-helix, (Stochastic Models (Mexico City, 2002). Stochastic Models (Mexico City, 2002), Contemp. Math., vol. 336 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 195-201 · Zbl 1046.60033 |
| [7] | Kahane, J.-P., Hélices et quasi-hélices, (Mathematical Analysis and Applications, Part B. Mathematical Analysis and Applications, Part B, Adv. in Math. Suppl. Stud., vol. 7 (1981), Academic Press: Academic Press New York-London), 417-433 · Zbl 0472.43005 |
| [8] | Kahane, J.-P., (Some Random Series of Functions. Some Random Series of Functions, Cambridge Studies in Advanced Mathematics, vol. 5 (1985), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0192.53801 |
| [9] | Li, W. V.; Shao, Q.-M., Gaussian processes: inequalities, small ball probabilities and applications, (Stochastic Processes: Theory and Methods. Stochastic Processes: Theory and Methods, Handbook of Statistics, vol. 19 (2001), North-Holland: North-Holland Amsterdam), 533-597 · Zbl 0987.60053 |
| [10] | Mémin, J.; Mishura, Y.; Valkeila, E., Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett., 51, 2, 197-206 (2001) · Zbl 0983.60052 |
| [11] | Mishura, Y. S., (Stochastic Calculus for Fractional Brownian Motion and Related Processes. Stochastic Calculus for Fractional Brownian Motion and Related Processes, Lecture Notes in Mathematics, vol. 1929 (2008), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1138.60006 |
| [12] | Mishura, Y.; Shevchenko, G.; Valkeila, E., Random variables as pathwise integrals with respect to fractional Brownian motion, Stochastic Process. Appl., 123, 6, 2353-2369 (2013) · Zbl 1328.60131 |
| [13] | Norros, I.; Valkeila, E.; Virtamo, J., An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli, 5, 4, 571-587 (1999) · Zbl 0955.60034 |
| [14] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives. Theory and Applications (1993), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Yverdon · Zbl 0818.26003 |
| [15] | Shevchenko, G.; Viitasaari, L., Integral representation with adapted continuous integrand with respect to fractional Brownian motion, Stoch. Anal. Appl., 32, 6, 934-943 (2014) · Zbl 1305.60041 |
| [16] | Shevchenko, G.; Viitasaari, L., Adapted integral representations of random variables, Int. J. Mod. Phys.: Conf. Ser., 36 (2015) · Zbl 1337.60110 |
| [17] | Viitasaari, L., Integral representation of random variables with respect to Gaussian processes, Bernoulli, 22, 1, 376-395 (2016) · Zbl 1359.60070 |
| [18] | Zähle, M., On the link between fractional and stochastic calculus, (Stochastic Dynamics (Bremen, 1997) (1999), Springer: Springer New York), 305-325 · Zbl 0947.60060 |
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