Ergodicity and parameter estimates for Infinite-dimensional fractional Ornstein-Uhlenbeck process. (English) Zbl 1176.35185

Maslowski, Bohdan; Pospíšil, Jan

Appl. Math. Optim. 57, No. 3, 401-429 (2008).

The authors consider a linear stochastic evolution equations in Hilbert spaces. The driving process is a cylindrical fractional Brownian motion with Hurst parameter \(H \in (0,1)\). It is shown that if the semigroup is exponentially stable there exists a strictly stationary solution and that this stationary solution is ergodic. Using these results parameter-dependent equations are considered with a multiplicative parameter in the drift. Based on these ergodic theorems the strong consistency of two families of estimators is proved.
Finally, the authors apply these general results to two particular cases: linear stochastic heat and wave equations with fractional noise.

Reviewer: Carles Rovira (Barcelona)

MSC:

35R60	PDEs with randomness, stochastic partial differential equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60J65	Brownian motion

Keywords:

stochastic partial differential equations; fractional Brownian motion, strictly stationary solution; ergodicity; parameter estimates

References:

[1]	Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29(2), 766–801 (2001) · Zbl 1015.60047
[2]	Caithamer, P.: The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise. Stoch. Dyn. 5(1), 45–64 (2005) · Zbl 1083.60053
[3]	Decreusefond, L., Üstünel, A.S.: Stochastic analysis of the fractional Brownian motion. Potential Anal. 10(2), 177–214 (1999) · Zbl 0924.60034
[4]	Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch. Dyn. 2(2), 225–250 (2002) · Zbl 1040.60054
[5]	Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stoch. Process. Appl. 115(8), 1357–1383 (2005) · Zbl 1076.60054
[6]	Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Linear stochastic equations in a Hilbert space with a fractional Brownian motion. In: International Series in Operations Research & Management Science, vol. 94, pp. 201–222. Springer, Berlin (2006) · Zbl 1133.60015
[7]	Goldys, B., Maslowski, B.: Parameter estimation for controlled semilinear stochastic systems: identifiability and consistency. J. Multivar. Anal. 80(2), 322–343 (2002) · Zbl 1006.62072
[8]	Grecksch, W., Anh, V.V.: A parabolic stochastic differential equation with fractional Brownian motion input. Stat. Probab. Lett. 41(4), 337–346 (1999) · Zbl 0937.60064
[9]	Hu, Y.: Heat equations with fractional white noise potentials. Appl. Math. Optim. 43(3), 221–243 (2001) · Zbl 0993.60065
[10]	Hu, Y.: Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Am. Math. Soc. 175(825), viii+127 (2005) · Zbl 1072.60044
[11]	Hu, Y., Øksendal, B., Zhang, T.: General fractional multiparameter white noise theory and stochastic partial differential equations. Commun. Partial Differ. Equ. 29(1–2), 1–23 (2004) · Zbl 1067.35161
[12]	Huebner, M., Rozovskii, B.L.: On asymptotic properties of maximum likelihood estimators for parabolic stochastic SPDE’s. Probab. Theory Relat. Fields 103, 143–163 (1995) · Zbl 0831.60070
[13]	Khasminskii, R., Milstein, G.N.: On estimation of the linearized drift for nonlinear stochastic differential equations. Stoch. Dyn. 1(1), 23–43 (2001) · Zbl 1064.62092
[14]	Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202(1), 277–305 (2003) · Zbl 1027.60060
[15]	Maslowski, B., Schmalfuss, B.: Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Stoch. Anal. Appl. 22(6), 1577–1607 (2004) · Zbl 1062.60060
[16]	Nualart, D., Vuillermot, P.-A.: Variational solutions for partial differential equations driven by a fractional noise. J. Funct. Anal. 232(2), 390–454 (2006) · Zbl 1089.35097
[17]	Pazy, A.: Semigroups of Linear Operators. Springer, New York (1983) · Zbl 0516.47023
[18]	Pipiras, V., Taqqu, M.S.: Integration questions related to fractional Brownian motion. Probab. Theory Relat. Fields 118(2), 251–291 (2000) · Zbl 0970.60058
[19]	Pospíšil, J.: On parameter estimates in stochastic evolution equations driven by fractional Brownian motion. Ph.D. Thesis, University of West Bohemia in Plzeň (2005), pp. iv+88
[20]	Pospíšil, J.: Numerical approaches to parameter estimates in stochastic differential equations driven by fractional Brownian motion. In: Proceedings of the Programs and Algorithms of Numerical Mathematics 13, Prague (2006)
[21]	Pospíšil, J.: Numerical parameter estimates in stochastic equations with fractional Brownian motion. In: Proceedings of the Numerical Analysis and Approximation Theory, pp. 353–364. Cluj-Napoca, Romania, July (2006) · Zbl 1119.65005
[22]	Prakasa Rao, B.L.S.: Estimation for some stochastic partial differential equations based on discrete observations. II. Calcutta Stat. Assoc. Bull. 54(215–216), 129–141 (2003)
[23]	Prakasa Rao, B.L.S.: Parametric estimation for linear stochastic differential equations driven by fractional Brownian motion. Random Oper. Stoch. Equ. 11(3), 229–242 (2003) · Zbl 1053.62089
[24]	Prakasa Rao, B.L.S.: Identification for linear stochastic systems driven by fractional Brownian motion. Stoch. Anal. Appl. 22(6), 1487–1509 (2004) · Zbl 1153.62351
[25]	Rozanov, Y.A.: Stationary Random Processes. Holden-Day, San Francisco (1966) · Zbl 0152.16302
[26]	Sinestrari, E.: On the abstract Cauchy problem of parabolic type in spaces of continuous functions. J. Math. Anal. Appl. 107(1), 16–66 (1985) · Zbl 0589.47042
[27]	Tindel, S., Tudor, C.A., Viens, F.: Stochastic evolution equations with fractional Brownian motion. Probab. Theory Relat. Fields 127(2), 186–204 (2003) · Zbl 1036.60056