Geodesic
Asymptotic behavior of the Unit Root Bilinear
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 5-33 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the Unit Root Bilinear model with a sequence of innovations given by the fractional Gaussian noise (increases of the fractional Brownian motion). For such a model we prove a variant of the Donsker–Prohorov limit theorem and obtain convergence of the model in probability to solution of a proper stochastic differential equation with fBm. The proof is based on the result about convergence of the Euler's scheme with ‘small perturbations’ for SDE with fBm, which is also proved. 
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T. Androshchuk. Asymptotic behavior of the Unit Root Bilinear. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 5-33. http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a0/

[1] W. W. Charemza, M. A. Lifshits, and S. B. Makarova, “Conditional testing for unit-root bilienarity in financial time series: some theoretical and empirical results”, Journal of Economic Dynamics and Control, 29 (2005), 63–96 | DOI | MR | Zbl

[2] M. A. Lifshits, “Printsip invariantnosti v bilineinoi modeli so slaboi nelineinostyu”, Zap. nauchn. semin. POMI, 320, 2004, 97–105 | Zbl

[3] J. Beran, Statistics for Long-Memory Processes, Chapman and Hall, New York, 1994 | MR | Zbl

[4] A. W. Lo and A. C. MacKinlay, “Stock Market Prices do not follow Random Walks: Evidence from a Simple Specification Test”, Rev. of Financial Studies 1, 1988, 41–66

[5] E. E. Peters, Fractal market analysis, Wiley, New York, 1994

[6] W. Willinger, M. S. Taqqu, and V. Teverovsky, “Stock Market Prices and Long-Range Dependence”, Finance and Stochastics, 3 (1999), 1–13 | DOI | Zbl

[7] A. N. Shiryaev, Essentials of Stochastic Finance, Word Scientific, Singapore, 1999 | MR | Zbl

[8] I. Adelman, “Long cycles – fact or artifact?”, Amer. Economic Rev., 60 (1965), 444–463

[9] C. W. J. Granger, “The typical spectral shape of an economic variable”, Econometrica, 34 (1966), 150–161 | DOI

[10] A. N. Shiryaev, “On arbitrage and replication for fractal models”, Proc. Workshop on Mathematical Finance (Paris, INRIA), 1998, 1–7 | MR

[11] T. Sottinen and E. Valkeila, “On arbitrage and replication in the fractional Black-Scholes pricing model”, Statistics and Decisions, 21 (2003), 93–107 | DOI | MR | Zbl

[12] T. Androshchuk and Yu. Mishura, “Mixed Brownian–Fractional Brownian Model: absence of arbitrage and related topics”, Stochastics, 78:5 (2006), 281–300 | MR | Zbl

[13] B. Mandelbrot and J. Van Ness, “Fractional Brownian motion, fractional noises and applications”, SIAM Review, 10:4 (1968), 422–437 | DOI | MR | Zbl

[14] T. Sottinen and E. Valkeila, Fractional Brownian motion as a model in Finance, Preprint No 302, 2001, 16 pp.

[15] M. Zähle, “Integration with respect to fractal functions and stochastic calculus, I”, Probability Theory and Related Fields, 111 (1998), 33–372 | MR

[16] D. Nualart and A. Rǎscanu, “Differential equations driven by fractional Brownian motion”, Collect. Math., 53:1 (2002), 55–81 | MR | Zbl

[17] P. Billingsli, Skhodimost veroyatnostnykh mer, Nauka, M., 1977, 352 pp. | MR

[18] S. J. Lin, “Stochastic analysis of fractional Brownian motion”, Stochastics and Stochastic Reports Collect, 55 (1995), 121–140 | MR | Zbl

[19] A. Neuenkirch and I. Nourdin, Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion, J. Theoret. Probab., 20, no. 4, 2006 | DOI | MR | Zbl

[20] Yu. Mishura and G. Shevchenko, “The rate of convergence for Euler approximation of solutions of stochastic differential equations driven by fractional Brownian motion”, Stochastics: An Internationa Journal of Probability and Stochastics Processes, 2006, submitted