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2011 | 9 | 1 | 65-84
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Karhunen-Loève expansions of α-Wiener bridges

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We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation $$ dX_t^{(\alpha )} = - \frac{\alpha } {{T - t}}X_t^{(\alpha )} dt + dB_t ,t \in [0,T) $$, with the initial condition X 0(α) = 0, where α > 0, T ∈ (0, ∞), and (B t)t≥0 is a standard Wiener process. This process is called an α-Wiener bridge or a scaled Brownian bridge, and in the special case of α = 1 the usual Wiener bridge. We present weighted and unweighted Karhunen-Loève expansions of X (α). As applications, we calculate the Laplace transform and the distribution function of the L 2[0, T]-norm square of X (α) studying also its asymptotic behavior (large and small deviation).
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  • University of Debrecen
  • University of Debrecen
  • [1] Abramowitz M., Stegun I.A. (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington D.C., 1964, 10th printing with corrections, 1972
  • [2] Adler R.J., An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, IMS Lecture Notes Monogr. Ser., 12, Institute of Mathematical Statistics, Hayward, 1990
  • [3] Ash R.B., Gardner M.F., Topics in Stochastic Processes, Probability and Mathematical Statistics, 27, Academic Press, New York-London, 1975
  • [4] Barczy M., Iglói E., Karhunen-Loève expansions of α-Wiener bridges, preprint available at http://arxivőrg/abs/1007.2904
  • [5] Barczy M., Pap G., α-Wiener bridges: singularity of induced measures and sample path properties, Stoch. Anal. Appl., 2010, 28(3), 447–466
  • [6] Barczy M., Pap G., Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions, preprint available at http://arxivőrg/abs/0810.2930
  • [7] Borodin A.N., Salminen P., Handbook of Brownian Motion - Facts and Formulae, 2nd ed., Probab. Appl., Birkhäuser, Basel-Boston-Berlin, 2002
  • [8] Bowman F., Introduction to Bessel Functions, Dover, New York, 1958
  • [9] Brennan M.J., Schwartz E.S., Arbitrage in stock index futures, Journal of Business, 1990, 63(1), S7–S31
  • [10] Corlay S., Pagès G., Functional quantization based stratified sampling methods, preprint available at http://arxivőrg/abs/1008.4441v1
  • [11] Csörgő M., Révész P., Strong Approximations in Probability and Statistics, Probab. Math. Statist., Academic Press, New York, 1981
  • [12] Deheuvels P., Karhunen-Loève expansions of mean-centered Wiener processes, In: High Dimensional Probability, IMS Lecture Notes Monogr. Ser., 51, Institute of Mathematical Statistics, Beachwood, 2006, 62–76
  • [13] Deheuvels P., A Karhunen-Loève expansion of a mean-centered Brownian bridge, Statist. Probab. Lett., 2007, 77(12), 1190–1200
  • [14] Deheuvels P., Martynov G., Karhunen-Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, In: High Dimensional Probability, III, Sandjberg, 2002, Progr. Probab., 55, Birkhäuser, Basel, 57–93
  • [15] Deheuvels P., Peccati G., Yor M., On quadratic functionals of the Brownian sheet and related processes, Stochastic Process. Appl., 2006, 116(3), 493–538
  • [16] Gutiérrez Jaimez R., Valderrama Bonnet M.J., On the Karhunen-Loève expansion for transformed processes, Trabajos Estadíst., 1987, 2(2), 81–90
  • [17] Hwang C.-R., Gaussian measure of large balls in a Hilbert space, Proc. Amer. Math. Soc., 1980, 78(1), 107–110
  • [18] Jacod J., Shiryaev A.N., Limit Theorems for Stochastic Processes, 2nd ed., Grundlehren Math. Wiss., 288, Springer, Berlin, 2003
  • [19] Korenev B.G., Bessel Functions and their Applications, Anal. Methods Spec. Funct., 8, Taylor & Francis, London, 2002
  • [20] Lévy P., Wiener's random function, and other Laplacian random functions, In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley-Los Angeles, 1951, 171–187
  • [21] Li W.V., Comparison results for the lower tail of Gaussian seminorms, J. Theoret. Probab., 1992, 5(1), 1–31
  • [22] Liu N., Ulukus S., Optimal distortion-power tradeoffs in sensor networks: Gauss-Markov random processes, preprint available at
  • [23] Liptser R.S., Shiryaev A.N., Statistics of Random Processes I. General Theory, 2nd ed., Appl. Math. (N.Y.), 5, Springer, Berlin-Heidelberg, 2001
  • [24] Luschgy H., Pagès G., Expansions for Gaussian processes and Parseval frames, Electron. J. Probab., 2009, 14(42), 1198–1221
  • [25] Mansuy R., On a one-parameter generalization of the Brownian bridge and associated quadratic functionals, J. Theoret. Probab., 2004, 17(4), 1021–1029
  • [26] Martynov G.V., Computation of distribution functions of quadratic forms of normally distributed random variables, Theory Probab. Appl., 1976, 20(4), 782–793
  • [27] Martynov G.V., A generalization of Smirnov's formula for the distribution functions of quadratic forms, Theory Probab. Appl., 1978, 22(3), 602–607
  • [28] Nazarov A.I., On the sharp constant in the small ball asymptotics of some Gaussian processes under L 2-norm, J. Math. Sci. (N.Y.), 2003, 117(3), 4185–4210
  • [29] Nazarov A.I., Nikitin Ya.Yu., Exact L 2-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems, Probab. Theory Related Fields, 2004, 129(4), 469–494
  • [30] Nazarov A.I., Pusev R.S., Exact small ball asymptotics in weighted L 2-norm for some Gaussian processes, J. Math. Sci. (N.Y.), 2009, 163(4), 409–429
  • [31] Øksendal B., Stochastic Differential Equations, 6th ed., Universitext, Springer, Berlin-Heidelberg-New York, 2003
  • [32] Papoulis A., Probability, Random Variables and Stochastic Processes, 3rd ed., McGraw-Hill, New York, 1991
  • [33] Smirnov N.V., On the distribution of Mises’ ω 2-test, Mat. Sb., 1937, 2(44)(5), 973–993 (in Russian)
  • [34] Sondermann D., Trede M., Wilfling B., Estimating the degree of interventionist policies in the run-up to EMU, Applied Economics (in press), DOI: 10.1080/00036840802481884
  • [35] Trede M., Wilfling B., Estimating exchange rate dynamics with diffusion processes: an application to Greek EMU data, Empirical Economics, 2007, 33(1), 23–39
  • [36] Watson G.N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1944
  • [37] Yor M., Some Aspects of Brownian Motion, Part II: Some Recent Martingale Problems, Lectures Math. ETH Zurich, Birkhäuser, Basel, 1997
  • [38] Zolotarev V.M., Concerning a certain probability problem, Theory Probab. Appl., 1961, 6(2), 201–204
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