Karhunen-Loève expansions of α-Wiener bridges

Mátyás Barczy; Endre Iglói

Open Mathematics (2011)

  • Volume: 9, Issue: 1, page 65-84
  • ISSN: 2391-5455

Abstract

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We study Karhunen-Loève expansions of the process(X t(α))t∈[0,T) given by the stochastic differential equation d X t ( α ) = - α T - t X t ( α ) d t + d B t , t ∈ [ 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).

How to cite

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Mátyás Barczy, and Endre Iglói. "Karhunen-Loève expansions of α-Wiener bridges." Open Mathematics 9.1 (2011): 65-84. <http://eudml.org/doc/269585>.

@article{MátyásBarczy2011,
abstract = {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).},
author = {Mátyás Barczy, Endre Iglói},
journal = {Open Mathematics},
keywords = {α-Wiener bridge; Scaled Brownian bridge; Karhunen-Loève expansion; Laplace transform; Large deviation; Small deviation; -Wiener bridge; scaled Brownian bridge; large deviation; small deviation},
language = {eng},
number = {1},
pages = {65-84},
title = {Karhunen-Loève expansions of α-Wiener bridges},
url = {http://eudml.org/doc/269585},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Mátyás Barczy
AU - Endre Iglói
TI - Karhunen-Loève expansions of α-Wiener bridges
JO - Open Mathematics
PY - 2011
VL - 9
IS - 1
SP - 65
EP - 84
AB - 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).
LA - eng
KW - α-Wiener bridge; Scaled Brownian bridge; Karhunen-Loève expansion; Laplace transform; Large deviation; Small deviation; -Wiener bridge; scaled Brownian bridge; large deviation; small deviation
UR - http://eudml.org/doc/269585
ER -

References

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  1. [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. [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 Zbl0747.60039
  3. [3] Ash R.B., Gardner M.F., Topics in Stochastic Processes, Probability and Mathematical Statistics, 27, Academic Press, New York-London, 1975 
  4. [4] Barczy M., Iglói E., Karhunen-Loève expansions of α-Wiener bridges, preprint available at http://arxivőrg/abs/1007.2904 Zbl1228.60047
  5. [5] Barczy M., Pap G., α-Wiener bridges: singularity of induced measures and sample path properties, Stoch. Anal. Appl., 2010, 28(3), 447–466 http://dx.doi.org/10.1080/07362991003704985 Zbl1195.60060
  6. [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 Zbl1215.60015
  7. [7] Borodin A.N., Salminen P., Handbook of Brownian Motion - Facts and Formulae, 2nd ed., Probab. Appl., Birkhäuser, Basel-Boston-Berlin, 2002 Zbl1012.60003
  8. [8] Bowman F., Introduction to Bessel Functions, Dover, New York, 1958 Zbl0083.05602
  9. [9] Brennan M.J., Schwartz E.S., Arbitrage in stock index futures, Journal of Business, 1990, 63(1), S7–S31 http://dx.doi.org/10.1086/296491 
  10. [10] Corlay S., Pagès G., Functional quantization based stratified sampling methods, preprint available at http://arxivőrg/abs/1008.4441v1 Zbl1310.65005
  11. [11] Csörgő M., Révész P., Strong Approximations in Probability and Statistics, Probab. Math. Statist., Academic Press, New York, 1981 Zbl0539.60029
  12. [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 http://dx.doi.org/10.1214/074921706000000761 Zbl1130.60045
  13. [13] Deheuvels P., A Karhunen-Loève expansion of a mean-centered Brownian bridge, Statist. Probab. Lett., 2007, 77(12), 1190–1200 http://dx.doi.org/10.1016/j.spl.2007.03.011 Zbl1274.62318
  14. [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 Zbl1048.60021
  15. [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 http://dx.doi.org/10.1016/j.spa.2005.10.004 Zbl1090.60020
  16. [16] Gutiérrez Jaimez R., Valderrama Bonnet M.J., On the Karhunen-Loève expansion for transformed processes, Trabajos Estadíst., 1987, 2(2), 81–90 http://dx.doi.org/10.1007/BF02863594 Zbl0734.60042
  17. [17] Hwang C.-R., Gaussian measure of large balls in a Hilbert space, Proc. Amer. Math. Soc., 1980, 78(1), 107–110 Zbl0393.60009
  18. [18] Jacod J., Shiryaev A.N., Limit Theorems for Stochastic Processes, 2nd ed., Grundlehren Math. Wiss., 288, Springer, Berlin, 2003 Zbl1018.60002
  19. [19] Korenev B.G., Bessel Functions and their Applications, Anal. Methods Spec. Funct., 8, Taylor & Francis, London, 2002 Zbl1065.33001
  20. [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. [21] Li W.V., Comparison results for the lower tail of Gaussian seminorms, J. Theoret. Probab., 1992, 5(1), 1–31 http://dx.doi.org/10.1007/BF01046776 Zbl0743.60009
  22. [22] Liu N., Ulukus S., Optimal distortion-power tradeoffs in sensor networks: Gauss-Markov random processes, preprint available at http://arxiv.org/PS_cache/cs/pdf/0604/0604040v1.pdf 
  23. [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 Zbl1008.62072
  24. [24] Luschgy H., Pagès G., Expansions for Gaussian processes and Parseval frames, Electron. J. Probab., 2009, 14(42), 1198–1221 Zbl1195.60056
  25. [25] Mansuy R., On a one-parameter generalization of the Brownian bridge and associated quadratic functionals, J. Theoret. Probab., 2004, 17(4), 1021–1029 http://dx.doi.org/10.1007/s10959-004-0588-8 Zbl1063.60049
  26. [26] Martynov G.V., Computation of distribution functions of quadratic forms of normally distributed random variables, Theory Probab. Appl., 1976, 20(4), 782–793 http://dx.doi.org/10.1137/1120085 Zbl0364.60043
  27. [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 http://dx.doi.org/10.1137/1122074 Zbl0392.62011
  28. [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 http://dx.doi.org/10.1023/A:1024868604219 
  29. [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 http://dx.doi.org/10.1007/s00440-004-0337-z Zbl1051.60041
  30. [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 http://dx.doi.org/10.1007/s10958-009-9683-9 Zbl1288.60045
  31. [31] Øksendal B., Stochastic Differential Equations, 6th ed., Universitext, Springer, Berlin-Heidelberg-New York, 2003 http://dx.doi.org/10.1007/978-3-642-14394-6 
  32. [32] Papoulis A., Probability, Random Variables and Stochastic Processes, 3rd ed., McGraw-Hill, New York, 1991 
  33. [33] Smirnov N.V., On the distribution of Mises’ ω 2-test, Mat. Sb., 1937, 2(44)(5), 973–993 (in Russian) 
  34. [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. [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 http://dx.doi.org/10.1007/s00181-006-0081-6 
  36. [36] Watson G.N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1944 Zbl0063.08184
  37. [37] Yor M., Some Aspects of Brownian Motion, Part II: Some Recent Martingale Problems, Lectures Math. ETH Zurich, Birkhäuser, Basel, 1997 
  38. [38] Zolotarev V.M., Concerning a certain probability problem, Theory Probab. Appl., 1961, 6(2), 201–204 http://dx.doi.org/10.1137/1106025 

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