Fractional Brownian motion as a weak limit of Poisson shot noise processes-with applications to finance

Stochastic Processes and their Applications

Volumn 113, Issue 2, 2004, Pages 333-351

  Klüppelberg, Claudia ^a   Kühn, Christoph ^b  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^b GOETHE UNIVERSITY   (Germany)

Author keywords

Alternative stock price models; Arbitrage; Fractional Brownian motion; Functional limit theorems; Non explosiveness of point processes; Shot noise process

Indexed keywords

EID: 4544374045     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2004.03.015     Document Type: Article

References (27)

1. E. Bayraktar U. Horst R. Sircar A limit theorem for financial markets with inert investors, 2003, submitted for publication. Available at http://www.princeton.edu/~sircar/
2. Billingsley P. Convergence of Probability Measures 2nd ed. Edition 1999 Wiley New York
3. Bingham N.H. Goldie C.M. Teugels J.L. Regular Variation 1987 Cambridge University Press Cambridge
4. Brémaud P. Point Processes and Queues, Martingale Dynamics 1981 Springer New York
5. Brody D.C. Syroka J. Zervos M. Dynamical pricing of weather derivatives J. Quant. Financ. 1 2001 1-10
6. Cheridito P. Mixed fractional Brownian motion Bernoulli 7 2001 913-934
7. Cheridito P. Arbitrage in fractional Brownian motion models Financ. Stochastics 7 2003 533-553
8. Daley D. Vere-Jones D. An Introduction to the Theory of Point Processes 1988 Springer New York
9. Davydov Y.A. The invariance principle for stationary processes Theory Probab. Appl. 15 1970 487-498
10. W. Feller An Introduction to Probability Theory and its Applications, 2nd Edition, Vol. 2, Wiley, New York, 1971
11. Granger C.W.J. Long memory relationships and the aggregation of dynamic models J. Econometrics 14 1980 227-238
12. Granger C.W.J. Joyeux R. An introduction to long-memory time series models and fractional differencing J. Time Ser. Anal. 1 1980 15-29
13. Gut A. Stopped Random Walks 1988 Springer New York
14. Hu Y. Øksendal B. Fractional white noise calculus and applications to finance Infinite Dimensional Anal., Quant. Probab. Rel. Topics 6 2003 1-32
15. Klüppelberg C. Mikosch T. Explosive poisson shot noise processes with applications to risk reserves Bernoulli 1 1995 125-147
16. Klüppelberg C. Mikosch T. Schärf A. Regular variation in the mean and stable limits for Poisson shot noise Bernoulli 9 2003 467-496
17. T. Konstantopoulos A. Sakhanenko, Convergence and convergence rate to fractional Brownian motion for weighted random sums, 2003, preprint
18. C. Kühn Shocks and choices-an analysis of incomplete market models, Ph.D. Thesis, Munich University of Technology. http://tumb1.biblio.tu-muenchen.de/publ/diss/ma/2002/kuehn.html
19. Lane J.A. The central limit theorem for the Poisson shot-noise process J. Appl. Probab. 21 1984 287-301
20. Mandelbrot B. Fractals and Scaling in Finance, Discontinuity, Concentration Risk 1997 Springer New York
21. T. Mikosch C. Starica Change of structure in financial time series, long range dependence and the GARCH model. Rev. Econom. Statist., to appear
22. Protter P. Stochastic Integration and Differential Equations: A New Approach 1990 Springer Berlin
23. Rogers L.C.G. Arbitrage with fractional Brownian motion Math. Financ. 7 1997 95-105
24. Samorodnitsky G. Taqqu M.S. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance 1994 Chapman & Hall New York
25. Sottinen T. Fractional Brownian motion, random walks and binary market models Financ. Stochastics 5 2001 343-355
26. W. Stute Jump diffusion processes with shot noise effects and their applications to finance, in: V. Núñez-Antón, E. Ferreira (Eds.), Statistical Modelling: Proceedings of the 15th International Workshop on Statistical Modelling, University of the Basque Country, Bilbao, Universidad del Pais Vasco, 2000, pp. 86-94
27. Westcott M. On the existence of a generalized shot-noise process in: Williams E.J. (Ed.) Studies in Probability and Statistics: Papers in Honour of Edwin J.G. Pitman 1976 73-88 North-Holland Amsterdam