Stochastic integration for uncoupled continuous-time random walks

Scalas, Enrico and Germano, Guido and Politi, Mauro and Schilling, René L. (2008): Stochastic integration for uncoupled continuous-time random walks.

Abstract

Continuous-time random walks are pure-jump processes with several applications in physics, but also in insurance, finance and economics. Based on heuristic considerations, a definition is given for the stochastic integral driven by continuous-time random walks. The martingale properties of the integral are investigated. Finally, it is shown how the definition can be used to easily compute the stochastic integral by means of Monte Carlo simulations.

• Item Type: MPRA Paper
• Language: English
• Keywords: Continuous-time random walks; models of tick-by-tick financial data; stochastic integration
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General

References

[1] E. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 (1965). [2] M. F. Shlesinger, Random processes, in Encyclopedia of Applied Physics, Vol. 16, edited by G. L. Trigg, VCH Publishers, New York, 1996, pp. 45–70. [3] J. Klafter and R. Metzler, Phys. Rep. 339, 1 (2000). [4] R. Metzler and J. Klafter, J. Phys. A: Math. Gen. 37, R161 (2004). [5] E. Scalas, Physica A 362, 225 (2006). [6] M. M. Meerschaert and H. P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice (Wiley, Hoboken, New Jersey, 2001). [7] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2 (Wiley, New York, 1971). [8] D. R. Cox, Renewal Theory (Methuen, London, 1967). [9] D. R. Cox and V. Isham, Point Processes (Chapman & Hall, London, 1979). [10] P. G. Hoel, S. C. Port, and J. Stone, Introduction to Stochastic Processes (Houghton Mifflin, Boston, 1972). [11] E. C¸inlar, Introduction to Stochastic Processes (Prentice-Hall, Englewood Cliffs, 1975). [12] J. Janssen and R. Manca, Semi-Markov Risk Models for Finance, Insurance and Reliability (Springer, New York, 2007). [13] P. Billingsley, Probability and Measure (Wiley, Chichester, New York, 1979). [14] K.-I. Sato, L´evy Processes and Infinitely Divisible Distributions (Cambridge University Press, Cambridge, UK, 1999). [15] J. Bertoin, L´evy Processes (Cambridge University Press, Cambridge, UK, 1996). [16] E.W.Montroll and H. Scher, J. Stat. Phys. 9, 101 (1973). [17] M. F. Shlesinger, J. Stat. Phys. 10, 421 (1974). [18] J. K. E. Tunaley, J. Stat. Phys. 11, 397 (1974). [19] J. K. E. Tunaley, J. Stat. Phys. 12, 1 (1975). [20] J. K. E. Tunaley, J. Stat. Phys. 14, 461 (1976). [21] M. F. Shlesinger, J. Klafter, and Y. M. Wong, J. Stat. Phys. 27, 499 (1982). [22] D. ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, UK, 2000). [23] V. Balakrishnan, Physica A 132, 569 (1985). [24] R. Hilfer and L. Anton, Phys. Rev. E 51, R848 (1995). [25] E. Scalas, R. Gorenflo, and F. Mainardi, Phys. Rev. E 69, 011107 (2004). [26] D. Fulger, E. Scalas, and G. Germano, Phys. Rev. E, in press (2008). [27] D. del-Castillo-Negrete, B. A. Carreras, and V. E. Lynch, Phys. Rev. Lett. 94, 065003 (2005). [28] J. L. A. Dubbeldam, A. Milchev, V. G. Rostiashvili, and T. A. Vilgis, Phys. Rev. E 76, 010801(R) (2007). [29] J. L. A. Dubbeldam, A. Milchev, V. G. Rostiashvili, and T. A. Vilgis, Europhys. Lett. 79, 18002 (2007). [30] J. Masoliver, M. Montero, J. Perell´o, G. H. Weiss, J. of Economic Behavior & Org. 61, 577 (2006). [31] P. Embrechts, C. Kl¨uppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, New York, 1997). [32] ´A. Cartea and D. del-Castillo-Negrete, Phys. Rev. E 76, 041105 (2007). [33] R. Zygad lo, Phys. Rev. E, 68, 046117 (2003). [34] R. C. Merton, J. of Financial Economics 3, 125 (1976). [35] I. M. Gel’fand and G. E. Shilov, Generalized Functions (Academic Press, New York, 1964). [36] P. Protter, Stochastic Integration and Differential Equations (Springer, Berlin, 1990). [37] C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985). [38] D. Williams, Probability with Martingales (Cambridge University Press, Cambridge, UK, 1991). [39] R. L. Schilling, Measures, Integrals and Martingales (Cambridge University Press, Cambridge, UK, 2005). [40] M. M. Meerschaert, D. A. Benson, H.-P. Scheffler, P. Becker-Kern, Phys. Rev. E 66, 060102(R) (2002). [41] M. M. Meerschaert and E. Scalas, Physica A 370, 114 (2006).