Abstract
Random walks are used to model various random processes in different fields. In this chapter we are only interested in random walks as approximating processes of some basic driving processes of stochastic differential equations discussed in the previous chapter. There is a vast literature (see, e.g., [GK54, Don52, Bil99, Taq75, GM98-1, GM01, MS01]) devoted to approximation of various basic stochastic processes like Brownian motion, fractional Brownian motion, Lévy processes, and their time-changed counterparts. In the context of approximation, the question in what sense a random walk approximates (or converges to) an associated stochastic process becomes important. We will be interested only in the convergence in the sense of finite-dimensional distributions, which is equivalent to the locally uniform convergence of corresponding characteristic functions (see, e.g., [Bil99]).
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Notes
- 1.
For the dimension in this chapter we use the letter d, since n is overloaded.
- 2.
The Gaussian density function with mean 0 and correlation matrix I evolving in time.
- 3.
The set (0, 2) can be replaced by (0, 2], but this requires an additional care (see [US06]).
References
Abdel-Rehim, E.A.: Explicit approximation solutions and proof of convergence of the space-time fractional advection dispersion equations. Appl. Math., 4, 1427–1440 (2013)
Becker-Kern, P., Meerschaert, M.M., Scheffler, H.-P.: Limit theorems for coupled continuous time random walks. Ann. Probab. 32 (1B), 730–756 (2004)
Billingsley, P.: Convergence of Probability Measures. 2nd ed. Wiley-Interscience publication (1999)
Donsker, M.D.: Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Stat., 23, 277–281 (1952)
Gillis, J.E., Weiss, G.H.: Expected number of distinct sites visited by a random walk with an infinite variance. J. Math. Phys. 11, 1307–1312 (1970)
Gnedenko, B.V., Kolmogorov, A.N.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley (1954)
Gorenflo, R., Abdel-Rehim, E.: Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion. J. Comp. and Appl. Math. 205 871–881 (2007)
Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal., 1, 167–191 (1998)
Gorenflo, R., Mainardi, F.: Approximation of Lévy-Feller diffusion by random walk. ZAA, 18 (2) 231–246 (1999)
Gorenflo, R., Mainardi, F.: Random walk models approximating symmetric space-fractional diffusion processes. In Elschner, Gohberg and Silbermann (eds): Problems in Mathematical Physics (Siegfried Prössdorf Memorial Volume). Birkhäuser Verlag, Boston-Basel-Berlin, 120–145 (2001)
Gorenflo, R., Mainardi, F.: Simply and multiply scaled diffusion limits for continuous time random walks. J. of Physics. Conference Series, 7, 1–16 (2005)
Gorenflo, R., Vivoli, A,: Fully discrete random walks for space-time fractional diffusion equations. Signal Processing, 83, 2411–2420 (2003)
Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G., Paradisi, P.: Discrete random walk models for space-time fractional diffusion, Chemical Physics, 284, 521–541 (2002)
Lawler, G., Limic, V.: Random walk: A Modern Introduction. Cambridge University Press (2010)
Liu, F., Shen, S., Anh, V., Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. ANZIAM J., 46, 488–504 (2005)
Lovász, L.: Random walks on graphs: a survey. Bolyai Society Math. Studies, Combinatorics 2, 1–46 (1993)
Meerschaert, M.M., Scheffler, H.-P.: Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice. John Wiley and Sons, Inc. (2001)
Meerschaert, M.M., Scheffler, H.-P.: Limit theorems for continuous time random walks with slowly varying waiting times. Stat. Probabil. Lett. 71 (1), 15–22 (2005)
Metzler, R,. Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339, 1–77 (2000)
Metzler, R., Klafter, J.: The restaurant in the end of random walk. Physics A: Mathematical and General. 37 (31), 161–208 (2004)
Montroll, E.W., Weiss, G.H.: Random walk on Lattices. II. J. Math. Phys. 6 (2), 167–181 (1965)
Pearson, K.: On the problem of random walk. Nature, 72, 294 (1905)
Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000)
Sottinen, T.: Fractional Brownian motion, random walks and binary market models. Finance and Stochastics. 5, 343–355 (2001)
Spitzer, F.: Principles of Random Walk. Springer (2001)
Taqqu, M.S.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31, 287–302 (1975)
Umarov, S.R.: Continuous time random walk models associated with distributed order diffusion equations. Frac. Calc. Appl. Anal. 18 (3), 821–837 (2015)
Umarov, S.R., Gorenflo, R.: On multi-dimensional symmetric random walk models approximating fractional diffusion processes. Frac. Calc. Appl. Anal., 8, 73–88 (2005)
Umarov, S.G., Steinberg, St. Random walk models associated with distributed fractional order differential equations. IMS Lecture Notes - Monograph Series. High Dimensional Probability, 51, 117–127 (2006)
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Umarov, S. (2015). Random walk approximants of mixed and time-changed Lévy processes. In: Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols. Developments in Mathematics, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-20771-1_8
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