Skip to main content
SpringerLink
Log in

Fractional Processes and Their Statistical Inference: An Overview

  • Review Article
  • Published:
Journal of the Indian Institute of Science Aims and scope

Abstract

We give an overview of properties of fractional processes such as fractional Brownian motion, mixed fractional Brownian motion, sub-fractional Brownian motion, fractional Lévy process , fractional Poisson process and present a short review of problems of statistical inference for processes driven by fractional processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alajmi S, Milki E (2021) On the long range dependence of the time-changed mixed fractional Brownian motion model. arXiv:2102.10180

  2. Aletti G, Leonenko N, Merzbach E (2017) Fractional Poisson fields and martingales. arXiv:1601.08136

  3. Alos E, Mazet O, Nualart D (2001) D. Stochastic calculus with respect to Gaussian processes. Ann Probab 29:766–801

    Google Scholar 

  4. Beghin L, Macci C (2012) Large deviations for fractional Poisson processes. arXiv:1204.1446

  5. Bender C, Sottinen TT, Valkeila E (2006) Arbitrage with fractional Brownian motion? Theory Stoch Process 12(28):1–12

    Google Scholar 

  6. Bender C, Lindner A, Schicks M (2012) Finite variation of fractional Lévy processes. J Theor Probab 25:594–612

    Google Scholar 

  7. Bender C, Sottinen T, Valkeila E (2011) Fractional processes as models in stochastic finance. In: Advanced mathematical methods in finance. Springer, pp 75–103

  8. Bertoin J (1996) Lévy processes. Cambridge University Press, Cambridge

    Google Scholar 

  9. Bender C, Knobloch RR, Oberacker P (2015) Maximal inequalities for fractional Lévy and related processes. Stoch Anal Appl 33:701–714

    Google Scholar 

  10. Bierme H, Lacaux C, Xiao Y (2009) hitting probabilities and the Haussdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull Lond Math Soc 41:253–273

    Google Scholar 

  11. Bingham NH (1971) Limit theorems for occupation times of Markov processes. Z Wahrsch Verw Gebiete 17:1–22

    Google Scholar 

  12. Bjork T, Hult H (2005) A note on Wick products and the fractional Black-Scholes model. Finance Stoch 9:197–209

    Google Scholar 

  13. Bojdecki T, Gorostiza AT (2004) Sub-fractional Brownian motion and its relation to occupation times. Stat Probab Lett 69:405–419

    Google Scholar 

  14. Cahoy DO, Polito F (2014) Parameter estimation for fractional birth and fractional death processes. Stat Comput 24:211–222

    Google Scholar 

  15. Cahoy DO, Uchaikin VV, Woyczynski W (2010) Parameter estimation for fractional Poisson processes. J Stat Plan Inference 140:3106–3120

    Google Scholar 

  16. Cai C, Chigansky P, Kleptsyna M (2016) Mixed Gaussian processes: a filtering approach. Ann Probab 44:3032–3075

    Google Scholar 

  17. Chen Y, Dong J, Ni H (2019) \(\epsilon \)-strong simulation of fractional Brownian motion and related stochastic differential equations. arXiv:1902.07824

  18. Cheng D, Liu P (2018) Extremes of spherical fractional Brownian motion. arXiv:1806.02965

  19. Cheridito P (2000) Regularizing fractional Brownian motion with a view toward stock price modeling. Ph.D. Dissertation, ETH, Zurich

  20. Cheridito P (2001) Mixed fractional Brownian motion. Bernoulli 7:913–934

    Google Scholar 

  21. Cheridito P (2003) Arbitrage in fractional Brownian motion models. Finance Stoch 7:533–553

    Google Scholar 

  22. Chigansky P, Kleptsyna M (2015) Statistical analysis of the mixed fractional Ornstein–Uhlenbeck process. arXiv:1507.04194

  23. Comte F (1996) Simulation and estimation of long memory continuous time models. J Time Ser Anal 17:19–36

    Google Scholar 

  24. De Oliveira Souza M, Rodriguez P (2021) On the fractional queueing model with catastrophies. arXiv:2012.09317

  25. Diedhiou A, Manga C, Mendy I (2011) Parametric estimation for SDEs with additive sub-fractional Brownian motion. J Numer Math Stoch 3:37–45

    Google Scholar 

  26. Dieker T (2004) Simulation of fractional Brownian motion. University of Twente, Enschede

    Google Scholar 

  27. Dean CR, Young AF, Cadden-Zimansky P (2011) Multicomponent fractional quantum Hall effect in graphene. Nat Phys 7(9):693–696

    CAS  Google Scholar 

  28. Doukhan P, Oppenheim G, Taqqu MS (2003) Theory of long-range dependence. Birkhauser, Boston

    Google Scholar 

  29. Dzhaparidze K, van Zanten H (2004) A series expansion of fractional Brownian motion. Probab Theory Relat Fields 130:39–55

    Google Scholar 

  30. Engelke S (2013) A unifying approach to fractional Lévy processes. Stoch Dyn 13:1250017

    Google Scholar 

  31. Fallahgoul HA, Focardi SM, Fabozzi FJ (2017) Fractional calculus and fractional processes with applications to financial economics: theory and applications. Elsevier/Academic Press, London

    Google Scholar 

  32. Fink H, Kluppelberg C (2011) Fractional Lévy driven Ornstein–Uhlenbeck processes and stochastic differential equations. Bernoulli 17:484–506

    Google Scholar 

  33. Foad S, Kilicman A (2014) Pricing currency option in a mixed fractional Brownian motion with jumps environment. Math Probl Eng 2014:13. https://doi.org/10.1155/2014/858210 (Article ID 858210)

    Article  Google Scholar 

  34. Garra R, Orsingher E, Polio F (2015) State dependent fractional point processes. J Appl Probab 52:18–36

    Google Scholar 

  35. Goldberger AL, West B (1987) Fractals in physiology and medicine. Yale J Med Biol 60:421–435

    CAS  Google Scholar 

  36. Gripenberg G, Norros I (1996) On the prediction of fractional Brownian motion. J Appl Prob 33:400–410

    Google Scholar 

  37. Hairer M (2005) Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann Probab 33:703–758

    Google Scholar 

  38. Henry MM, Zafforoni P (2003) The long-range dependence paradigm for macroeconomics and finance. In: Doukhan P, Oppenheim G, Taqqu MS (eds) Theory of long-range dependence. Birkhauser, Boston, pp 417–438

    Google Scholar 

  39. Hurst HE (1951) Long term storage capacity of reservoirs (with discussion). Trans Am Soc Civ Eng 116:770–808

    Google Scholar 

  40. Ichiba T, Pang G, Taqqu MS (2020) Semimartingale properties of a generalized fractional Brownian motion and its mixture with applications in finance. arXiv:2012.00975

  41. Ichiba T, Pang G, Taqqu MS (2020) Path properties of a generalized fractional Brownian motion. J Theor Prob (to appear)

  42. Istas J (2005) Spherical and hyperbolic fractional Brownian motion. Electron Commun Probab 10:254–262

    Google Scholar 

  43. Istas J (2006) Karhunen–Loeve expansion of spherical fractional Brownian motion. Stat Probab Lett 76:1578–1583

    Google Scholar 

  44. Kataria KK, Vellaisamy P (2018) On distributions of certain state dependent fractional point processes. arXiv:1709.01346

  45. Kataria KK, Khandakar M (2021) Fractional Skellam process of order \(k.\)arXiv:2103.09187

  46. Kleptsyna ML, Le Breton A (2002) Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Stat Inference Stoch Process 5:229–248

    Google Scholar 

  47. Kleptsyna ML, Le Breton AA, Roubaud M-C (2000) Parameter estimation and optimal filtering for fractional type stochastic systems. Stat Inference Stoch Process 3:173–182

    Google Scholar 

  48. Kolmogorov AN (1940) Wienersche Spiralen und einige und andere interessante Kurven im Hilbertschen Raum. C R (Doklady) Acad Sci URSS (N.S.) 26:115–118

    Google Scholar 

  49. Kuang N, Liu B (2015) Parameter estimations for the sub-fractional Brownian motion with drift at discrete observation. Braz J Probab Stat 29:778–789

    Google Scholar 

  50. Kuang N, Xie H (2015) Maximum likelihood estimator for the sub-fractional Brownian motion approximated by a random walk. Ann Inst Stat Math 67:75–91

    Google Scholar 

  51. Kuhn T, Linde W (2002) Optimal series representation of fractional Brownian sheets. Bernoulli 8:669–696

    Google Scholar 

  52. Kurchenko OO (2003) A consistent estimator of the Hurst parameter for a fractional Brownian motion. Theor Probab Math Stat 67:97–106

    Google Scholar 

  53. Kuznetsov Y (1999) The absence of arbitrage in a model with fractal Brownian motion. Russ Math Surv 54:847–848

    Google Scholar 

  54. Lahiri AA, Sen R (2020) Fractional Brownian motion with time-varying volatility and high frequency data. Econom Stat 16:91–107

    Google Scholar 

  55. Lamperti J (1962) Semi-stable stochastic processes. Trans Am Math Soc 104:62–78

    Google Scholar 

  56. Laskin N (2003) Fractional Poisson process. Commun Nonlinear Sci Numer Simul 8:201–203

    Google Scholar 

  57. Le Breton A (1998) Filtering and parameter estimation in a simple linear model driven by a fractional Brownian motion. Stat Probab Lett 38:263–274

    Google Scholar 

  58. Leland WE, Taqqu MS, Willinger W, Wilson DV (1994) On the self-similar mature of Ethernet traffic (extended version). IEEE/ACM Trans Netw 2:1–15

    Google Scholar 

  59. Liptser RS (1980) A strong law of large numbers. Stochastics 3:217–228

    Google Scholar 

  60. Liptser RS, Shiryayev AN (1989) The theory of martingales. Kluwer, Dordrecht

    Google Scholar 

  61. Maheswari A, Vellaisamy P (2016) On the long range dependence of fractional Poisson and negative binomial processes. J Appl Probab 53:989–1000

    Google Scholar 

  62. Maheswari A, Vellaisamy P (2018) Non-homogeneous space-time fractional Poisson processes. Stoch Anal Appl. https://doi.org/10.1080/07362994.2018.1541749

    Article  Google Scholar 

  63. Maheswari A, Vellaisamy P (2019) Fractional Poisson process time-changed by Lévy subordinator and its inverse. J Theor Probab 32:1278–1305

    Google Scholar 

  64. Mainardi F, Gorenflo R, Scalas E (2004) A fractional generalization of the Poisson process. Vietnam J Math 32:53–64

    Google Scholar 

  65. Mainardi F, Gorenflo R, Vivoli A (2007) Beyond the Poisson renewal process: a tutorial survey. J Comput Appl Math 205:725–735

    Google Scholar 

  66. Mandelbrot BB (1982) The fractal geometry of nature. W.H. Freeman, San Fransisco

    Google Scholar 

  67. Mandelbrot BB, Van Ness J (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422–437

    Google Scholar 

  68. Mao Z, Liang Z (2014) Evaluation of geometric Asian options under fractional Brownian motion. J Math Econ 4:1–9

    Google Scholar 

  69. Marquardt T (2006) Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12:1109–1126

    Google Scholar 

  70. Marushkevych D (2016) Large deviations for drift parameter estimator of mixed fractional Ornstein–Uhlenbeck process. Mod Stoch Theory Appl 3:107–117

    Google Scholar 

  71. Meerschaert MR, Nane E, Vellaisamy P (2011) The fractional Poisson process and the inverse stable subordinator. Electron J Stat 16:1600–1620

    Google Scholar 

  72. Memin J, Mishura Y, Valkeila E (2001) Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Stat Probab Lett 51:197–206

    Google Scholar 

  73. Mendy I (2013) Parametric estimation for sub-fractional Ornstein–Uhlenbeck process. J Stat Plan Inference 143:663–674

    Google Scholar 

  74. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77

    CAS  Google Scholar 

  75. Miao Y, Ren W, Ren Z (2008) On the fractional mixed fractional Brownian motion. Appl Math Sci 2:1729–1738

    Google Scholar 

  76. Mishra MN, Prakasa Rao BLS (2011) Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion. Stat Inference Stoch Process 14:101–109

    Google Scholar 

  77. Mishra MN, Prakasa Rao BLS (2011) Nonparametric estimation of linear multiplier for fractional diffusion processes. Stoch Anal Appl 29:706–712

    Google Scholar 

  78. Mishra MN, Prakasa Rao BLS (2014) Estimation of drift parameter and change point for switching fractional diffusion processes. Stoch Anal Appl 32:664–686

    Google Scholar 

  79. Mishra MN, Prakasa Rao BLS (2014) Estimation of change point for switching fractional diffusion processes. Stoch Int J Probab Stoch Process 86:429–449

    Google Scholar 

  80. Mishra MN, Prakasa Rao BLS (2016) Local asymptotic normality and estimation via Kalman–Bucy filter for linear system when signal driven by a fractional Brownian motion and observation driven by a Brownian motion. J Indian Stat Assoc 54:21–42

    Google Scholar 

  81. Mishra MN, Prakasa Rao BLS (2016) Estimation of change point via Kalman–Bucy filter for linear systems driven by fractional Brownian motions. Commun Stoch Anal 10:219–238

    Google Scholar 

  82. Mishra MN, Prakasa Rao BLS (2016) Local asymptotic normality and estimation via Kalman–Bucy filter for linear systems driven by fractional Brownian motions. Stoch Anal Appl 34:707–721

    Google Scholar 

  83. Mishra MN, Prakasa Rao BLS (2017) Large deviation probabilities for maximum likelihood estimator and Bayes estimator of a parameter for mixed fractional Ornstein-Uhlenbeck type process. Bull Inform Cyber 49:67–80

    Google Scholar 

  84. Mishra MN, Prakasa Rao BLS (2018) Estimation of drift parameter and change point via Kalman–Bucy filter for linear systems with signal driven by a fractional Brownian motion and observation driven by a Brownian motion. J Korean Math Soc 55:1063–1073

    Google Scholar 

  85. Mishra MN, Prakasa Rao BLS (2019) Berry–Esseen type bound for fractional Ornstein–Uhlenbeck type process driven by a mixed fractional Brownian motion. J Indian Stat Assoc 57(2019):1–18

    Google Scholar 

  86. Mishra MN, Prakasa Rao BLS (2020) Parametric estimation for cusp-type signal driven by fractional Brownian motion. Stoch Anal Appl 38:62–75

    Google Scholar 

  87. Mishura Y (2008) Stochastic calculus for fractional Brownian motion and related processes. Springer, Berlin

    Google Scholar 

  88. Mishura Y, Zili M (2018) Stochastic analysis of mixed fractional Gaussian processes. ISTE Press and Elsevier, London

    Google Scholar 

  89. Montanari A (2003) Long-range dependence in hydrology. In: Doukhan P, Oppenheim G, Taqqu MS (eds) Theory of long-range dependence. Birkhauser, Boston, pp 461–472

    Google Scholar 

  90. Necula C (2002) Option pricing in a fractional Brownian motion environment. Preprint, DOFIN, Academy of Economic Studies, Bucharest

  91. Norros I (2003) Large deviations of queues with long-range dependent input. In: Doukhan P, Oppenheim G, Taqqu MS (eds) Theory of long-range dependence. Birkhauser, Boston, pp 409–415

    Google Scholar 

  92. Norros I, Valkeila E, Virtamo J (1999) An elementary approach to a Girsanov type formula and other analytical results on fractional Brownian motion. Bernoulli 5:571–587

    Google Scholar 

  93. Novikov AA, Valkeila E (1999) On some maximal inequalities for fractional Brownian motion. Stat Probab Lett 44:47–54

    Google Scholar 

  94. Nualart D, Rascanu A (2002) Differential equations driven by fractional Brownian motion. Collect Math 53(1):55–81

    Google Scholar 

  95. Nuzman CJ, Poor HV (2000) Linear estimation of self-similar processes via Lamperti’s transformation. J Appl Probab 37:429–452

    Google Scholar 

  96. Nuzman CJ, Poor HV (2001) Reproducing kernel Hilbert space methods for wide-sense self-similar processes. Ann Appl Probab 11:1199–1219

    Google Scholar 

  97. Oliveira G, Barreto-Souza W, Silva RWC (2021) Fractional Poisson random sum and its associated normal variance mixture. arXiv:2103.08691

  98. Parzen E (1962) Stochastic processes. Holden-Day Inc., San Francisco

    Google Scholar 

  99. Peng C-K, Buldyrev SV, Goldberger AL, Havlin S, Sciortino F, Simons M, Stanley HE (1992) Long-range correlation in nucleotide sequences. Nature 356:168–170

    CAS  Google Scholar 

  100. Peng C-K, Hausdorff JM, Mietus JE, Havlin S, Stanley HE, Goldberger AL (1995) Fractals in physiological control from heartbeat to gait. In: Shlesinger MF, Zaslavsky GM, Frisch U (eds) Lévy flights and related phenomena in physics, Proceedings of the 1994 international conference on Lévy flights. Springer, Berlin, pp 315–330

  101. Peng C-K, Havlin S, Stanley HE, Goldberger AL (1995) Quantification of scaling exponents and crossover phenomena in non-stationary heartbeat time series. Chaos 5:82–87

    CAS  Google Scholar 

  102. Percival DP, Guttorp P (1994) Long-memory processes, the Allan variance and wavelets. In: Foufoula-Georgiou E, Kumar P (eds) Wavelets in geophysics. Academic Press, New York, pp 325–357

    Google Scholar 

  103. Pillai RN (1990) On Mittag–Leffler functions and related distributions. Ann Inst Stat Math 42:157–161

    Google Scholar 

  104. Pipiras V, Taqqu MS (2002) Deconvolution of fractional Brownian motion. J Time Ser Anal 23:487–501

    Google Scholar 

  105. Prakasa Rao BLS (1966) Asymptotic distributions in some non-regular statistical problems. Ph.D. Dissertation, Michigan State University

  106. Prakasa Rao BLS (1968) Estimation of the location of the cusp of a continuous density. Ann Math Stat 39:76–87

    Google Scholar 

  107. Prakasa Rao BLS (1983) Nonparametric functional estimation. Academic Press, New York

    Google Scholar 

  108. Prakasa Rao BLS (1987) Asymptotic theory of statistical inference. Wiley, New York

    Google Scholar 

  109. Prakasa Rao BLS (1999) Statistical inference for diffusion type processes. Arnold, London and Oxford University Press, New York

    Google Scholar 

  110. Prakasa Rao BLS (1999) Semimartingales and their statistical inference. CRC Press, Boca Raton and Chapman and Hall, London

    Google Scholar 

  111. Prakasa Rao BLS (2003) Parameter estimation for linear stochastic differential equations driven by fractional Brownian motion. Random Oper Stoch Equ 11:229–242

    Google Scholar 

  112. Prakasa Rao BLS (2004) Minimum \(L_1\)-norm estimation for fractional Ornstein–Uhlenbeck type process. Theory Probab Math Stat 71:181–189

    Google Scholar 

  113. Prakasa Rao BLS (2005) Berry–Esseen bound for MLE for linear stochastic differential equations driven by fractional Brownian motion. J Korean Stat Soc 34:281–295

    Google Scholar 

  114. Prakasa Rao BLS (2008) Singularity of fractional motions with different Hurst indices. Stoch Anal Appl 26:334–337

    Google Scholar 

  115. Prakasa Rao BLS (2009) Estimation for stochastic differential equations driven by mixed fractional Brownian motion. Calcutta Stat Assoc Bull 61:143–153

    Google Scholar 

  116. Prakasa Rao BLS (2010) Statistical inference for fractional diffusion processes. Wiley, London

    Google Scholar 

  117. Prakasa Rao BLS (2012) Singularity of subfractional Brownian motions with different Hurst indices. Stoch Anal Appl 30:538–542

    Google Scholar 

  118. Prakasa Rao BLS (2013) Introduction to statistics in finance. Lecture Notes, CR Rao Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad, p 156

  119. Prakasa Rao BLS (2013) Some maximal inequalities for fractional Brownian motion with polynomial drift. Stoch Anal Appl 31:785–799

    Google Scholar 

  120. Prakasa Rao BLS (2014) Maximal inequalities for fractional Brownian motion: An overview. Stoch Anal Appl 32:450–479

    Google Scholar 

  121. Prakasa Rao BLS (2015) Option pricing for processes driven by mixed fractional Brownian motion with superimposed jumps. Probab Eng Inf Sci 29:589–596

    Google Scholar 

  122. Prakasa Rao BLS (2015) Pricing geometric Asian power options under mixed fractional Brownian motion environment. Phys A 446:92–99

    Google Scholar 

  123. Prakasa Rao BLS (2015) Filtered fractional Poisson processes. Stat Methodol 26:124–134

    Google Scholar 

  124. Prakasa Rao BLS (2017) On some maximal and integral inequalities for sub-fractional Brownian motion. Stoch Anal Appl 35:279–287

    Google Scholar 

  125. Prakasa Rao BLS (2017) Optimal estimation of a signal perturbed by a sub-fractional Brownian motion. Stoch Anal Appl 35:533–541

    Google Scholar 

  126. Prakasa Rao BLS (2017) Parameter estimation for linear stochastic differential equations driven by sub-fractional Brownian motion. Random Oper Stoch Equ 25:235–247

    Google Scholar 

  127. Prakasa Rao BLS (2017) Instrumental variable estimation for a linear stochastic differential equation driven by a mixed fractional Brownian motion. Stoch Anal Appl 35:943–953

    Google Scholar 

  128. Prakasa Rao BLS (2017) Optimal estimation of a signal perturbed by a mixed fractional Brownian motion. Theory Stoch Process 22(38):62–68

    Google Scholar 

  129. Prakasa Rao BLS (2018) Parametric estimation for linear stochastic differential equations driven by mixed fractional Brownian motion. Stoch Anal Appl 36:767–781

    Google Scholar 

  130. Prakasa Rao BLS (2018) Berry–Esseen type bound for fractional Ornstein–Uhlenbeck type process driven by sub-fractional Brownian motion. Theory Stoch Process 23(39):82–92

    Google Scholar 

  131. Prakasa Rao BLS (2018) Instrumental variable estimation for stochastic differential equations linear in drift parameter and driven by a sub-fractional Brownian motion. Stoch Anal Appl 36:600–612

    Google Scholar 

  132. Prakasa Rao BLS (2018) Pricing geometric Asian options under mixed fractional Brownian motion environment with superimposed jumps. Calcutta Stat Assoc Bull 70:1–6

    Google Scholar 

  133. Prakasa Rao BLS (2019) Nonparametric estimation of linear multiplier for processes driven by sub-fractional Brownian motion. Stoch Anal Appl 37:799–810

    Google Scholar 

  134. Prakasa Rao BLS (2019) Nonparametric estimation of trend for stochastic differential equations driven by mixed fractional Brownian motion. Stoch Anal Appl 37:271–280

    Google Scholar 

  135. Prakasa Rao BLS (2020) More on maximal inequalities for sub-fractional Brownian motion. Stoch Anal Appl 38:238–247

    Google Scholar 

  136. Prakasa Rao BLS (2021) Maximum likelihood estimation in the mixed fractional Vasicek model. J Indian Soc Probab Stat 22:9–25

    Google Scholar 

  137. Prakasa Rao BLS (2020) Nonparametric estimation of linear multiplier in stochastic differential equations driven by \(\alpha \)-stable noise. J Indian Stat Assoc (to appear)

  138. Prakasa Rao BLS (2021) Nonparametric estimation of trend for stochastic differential equations driven by fractional Levy process, In the Special Issue in honour of CR Rao Birth Centenary. J Stat Theory Pract 15:13 (Paper no. 7)

  139. Prakasa Rao BLS (2021) Nonparametric estimation for stochastic differential equations driven by mixed fractional Brownian motion with random effects. In the Special Issue in honour of CR Rao Birth Centenary. Sankhya Ser A 83:554–568

  140. Prakasa Rao BLS (2020) Nonparametric estimation of linear multiplier for stochastic differential equations driven by fractional Levy process with small noise. Bull Inform Cybern 52:1–13

    Google Scholar 

  141. Prakasa Rao BLS (2020) Nonparametric estimation of trend for stochastic differential equations driven by sub-fractional Brownian motion. Random Oper Stoch Equ 28:113–122

    Google Scholar 

  142. Prakasa Rao BLS (2021) Nonparametric estimation of linear multiplier for processes driven by mixed fractional Brownian motion In the Special Issue in memory of Aloke Dey. Stat Appl 19(1):67–76

    Google Scholar 

  143. Rajput B, Rosinski J (1989) Spectral representations of infinitely divisible processes. Probab Theory Relat Fields 82:451–487

    Google Scholar 

  144. Repin ON, Saichev AT (2000) Fractional Poisson law. Radiophys Quantum Electron 43:738–741

    Google Scholar 

  145. Rogers LCG (1997) Arbitrage with fractional Brownian motion. Math Financ 7:95–105

    Google Scholar 

  146. Rudomino-Dusyatska N (2003) Properties of maximum likelihood estimates in diffusion and fractional Brownian models. Theor Probab Math Stat 68:139–146

    Google Scholar 

  147. Sadhu T, Wiese KJ (2021) Functionals of fractional Brownian motion and the three arcsine laws. arXiv:2103.09032

  148. Saichev AT, Zaslavsky GM (1997) Fractional kinetic equations: solutions and applications. Chaos 7:753–764

    CAS  Google Scholar 

  149. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Gordon and Breach Science, Yverdon

    Google Scholar 

  150. Samorodnitsky G (2016) Stochastic processes and long range dependence. Springer International, Cham

    Google Scholar 

  151. Samorodnitsky G, Taqqu MS (1994) Stable non-Gaussian processes: stochastic models with infinite variance. Chapman and Hall, London

    Google Scholar 

  152. Shen GJ, Yan LT (2014) Estimators for the drift of subfractional Brownian motion. Commun Stat Theory Methods 43:1601–1612

    Google Scholar 

  153. Shen GJ, Li Y, Gao Z (2018) Parameter estimation for Ornstein–Uhlenbeck processes driven by fractional Lévy process. J Inequal Appl 2018:356. https://doi.org/10.1186/s13660-018-1951-0

    Article  Google Scholar 

  154. Shevchenko G (2014) Mixed stochastic delay differential equations. Theory Probab Math Stat 89:181–195

    Google Scholar 

  155. Skellam JG (1946) The frequency distribution of the difference between two Poisson variates belonging to different populations. J R Stat Soc (N.S.) 109:206

    Google Scholar 

  156. Song N, Liu Z (2014) Parameter estimation for stochastic differential equations driven by mixed fractional Brownian motion. Abstr Appl Anal 2014:6 (Article ID 942307)

    Google Scholar 

  157. Smith HF (1938) An empirical law describing heterogenity in the yields of agricultural crops. J Agric Sci 28:1–23

    Google Scholar 

  158. Stein EM (1971) Singular integrals and differentiability. Princeton University Press, Princeton

    Google Scholar 

  159. Stewart CV, Moghaddam B, Hintz KJ, Novak LM (1993) Fractional Brownian motion models for synthetic aperture radar imagery scene segmentation. Proc IEEE 81:1511–1521

    Google Scholar 

  160. Sun L (2013) Pricing currency options in the mixed fractional Brownian motion. Phys A 392:3441–3458

    Google Scholar 

  161. Sun X, Yan L (2012) Mixed-fractional models in credit risk pricing. J Stat Econ Methods 1:79–96

    Google Scholar 

  162. Taqqu MS (2003) Fractional Brownian motion and long-range dependence. In: Doukhan P, Oppenheim G, Taqqu MS (eds) Theory of long-range dependence. Birkhauser, Boston, pp 5–38

    Google Scholar 

  163. Tudor C (2007) Some properties of the sub-fractional Brownian motion. Stochastics 79:431–448

    Google Scholar 

  164. Tudor C (2007) Prediction and linear filtering with sub-fractional Brownian motion (preprint)

  165. Tudor C (2008) Some aspects of stochastic calculus for the sub-fractional Brownian motion. Analele Universitat ii Bucaresti, Matematica, Anul LVII, pp 199–230

  166. Tudor C (2009) On the Wiener integral with respect to a sub-fractional Brownian motion on an interval. J Math Anal Appl 351:456–468

    Google Scholar 

  167. Willinger W, Taqqu MS, Sherman R, Wilson D (1997) Self-similarity through high variability: statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Trans Netw 5:71–86

    Google Scholar 

  168. Willinger W, Taqqu MS, Teverovsky V (1999) Stock market prices and long-range dependence. Finance Stoch 3:1–13

    Google Scholar 

  169. Willinger W, Paxson V, Riedi RH, Taqqu MS (2003) Long-range dependence and data network traffic. In: Doukhan P, Oppenheim G, Taqqu MS (eds) Theory of long-range dependence. Birkhauser, Boston, pp 373–407

    Google Scholar 

  170. Xiao Y (2009) Sample path properties of anisotropic Gaussian random fields. In: A minicourse on stochastic partial differential equations, Lecture notes in mathematics, vol 1962. Springer, Berlin, p 9

  171. Xiao WL, Zhang WG, Zhang XL (2012) Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm. Phys A 391:6418–6431

    Google Scholar 

  172. Yan L, Shen G, He K (2011) Ito’s formula for a sub-fractional Brownian motion. Commun Stoch Anal 5:135–159

    Google Scholar 

  173. Zahle M (1998) Integration with respect to fractal functions and stochastic calculus I. Probab Theory Relat Fields 111:333–374

    Google Scholar 

  174. Zaslavsky GM (2002) Chaos, fractional kinetics and anomalous transport. Phys Rep 371:461–580

    Google Scholar 

  175. Zhang X, Haoran Y, Shu H (2019) Nonparametric estimation of the trend for stochastic differential equations driven by small \(\alpha \)-stable noises. Stat Probab Lett 151:8–16

    Google Scholar 

  176. Zili M (2006) On the mixed fractional Brownian motion. J Appl Math Stoch Anal 2006:1–9. https://doi.org/10.1155/JAMSA/2006/32435 (Article ID 32435)

    Article  Google Scholar 

Download references

Acknowledgements

Work presented in this paper is supported under the “INSA Senior Scientist” scheme at the CR Rao Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad, India. The author thanks the referees for their exhaustive review and Prof. Arni Srinivasa Rao for inviting him to contribute to this Special Issue of the Journal of Indian Institute of Science on Probability and Statistics.

Author information

Authors and Affiliations

  1. CR Rao Advanced Institute of Mathematics, Statistics and Computer Science, Hyderabad, India

    B. L. S. Prakasa Rao

Authors
  1. B. L. S. Prakasa Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. L. S. Prakasa Rao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Prakasa Rao, B.L.S. Fractional Processes and Their Statistical Inference: An Overview. J Indian Inst Sci 102, 1145–1175 (2022). https://doi.org/10.1007/s41745-021-00271-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41745-021-00271-z

Keywords

Mathematics Subject Classification

Search

Navigation