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Financial Applications of Fractional Brownian Motion

Part of the Lecture Notes in Mathematics book series (LNM, volume 1929)

Long-range Dependence in Economics and Finance

As mentioned in the paper (WTT99), long-range dependence in economics and finance has a long history and is an area of active research (e.g., see (Lo91), (CKW95)). The importance of long-range dependent processes as stochastic models lies in the fact that they provide an explanation and interpretation of an empirical law that is commonly referred to as the Hurst law or Hurst effect. In short, for a given set of observations \(\left\{ {X_{i,} i \ge 1} \right\}\) with partial sum \(Y\left( n \right) = \sum\limits_{i = 1}^n {X_i ,n \ge 1,} \) and sample variance \(S^2 \left( n \right) = n^{ - 1} \sum\limits_{i = 1}^n {\left( {X_i - n^{ - 1} Y\left( n \right)} \right)^2 ,n \ge 1,} \) the rescaled adjusted range statistic or R/S-statistic is defined by
$$\frac{R}{S}\left( n \right) = \frac{1}{{S\left( n \right)}}\left( {\mathop {\max }\limits_{0 \le t \le n} \left( {Y\left( t \right) - \frac{t}{n}Y\left( n \right)} \right) - \mathop {\min }\limits_{0 \le t \le n} \left( {Y\left( t \right) - \frac{t}{n}Y\left( n \right)} \right)} \right),n \ge 1.$$

Hurst in (Hur51) found that many naturally occurring empirical records appear to be well represented by the relation \(E\left( {\left( {R/S} \right)\left( n \right)} \right) \sim c_1 n^H \) as \(n \to \infty \) with typical values of the Hurst parameter \(H \in \left( {1/2,1} \right)\), and c 1 a finite positive constant not depending on n. But in the case when the observations come from a short-range dependent model, then \(E\left( {R/S\left( n \right)} \right) \sim c_2 n^{1/2} \) as \(n \to \infty \), where c 2 does not depend on n. The discrepancy between these two relations is called the Hurst effect or Hurst phenomenon. The analysis of the R/S-statistic, provided in (WTT99), (TTW95) and (TT97), leads to the recommendation to use a diverse portfolio of time-domain-based and frequency-domain-based graphics and statistical methods, including the graphical R/S-method, the modified R/S-statistic (Lo91) and Whittle’s approach. Also, another (possibly, surprising) recommendation is: in the case when statistical analysis cannot be expected to provide a definitive answer concerning the presence or absence of long-range dependence in asset price returns, a more revealing and also much more challenging approach to tackle this problem consists of providing a mathematically rigorous physical “explanation” for the presence or absence of the long-range dependence phenomenon in stock returns.

Keywords

Risky Asset Fractional Brownian Motion Burger Equation Financial Application Hurst Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Arino, O., Sánchez, E., Bravo de la Parra, R., Auger, P.: A singular perturbation in an age-structured population model. SIAM J. Appl. Math., 60, 408–436 (1999)Google Scholar
  2. 2.
    Arino, O., Sánchez, E., Bravo de la Parra, R.: A model of an age-structured population in a multipatch environment. Math. Comput. Model., 27, 137–150 (1998)CrossRefzbMATHGoogle Scholar
  3. 3.
    Auger, P.: Dynamics and thermodynamics in hierarchically organized systems. Pergamon Press, Oxford (1989)Google Scholar
  4. 4.
    Auger, P., Benoit, E.: A prey–predator model in a multi-patch environment with different time scales. J. Biol. Syst., 1(2), 187–197 (1993)CrossRefGoogle Scholar
  5. 5.
    Auger, P., Kooi, B., Bravo de la Parra, R., Poggiale, J.C.: Bifurcation analysis of a predator–prey model with predators using hawk and dove tactics. J. Theor. Biol., 238, 597–607 (2006)CrossRefGoogle Scholar
  6. 6.
    Auger, P., Lett. C.: Integrative biology: linking levels of organization. C. R. Acad. Sci. Paris, Biol., 326, 517–522 (2003)Google Scholar
  7. 7.
    Auger, P., Bravo de la Parra, R., Morand, S., Sánchez, E.: A predator–prey model with predators using hawk and dove tactics. Math. Biosci., 177/178, 185–200 (2002)CrossRefGoogle Scholar
  8. 8.
    Auger, P., Bravo de la Parra, R.: Methods of aggregation of variables in population dynamics. C. R. Acad. Sci. Paris, Sciences de la vie, 323, 665–674 (2000)Google Scholar
  9. 9.
    Auger, P., Charles, S., Viala, M., Poggiale, J.C.: Aggregation and emergence in ecological modelling: integration of the ecological levels. Ecol. Model., 127, 11–20 (2000)CrossRefGoogle Scholar
  10. 10.
    Auger, P., Poggiale, J.C., Charles, S.: Emergence of individual behaviour at the population level: effects of density dependent migration on population dynamics. C. R. Acad. Sci. Paris, Sciences de la Vie, 323, 119–127 (2000)Google Scholar
  11. 11.
    Auger, P., Chiorino, G., Poggiale, J.C.: Aggregation, emergence and immergence in hierarchically organized systems. Int. J. Gen. Syst., 27(4–5), 349–371 (1999)zbMATHCrossRefGoogle Scholar
  12. 12.
    Auger, P., Poggiale, J.C.: Aggregation and Emergence in Systems of Ordinary Differential Equations. Math. Comput. Model., 27(4), 1–22 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Auger, P., Poggiale, J.C.: Aggregation and emergence in hierarchically organized systems: population dynamics. Acta Biotheor., 44, 301–316 (1996)CrossRefGoogle Scholar
  14. 14.
    Auger, P., Poggiale, J.C.: Emergence of population growth models: fast migration and slow growth. J. Theor. Biol., 182, 99–108 (1996)CrossRefGoogle Scholar
  15. 15.
    Auger, P., Poggiale, J.C.: Emerging properties in population dynamics with different time scales. J. Biol. Syst., 3(2), 591–602 (1995)CrossRefGoogle Scholar
  16. 16.
    Auger, P., Roussarie, R.: Complex ecological models with simple dynamics: from individuals to populations. Acta Biotheor., 42, 111–136 (1994)CrossRefGoogle Scholar
  17. 17.
    Auger, P., Pontier, D.: Fast game theory coupled to slow population dynamics: the case of domestic cat populations. Math. Biosci., 148, 65–82 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Auger, P., Bravo de la Parra, R., Sánchez, E.: Hawk-dove game and competition dynamics. Math. Comput. Model., Special issue Aggregation and emergence in population dynamics. Antonelli, P., Auger, P., guest-Editors, 27(4), 89–98 (1998)Google Scholar
  19. 19.
    Bates, P.W., Lu, K., Zeng, C.: Invariant foliations near normally hyperbolic invariant manifolds for semiflows. Trans. Am. Math. Soc., 352, 4641–4676 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Bates, P.W., Lu, K., Zeng, C.: Existence and persistence of invariant manifolds for semiflow in Banach space. Memoir. Am. Math. Soc., 135, 129 (1998)MathSciNetGoogle Scholar
  21. 21.
    Benoît, E.: Canards et enlacements. Extraits des Publications Mathématiques de l’IHES, 72, 63–91 (1990)zbMATHGoogle Scholar
  22. 22.
    Benoît, E.: Systèmes lents-rapides dans R3 et leurs canards. Astérisque, 109/110, 159–191 (1983)Google Scholar
  23. 23.
    Benoît, E., Callot, J.L., Diener, F., Diener, M.: Chasse au canard. Collection Mathématique, 31/32(1–3), 37–119 (1981)Google Scholar
  24. 24.
    Bernstein, C., Auger, P.M., Poggiale, J.C.: Predator migration decisions, the ideal free distribution and predator–prey dynamics. Am. Nat., (1999), 153(3), 267–281 (1999)Google Scholar
  25. 25.
    Blasco, A., Sanz, L., Auger, P., Bravo de la Parra, R.: Linear discrete population models with two time scales in fast changing environments II: non autonomous case. Acta Biotheor., 50(1), 15–38 (2002)CrossRefGoogle Scholar
  26. 26.
    Blasco, A., Sanz, L., Auger, P., Bravo de la Parra, R.: Linear discrete population models with two time scales in fast changing environments I: autonomous case. Acta Biotheor., 49, 261–276 (2001)CrossRefGoogle Scholar
  27. 27.
    Bravo de la parra, R., Arino, O., Sánchez, E., Auger, P.: A model of an age-structured population with two time scales. Math. Comput. Model., 31, 17–26 (2000)zbMATHGoogle Scholar
  28. 28.
    Bravo de la Parra, R., Sánchez, E., Auger, P.: Time scales in density dependent discrete models. J. Biol. Syst., 5, 111–129 (1997)zbMATHCrossRefGoogle Scholar
  29. 29.
    Bravo de la Parra, R., Auger, P., Sánchez, E.: Aggregation methods in discrete models. J. Biol. Syst., 3, 603–612 (1995)CrossRefGoogle Scholar
  30. 30.
    Bravo de la Parra, R., Sánchez, E.: Aggregation methods in population dynamics discrete models. Math. Comput. Model., 27(4), 23–39 (1998)CrossRefzbMATHGoogle Scholar
  31. 31.
    Bravo de la Parra, R., Sánchez, E., Arino, O., Auger, P.: A Discrete Model with Density Dependent Fast Migration. Math. Biosci., 157, 91–110 (1999)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Carr, J.: Applications of centre manifold theory. Springer, Berlin Heidelberg New York (1981)zbMATHGoogle Scholar
  33. 33.
    Caswell, H.: Matrix population models. Sinauer Associates, Sunderland, MA, USA (2001)Google Scholar
  34. 34.
    Charles, S., Bravo de la Parra, R., Mallet, J.P., Persat, H., Auger, P.: Population dynamics modelling in an hierarchical arborescent river network: an attempt with Salmo trutta. Acta Biotheor., 46, 223–234 (1998)CrossRefGoogle Scholar
  35. 35.
    Charles, S., Bravo de la Parra, R., Mallet, J.P., Persat, H., Auger, P.: A density dependent model describing Salmo trutta population dynamics in an arborescent river network: effects of dams and channelling. C. R. Acad. Sci. Paris, Sciences de la vie, 321, 979–990 (1998)Google Scholar
  36. 36.
    Charles, S., Bravo de la Parra, R., Mallet, J.P., Persat, H., Auger, P.: Annual spawning migrations in modeling brown trout population dynamics inside an arborescent river network. Ecol. Model., 133, 15–31 (2000)CrossRefGoogle Scholar
  37. 37.
    Chaumot, A., Charles, S., Flammarion, P., Garric, J., Auger, P.: Using aggregation methods to assess toxicant effects on population dynamics in spatial systems. Ecol. Appl., 12(6), 1771–1784 (2002)CrossRefGoogle Scholar
  38. 38.
    Chaumot, A., Charles, S., Flammarion, P., Auger, P.: Ecotoxicology and spatial modeling in population dynamics: an attempt with brown trout. Environ. Toxicol. Chem., 22(5), 958–969 (2003)CrossRefGoogle Scholar
  39. 39.
    Chaumot, A., Charles, S., Flammarion, P., Auger, P.: Do migratory or demographic disruptions rule the population impact of pollution in spatial networks? Theor. Pop. Biol., 64, 473–480 (2003)zbMATHCrossRefGoogle Scholar
  40. 40.
    Chiorino, O., Auger, P., Chasse, J.L., Charles, S.: Behavioral choices based on patch selection: a model using aggregation methods. Math. Biosci., 157, 189–216 (1999)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Comins, H.N., Hassell, M.P., May, R.M.: The spatial dynamics of host–parasitoid systems. J. Anim. Ecol., 61, 735–748 (1992)CrossRefGoogle Scholar
  42. 42.
    De Feo, O., Rinaldi, S.: Singular homoclinic bifurcations in tritrophic food chains. Math. biosci., 148, 7–20 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Diener, M.: Canards et bifurcations. In: Outils et modèles mathématiques pour l’automatique, l’analyse des systèmes et le traitement du signal, vol. 3, Publication du CNRS, 289–313 (1983)Google Scholar
  44. 44.
    Diener, M.: Etude générique des canards. Thesis, Université de Strasbourg (1981)Google Scholar
  45. 45.
    Dubreuil, E., Auger, P., Gaillard, J.M., Khaladi, M.: Effects of aggressive behaviour on age structured population dynamics. Ecol. Model., 193, 777–786 (2006)CrossRefGoogle Scholar
  46. 46.
    Dumortier, F., Roussarie, R.: Geometric singular perturbation theory beyond normal hyperbolicity. In: Jones, C.K.R.T., Khibnik, A.I. (eds) Multiple time scale dynamical systems. Springer, Berlin Heidelberg New York (2000)Google Scholar
  47. 47.
    Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Memoir. Am. Math. Soc., 121(577), 1–100 (1996)MathSciNetGoogle Scholar
  48. 48.
    Edelstein-Keshet, L.: Mathematical models in biology. Random House, New York (1989)Google Scholar
  49. 49.
    Fenichel, N.: Persistence and Smoothness of Invariant Manifolds for Flows. Indiana Univ. Math. J., 21(3), 193–226 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture notes in mathematics vol. 583. Springer, Berlin Heidelberg New York (1977)Google Scholar
  51. 51.
    Hofbauer, J., Sigmund, K.: Evolutionary games and population dynamics. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  52. 52.
    Iwasa, Y., Andreasen, V., Levin, S.: Aggregation in model ecosystems. I. Perfect aggregation. Ecol. Model., 37, 287–302 (1987)CrossRefGoogle Scholar
  53. 53.
    Iwasa, Y., Levin, S., Andreasen, V.: Aggregation in model ecosystems. II. Approximate Aggregation. IMA. J. Math. Appl. Med. Biol., 6, 1–23 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Kaper, T.J., Jones, C.K.R.T.: A primer on the exchange lemma for fast-slow systems. In: Jones, C.K.R.T., Khibnik, A.I. (eds) Multiple time scale dynamical systems. Springer, Berlin Heidelberg New York (2000)Google Scholar
  55. 55.
    Kooi, B.W., Poggiale, J.C., Auger, P.M.: Aggregation methods in food chains. Math. Comput. Model., 27(4), 109–120 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    Krylov, N., Bogoliubov, N.: The application of methods of nonlinear mechanics to the theory of stationary oscillations. Publication 8 of the Ukrainian Academy of Science, Kiev (1934)Google Scholar
  57. 57.
    Lett, C., Auger, P., Bravo de la Parra, R.: Migration frequency and the persistence of host–parasitoid interactions. J. Theor. Biol., 221, 639–654 (2003)CrossRefMathSciNetGoogle Scholar
  58. 58.
    Lett, C., Auger, P., Fleury, F.: Effects of asymmetric dispersal and environmental gradients on the stability of host–parasitoid systems. Oikos, 109, 603–613 (2005)CrossRefGoogle Scholar
  59. 59.
    Lotka, A.J.: Undamped oscillations derived from the mass action law. J. Am. Chem. Soc., 42, 1595–1599 (1920)CrossRefGoogle Scholar
  60. 60.
    Lotka, A.J.: Elements of physical biology. William and Wilkins, Baltimore (1925)zbMATHGoogle Scholar
  61. 61.
    Michalski, J., Poggiale, J.C., Arditi, R., Auger, P.: Effects of migrations modes on patchy predator–prey systems. J. Theor. Biol., 185, 459–474 (1997)CrossRefGoogle Scholar
  62. 62.
    Mchich, R., Auger, P., Poggiale, J.C.: Effect of predator density dependent dispersal of prey on stability of a predator–prey system. Math. Biosci., 206, 343–356 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    Mchich, R., Auger, P., Raïssi, N.: The stabilizability of a controlled system describing the dynamics of a fishery. C. R. Acad. Sci. Paris, Biol., 329, 337–350 (2005)Google Scholar
  64. 64.
    Mchich, R., Auger, P., Bravo de la Parra, R., Raïssi, N.: Dynamics of a fishery on two fishing zones with fish stock dependent migrations: aggregation and control. Ecol. Model., 158, 51–62 (2002)CrossRefGoogle Scholar
  65. 65.
    Muratori, S., Rinaldi, S.: Low and high frequency oscillations in three dimensional food chain systems. SIAM J. Appl. Math., 52(6), 1688–1706 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    Murray, J.D.: Mathematical biology. Springer, Berlin Heidelberg New York (1989)zbMATHGoogle Scholar
  67. 67.
    Nguyen Huu, T., Lett, C., Poggiale J.C., Auger, P.: Effects of migration frequency on global host–parasitoid spatial dynamics with unstable local dynamics. Ecol. Model., 177, 290–295 (2006)CrossRefGoogle Scholar
  68. 68.
    Nguyen-Huu, T., Lett, C., Auger, P., Poggiale, J.C.: Spatial synchrony in host–parasitoid models using aggregation of variables. Math. Biosci., 203, 204–221 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    Nicholson, A.J.: The balance of animal populations. J. Anim. Ecol., 2, 132–178 (1933)Google Scholar
  70. 70.
    Nicholson, A.J., Bailey, V.A.: The balance of animal populations, part I. Proc. Zool. Soc. Lond., 3, 551–598 (1935)Google Scholar
  71. 71.
    Pichancourt, J.B., Burel, F., Auger, P.: Assessing the effect of habitat fragmentation on population dynamics: an implicit modelling approach. Ecol. Model., 192, 543–556 (2006)CrossRefGoogle Scholar
  72. 72.
    Pliss, V.A., Sell, G.R.: Perturbations of normally hyperbolic manifolds with applications to the Navier–Stokes equations. J. Differ. Equat., 169, 396–492 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    Poggiale, J.C.: Lotka–Volterra’s model and migrations: breaking of the well-known center. Math. Comput. Model., 27(4), 51–62 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  74. 74.
    Poggiale, J.C.: From behavioural to population level: growth and competition. Math. Comput. Model., 27(4), 41–50 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  75. 75.
    Poggiale, J.C.: Predator–prey models in heterogeneous environment: emergence of functional response. Math. Comp. Model., 27(4), 63–71 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  76. 76.
    Poggiale, J.C., Michalski, J., Arditi, R.: Emergence of donor control in patchy predator–prey systems. Bull. Math. Biol., 60(6), 1149–1166 (1998)zbMATHCrossRefGoogle Scholar
  77. 77.
    Poggiale, J.C., Auger, P.: Impact of spatial heterogeneity on a predator–prey system dynamics. C. R. Biol., 327, 1058–1063 (2004)CrossRefGoogle Scholar
  78. 78.
    Poggiale, J.C., Auger, P.: Fast oscillating migrations in a predator–prey model. Methods Model. Meth. Appl. Sci., 6(2), 217–226 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    Poggiale, J.C., Auger, P., Roussarie, R.: Perturbations of the classical Lotka–Volterra system by behavioural sequences. Acta Biotheor., 43, 27–39 (1995)CrossRefGoogle Scholar
  80. 80.
    Sakamoto, K.: Invariant manifolds in singular perturbations problems for ordinary differential equations. Proc. Roy. Soc. Ed., 116A, 45–78 (1990)MathSciNetGoogle Scholar
  81. 81.
    Sánchez, E., Bravo de la Parra, R., Auger, P., Gómez-Mourelo, P.: Time scales in linear delayed differential equations. J. math. Anal. Appl., 323, 680–699 (2006)CrossRefGoogle Scholar
  82. 82.
    Sánchez, E., Bravo de la Parra, R., Auger, P.: Discrete models with different time-scales. Acta Biotheor., 43, 465–479 (1995)CrossRefGoogle Scholar
  83. 83.
    Sánchez, E., Auger, P., Bravo de la Parra, R.: Influence of individual aggressiveness on the dynamics of competitive populations. Acta Biotheor., 45, 321–333 (1997)CrossRefGoogle Scholar
  84. 84.
    Sanz, L., Bravo de la Parra, R.: Variables aggregation in time varying discrete systems. Acta Biotheor., 46, 273–297 (1998)CrossRefGoogle Scholar
  85. 85.
    Sanz, L., Bravo de la Parra, R.: Variables aggregation in a time discrete linear model. Math. Biosci., 157, 111–146 (1999)CrossRefMathSciNetGoogle Scholar
  86. 86.
    Sanz, L., Bravo de la Parra, R.: Time scales in stochastic multiregional models. Nonlinear Anal. R. World Appl., 1, 89–122 (2000)zbMATHCrossRefGoogle Scholar
  87. 87.
    Sanz, L., Bravo de la Parra, R.: Time scales in a non autonomous linear discrete model. Math. Model. Meth. Appl. Sci., 11(7), 1203–1235 (2001)zbMATHCrossRefGoogle Scholar
  88. 88.
    Sanz, L., Bravo de la Parra, R.: Approximate reduction techniques in population models with two time scales: study of the approximation. Acta Biotheor., 50(4), 297–322 (2002)CrossRefGoogle Scholar
  89. 89.
    Sanz, L., Bravo de la Parra, R.: Approximate reduction of multiregional models with environmental stochasticity. Math. Biosci., 206, 134–154 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  90. 90.
    Sanz, L., Bravo de la Parra, R., Sánchez, E.: Approximate reduction of nonlinear discrete models with two time scales. J. Differ. Equ. Appl., DOI: 10.1080/10236190701709036 (2008)Google Scholar
  91. 91.
    Scheffer, M., Rinaldi, S., Kuztnetsov, Y.A., Van Nes, E.H.: Seasonal dynamics of Daphnia and algae explained as a periodically forced predator–prey system. Oikos, 80(3), 519–532 (1997)CrossRefGoogle Scholar
  92. 92.
    Scheffer, M., De Boer, R.J.: Implications of spatial heterogeneity for the paradox of enrichment. Ecol., 76(7), 2270–2277 (1995)CrossRefGoogle Scholar
  93. 93.
    Stewart, G.W, Guang Sun, J.I.: Matrix perturbation theory. Academic Press, Boston (1990)zbMATHGoogle Scholar
  94. 94.
    Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI, 2, 31–113 (1926)Google Scholar
  95. 95.
    Wiggins, S.: Normally Hyperbolic invariant manifolds in dynamical systems. Springer, Berlin Heidelberg New York (1994)zbMATHGoogle Scholar

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