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Statistical Inference with Fractional Brownian Motion

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

Testing Problems for the Density Process for fBm with Different Drifts

As we have seen in Subsection 5.2.2, the form of geometric fBm (5.2.6) depends on the kind of integral that is used in its calculations: if we use the Riemann–Stieltjes integral,
$$S_t^{\left( 1 \right)} = S_0^{\left( 1 \right)} + \mu \int_0^1 {S_s^{\left( 1 \right)} ds} + \sigma \int_0^t {S_s^{\left( 1 \right)} dB_s^H ,} $$
then \(S_t^{\left( 1 \right)} = S_0^{\left( 1 \right)} \exp \left\{ {\mu t + \sigma B_t^H } \right\},\) and if the behavior of geometric process is guided by the Wick integral,
$$S_t^{\left( 2 \right)} = S_0^{\left( 2 \right)} + \mu \int_0^2 {S_s^{\left( 2 \right)} ds} + \sigma \int_0^t {S_s^{\left( 2 \right)} \diamondsuit dB_s^H ,} $$
then \(S_t^{\left( 2 \right)} = S_0^{\left( 2 \right)} \exp \left\{ {\mu t + \sigma B_t^H - \frac{1}{2}\sigma ^2 t^{2H} } \right\}\) So, the natural question arises: what trend actually has geometric fBm? This question was considered in the paper (KMV05), and here we present a solution of this problem. In what follows the notation \(X_n = o_P \left( 1 \right)\) means that \(X_n \rightarrow{P}0,X_n = O_P \left( 1 \right)\) means that \(\mathop {\lim }\limits_{C \to \infty } \mathop {\lim \sup }\limits_n P\left\{ {\left| {X_n } \right| \ge C} \right\} = 0.\) Assume that \(H \in \left( {1/2,1} \right)\). For a fixed \(\mu \in {\rm R}\) let \(P_{\mu ,\sigma } \) be the distribution of the process
$$X_t : = \sigma B_t^H + \mu t - \frac{{\sigma ^2 }}{2}t^{2H} ,0 \le t \le T$$
in the space \(C_{\left[ {0,T} \right]} \) of continuous functions. Similarly, \(P_{\mu ,\sigma } \) is the distribution of the process
$$X_t : = \sigma B_t^H + \mu t,0 \le t \le T$$
in the space \(C_{\left[ {0,T} \right]} \)

Keywords

Likelihood Ratio Probability Measure Maximum Likelihood Estimate Statistical Inference Wiener Process 
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.
    R. Aris: Mathematical Modeling, a Chemical Engineer’s Perspective (Academic, New York 1999)zbMATHGoogle Scholar
  2. 2.
    J. Bailey, D. Ollis: Biochemical Engineering Fundamentals, 2nd edn (McGraw Hill, New York 1986)Google Scholar
  3. 3.
    M. Ballyk, D. Jones, D. Le, H.L. Smith: Effects of random motility on microbial growth and competition in a flow reactor, SIAM J. Appl. Math. 59, 2 (1998) pp 573–596Google Scholar
  4. 4.
    M. Ballyk, D. Jones, H.L. Smith: Microbial competition in reactors with wall attachment: a comparison of chemostat and plug flow models, Microb. Ecol. 41 (2001) pp 210–221Google Scholar
  5. 5.
    M. Ballyk, H.L. Smith: A Flow reactor with wall growth. In: Mathematical Models in Medical and Health Sciences, ed by M. Horn, G. Simonett, G. Webb (Vanderbilt University Press, Nashville, TN 1998)Google Scholar
  6. 6.
    M. Ballyk, H.L. Smith: A model of microbial growth in a plug flow reactor with wall attachment, Math. Biosci. 158 (1999) pp 95–126zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    B. Baltzis, A. Fredrickson: Competition of two microbial populations for a single resource in a chemostat when one of them exhibits wall attachment, Biotechnol. Bioeng. 25 (1983) pp 2419–2439CrossRefGoogle Scholar
  8. 8.
    R. Bakke, M.G. Trulear, J.A. Robinson, W.G. Characklis: Activity of Pseudomonas aeruginosa in biofilms: steady state, Biotechnol. Bioeng. 26 (1984) pp 1418–1424CrossRefGoogle Scholar
  9. 9.
    H. Berg: Random Walks in Biology (Princeton University Press, Princeton, NJ 1983)Google Scholar
  10. 10.
    A. Berman, R. Plemmons: Nonnegative Matrices in the Mathematical Sciences (Academic, New York 1979)zbMATHGoogle Scholar
  11. 11.
    J. Bryers, ed: Biofilms II, Process Analysis and Applications, Wiley series in Ecological and Applied Microbiology (Wiley-Liss, NY 2000)Google Scholar
  12. 12.
    W. Characklis, K. Marshall (eds): Biofilms, Wiley Series in Ecological and Applied Microbiology (Wiley, New York 1990)Google Scholar
  13. 13.
    N.G. Cogan: Effects of persister formation on bacterial response to dosing, J. Theor. Biol. 238, 3 (2006) pp 694–703Google Scholar
  14. 14.
    N.G. Cogan and J.P. Keener: The role of the biofilm matrix in structural development, Math. Med. Biol. 21 (2004) pp 147–166zbMATHCrossRefGoogle Scholar
  15. 15.
    J. Costerton: Overview of microbial biofilms, J. Indust. Microbiol. 15 (1995) pp 137–140CrossRefGoogle Scholar
  16. 16.
    J. Costerton, P. Stewart, E. Greenberg: Bacterial biofilms: a common cause of persistent infections, Science 284 (1999) pp 1318–1322CrossRefGoogle Scholar
  17. 17.
    J. Costerton, Z. Lewandowski, D. Debeer, D. Caldwell, D. Korber, G. James: Biofilms, the customized microniche, J. Bacteriol. 176 (1994) pp 2137–2142Google Scholar
  18. 18.
    J. Costerton, Z. Lewandowski, D. Caldwell, D. Korber, H. Lappin-Scott: Microbial biofilms, Annu. Rev. Microbiol. 49 (1995) pp 711–745CrossRefGoogle Scholar
  19. 19.
    O. Diekmann, J. Heesterbeek: Mathematical Epidemiology of Infectious Diseases, Model Building, Analysis and Interpretation (Wiley, Chichester 2000)Google Scholar
  20. 20.
    R. Dillon, L. Fauci, A. Fogelson, D. Gaver: Modeling biofilm processes using the immersed boundary method, J. Comput. Phys. 129 (1996) pp 57–73zbMATHCrossRefGoogle Scholar
  21. 21.
    J. Dockery, I. Klapper: Finger formation in biofilm layers, SIAM J. Appl. Math. 62 (2001) pp 853–869zbMATHMathSciNetGoogle Scholar
  22. 22.
    H.J. Eberl, D.F. Parker, M.C.M. van Loosdrecht: A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med. 3 (3) (2001) pp 161–175zbMATHGoogle Scholar
  23. 23.
    R. Freter: Interdependence of mechanisms that control bacterial colonization of the large intestine, Microecol. Ther. 14 (1984) pp 89–96Google Scholar
  24. 24.
    R. Freter: Mechanisms that control the microflora in the large intestine. In: Human Intestinal Microflora in Health and Disease, ed by D. Hentges (Academic, New York 1983)Google Scholar
  25. 25.
    R. Freter, H. Brickner, J. Fekete, M. Vickerman, K. Carey: Survival and implantation of Escherichia coli in the intestinal tract, Infect. Immun. 39 (1983) pp 686–703Google Scholar
  26. 26.
    R. Freter, H. Brickner, S. Temme: An understanding of colonization resistance of the mammalian large intestine requires mathematical analysis, Microecol. Ther. 16 (1986) pp 147–155Google Scholar
  27. 27.
    C.A. Fux, J.W. Costerton, P.S. Stewart, and P. Stoodley, Survival strategies of infectious biofilms, Trends Microbiol., 13 (2005) pp 34–40CrossRefGoogle Scholar
  28. 28.
    D. Herbert, R. Elsworth, R. Telling: The continuous culture of bacteria; a theoretical and experimental study, J. Can. Microbiol. 14 (1956) pp 601–622Google Scholar
  29. 29.
    M. Imran, D. Jones, H.L. Smith: Biofilms and the plasmid maintenance question, Math. Biosci. 193 (2005) pp 183–204zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    M. Imran, H.L. Smith: A mathematical model of gene transfer in a biofilm. In: Mathematics for Ecology and Environmental Sciences, Vol.1 (Springer, Berlin Heidelberg New York 2006)Google Scholar
  31. 31.
    D. Jones, H. Kojouharov, D. Le, H.L. Smith: Bacterial wall attachment in a flow reactor: mixed culture, Can. Appl. Math. Q. 10 (2004) pp 111–138MathSciNetGoogle Scholar
  32. 32.
    D. Jones, H. Kojouharov, D. Le, H.L. Smith: Bacterial wall attachment in a flow reactor, SIAM J. Appl. Math. 62 (2002) pp 1728–1771zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    D. Jones, H. Kojouharov, D. Le, H.L. Smith: Microbial Competition for Nutrient in a 3D Flow Reactor, Dynamics of Continuous, Discrete Impulsive Dynamical Syst. 10 (2003) pp 57–67zbMATHMathSciNetGoogle Scholar
  34. 34.
    D. Jones, H. Kojouharov, D. Le, H.L. Smith: The Freter model: a simple model of biofilm formation, J. Math. Biol. 47 (2003) pp 137–152zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    D. Jones, H.L. Smith: Microbial competition for nutrient and wall sites in plug flow, SIAM J. Appl. Math. 60 (2000) pp 1576–1600zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    J.-U. Kreft, C. Picioreanu, J. Wimpenny, M. van Loosdrecht: Individual-based modelling of biofilms, Microbiology 147 (2001) pp 2897–2912Google Scholar
  37. 37.
    C.M. Kung, B. Baltzis: The growth of pure and simple microbial competitors in a moving distributed medium, Math. Biosci. 111 (1992) pp 295–313zbMATHCrossRefGoogle Scholar
  38. 38.
    C.S. Laspidou and B.E. Rittmann: Modeling the development of biofilm density including active bacteria, inert biomass, and extracellular polymeric substances, Water Res. 38 (2004) pp 3349–3361CrossRefGoogle Scholar
  39. 39.
    D. Noguera, S. Okabe, C. Picioreanu: Biofilm modeling: present status and future directions, Water Sci. Technol. 39 (1999) pp 273–278Google Scholar
  40. 40.
    D. Noguera, G. Pizarro, D. Stahl, B. Rittman: Simulation of multispecies biofilm development in three dimensions, Wat.Sci. Tech. 39 (1999) 123–130Google Scholar
  41. 41.
    C. Picioreanu, M.C.M. van Loosdrecht, J. Heijnen: Mathematical Modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach, Biotechnol. Bioeng. 58 (1998) pp 101–116CrossRefGoogle Scholar
  42. 42.
    S. Pilyugin and P. Waltman: The simple chemostat with wall growth, SIAM J. Appl. Math. 59 (1999) pp 1552–1572zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    H.L. Smith: A semilinear hyperbolic system, Proceedings of the Mathematics Conference, ed by S. Elyadi, F. Allan, A. Elkhader, T. Muhgrabi, M. Saleh (World Scientific, Singapore 2000)Google Scholar
  44. 44.
    H.L. Smith, P. Waltman: The Theory of the Chemostat (Cambridge University Press, New York 1995)zbMATHGoogle Scholar
  45. 45.
    H.L. Smith, X.-Q. Zhao: Microbial growth in a plug flow reactor with wall attachment and cell motility, JMAA 241 (2000) pp 134–155zbMATHMathSciNetGoogle Scholar
  46. 46.
    E. Stemmons, H.L. Smith, H.L.: Competition in a chemostat with wall attachment, SIAM J. Appl. Math. 61 (2000) pp 567–595Google Scholar
  47. 47.
    P. Stewart, G. Mcfeters, C.-T. Huang: Biofilm control by antimicrobial agents, Chapter 11. In: Biofilms II: Process Analysis and Applications, ed by J. Bryers (Wiley-Liss, New York 2000)Google Scholar
  48. 48.
    S.M. Hunt, M.A. Hamilton, P.S. Stewart: A 3D model of antimicrobial action on biofilms, Water Sci. Technol. 52, 7 (2005) 143–148Google Scholar
  49. 49.
    H. Topiwala and G. Hamer: Effect of wall growth in steady-state continuous cultures, Biotechnol. Bioeng. 13 (1971) pp 919–922CrossRefGoogle Scholar
  50. 50.
    C.Y. Wen, L.T. Fan: Models for flow systems and chemical reactors. In: Chemical Processing and Engineering, vol. 3 (Marcel Dekker, New York 1975)Google Scholar
  51. 51.
    J.W.T. Wimpenny, R. Colasanti: A unifying hypothesis for the structure of microbial biofilms based on cellular automaton models, FEMS Microb. Ecol. 22 (1997) pp 1–16CrossRefGoogle Scholar

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