Stochastic Integration with Respect to fBm and Related Topics

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

Pathwise Stochastic Integration in the Fractional Sobolev-type Spaces

In this subsection we consider pathwise integrals \(\int\limits_0^T {f\left( t \right)dB_t^H } \) for processes f from the fractional Sobolev type spaces \(I_{a + }^\alpha \left( {L^p } \right)\) for some p > 1. This approach was developed by Zähle (Zah98), (Zah99), (Zah01).

Consider two nonrandom functions f and g defined on some interval \(\left[ {a,b} \right] \subset {\rm R}\) and suppose that the limits \(f\left( {u + } \right): = \lim _{\delta \downarrow 0} f\left( {u + \delta } \right)\) and \(g\left( {u - } \right): = \lim _{\delta \downarrow 0} g\left( {u - \delta } \right),a \le u \le b,\) exist. Put \(f_{a + } \left( x \right): = \left( {f\left( x \right) - f\left( {a + } \right)} \right)1_{\left( {a,b} \right)} \left( x \right),g_{b - } \left( x \right): = \left( {g\left( {b - } \right) - g\left( x \right)} \right)1_{\left( {a,b} \right)} \left( x \right).\) Suppose also that \(f_{a + } \in I_{a + }^\alpha \left( {L_p \left[ {a,b} \right]} \right),g_{b - } \in I_{b - }^{1 - \alpha } \left( {L_p \left[ {a,b} \right]} \right)\) for some \(p \ge 1,q \ge 1,1/p + 1/q \le 1,0 \le \alpha \le 1.\) Then evidently, \(D_{a + }^\alpha f_{a + } \in L_p \left[ {a,b} \right],D_{b - }^{1 - \alpha } g_{b - } \in L_q \left[ {a,b} \right].\)


Related Topic Fractional Brownian Motion Standard Brownian Motion Stochastic Integration Predictable 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Pearson Education, Upper Saddle River 2003Google Scholar
  2. 2.
    F. Arrigoni, Deterministic approximation of a stochastic metapopulation model, Adv. Appl. Prob. 35 (2003) 691–720zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J.M. Ayerbe Toledano, T. Dominguez Benavides, and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser, Basel 1997zbMATHGoogle Scholar
  4. 4.
    J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer, London 2006zbMATHGoogle Scholar
  5. 5.
    A.D. Barbour and A. Pugliese, Asymptotic behaviour of a metapopulation model, Ann. Appl. Prob. 15 (2005) 1306–1338zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    E. Born and K. Dietz, Parasite population dynamics within a dynamic host population. Prob. Theor. Rel. Fields 83 (1989) 67–85zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R. Casagrandi and M. Gatto, A persistence criterion for metapopulations, Theor. Pop. Biol. 61 (2002) 115–125zbMATHCrossRefGoogle Scholar
  8. 8.
    S.-N. Chow and J.K. Hale, Methods of Bifurcation Theory, Springer, New York 1982zbMATHGoogle Scholar
  9. 9.
    Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, and B. de Pagter, One-Parameter Semigroups, North-Holland, Amsterdam 1987Google Scholar
  10. 10.
    M.D. Clinchy, D.T. Haydon, and A.T. Smith, Pattern does not equal process: what does patch occupancy really tell us about metapopulation dynamics? Am. Nat. 159 (2002) 351–362CrossRefGoogle Scholar
  11. 11.
    E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York 1955zbMATHGoogle Scholar
  12. 12.
    W. Desch and W. Schappacher, Linearized stability for nonlinear semigroups, Differential Equations in Banach Spaces (A. Favini, E. Obrecht, eds.) 61–73, LNiM 1223, Springer, Berlin Heidelberg New York 1986CrossRefGoogle Scholar
  13. 13.
    K. Dietz, Overall population patterns in the transmission cycle of infectious disease agents, Population Biology of Infectious Diseases (R.M. Anderson, R.M. May, eds.), 87–102, Life Sciences Report 25, Springer, Berlin Heidelberg New York 1982Google Scholar
  14. 14.
    K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, Berlin Heidelberg New York 2000zbMATHGoogle Scholar
  15. 15.
    H. Engler, J. Prüss, and G.F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl. 324 (2006) 98–117zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    W. Feller, On the integro-differential equations of purely discontinuous Markoff processes, Trans. AMS 48 (1940) 488–515, Errata ibid., 58 (1945) 474Google Scholar
  17. 17.
    W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edition, Wiley, New York 1968zbMATHGoogle Scholar
  18. 18.
    W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York 1965Google Scholar
  19. 19.
    Z. Feng, L. Rong, and R.K. Swihart, Dynamics of an age-structured metapopulation model, Nat. Res. Mod. 18 (2005) 415–440zbMATHMathSciNetGoogle Scholar
  20. 20.
    M. Gilpin and I. Hanski, eds., Metapopulation Dynamics. Empirical and Theoretical Investigations, Academic Press, New York 1991Google Scholar
  21. 21.
    M.L. Greer, L. Pujo-Menjouet, and G.F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theor. Biol. 242 (2006) 598–606CrossRefMathSciNetGoogle Scholar
  22. 22.
    M.L. Greer, P. van den Driessche, L. Wang, and G.F. Webb, Effects of general incidence and polymer joining on nucleated polymerization in a prion disease model, SIAM J. Appl. Math. 68(1) (2007) 154–170zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    M. Gyllenberg and I. Hanski, Single species metapopulation dynamics: a structured model, Theor. Pop. Biol. 42 (1992) 35–61zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    K.P. Hadeler and K. Dietz, An integral equation for helminthic infections: global existence of solutions, Recent Trends in Mathematics, Reinhardsbrunn 1982 (H. Kurke, J. Mecke, H. Triebel, R. Thiele, eds.) 153–163, Teubner, Leipzig 1982Google Scholar
  25. 25.
    K.P. Hadeler and K. Dietz, Population dynamics of killing parasites which reproduce in the host, J. Math. Biol. 21 (1984) 45–65zbMATHMathSciNetGoogle Scholar
  26. 26.
    J.K. Hale, Asymptotic Behavior of Dissipative Systems. AMS, Providence 1988zbMATHGoogle Scholar
  27. 27.
    J.K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989) 388–395zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    I. Hanski and M.E. Gilpin, eds., Metapopulation Biology: Ecology, Genetics, and Evolution, Academic Press, San Diego 1996Google Scholar
  29. 29.
    E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups, AMS, Providence 1957Google Scholar
  30. 30.
    T. Kato, On the semi-groups generated by Kolmogoroff’s differential equations, J. Math. Soc. Japan 6 (1954) 12–15CrossRefGoogle Scholar
  31. 31.
    A.N. Kolmogorov, Über die analytischen Methoden der Wahrscheinlichkeitrechnung, Math. Annalen 104 (1931) 415–458zbMATHCrossRefGoogle Scholar
  32. 32.
    T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin Heidelberg New York 1976zbMATHGoogle Scholar
  33. 33.
    V.A. Kostizin, Symbiose, parasitisme et évolution (étude mathématique), Herman, Paris 1934, translated in The Golden Age of Theoretical Ecology (F. Scudo, J. Ziegler, eds.), 369–408, Lecture Notes in Biomathematics 22, Springer, Berlin Heidelberg New York 1978Google Scholar
  34. 34.
    M. Kretzschmar, A renewal equation with a birth-death process as a model for parasitic infections, J. Math. Biol. 27 (1989) 191–221zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    P. Laurençot and C. Walker, Well-posedness for a model of prion proliferation dynamics, J. Evol. Eqn. 7 (2007) 241–264zbMATHCrossRefGoogle Scholar
  36. 36.
    R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bull. Entom. Soc. Am. 15 (1969) 237–240Google Scholar
  37. 37.
    R. Levins, Extinction, Some Mathematical Questions in Biology (M.L. Gerstenhaber, ed.) 75–107, Lectures on Mathematics in the Life Sciences 2, AMS 1970Google Scholar
  38. 38.
    M. Martcheva and H.R. Thieme, A metapopulation model with discrete patch-size structure, Nat. Res. Mod. 18 (2005), 379–413zbMATHMathSciNetGoogle Scholar
  39. 39.
    M. Martcheva, H.R. Thieme, and T. Dhirasakdanon, Kolmogorov’s differential equations and positive semigroups on first moment sequence spaces, J. Math. Biol. 53 (2006) 642–671zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    R.H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York 1976zbMATHGoogle Scholar
  41. 41.
    J. Masel, V.A.A. Jansen, and M.A. Nowak, Quantifying the kinetic parameters of prion replication, Biophys. Chem. 77 (1999) 139–152CrossRefGoogle Scholar
  42. 42.
    J.A.J. Metz and M. Gyllenberg, How should we define fitness in structured metapopulation models? Including an application to the calculation of evolutionary stable dispersal strategies, Proc. R. Soc. Lond. B 268 (2001) 499–508Google Scholar
  43. 43.
    A. Moilanen, A.T. Smith, and I. Hanski, Long term dynamics in a metapopulation of the American pika, Am. Nat. 152 (1998) 530–542CrossRefGoogle Scholar
  44. 44.
    J. Nagy, Evolutionary Attracting Dispersal Strategies in Vertebrate Metapopulations, Ph.D. Thesis, Arizona State University, Tempe 1996Google Scholar
  45. 45.
    M.A. Nowak, D.C. Krakauer, A. Klug, and R.M. May, Prion infection dynamics, Integr Biol 1 (1998) 3–15CrossRefGoogle Scholar
  46. 46.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin Heidelberg New York 1982Google Scholar
  47. 47.
    J. Prüss, L. Pujo-Menjouet, G.F. Webb, and R. Zacher, Analysis of a model for the dynamics of prions, Discr. Cont. Dyn. Syst. B 6 (2006) 215–225Google Scholar
  48. 48.
    A. Pugliese, Coexistence of macroparasites without direct interactions, Theor. Pop. Biol. 57 (2000) 145–165zbMATHCrossRefGoogle Scholar
  49. 49.
    A. Pugliese, Virulence evolution in macro-parasites, Mathematical Approaches for Emerging and Reemerging Infectious Diseases (C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, A.-A. Yakubo, eds.) 193–213, Springer, Berlin Heidelberg New York 2002Google Scholar
  50. 50.
    G.E.H. Reuter, Denumerable Markov processes and the associated contraction semigroups on , Acta Mathematica 97 (1957), 1–46zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    G.E.H. Reuter and W. Ledermann, On the differential equations for the transition probabilities of Markov processes with enumerable many states, Proc. Camb. Phil. Soc. 49 (1953) 247–262zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    G.R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, Berlin Heidelberg New York 2002zbMATHGoogle Scholar
  53. 53.
    G. Simonett and C. Walker, On the solvability of a mathematical model for prion proliferation, J. Math. Anal. Appl. 324 (2006) 580–603zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    A.T. Smith and M.E. Gilpin: Spatially correlated dynamics in a pika metapopulation, [28], 407–428Google Scholar
  55. 55.
    H.R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations 3 (1990) 1035–1066zbMATHMathSciNetGoogle Scholar
  56. 56.
    H.R. Thieme, Persistence under relaxed point-dissipativity (with applications to an epidemic model), SIAM J. Math. Anal. 24 (1993) 407–435zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    H.R. Thieme, Remarks on resolvent positive operators and their perturbation, Disc. Cont. Dyn. Sys. 4 (1998) 73–90zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003zbMATHGoogle Scholar
  59. 59.
    H.R. Thieme and J. Voigt, Stochastic semigroups: their construction by perturbation and approximation, Positivity IV – Theory and Applications (M.R. Weber, J. Voigt, eds.), 135–146, Technical University of Dresden, Dresden 2006Google Scholar
  60. 60.
    J. Voigt, On substochastic C 0-semigroups and their generators, Transp. Theory Stat. Phys. 16 (1987) 453–466zbMATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    C. Walker, Prion proliferation with unbounded polymerization rates, Electron. J. Diff. Equations, Conference 15 (2007) 387–397Google Scholar
  62. 62.
    X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, Berlin Heidelberg New York 2003zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Personalised recommendations