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On Fractional Lévy Processes: Tempering, Sample Path Properties and Stochastic Integration

Abstract

We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP II, respectively). TFLP and TFLP II make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Lévy process. Accordingly, the increment processes of TFLP and TFLP II display semi-long range dependence. We establish the sample path properties of TFLP and TFLP II. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP II, which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.

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References

  1. 1.

    Giraitis, L., Kokoszka, P., Leipus, R.: Stationary ARCH models: dependence structure and central limit theorem. Econom. Theory 16(1), 3–22 (2000)

  2. 2.

    Mandelbrot, B., Van Ness, J.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968)

  3. 3.

    Ciuciu, P., Abry, P., He, B.: Interplay between functional connectivity and scale-free dynamics in intrinsic fMRI networks. Neuroimage 95, 248–263 (2014)

  4. 4.

    Foufoula-Georgiou, E., Kumar, P.: Wavelets in Geophysics, vol. 4. Academic Press, Cambridge (2014)

  5. 5.

    Ivanov, P., Nunes Amaral, L., Goldberger, A., Havlin, S., Rosenblum, M., Struzik, Z., Stanley, H.: Multifractality in human heartbeat dynamics. Nature 399(6735), 461–465 (1999)

  6. 6.

    Mandelbrot, B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974)

  7. 7.

    Taqqu, M., Willinger, W., Sherman, R.: Proof of a fundamental result in self-similar traffic modeling. ACM SIGCOMM Comput. Commun. Rev. 27(2), 5–23 (1997)

  8. 8.

    Flandrin, P.: Wavelet analysis and synthesis of fractional brownian motion. IEEE Trans. Inf. Theory 38, 910–917 (1992)

  9. 9.

    Wornell, G., Oppenheim, A.: Estimation of fractal signals from noisy measurements using wavelets. IEEE Trans. Signal Process. 40(3), 611–623 (1992)

  10. 10.

    Embrechts, P., Maejima, M.: Selfsimilar Processes. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ (2002)

  11. 11.

    Pipiras, V., Taqqu, M.S.: Long-Range Dependence and Self-similarity. Cambridge University Press, Cambridge (2017)

  12. 12.

    Beran, J., Feng, Y., Ghosh, S., Kulik, R.: Long Memory Processes: Probabilistic Properties and Statistical Models. Springer, Heidelberg (2013)

  13. 13.

    Dobrushin, R., Major, P.: Non-central limit theorems for non-linear functional of Gaussian fields. Probab. Theory Relat. Fields 50(1), 27–52 (1979)

  14. 14.

    Granger, C., Joyeux, R.: An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1(1), 15–29 (1980)

  15. 15.

    Moulines, E., Roueff, F., Taqqu, M.: A wavelet Whittle estimator of the memory parameter of a nonstationary Gaussian time series. Ann. Stat. 36, 1925–1956 (2008)

  16. 16.

    Taqqu, M.S.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Probab. Theory Relat. Fields 31(4), 287–302 (1975)

  17. 17.

    Taqqu, M.S.: Convergence of integrated processes of arbitrary Hermite rank. Probab. Theory Relat. Fields 50(1), 53–83 (1979)

  18. 18.

    Samorodnitsky, G., Taqqu, M.: Stable non-Gaussian random processes. Chapman and Hall, New York (1994)

  19. 19.

    Bardet, J.-M., Tudor, C.: Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process. J. Multivar. Anal. 131, 1–16 (2014)

  20. 20.

    Clausel, M., Roueff, F., Taqqu, M.S., Tudor, C.: Wavelet estimation of the long memory parameter for Hermite polynomial of Gaussian processes. ESAIM: Probab. Stat. 18, 42–76 (2014)

  21. 21.

    Kolmogorov, A.N.: The Wiener spiral and some other interesting curves in Hilbert space. Dokl. Akad. Nauk SSSR 26, 115–118 (1940)

  22. 22.

    Kolmogorov, A.N.: The local structure of turbulence in an incompressible fluid at very high Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303 (1941)

  23. 23.

    Friedlander, S.K., Topper, L.: Turbulence: Classic Papers on Statistical Theory. Interscience Publishers, Geneva (1961)

  24. 24.

    Shiryaev, A.N.: Kolmogorov and the Turbulence. Centre for Mathematical Physics and Stochastics, University of Aarhus, Aarhus (1999)

  25. 25.

    Von Kármán, T.: Progress in the statistical theory of turbulence. Proc. Natl. Acad. Sci. USA 34(11), 530 (1948)

  26. 26.

    U.S. Department of Defense: Flying qualities of piloted aircraft, military standard MIL-STD-1797A (2004)

  27. 27.

    Penner, S., Williams, F., Libby, P., Nemat-Nasser, S.: Von Kármán’s work: the later years (1952 to 1963) and legacy. Ann. Rev. Fluid Mech. 41, 1–15 (2009)

  28. 28.

    Beaupuits, J.P., Otárola, A., Rantakyrö, F., Rivera, R., Radford, S., Nyman, L.: Analysis of Wind Data Gathered at Chajnantor. ALMA Memo 497, pp. 1–20. National Radio Astronomy Observatory, Charlottesville (2004)

  29. 29.

    Jang, J.-J., Guo, J.-S.: Analysis of maximum wind force for offshore structure design. J. Mar. Sci. Technol. 7(1), 43–51 (1999)

  30. 30.

    Norton, D.J., Wolff, C.V., et al.: Mobile offshore platform wind loads. In: Offshore Technology Conference. Offshore Technology Conference (1981)

  31. 31.

    Davenport, A.: The spectrum of horizontal gustiness near the ground in high winds. Q. J. R. Meteorol. Soc. 87(372), 194–211 (1961)

  32. 32.

    Norton, D., Wolff, C.: Mobile offshore platform wind loads. In: Offshore Technology Conference. Offshore Technology Conference (1981)

  33. 33.

    Li, Y., Kareem, A.: ARMA systems in wind engineering. Probab. Eng. Mech. 5(2), 49–59 (1990)

  34. 34.

    Beaupuits, J., Otárola, A., Rantakyrö, F.T., Rivera, R.C., Radford, S.J.E., Nyman, L.: Analysis of Wind Data Gathered at Chajnantor. ALMA Memo 497. National Radio Astronomy Observatory, Charlottesville (2004)

  35. 35.

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

  36. 36.

    Kou, S.: Stochastic modeling in nanoscale biophysics: subdiffusion within proteins. Ann. Appl. Stat. 2, 501–535 (2008)

  37. 37.

    Sokolov, I.: Statistics and the single molecule. Physics 1, 8 (2008)

  38. 38.

    Didier, G., McKinley, S.A., Hill, D.B., Fricks, J.: Statistical challenges in microrheology. J. Time Ser. Anal. 33(5), 724–743 (2012)

  39. 39.

    Grebenkov, D.S., Vahabi, M., Bertseva, E., Forró, L., Jeney, S.: Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium. Phys. Rev. E 88(4), 040701 (2013)

  40. 40.

    Zhang, K., Crizer, K., Schoenfisch, M.H., Hill, D.B., Didier, G.: Fluid heterogeneity detection based on the asymptotic distribution of the time-averaged mean squared displacement in single particle tracking experiments. J. Phys. A 51, 445601 (2018)

  41. 41.

    Piryatinska, A., Sanchev, A., Woyczynski, W.A.: Models of anomalous diffusion: the subdiffusive case. Physica A 349, 375–420 (2005)

  42. 42.

    Stanislavsky, A., Weron, K., Weron, A.: Diffusion and relaxation controlled by tempered \(\alpha \)-stable processes. Phys. Rev. E 78(5), 051106 (2008)

  43. 43.

    Baeumer, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233(10), 2438–2448 (2010)

  44. 44.

    Sandev, T., Chechkin, A., Kantz, H., Metzler, R.: Diffusion and Fokker–Planck–Smoluchowski equations with generalized memory kernel. Fract. Calc. Appl. Anal. 18(4), 1006–1038 (2015)

  45. 45.

    Wu, X., Deng, W., Barkai, E.: Tempered fractional Feynman–Kac equation: theory and examples. Phys. Rev. E 93(3), 032151 (2016)

  46. 46.

    Chen, Y., Wang, X., Deng, W.: Localization and ballistic diffusion for the tempered fractional Brownian–Langevin motion. J. Stat. Phys. 169, 18–37 (2017)

  47. 47.

    Liemert, A., Sandev, T., Kantz, H.: Generalized Langevin equation with tempered memory kernel. Physica A 466, 356–369 (2017)

  48. 48.

    Chen, Y., Wang, X., Deng, W.: Resonant behavior of the generalized Langevin system with tempered Mittag–Leffler memory kernel. J. Phys. A 51(18), 185201 (2018)

  49. 49.

    Saxton, M.J.: A biological interpretation of transient anomalous subdiffusion. I. Qualitative model. Biophys. J. 92(4), 1178–1191 (2007)

  50. 50.

    Molina-Garcia, D., Sandev, T., Safdari, H., Pagnini, G., Chechkin, A., Metzler, R.: Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers. New J. Phys. 20(10), 103027 (2018)

  51. 51.

    Taylor, G.I.: Diffusion by continuous movements. Proc. Lond. Math. Soc. s2–20(1), 196–212 (1922)

  52. 52.

    Xia, H., Francois, N., Punzmann, H., Shats, M.: Lagrangian scale of particle dispersion in turbulence. Nat. Commun. 4, 1–8 (2013)

  53. 53.

    Boniece, B.C., Didier, G., Sabzikar, F.: Tempered fractional Brownian motion: wavelet estimation, modeling and testing. To appear in Appl. Comput. Harmon. Anal. 1–51 (2019)

  54. 54.

    Meerschaert, M., Sabzikar, F.: Tempered fractional Brownian motion. Stat. Probab. Lett. 83(10), 2269–2275 (2013)

  55. 55.

    Sabzikar, F., Surgailis, D.: Tempered fractional Brownian and stable motions of second kind. Stat. Probab. Lett. 132, 17–27 (2018)

  56. 56.

    Meerschaert, M.M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, 17 (2008)

  57. 57.

    Meerschaert, M., Sabzikar, F., Phanikumar, M., Zeleke, A.: Tempered fractional time series model for turbulence in geophysical flows. J. Stat. Mech. Theory Exp. 2014(9), P09023 (2014)

  58. 58.

    Fricks, J., Yao, L., Elston, T., Forest, M.G.: Time-domain methods for diffusive transport in soft matter. SIAM J. Appl. Math. 69(5), 1277–1308 (2009)

  59. 59.

    Francois, N., Xia, H., Punzmann, H., Combriat, T., Shats, M.: Inhibition of wave-driven two-dimensional turbulence by viscoelastic films of proteins. Phys. Rev. E 92, 023027 (2015)

  60. 60.

    Xia, H., Francois, N., Punzmann, H., Shats, M.: Taylor particle dispersion during transition to fully developed two-dimensional turbulence. Phys. Rev. Lett. 112, 104501 (2014)

  61. 61.

    Meerschaert, M., Sabzikar, F.: Stochastic integration with respect to tempered fractional Brownian motion. Stoch. Process. Appl. 124(7), 2363–2387 (2014)

  62. 62.

    Zeng, C., Yang, Q., Chen, Y.: Bifurcation dynamics of the tempered fractional Langevin equation. Chaos 26(8), 084310 (2016)

  63. 63.

    Boniece, B.C., Sabzikar, F., Didier, G.: Tempered fractional Brownian motion: wavelet estimation and modeling of geophysical flows. In: IEEE Statistical Signal Processing Workshop—Freiburg, Germany. IEEE, pp. 1–5 (2018)

  64. 64.

    Barndorff-Nielsen, O.: Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lond. A 353(1674), 401–419 (1977)

  65. 65.

    Barndorff-Nielsen, O.: Models for non-Gaussian variation, with applications to turbulence. Proc. R. Soc. Lond. A 368(1735), 501–520 (1979)

  66. 66.

    Barndorff-Nielsen, O., Jensen, J.L., Sørensen, M.: Wind shear and hyperbolic distributions. Bound. Layer Meteorol. 49(4), 417–431 (1989)

  67. 67.

    Barndorff-Nielsen, O., Jensen, J.L., Sørensen, M.: Parametric modelling of turbulence. Philos. Trans. R. Soc. Lond. A 332(1627), 439–455 (1990)

  68. 68.

    Barndorff-Nielsen, O., Jensen, J.L., Sørensen, M.: A statistical model for the streamwise component of a turbulent velocity field. Ann. Geophys. 11, 99–103 (1993)

  69. 69.

    Skyum, P., Christiansen, C., Blaesild, P.: Hyperbolic distributed wind, sea-level and wave data. J. Coast. Res. 6, 883–889 (1996)

  70. 70.

    Barndorff-Nielsen, O.: Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Stat. 24(1), 1–13 (1997)

  71. 71.

    Sabzikar, F.: Tempered Hermite process. Mod. Stoch. Theory Appl. 2, 327–341 (2015)

  72. 72.

    Rosiński, J.: Tempering stable processes. Stoch. Process. Appl. 117(6), 677–707 (2007)

  73. 73.

    Bianchi, M.L., Rachev, S.T., Kim, Y.S., Fabozzi, F.J.: Tempered stable distributions and processes in finance: numerical analysis. In: Mathematical and Statistical Methods for Actuarial Sciences and Finance, pp. 33–42 (2010)

  74. 74.

    Gajda, J., Magdziarz, M.: Fractional Fokker–Planck equation with tempered \(\alpha \)-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E 82, 011117 (2010)

  75. 75.

    Rosiński, J., Sinclair, J.: Generalized tempered stable processes. Stabil. Probab. 90, 153–170 (2010)

  76. 76.

    Kawai, R., Masuda, H.: Infinite variation tempered stable Ornstein–Uhlenbeck processes with discrete observations. Commun. Stat. Simul. Comput. 41(1), 125–139 (2012)

  77. 77.

    Küchler, U., Tappe, S.: Tempered stable distributions and processes. Stoch. Process. Appl. 123(12), 4256–4293 (2013)

  78. 78.

    Mantegna, R.N., Stanley, H.E.: Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight. Phys. Rev. Lett. 73(22), 2946 (1994)

  79. 79.

    Chechkin, A.V., Gonchar, V.Y., Klafter, J., Metzler, R.: Natural cutoff in Lévy flights caused by dissipative nonlinearity. Phys. Rev. E 72(1), 010101 (2005)

  80. 80.

    Benassi, A., Cohen, S., Istas, J.: Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8(1), 97–115 (2002)

  81. 81.

    Brockwell, P.J., Marquardt, T.: Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Stat. Sin. 15, 477–494 (2005)

  82. 82.

    Marquardt, T.: Fractional Lévy processes with an application to long memory moving average processes. Bernoulli 12(6), 1099–1126 (2006)

  83. 83.

    Lacaux, C., Loubes, J.-M.: Hurst exponent estimation of fractional Lévy motion. ALEA: Latin Am. J. Probab. Math. Stat. 3, 143–164 (2007)

  84. 84.

    Bender, C., Marquardt, T.: Stochastic calculus for convoluted Lévy processes. Bernoulli 14(2), 499–518 (2008)

  85. 85.

    Barndorff-Nielsen, O.E., Schmiegel, J.: Time change, volatility, and turbulence. In: Mathematical Control Theory and Finance, pp. 29–53. Springer, New York (2008)

  86. 86.

    Suciu, N.: Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields. Phys. Rev. E 81(5), 056301 (2010)

  87. 87.

    Magdziarz, M., Weron, A.: Ergodic properties of anomalous diffusion processes. Ann. Phys. 326(9), 2431–2443 (2011)

  88. 88.

    Zhang, S., Lin, Z., Zhang, X.: A least squares estimator for Lévy-driven moving averages based on discrete time observations. Commun. Stat. Theory Methods 44(6), 1111–1129 (2015)

  89. 89.

    Xu, Y., Li, Y., Zhang, H., Li, X., Kurths, J.: The switch in a genetic toggle system with Lévy noise. Sci. Rep. 6, 31505 (2016)

  90. 90.

    Fink, H.: Conditional distributions of Mandelbrot–Van Ness fractional Lévy processes and continuous-time ARMA-GARCH-type models with long memory. J. Time Ser. Anal. 37(1), 30–45 (2016)

  91. 91.

    Bender, C., Knobloch, R., Oberacker, P.: Maximal inequalities for fractional Lévy and related processes. Stoch. Anal. Appl. 33(4), 701–714 (2015)

  92. 92.

    Chevillard, L.: Regularized fractional Ornstein–Uhlenbeck processes and their relevance to the modeling of fluid turbulence. Phys. Rev. E 96, 033111 (2017)

  93. 93.

    Pipiras, V., Taqqu, M.S.: Integration questions related to fractional Brownian motion. Probab. Theory Relat. Fields 118(2), 251–291 (2000)

  94. 94.

    Meerschaert, M.M., Sikorskii, A.: Stoch. Models Fract. Calc., vol. 43. Walter de Gruyter, Berlin (2011)

  95. 95.

    Oldham, K., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

  96. 96.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. CRC Press, Boca Raton (1993)

  97. 97.

    Cartea, Á., del Castillo-Negrete, D.: Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76(4), 041105 (2007)

  98. 98.

    Rozanov, Y.A.: Stationary Random Processes. Holden-Day, San Francisco (1967)

  99. 99.

    Sato, K.-I., Ken-Iti, S.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

  100. 100.

    Rajput, B.S., Rosinski, J.: Spectral representations of infinitely divisible processes. Probab. Theory Relat. Fields 82(3), 451–487 (1989)

  101. 101.

    Klüppelberg, C., Matsui, M.: Generalized fractional Lévy processes with fractional Brownian motion limit. Adv. Appl. Probab. 47(4), 1108–1131 (2015)

  102. 102.

    Barndorff-Nielsen, J., Schmiegel, O.E.: Brownian semistationary processes and volatility/intermittency. Radon Ser. Comput. Appl. Math. 8, 1–26 (2009)

  103. 103.

    Barndorff-Nielsen, O.E.: Assessing gamma kernels and BSS/LSS processes. CREATES Res. Pap. 2016–9, 1–17 (2016)

  104. 104.

    Marquardt, T.M.: Fractional Lévy Processes, CARMA Processes and Related Topics. PhD thesis, Technische Universität München (2006)

  105. 105.

    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, vol. 113. Springer, New York (2012)

  106. 106.

    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2006)

  107. 107.

    Sabzikar, F., Wang, Q., Phillips, P.C.: Asymptotic theory for near integrated process driven by tempered linear process. Submitted (2019)

  108. 108.

    Rosinski, J.: On path properties of certain infinitely divisible processes. Stoch. Process. Appl. 33(1), 73–87 (1989)

  109. 109.

    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, New York (2007)

  110. 110.

    Basse, A., Pedersen, J.: Lévy driven moving averages and semimartingales. Stoch. Process. Appl. 119(9), 2970–2991 (2009)

  111. 111.

    Cheridito, P.: Gaussian moving averages, semimartingales and option pricing. Stoch. Process. Appl. 109(1), 47–68 (2004)

  112. 112.

    Protter, P.E.: Stochastic differential equations. In: Stochastic Integration and Differential Equations. Springer, New York (2003)

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Acknowledgements

Farzad Sabzikar would like to thank Alex Lindner for fruitful discussions and for providing the proofs of Proposition 2.5 and Theorem 2.6. The authors are also grateful to an anonymous reviewer for the comments and suggestions. Gustavo Didier was partially supported by the Prime Award No. W911NF–14–1–0475 from the Biomathematics subdivision of the Army Research Office, USA.

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Correspondence to Gustavo Didier.

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Communicated by Eric A. Carlen.

Appendix: Proofs

Appendix: Proofs

Proof of Proposition 2.3

The proof of (2.7) follows by a similar argument of Proposition 2.3 in [54] and hence we omit the details. To show (2.9), apply the covariance function formula (2.7) in Proposition 2.3 for \(s=t\) to arrive at

$$\begin{aligned} \mathrm{Var}\big [S^{I}_{d,\lambda }(t) \big ] = \frac{ {\mathbb {E}}( L(1)^2 ) }{ \Gamma (1+d)^2 } \Bigg [ \frac{ 2\Gamma (1+2d) }{ (2\lambda )^{1+2d} } - \frac{2\Gamma (1+d)}{ \sqrt{\pi } } \Big ( \frac{1}{ 2\lambda }\Big )^{d+\frac{1}{2} } |t|^{d+\frac{1}{2}} K_{d+\frac{1}{2}} (\lambda t)\Bigg ]. \end{aligned}$$
(A.1)

The second term inside the bracket tends to zero as \(t\rightarrow \infty \), since

$$\begin{aligned} K_{d+\frac{1}{2}} (\lambda t) \sim \sqrt{\frac{\pi }{2\lambda t}} e^{-\lambda t}. \end{aligned}$$

Hence, relation (2.9) holds, as claimed. \(\square \)

Proof of Proposition 2.5

Starting from the definition of TFLP, we can use integration by parts (see [82, p. 1106]) to write

$$\begin{aligned} \begin{aligned} \Gamma (d+1) S^{I}_{d,\lambda }(t)&= \int _{{\mathbb {R}}}\big [ e^{-\lambda (t-x)_{+}} (t-x)_{+}^{d} - e^{-\lambda (-x)_{+}} (-x)_{+}^{d} \big ]\ dL(x)\\&= \int _{-\infty }^{t-} e^{-\lambda (t-x)} (t-x)^{d} dL(x) - \int _{-\infty }^{0-} e^{\lambda x} (-x)^{d} dL(x) \\&= \lim _{u\uparrow t}\Big ( e^{-\lambda (t-u)} (t-u)^{d} L(u) - \int _{-\infty }^{u} L(u) d(e^{-\lambda (t-u)} (t-u)^{d}) \Big ) \\&\quad - \lim _{u\uparrow 0}\Big ( e^{\lambda u} (-u)^{d} L(u) - \int _{-\infty }^{u} L(u) d(e^{\lambda u} (-u)^{d}) \Big ). \end{aligned} \end{aligned}$$
(A.2)

Using [99, Proposition 47.11], we have \(e^{\lambda v}L(v)\rightarrow 0\) as \(v \rightarrow 0\). Hence, for \(d>0\),

$$\begin{aligned} \lim _{u\uparrow t} e^{-\lambda (t-u)} (t-u)^{d} L(u) = \lim _{u\uparrow 0} e^{\lambda u} (-u)^{d} L(u) = 0. \end{aligned}$$

Therefore, we can reexpress (A.2) as

$$\begin{aligned} \begin{aligned}&\quad \quad - \lim _{u\uparrow t} \int _{-\infty }^{u} L(u) \Big (-d e^{-\lambda (t-u)} (t-u)^{d-1} + \lambda e^{-\lambda (t-u)} (t-u)^{d} \Big ) du\\&\quad \quad + \lim _{u\uparrow 0} \int _{-\infty }^{u} L(u) \Big (-d e^{\lambda u} (-u)^{d-1} + \lambda e^{\lambda u} (-u)^{d} \Big ) du\\&= d \int _{{\mathbb {R}}} L(u) \big [ e^{-\lambda (t-u)_{+}} (t-u)_{+}^{d-1} - e^{-\lambda (-u)_{+}} (-u)_{+}^{d-1} \big ]\ du\\&\quad \quad -\lambda \int _{{\mathbb {R}}} L(u) \big [ e^{-\lambda (t-u)_{+}} (t-u)_{+}^{d} - e^{-\lambda (-u)_{+}} (-u)_{+}^{d} \big ]\ du.\\ \end{aligned} \end{aligned}$$

Hence, (2.11) holds.

To show the continuity of the process (2.11), without loss of generality fix \(t\in (a,b) \subseteq {\mathbb {R}}_+\). Rewrite the first term in the expression (2.11) as

$$\begin{aligned} \Big \{\int ^{a}_{-\infty } +\int ^{t}_a \Big \}\Big ( e^{-\lambda (t-x)_{+}}{(t-x)_{+}^{d-1}}-e^{-\lambda (-x)_{+}}{(-x)_{+}^{d-1}}\Big )\ L(x)\ dx. \end{aligned}$$

We want to show that this expression is continuous as a function of t. On one hand, the mapping \(t\mapsto \int _{-\infty }^a L(u) \big [ e^{-\lambda (t-u)_{+}} (t-u)_{+}^{d-1} - e^{-\lambda (-u)_{+}} (-u)_{+}^{d-1} \big ]\ du\) is continuous. This is a consequence of the dominated convergence theorem, since

$$\begin{aligned}&{\mathbf {1}}_{(-\infty ,a]}(u)| L(u)|\left| \big [ e^{-\lambda (t-u)_{+}} (t-u)_{+}^{d-1} - e^{-\lambda (-u)_{+}} (-u)_{+}^{d-1}\right| \\&\quad \le {\mathbf {1}}_{(-\infty ,a]}(u)| L(u)| \Big ( e^{-\lambda (a-u)} (b-u)_{+}^{d-1}+ e^{-\lambda (-u)_{+}} (-u)_{+}^{d-1}\Big )\in L^1({\mathbb {R}}), \end{aligned}$$

where we use the fact that L is locally bounded. On the other hand, by making the change of variable \(z = t-u\),

$$\begin{aligned} \int ^t_a L(u) e^{-\lambda (t-u)_{+}} (t-u)_{+}^{d-1} \ du = \int _{{\mathbb {R}}} 1_{[0,t-a]}(z) L(t-z) e^{-\lambda z} z^{d-1} \ dz. \end{aligned}$$
(A.3)

However, the integrand in (A.3) is bounded in absolute value by

$$\begin{aligned} \sup _{w \in (a,b)}|L(w)| 1_{[0,b-a]}(z) e^{-\lambda z} z^{d-1} \in L^1({\mathbb {R}}). \end{aligned}$$

Therefore, by the dominated convergence theorem, the mapping \( t \mapsto \int _a^t L(u) \big [ e^{-\lambda (t-u)_{+}} (t-u)_{+}^{d-1} - e^{-\lambda (-u)_{+}} (-u)_{+}^{d-1} \big ]\ du\) is also continuous. Hence, the first term in the expression (2.11) is continuous as a function of t, as claimed. Again by the dominated convergence theorem, the second term in the expression (2.11) is also continuous as a function of t. This establishes that the process (2.11) is continuous. \(\square \)

Proof of Theorem 2.6

First, we establish (a). We use the modification of \(S^{I}_{d,\lambda }\) given in Theorem 2.5 to write

$$\begin{aligned} \begin{aligned} |S^{I}_{d,\lambda }(t) - S^{I}_{d,\lambda }(s)|&\le \frac{1}{\Gamma (d)} \int _{{\mathbb {R}}} \Big | e^{-\lambda (t-u)_{+}} (t-u)_{+}^{d-1} - e^{-\lambda (s-u)_{+}} (s-u)_{+}^{d-1} \Big | \Big |L(u)\Big | \ du\\&\quad + \frac{\lambda }{\Gamma (d+1)} \int _{{\mathbb {R}}} \Big | e^{-\lambda (t-u)_{+}} (t-u)_{+}^{d} - e^{-\lambda (s-u)_{+}} (s-u)_{+}^{d} \Big | \Big |L(u)\Big | \ du. \end{aligned} \end{aligned}$$
(A.4)

Recall that \(0 < d \le 1/2\). For notational simplicity, consider a parameter \(\beta \), which can be interpreted either as d or \(d-1\), i.e. \(\beta \in (-1,-1/2]\cup (0,1/2]\). Define

$$\begin{aligned} W_{\beta }(s,t) = \int _{{\mathbb {R}}} \Big | e^{-\lambda (t-u)_{+}} (t-u)_{+}^{\beta } - e^{-\lambda (s-u)_{+}} (s-u)_{+}^{\beta } \Big | \Big |L(u)\Big | \ du. \end{aligned}$$

For s, t satisfying \(-T \le s \le t \le T\), we obtain

$$\begin{aligned} \begin{aligned} W_{\beta }(s,t) =&\int _{s}^{t} e^{-\lambda (t-u)} (t-u)^{\beta } \Big |L(u)\Big | \ du\\&+ \int _{-\infty }^{s} \Big | e^{-\lambda (t-u)} (t-u)^{\beta } - e^{-\lambda (s-u)} (s-u)^{\beta } \Big | \Big |L(u)\Big | \ du\\ \le&\sup _{|u|\le T} |L(u)| \int _{s}^{t} e^{-\lambda (t-u)} (t-u)^{\beta } \ du \\&+ \int _{-\infty }^{s} e^{-\lambda (t-u)} \Big | (t-u)^{\beta } - (s-u)^{\beta } \Big | \Big |L(u)\Big | \ du\\&+ \int _{-\infty }^{s} (s-u)^{\beta } \Big | e^{-\lambda (t-u)} - e^{-\lambda (s-u)} \Big | \Big |L(u)\Big | \ du. \end{aligned} \end{aligned}$$

Using the substitution \(h=t-s\), we get

$$\begin{aligned} \begin{aligned} W_{\beta }(s,t)&\le \frac{h^{\beta +1}}{\beta +1}\sup _{|u|\le T} |L(u)| + e^{-\lambda h}\int _{-\infty }^{s} e^{-\lambda (s-u)} \Big | (h+s-u)^{\beta } - (s-u)^{\beta } \Big | \Big |L(u)\Big | \ du\\&\quad + \int _{-\infty }^{s} | e^{-\lambda h}-1 | (s-u)^{\beta } e^{-\lambda (s-u)} \Big |L(u)\Big | \ du\\&= \frac{h^{\beta +1}}{\beta +1}\sup _{|u|\le T} |L(u)| + e^{-\lambda h}\int _{0}^{\infty } e^{-\lambda v} \Big | (h+v)^{\beta } - v^{\beta } \Big | \Big |L(s-v)\Big | \ dv\\&\quad + ( 1- e^{-\lambda h} ) \int _{0}^{\infty } v^{\beta } e^{-\lambda v} \Big | L(s-v) \Big | \ dv\\&=: I_1 + I_2 + I_3. \end{aligned} \end{aligned}$$
(A.5)

Since L is locally bounded, then

$$\begin{aligned} I_1 \le C_{1}(\omega ) h^{\beta +1} \end{aligned}$$
(A.6)

for an almost surely finite random variable \(C_1\). Next, observe that

$$\begin{aligned} \limsup _{|v|\rightarrow \infty }\quad \frac{|L(v)|}{|v|} = 0 \end{aligned}$$
(A.7)

by [99, Proposition 48.9]. In particular, the integrands appearing in \(I_2\) and \(I_3\) are finite almost surely (since \(\lambda >0\)). Since \(( 1- e^{-\lambda h} ) \le \lambda h\) for \(h>0\), we conclude that there is an almost sure finite continuous random variable \(C_3(\omega )\) such that

$$\begin{aligned} I_{3}(\omega ) \le C_{3}(\omega ) h \end{aligned}$$
(A.8)

for all \(-T\le s\le t\le T\).

In regard to \(I_2\), consider the decomposition

$$\begin{aligned} \Big \{\int _{0}^{1} + \int _{1}^{\infty } \Big \} \quad e^{-\lambda v} | (h+v)^{\beta } - v^{\beta } | |L(s-v)| \ dv. \end{aligned}$$
(A.9)

By the mean value theorem, for each \(v>0\) there exists some \(v_{h}\in [v,v+h]\) such that \((h+v)^{\beta } - v^{\beta } = h\beta {v_{h}}^{\beta -1}\). Thus, we can bound the second integral in (A.9) by

$$\begin{aligned} \begin{aligned}&\int _{1}^{\infty } e^{-\lambda v} | (h+v)^{\beta } - v^{\beta } | |L(s-v)| \ dv \\&\quad \le \int _{1}^{\infty } e^{-\lambda v} h \beta \max \{v^{\beta }, (v+h)^{\beta }\} |L(s-v)| \ dv \le C_{2,1} h, \end{aligned} \end{aligned}$$
(A.10)

\(-T \le s\le t \le T\). In (A.10), \(C_{2,1}\) is an almost surely finite random variable as a consequence of (A.7). On the other hand, the first integral in (A.9) can be bounded by

$$\begin{aligned} \begin{aligned}&\int _{0}^{1} e^{-\lambda v} \Big | (h+v)^{\beta } - v^{\beta } \Big | \Big |L(s-v)\Big | \ dv \\&\quad \le \sup _{v\in [-T-1,T]} |L(v)| \int _{0}^{1} \Big | (h+v)^{\beta } - v^{\beta } \Big | \ dv \\&\quad = \sup _{v\in [-T-1,T]} |L(v)| \Big | \int _{0}^{1} (h+v)^{\beta }\ dv - \int _0^1 v^{\beta } \ dv \Big | \\&\quad = \sup _{v\in [-T-1,T]} |L(v)|\quad \frac{1}{\beta +1} \Big | (1+h)^{\beta +1} - h^{\beta +1} -1 \Big | \\&\quad \le \sup _{v\in [-T-1,T]} |L(v)| \quad \frac{1}{\beta +1} \Big ( |(1+h)^{\beta +1}-1| + h^{\beta +1} \Big ). \end{aligned} \end{aligned}$$

Using a Taylor expansion, it follows that there is an almost surely finite random variable \(C_{2,2}\) such that

$$\begin{aligned} \begin{aligned}&\int _{0}^{1} e^{-\lambda v} \Big | (h+v)^{\beta } - v^{\beta } \Big | \Big |L(s-v)\Big | \ dv \le C_{2,2}|h|^{\min (1,\beta +1)} \end{aligned} \end{aligned}$$
(A.11)

for \(s,t\in [-T,T]\). Combining (A.5), (A.6), (A.8), (A.10), and (A.11), we see that

$$\begin{aligned} |W_{\beta }(s,t)| \le C_{\beta } h^{\min (1,\beta +1)} \end{aligned}$$
(A.12)

for \(s,t\in [-T,T]\), where \(C_{\beta }\) is an almost surely finite random variable. Applying (A.12) to (A.4) with \(\beta = d\) and \(\beta = d-1\) yields

$$\begin{aligned} |S^{I}_{d,\lambda }(t) - S^{I}_{d,\lambda }(s)| \le C_{T} |t-s|^{d}, \quad s,t\in [-T,T], \end{aligned}$$

which establishes (2.20).

To show (b), let \(-\frac{1}{2}< d < 0\). In this case, the kernel function \(g^{I}_{d,\lambda ,\cdot }(s)\) is not locally bounded and in fact the mapping \(t\longmapsto g^{I}_{d,\lambda ,t}(s)\), \(t\in {\mathbb {R}}\), is unbounded and discontinuous for all s. Therefore, Theorem 4 in [108] implies that the sample paths of \(S^{I}_{d,\lambda }\) are unbounded and discontinuous with positive probability, as claimed. \(\square \)

Proof of Proposition 2.7

To prove (a), note that TFLN has the same covariance structure as tempered fractional Gaussian noise (TFGN), up to a constant. Expression (2.15) can be obtained by following the same argument as in Chen et al. [46, Appendix 2] for the asymptotic behavior of TFGN over large covariance lags.

To show (b), let a(t) be the time domain kernel of the moving average representation (2.14) of TFLN. Then, the spectral density is given by

$$\begin{aligned} h^{I}(\omega ) = \frac{1}{2\pi } \quad \Big | \int _{{\mathbb {R}}}e^{-i\omega t} a(t) dt\Big |^2 = \frac{1}{2\pi } \quad \Big | \frac{ e^{i\omega } -1 }{ (\lambda + i\omega )^{d+1} } \Big |^2. \end{aligned}$$

This establishes (2.16). \(\square \)

The next lemma is mentioned in Sect. 2.2. As a consequence of the lemma, \(S^{I\!I}_{d,\lambda }(t)\) is well defined for any \(t>0\).

Lemma A.1

Let \(g^{I\! I}_{d,\lambda ,t}(y)\) be the function (2.8). Then,

$$\begin{aligned} g^{I\! I}_{d,\lambda ,t}(y) \in L^2({\mathbb {R}}) \end{aligned}$$
(A.13)

for any \(t \in {\mathbb {R}}\) and any \(\lambda >0\), \(d > -\frac{1}{2}\).

Proof of Lemma A.1

Let \(t >0\). By applying Minkowski’s inequality to (2.8), we arrive at

$$\begin{aligned}&\Vert g^{I\! I}_{d,\lambda ,t}(\cdot )\Vert _{2} \le \Big (\int _{\mathbb {R}}(t-y)_+^{2d} \ e^{-2 \lambda (t-y)_+} \ dy\Big )^{1/2} + \Big (\int _{\mathbb {R}}(-y)_+^{2d} \ e^{-2 \lambda (-y)_+} \ dy\Big )^{1/2}\\&\quad + \, \lambda \Big ( \int _{\mathbb {R}}\ \Big \{\int ^t_0 (s-y)_+^{d} \ e^{-\lambda (s-y)_+} \ ds \Big \}^{2}dy \Big )^{1/2} < \infty , \end{aligned}$$

where finiteness is a consequence of the facts that \(2d+1>0\) and \(\lambda >0\). Since \(g^{I\! I}_{d,\lambda ,-t}(y) = -g^{I\! I}_{d,\lambda ,t}(y+t) \) for any \(t, y \in {\mathbb {R}}\), (A.13) holds. \(\square \)

Proof of Proposition 2.9

We first note that \(g^{I\! I}_{d,\lambda ,t}(y) = d\int _{0}^{t} (s-y)_+^{d-1} e^{-\lambda (s-y)_+} ds \), where \(g^{I\! I}_{d,\lambda ,t}(y)\) is the function given by (2.8). Hence,

$$\begin{aligned} S^{I\!I}_{d,\lambda }(t)=\frac{1}{\Gamma (d+1)} \int _{{\mathbb {R}}} g^{I\! I}_{d,\lambda ,t}(y)\ dL(y) = \frac{1}{\Gamma (d)} \int _{{\mathbb {R}}}\int _{0}^{t} (s-y)_+^{d-1} e^{-\lambda (s-y)_+} ds \ dL(y) \end{aligned}$$
(A.14)

From Proposition 2.1,

$$\begin{aligned} {\text{ Cov }}\Big (\int _{{\mathbb {R}}}f(y)\ dL(y), \int _{{\mathbb {R}}}g(y)\ dL(y)\Big )={\mathbb {E}}[L(1)^2] \int _{{\mathbb {R}}} f(y)g(y) dy \end{aligned}$$
(A.15)

Now, by Lemma A.1, we can apply (A.15) to TFLP \(I\!I\) in (A.14) to write

$$\begin{aligned} \begin{aligned}&{\text {Cov}}\Big (S^{I\!I}_{d,\lambda }(t),S^{I\!I}_{d,\lambda }(s)\Big ) \\&\quad \quad =\frac{{\mathbb {E}}[L(1)^2]}{(\Gamma (d))^{2}}\int _{{\mathbb {R}}} g^{I\! I}_{d,\lambda ,t}(y)g^{I\! I}_{d,\lambda ,s}(y) dy\\&\quad \quad =\frac{{\mathbb {E}}[L(1)^2]}{(\Gamma (d))^{2}}\int _{{\mathbb {R}}}\Big (\int _{0}^{t}\int _{0}^{s}(u-y)_{+}^{d-1}(v-y)_{+}^{d-1}e^{-\lambda (u-y)_{+}}e^{-\lambda (v-y)_{+}}dv\ du\Big )dy\\&\quad \quad =\frac{{\mathbb {E}}[L(1)^2]}{(\Gamma (d))^{2}}\int _{0}^{t}\int _{0}^{s}\Bigg [\int _{-\infty }^{\min (u,v)} (u-y)^{d-1}(v-y)^{d-1}e^{-\lambda (u-y)}e^{-\lambda (v-y)}dy\Bigg ]dv\ du. \end{aligned} \end{aligned}$$
(A.16)

Using the relation

$$\begin{aligned} \int _{0}^{\infty }x^{\nu -1}(x+\beta )^{\nu -1}e^{-\mu x}dx=\frac{1}{\sqrt{\pi }}\left( \frac{\beta }{\mu }\right) ^{\nu -\frac{1}{2}}e^{\frac{\beta \mu }{2}}\ \Gamma (\nu )K_{\frac{1}{2}-\nu }\Big (\frac{\beta \mu }{2}\Big ), \end{aligned}$$
(A.17)

(see [109, p. 348]), we have

$$\begin{aligned} \int _{-\infty }^{\min (u,v)}(u-y)^{d-1}(v-y)^{d-1}e^{-\lambda (u-y)}e^{-\lambda (v-y)}dy =\frac{\Gamma (d)}{\sqrt{\pi }}\Big (\frac{|u-v|}{2\lambda }\Big )^{d-\frac{1}{2}}K_{d-\frac{1}{2}}(\lambda |u-v|). \end{aligned}$$
(A.18)

Therefore, from (A.16) and (A.18), we have

$$\begin{aligned} {\text{ Cov }}\Big (S^{I\!I}_{d,\lambda }(t),S^{I\!I}_{d,\lambda }(s)\Big )=\frac{{\mathbb {E}}[L(1)^2]}{\sqrt{\pi }\Gamma (d)(2\lambda )^{d-\frac{1}{2}}}\int _{0}^{t}\int _{0}^{s} |u-v|^{d-\frac{1}{2}}K_{d-\frac{1}{2}}(\lambda |u-v|)dv\ du \end{aligned}$$

for any \(d>0\) and \(\lambda >0\), as claimed. \(\square \)

Proof of Proposition 2.11

The proof follows the similar technique that was employed in Theorem 2.5 and hence we omit it. \(\square \)

Proof of Theorem 2.12

We use the Kolmogorov-\(\breve{\text {C}}\)entsov theorem (e.g., [105, p. 53]) to establish the claim. Since \(\lambda >0\) is fixed, we can assume \(\lambda =1 \) without loss of generality. Since the increments of \(S^{I\!I}_{d,1}(t)\) are stationary, it suffices to show that

$$\begin{aligned} {\mathbb {E}}|S^{I\!I}_{d,1}(t)|^2 \le C t^{1+\beta } \end{aligned}$$
(A.19)

for some \( \beta > 0\) and all \( 0< t < 1 \). Consider \(g^{I\! I}_{d,1,t}\) as in (2.17). By (A.15),

$$\begin{aligned} {\mathbb {E}}|S^{I\!I}_{d,1}(t)|^2 = C \int _{-\infty }^t (g^{I\! I}_{d,1,t}(y))^2 \ dy =: C (I_1 + I_2), \end{aligned}$$

where

$$\begin{aligned} I_1= & {} \int _{-t}^t (g^{I\! I}_{d,1,t}(y))^2 \ dy= \frac{1}{\Gamma (d)}\int _{-t}^t\left( \int _{0}^{t} (s-x)^{d-1} e^{-(s-x)} ds\right) ^2dx\\\le & {} C\int _{-t}^{t} (t - y)^{2d} \ dy \le C t^{2d+1} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} I_2 =&\int _{-\infty }^{-t} (g^{I\! I}_{d,1,t}(y))^2 \ dy \le C\int _t^\infty ((t + y)^{d} \ e^{-t-y} - y^{d} \ e^{-y})^2 \ dy\\&\ + C\int _t^\infty \Big \{\int _0^t (s+y)^{d} \ e^{-s-y} \ ds \Big \}^2 \ dy = C(I_2' + I_2''). \end{aligned} \end{aligned}$$

Using \(|(t + y)^{d} \ e^{-(t-y)} - y^{d} \ e^{-y}| \le |\ e^{-t}-1|\, \ e^{-y} (t + y)^{d} + \ e^{-y}\, |(t + y)^{d} - y^{d} | \le C t \, \ e^{-y} (t + y)^{d} + C t \, \ e^{-y} y^{d} \) we obtain \(I'_2 \le C t^2 \) and, similarly, \(I''_2 \le C t^2 \), implying \(I_1 + I_2 \le C(t^{2d+1} + t^2) \) and \( {\mathbb {E}}|S^{I\!I}_{d,1}(t)|^2 \le C(t^{2d+1} + t^2)\le t^{2d+1}\) since \(d\in (0,1/2]\) and \(0< t < 1\). Hence, (A.19) is satisfied with \(\beta = 2d\). This completes the proof. To show (b), note that when \(-\frac{1}{2}< d < 0\)\(g^{I\! I}_{d,\lambda ,\cdot }(s)\) is not locally bounded and \(t\longmapsto g^{I\! I}_{d,\lambda ,t}(s)\), \(t\in {\mathbb {R}}\) is unbounded and discontinuous for all s, and so the same proof in part (b) of Theorem 2.6 applies. \(\square \)

Proof of Proposition 2.13

To show (a), note that the autocovariance function of a TFGN \(I\! I\) satisfies \(\gamma (h) \asymp e^{-\lambda h } h^{ d -1 }\) as \(h\rightarrow \infty \) (see [55]). From (2.3), TFBM \(I\! I\) and TFLP \(I\! I\) have the same second order structure up to constants. Hence, (2.23) holds.

To show (b), let \(\Big ({\mathbb {I}}^{d,\lambda }_{-} f\Big )(x)\) be as in (3.4) with \(\kappa = d\). Note that the process \(X^{I\!I}_{d,\lambda }\) as in (2.14) has the integral representation

$$\begin{aligned} X^{I\!I}_{d,\lambda }(t) = \int _{{\mathbb {R}}} \Big ( {\mathbb {I}}^{d,\lambda }_{-}{} \mathbf{1}_{[t,t+1]}(x) \Big ) \ dL(x). \end{aligned}$$

Therefore, its spectral density is given by

$$\begin{aligned} \begin{aligned} h^{I\!I}(\omega )&= \frac{1}{2\pi } \Big |\int _{{\mathbb {R}}} e^{-i\omega t} \Big ( {\mathbb {I}}^{d,\lambda }_{-}{} \mathbf{1}_{[t,t+1]}(\omega ) \Big ) dt \Big |^2\\&= \frac{1}{2\pi } \Big | (\lambda + i\omega )^{-d}\int _{t}^{t+1} e^{-i\omega x} dx \Big |^2 = \frac{1}{2\pi } \frac{ 2( 1- \cos (\omega ) ) }{(\lambda ^2 + \omega ^2)^{d}\ \omega ^2}, \end{aligned} \end{aligned}$$

as claimed. \(\square \)

Proof of Proposition 2.14

Write

$$\begin{aligned} \phi ^I(x)= x_+^de^{-\lambda x_+}, \quad \phi ^{II}(x)= x_+^d e^{-\lambda x_+} + \lambda \int _0^{x} u_+^d e^{-\lambda u_+} du, \end{aligned}$$
(A.20)

and note that

$$\begin{aligned} S^I_{d,\lambda }(t)= & {} \frac{1}{\Gamma (d+1)}\int _{{\mathbb {R}}} \{\phi ^I(t-x)-\phi ^I(-x)\}dL(x),\nonumber \\ S^{I\! I}_{d,\lambda }(t)= & {} \frac{1}{\Gamma (d+1)}\int _{{\mathbb {R}}} \{\phi ^{II}(t-x)-\phi ^{II}(-x)\}dL(x). \end{aligned}$$
(A.21)

For \(x>0\), the derivatives \(\eta ^I(x):=\frac{d}{dx}\phi ^I(x)\), \(\eta ^{II}(x):=\frac{d}{dx}\phi ^{II}(x)\) exist and satisfy \(\eta ^I(x)\sim \eta ^{II}(x)\sim d x^{d-1}\), \(x \rightarrow 0^+\). Hence

$$\begin{aligned} \int _a^b \left| \eta ^I(x)\right| ^\alpha dx =\infty ,\quad \int _a^b \left| \eta ^{II}(x)\right| ^\alpha dx=\infty \end{aligned}$$

for any interval [ab) containing 0 whenever \(\alpha (d-1)+1<0\), i.e., whenever \(d+\frac{1}{\alpha }<1\). Hence, by Corollary 3.4 in [110], the processes

$$\begin{aligned} \int _0^t\phi ^I(t-x)dL(x), \quad \int _0^t\phi ^{II}(t-x)dL(x), \quad t\ge 0 \end{aligned}$$

are not \(({\mathcal {F}}^{L}_t)_{t\ge 0}\)-semimartingales, where \(({\mathcal {F}}^{L}_t)_{t\ge 0}=\sigma \{L(s); 0\le s \le t \}\). Thus, since L is symmetric, in view of the representations (A.21), by Lemma 5.2 of [110] \(S^I_{d,\lambda }\) and \(S^{I\! I}_{d,\lambda }\) are not \(({\mathcal {F}}^{L,\infty }_t)_{t\ge 0}\)-semimartingales. \(\square \)

Proof of Proposition 3.1

The proof is similar to that of Theorem 3.9 in [111]. Write \(\eta ^I(x)=\frac{d}{dx}\phi ^I(x)\) where \(\phi ^I\) is given in (A.20). Note since \(d>1/2\), \(\eta ^I\in L^2({\mathbb {R}})\), and hence the integral \(\int _{\mathbb {R}}\eta ^I(x) dL(x)\) is well-defined. Now,

$$\begin{aligned} \Gamma (d+1)S^I_{d,\lambda }(t)= & {} \int _{-\infty }^t \{\phi ^I(t-x) - \phi ^I(-x) \}dL(x)\\= & {} \int _{-\infty }^0 \{\phi ^I(t-x) - \phi ^I(-x)\} dL(x) + \int _{0}^t \phi ^I(t-x) dL(x)\\= & {} \int _{-\infty }^0 \int _0^t \eta ^I(s-x) ds dL(x) + \int _{0}^t \int _{x}^t\eta ^I(s-x)ds dL(x). \end{aligned}$$

Hence, by a stochastic version of the Fubini theorem (e.g. [112, Theorem 65]), the above process has a version that is equal to

$$\begin{aligned} \int _0^t\int _{-\infty }^0 \eta ^I(s-x) dL(x)ds + \int _{0}^t \int _{0}^s\eta ^I(s-x) dL(x)ds = \int _0^t\int _{-\infty }^s \eta ^I(s-x) dL(x)ds. \end{aligned}$$

This establishes (i).

We now turn to (ii). First note that

$$\begin{aligned} S^{I\! I}_{d,\lambda }(t) = \frac{1}{\Gamma (d+1)}\int _{{\mathbb {R}}}\{ \phi ^{II}(t-x)-\phi ^{II}(-x)\}dL(x), \end{aligned}$$

where \(\phi ^{II}\) is given in (A.20). Since \(d>1/2\), \(\frac{d}{dx}\phi ^{II}(x)\in L^2({\mathbb {R}})\), and the rest of the proof can be done similarly to that of part (i). \(\square \)

The following lemma is used in Sect. 3.

Lemma A.2

Let \( S^{I}_{d,\lambda } \) and \(S^{I\!I}_{d,\lambda }(t) \) be a TFLP and TFLP \(I\!I\) given by (2.6) and (2.17), respectively. Then, for every \(t\in {\mathbb {R}}\),

  1. (a)

    when \(d>0\), expressions (3.9) and (3.10) hold;

  2. (b)

    when \(-\frac{1}{2}<d<0\), expressions (3.11) and (3.12) hold.

Proof of Lemma A.2

The proofs can be developed along the same lines of that of Lemma 3.4 in [61] for TFLP, and of Proposition 2.5 in [55] for TFLP \(I\!I\). \(\square \)

Proof of Theorem 3.5

To show that \({{\mathcal {A}}}_{1}\) is an inner product space, it suffices to establish that \({\langle f,f \rangle }_{{{\mathcal {A}}}_{1}}=0\) implies \(f=0\)dx–a.e. If \({\langle f,f \rangle }_{{{\mathcal {A}}}_{1}}=0\), then in view of (3.17) and (3.18) we have \({\langle F,F\rangle }_{2}=0\), so \(F(x)=\Big ({{\mathbb {I}}}^{d,\lambda }_{-}f\Big )(x)=0\)dx–a.e. Then,

$$\begin{aligned} \Big ({{\mathbb {I}}}^{d,\lambda }_{-}f\Big )(x)=0\quad dx-\text {a.e.} \end{aligned}$$
(A.22)

Apply \({{\mathbb {D}}}^{d,\lambda }_{-}\) to both sides of Eq. (A.22) and use Lemma 2.14 in [61] to get \(f(x)=0\)dx–a.e. Hence, \({{\mathcal {A}}}_{1}\) is an inner product space, as claimed. \(\square \)

Next, we want to show that the set of elementary functions \({{\mathcal {E}}}\) is dense in \({{\mathcal {A}}}_{1} \subseteq L^2({\mathbb {R}})\). For any \(f\in {{\mathcal {A}}}_{1}\), we also have \(f\in {L}^{2}({\mathbb {R}})\), and hence there exists a sequence of elementary functions \((f_n)_{n \in {\mathbb {N}}}\) in \(L^2({\mathbb {R}})\) such that \(\Vert f-f_n\Vert _2\rightarrow 0\) as \(n \rightarrow \infty \). However,

$$\begin{aligned} \Vert f-f_n\Vert ^2_{{{\mathcal {A}}}_{1}} ={\langle f-f_n,f-f_n \rangle }_{{{\mathcal {A}}}_{1}}={\langle F-F_n,F-F_n\rangle }_2= \Vert F-F_n\Vert ^2_2, \end{aligned}$$

where \( F_n(x)=\Big ({{\mathbb {I}}}^{d,\lambda }_{-}{f_n}\Big )(x) \) and F(x) is given by (3.18). It can be further shown that \(\Vert {{\mathbb {I}}}^{\kappa ,\lambda }_{-}(f)\Vert _{2}\le C\Vert f\Vert _{2}\) for some constant C. Then,

$$\begin{aligned} \Vert f-f_n\Vert _{{{\mathcal {A}}}_{1}}=\left\| F-F_n\right\| _{2}=\Vert {{\mathbb {I}}}^{d,\lambda }_{-}(f-f_n)\Vert _{2}\le C\Vert f-f_n\Vert _{2}. \end{aligned}$$

Since \(\Vert f-f_n\Vert _2\rightarrow 0\) as \(n \rightarrow \infty \), it follows that the set of elementary functions is dense in \({{{\mathcal {A}}}_{1}}\). Finally, using the example provided in the [93, Theorem 3.1], one can show that \({{\mathcal {A}}}_{1}\) is not complete.

The following proposition can be established by a direct adaptation of the proof of Proposition 2.1 in [93].

Proposition A.3

For \(d > -1/2\), \(\lambda >0\), let \({{\mathcal {E}}}\) be the set of elementary functions, let \({{\mathcal {I}}}^{d,\lambda }(f)\) be an integral (3.19) of \(f \in {{\mathcal {E}}}\) with respect to the Lévy process L as in (2.1). Suppose \({{\mathcal {D}}}\) is a set of deterministic functions on \({\mathbb {R}}\) such that: (i) \({\mathcal D}\) is an inner product space with an inner product \(\langle f,g\rangle _{{{\mathcal {D}}}}\) for \(f,g \in {{\mathcal {D}}}\); (ii) \({{\mathcal {E}}} \subseteq {{\mathcal {D}}}\) and \(\langle f,g\rangle _{{{\mathcal {D}}}} = \langle {\mathcal I}^{d,\lambda }(f),{{\mathcal {I}}}^{d,\lambda }(g)\rangle _{L^2(\Omega )}\), \(f,g \in {{\mathcal {E}}}\); (iii) the set is dense in \({{\mathcal {D}}}\). Then,

  1. (a)

    there is an isometry between the space \({{\mathcal {D}}}\) and a linear subspace of \(\overline{\text {Sp}}(S^{I\!I}_{d,\lambda } )\) which is an extension of the mapping \(f \mapsto {{\mathcal {I}}}^{d,\lambda }(f)\), \(f \in {{\mathcal {E}}}\);

  2. (b)

    \({{\mathcal {D}}}\) is isometric to \(\overline{\text {Sp}}(S^{I\!I}_{d,\lambda } )\) itself if and only if \({{\mathcal {D}}}\) is complete.

We are now in a position to prove Theorem 3.8.

Proof of Theorem 3.8 :

Since \(\Vert {{\mathbb {I}}}^{\kappa ,\lambda }_{-}(f)\Vert _{2}\le C\Vert f\Vert _{2}\) then the stochastic integral (3.19) is well-defined for any \(f\in {{\mathcal {A}}}_{1}\). By using the isometry (2.3) and expression (3.19), it follows from Proposition A.3 and (3.17) that, for any \(f,g\in {{{\mathcal {A}}}_{1}}\),

$$\begin{aligned} {\langle f,g \rangle }_{{{\mathcal {A}}}_{1}}={\langle F,G\rangle }_{L^{2}({\mathbb {R}})} ={\langle {{{\mathcal {I}}}}^{d,\lambda }(f),{{{\mathcal {I}}}}^{d,\lambda }(g)\rangle }_{L^2(\Omega )}. \end{aligned}$$

Then, Theorem 3.5 implies that \({{\mathcal {A}}}_{1}\) is isometric to a subset of \(\overline{\mathrm{Sp}}(S^{I\!I}_{d,\lambda })\), as claimed. However, again by Theorem 3.5, \({{\mathcal {A}}}_{1}\) is not complete. Therefore, \({{\mathcal {A}}}_{1}\) is isometric to a strict subset of \(\overline{\mathrm{Sp}}(S^{I\!I}_{d,\lambda })\). \(\square \)

Lemmas A.4 and A.5, stated and proved next, are used in the proof of Theorem 3.9.

Lemma A.4

Under the assumptions of Theorem 3.9, every \(f\in W^{-d,2}({\mathbb {R}})\) is an element of \({{\mathcal {A}}}_{2}\) for \(-\frac{1}{2}< d <0\) and \(\lambda >0 \), i.e., as sets, \(W^{-d,2}({\mathbb {R}})={\mathcal {A}}_2\).

Proof of Lemma A.4

Given \(f\in W^{-d,2}({\mathbb {R}})\), we need to show that

$$\begin{aligned} \varphi _f={\mathbb {D}}^{-d,\lambda }_{-} f \end{aligned}$$
(A.23)

for some \(\varphi _f\in L^{2}( {\mathbb {R}})\). From the definition (3.8) we see that \(\int (\lambda ^2+\omega ^2)^{-d}|\hat{f}(\omega )|^2\ d\omega <\infty \). Define \(h_1(\omega )=(\lambda -i\omega )^{-d}\hat{f}(\omega )\) and note that \(h_1\) is the Fourier transform of some function \(\varphi _1\in L^{2}( {\mathbb {R}})\). Define \(\varphi _f:=\varphi _{1}\) so that

$$\begin{aligned} \widehat{\varphi _f}(\omega ) = \widehat{\varphi _1}(\omega ) = {\widehat{f}}(\omega )(\lambda -i\omega )^{-d}. \end{aligned}$$
(A.24)

Since \(f\in W^{-d,2}({\mathbb {R}})\subset L^2({\mathbb {R}})\), we can apply Definition 3.4 to get the desired result. \(\square \)

We state the following lemma that will be used to proof Theorem 3.9. We refer the reader to [61, Lemma 3.12] for the proof of the Lemma.

Lemma A.5

Suppose the assumptions of Theorem 3.9 hold. If \(f\in W^{-d,2}({\mathbb {R}})\), then there exists a sequence of functions \((f_n)_{n \in {\mathbb {N}}} \subseteq {{\mathcal {E}}}\) such that \(\Vert f_n - f\Vert _{L^2({\mathbb {R}})}\). Moreover, when \(-\frac{1}{2}< d <0\),

$$\begin{aligned} \int _{\mathbb {R}}|{\widehat{f}}_n(\omega )-\widehat{f}(\omega )|^2|\omega |^{-2d}d\omega \rightarrow 0, \quad n\rightarrow \infty . \end{aligned}$$
(A.25)

Proof of Theorem 3.9

For \(f\in {{\mathcal {A}}}_{2}\) we define

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {A}}}_{2}}=\sqrt{{\langle f,f\rangle }_{{{\mathcal {A}}}_{2}}}=\sqrt{{\langle \varphi _f,\varphi _f\rangle }_2}=\Vert {\varphi _f}\Vert _{2}, \end{aligned}$$
(A.26)

where \(\varphi _f\) is given by (A.23). Next, use (A.24) to see that

$$\begin{aligned} \widehat{\varphi _f}(\omega ) = (\lambda -i\omega )^{-d}{\widehat{f}}(\omega ) . \end{aligned}$$
(A.27)

To verify that (3.23) is an inner product, it suffices to show that, if \({\langle f,f \rangle }_{{{\mathcal {A}}}_{2}}=0\), then

$$\begin{aligned} f=0 \quad dx-\text {a.e.} \end{aligned}$$
(A.28)

In fact,

$$\begin{aligned} 0 = \Vert f\Vert _{{{\mathcal {A}}}_{2}}^2=\Vert {\varphi _f}\Vert _{2}^2=\Vert {\widehat{\varphi _f}}\Vert ^{2}_{2} =\int _{\mathbb {R}}|{\widehat{f}}(\omega )|^{2} (\lambda ^2+\omega ^2)^{-d}\ d\omega \end{aligned}$$
(A.29)

implies that \({\widehat{f}}(\omega )=0\)\(d\omega \)—a.e. Hence, (A.28) holds.

We now show that \({{\mathcal {E}}}\) is dense in \({{\mathcal {A}}}_2\). By Lemma A.5, there is a sequence \((f_n)_{n \in {\mathbb {N}}} \subseteq {{\mathcal {E}} }\) such that

$$\begin{aligned} \Vert f_n-f\Vert _2\rightarrow 0, \quad n \rightarrow \infty , \end{aligned}$$
(A.30)

and (A.25) holds. On the other hand, by Lemma A.4, \({{\mathcal {E}}} \subseteq W^{-d,2}({\mathbb {R}}) \subseteq {{\mathcal {A}}}_2\). By (A.29), we can write

$$\begin{aligned} \Vert f_n-f\Vert ^{2}_{{{\mathcal {A}}}_2} =\int _{\mathbb {R}}\Big |{\widehat{f}}_n(\omega )-{\widehat{f}}(\omega )\Big |^{2}(\lambda ^2 + \omega ^2)^{-d}\ d\omega =: I_{1} + I_{2}, \end{aligned}$$

where

$$\begin{aligned} I_1 = \int _{|\omega | < \lambda }\Big |{\widehat{f}}_n(\omega )-{\widehat{f}}(\omega )\Big |^{2} (\lambda ^2+\omega ^2)^{-d}\ d\omega , \quad I_2 = \int _{|\omega | \ge \lambda }\Big |{\widehat{f}}_n(\omega )-{\widehat{f}}(\omega )\Big |^{2} (\lambda ^2+\omega ^2)^{-d}\ d\omega . \end{aligned}$$

Since \(|\omega | < \lambda \), then \(I_1 \le 2\lambda ^{-2d} \int _{{\mathbb {R}}} |{\widehat{f}}_n(\omega )-{\widehat{f}}(\omega )\Big |^{2} \ d\omega \rightarrow 0\) as \(n\rightarrow \infty \), where convergence is a consequence of (A.30). Moreover, by (A.25), \(I_2 \le 2^{-d} \int _{{\mathbb {R}}} \Big |{\widehat{f}}_n(\omega )-{\widehat{f}}(\omega )\Big | |\omega |^{-2d}\ d\omega \rightarrow 0\) as \(n\rightarrow \infty \). Hence, \(\Vert f_n-f\Vert ^{2}_{{{\mathcal {A}}}_2} \rightarrow 0\) as \(n\rightarrow \infty \), namely, \({{\mathcal {E}}}\) is dense in \({{\mathcal {A}}}_2\).

It only remains to show that \({{\mathcal {A}}}_{2}\) is complete. In fact, let \(\big ( f_n \big )_{n \in {\mathbb {N}}}\) be a Cauchy sequence in \({{\mathcal {A}}}_{2}\). Then, by using the inner product (3.23), the corresponding sequence \(\big ( { \varphi _{f_n} } \big )_{n \in {\mathbb {N}}}\) is Cauchy in \(L^2({\mathbb {R}})\). Again by the inner product (3.23), and since \(L^2({\mathbb {R}})\) is complete, there exists \(\varphi _{f^*}\) such that \(\Vert f_n- f^*\Vert _{{{\mathcal {A}}}_{2}} = \Vert {\varphi _{f_n} } - {\varphi _{f^*} }\Vert _{2}\rightarrow 0\), \(n\rightarrow \infty \). Hence, \(f^* \in {{\mathcal {A}}}_2\) and \({{\mathcal {A}}}_{2}\) is complete.

\(\square \)

Proof of Theorem 3.11

By Lemma A.4, the stochastic integral (3.24) is well-defined for any \(f\in {{\mathcal {A}}}_{2}\). Since \({{\mathcal {A}}}_{2}\) is a complete space with inner product (3.23) and \({\mathcal {E}}\) is dense, then Proposition A.3 implies that \({{\mathcal {A}}}_{2}\) is isometric to \(\overline{\mathrm{Sp}}( S^{I\!I}_{d,\lambda } )\). This completes the proof. \(\square \)

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Boniece, B.C., Didier, G. & Sabzikar, F. On Fractional Lévy Processes: Tempering, Sample Path Properties and Stochastic Integration. J Stat Phys 178, 954–985 (2020). https://doi.org/10.1007/s10955-019-02475-1

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Keywords

  • Lévy processes
  • Fractional processes
  • Statistical turbulence
  • Anomalous diffusion
  • Stochastic analysis