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Stochastics
An International Journal of Probability and Stochastic Processes
Volume 89, 2017 - Issue 6-7: Proceedings of the Hammamet Conference, 19-23 October 2015
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Articles

Strong approximation of stochastic processes at random times and application to their exact simulation

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Pages 883-895 | Received 27 Apr 2016, Accepted 28 Nov 2016, Published online: 22 Dec 2016

References

  • S. Asmussen, P. Glynn, and J. Pitman, Discretization error in simulation of one-dimensional reflecting Brownian motion, Ann. Appl. Probab. 5(4) (1995), pp. 875–896.
  • H. Allouba, Brownian-time processes: The PDE connection and the corresponding Feynman--Kac formula, Trans. Am. Math. Soc. 354(11) (2002), pp. 4627–4637.
  • H. Allouba and W. Zheng, Brownian-time processes: The PDE connection and the half-derivative generator, Ann. Probab. 29(2) (2001), pp. 1780–1795.
  • P. Andersson and A. Kohatsu-Higa, Exact simulation of stochastic differential equations using parametrix expansions, preprint (2016), to appear in Bernoulli.
  • B. von Bahr and C.-G. Esseen, Inequalities for the rth absolute moment of a sum of random variables, 1 ≤ r ≤ 2, Ann. Math. Stat. 36(1) (1965), pp. 299–303.
  • B. Bouchard, S. Geiss, and E. Gobet, First time to exit of a continuous Itô process: General moment estimates and L1-convergence rate for discrete time approximations, preprint (2016), to appear in Bernoulli.
  • K. Burdzy, and D. Khoshnevisan, Brownian motion in a Brownian crack, Ann. Appl. Probab. 8(3) (1998), pp. 708–748.
  • K. Burdzy, Some path properties of iterated Brownian motion, in Seminar on Stochastic Processes, 1992 Vol. 33, Birkhäuser Boston, Boston, MA, 1993, pp. 67–87.
  • S. Cohen and J. Istas, Fractional Fields and Applications, Springer, Heidelberg, 2013.
  • N. Curien and T. Konstantopoulos, Iterating Brownian motions, ad libitum, J. Theoret. Probab. 27(2) (2014), pp. 433–448.
  • F. Comte and E. Renault, Long memory in continuous-time stochastic volatility models, Math. Finance 8(4) (1998), pp. 291–323.
  • H. Föllmer and P. Protter, Local martingales and filtration shrinkage, ESAIM: Probab. Stat. 15 (2011), pp. S25–S38.
  • T. Funaki, Probabilistic construction of the solution of some higher order parabolic differential equation, Proc. Japan Acad. Ser. A, Math. Sci. 55(5) (1979), pp. 176–179.
  • M.B. Giles, Multilevel Monte Carlo path simulation, Oper. Res. 56 (2008), pp. 607–617.
  • E. Gobet and M. Mrad, Convergence rate of strong approximations of compound random maps, preprint (2015), hal-01141320.
  • S. Heinrich, Multilevel Monte Carlo Methods, in LSSC ’01 Proceedings of the Third International Conference on Large-scale Scientific Computing, Lecture Notes in Computer Science Vol. 2179, Springer-Verlag, Heidelberg, 2001, pp. 58–67.
  • D. Hong, S. Man, J.-C. Birget, and D.S. Lun, A wavelet-based almost-sure uniform approximation of fractional Brownian motion with a parallel algorithm, J. Appl. Probab. 51(1) (2014), pp. 1–18.
  • D. Khoshnevisan, Exact rates of convergence to Brownian local time, Ann. Probab. 22(3) (1994), pp. 1295–1330.
  • B.B. Mandelbrot, and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10(4) (1968), pp. 422–437.
  • I. Nourdin and R. Zeineddine, An Itô-type formula for the fractional Brownian motion in Brownian time, Electron. J. Probab 19(99) (2014), pp. 1–15.
  • J. Obłój, The Skorokhod embedding problem and its offspring, Probab. Surv. 1 (2004), pp. 321–392.
  • A. Ohashi, and A.B. Simas, A note on the sharp Lp-convergence rate of upcrossings to the Brownian local time, Stat. Probab. Lett. 100 (2015), pp. 137–141.
  • C. Rhee and P.W. Glynn, A new approach to unbiased estimation for SDEs, in Proceedings of the 2012 Winter Simulation Conference, C. Laroque, J. Himmelspach, R. Pasupathy, O. Rose, and A.M. Uhrmacher, eds., 2012, pp. 495–503.
  • D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd ed., Comprehensive Studies in Mathematics, Springer-Verlag, Berlin, 1999.
  • T. Szabados, Strong approximation of fractional Brownian motion by moving averages of simple random walks, Stoch. Process. Appl. 92(1) (2001), pp. 31–60.

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