Fractional and Multifractional Brownian Motions
       Stochastic integral w.r.t. mBm White Noise Theory
            Stochastic Calculus w.r.t. Multifractional Brownian
                                                     Motion
                   Séminaire Cristolien d’Analyse Multifractale
                                           Créteil, le 21 novembre 2013.
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            Joachim Lebovits, Université de Paris 13 Nord     Stochastic Calculus w.r.t. mBm
          Fractional and Multifractional Brownian Motions
       Stochastic integral w.r.t. mBm White Noise Theory
     Outline of the presentation
          1   Fractional and Multifractional Brownian Motions
                  Fractional and multifractional Brownian motions
                  The mBm as limit of lumped fBm
                  Non semi-martingales versus integration
          2   Stochastic integral w.r.t. mBm in the White Noise Theory sense
                  Background on White Noise Theory
                  Stochastic Integral with respect to mBm of functional
                  parameter h
                  Miscellaneous formulas
                                                                                                             2/47
            Joachim Lebovits, Université de Paris 13 Nord     Stochastic Calculus w.r.t. mBm
                                                               Fractional and multifractional Brownian motions
          Fractional and Multifractional Brownian Motions      The Multifractional Brownian Motion (mBm)
       Stochastic integral w.r.t. mBm White Noise Theory       The mBm as limit of lumped fBm
                                                               Non semi-martingales versus integration
     Outline of the presentation
          1   Fractional and Multifractional Brownian Motions
                  Fractional and multifractional Brownian motions
                  The mBm as limit of lumped fBm
                  Non semi-martingales versus integration
          2   Stochastic integral w.r.t. mBm in the White Noise Theory sense
                  Background on White Noise Theory
                  Stochastic Integral with respect to mBm of functional
                  parameter h
                  Miscellaneous formulas
                                                                                                               3/47
            Joachim Lebovits, Université de Paris 13 Nord      Stochastic Calculus w.r.t. mBm
                                                               Fractional and multifractional Brownian motions
          Fractional and Multifractional Brownian Motions      The Multifractional Brownian Motion (mBm)
       Stochastic integral w.r.t. mBm White Noise Theory       The mBm as limit of lumped fBm
                                                               Non semi-martingales versus integration
     AgaussianprocessmoreflexiblethanstandardBrownian
     motion (A. Kolmogorov, 1949)
          Definition
          Let H 2 (0,1) be a real constant. A process BH := (BH;t 2 R+) is an fBm if it is
                                                                          t
          centred, Gaussian, with covariance function given by:
                                    E[BHBH]=1/2(t2H +s2H |t s|2H)..
                                        t   s
          Properties
          The process BH verifies
                 BH =0,a.s.
                   0
                                        H       H                                  2H
                 For all t > s > 0,B Bs follows the law N(0,(t s)                   ).
                                        t
                 The trajectoiries de BH are contiuous.
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            Joachim Lebovits, Université de Paris 13 Nord      Stochastic Calculus w.r.t. mBm