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with a0 = and aK+1 = + , and where ai is an increasing sequence of realities By taking (647), we arrive at the following de nition
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k DEFINITION 68 Let + = { 2j , for k Z, j N} By SFBM we mean the process of multifractional function h de ned by (652):
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where functions are de ned by (648) In the preceding de nition, there are some differences as compared to (647) Some of them are technical, like the suppression of standardization v(h(t)) or the absence of negative frequencies On the other hand, the SFBM has continuous trajectories whereas the MBM which corresponds to a piecewise multifractal function is discontinuous This phenomenon occurs due to the replacement of (t, h(t)) by (t, h( )) Indeed, the rst function is discontinuous as h at points ai , this jump disappearing in (t, h( )) However, the fast decay property of functions t (t, h( )) when |t | + causes the SFBMs to have local properties very close to those of the MBM outside the jump moments of the multifractional function The following theorem, which more precisely describes the regularity of the SFBM, can be found in [BEN 00] THEOREM 65 For any open interval I of , we suppose: H (I) = inf{h(t), for t I}
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If Qh is a SFBM of the multifractional function h, then Qh is the overall H lderian of exponent H for all 0 < H < H (I), on any compact interval J I Thus, in terms of regularity, the SFBM is a satisfactory model We will see, in section 65, that we can completely identify the multifractional function: moments and amplitudes of the jumps 6432 Generalized multifractional Brownian motion Let us now outline the work of [AYA 99] The authors propose to circumvent the low frequency problems encountered within the de nition of MBM, by replacing the multifractional function h with a sequence of regular functions hn , whose limit, which will play the role of the multifractional function, can be very irregular Let us rst specify the technical conditions relating to the sequence (hn )n N DEFINITION 69 A function h is said to be locally H lderian of exponent r and of constant c > 0 on if, for all t1 and every t2 , such that, |t1 t2 | 1, we have: |h(t1 ) h(t2 )| c|t1 t2 |r
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Such a function will be called (r, c) H lderian We can consequently de ne the multifractional sequences which generalize the multifractional functions for the GMBM DEFINITION 610 We will call a multifractional sequence a sequence (hn )n N of H lderian functions (r, cn ) with values in an interval [a, b] ]0, 1[ and we will call its lower limit a generalized multifractional function (GMF): h(t) = lim inf hn (t)
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if (hn )n N veri es the following properties: for all and all t0 , there exists n0 (t0 , ) and h0 (t0 , ) > 0 such that, for all n > n0 and, |h| < h0 we have: hn (t0 + h) > h(t0 ) for all t, we have h(t) < r and cn = O(n) In the preceding de nition, it is essential that the generalized multifractional function is a limit when the index n tends towards + ; we will see that this translates the high frequency portion of the information contained in the multifractional sequence In addition, the GMF set contains very irregular functions like, for example, 0 < a < b < 1: t b + (a b)1F (t)
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