By Ivan Nourdin
Preliminaries -- Fractional Brownian movement -- Integration with recognize to Fractional Brownian movement -- Supremum of the Fractional Brownian movement -- Malliavin Calculus in a Nutshell -- vital restrict Theorem at the Wiener area -- susceptible Convergence of Partial Sums of desk bound Sequences -- Non-Commutative Fractional Brownian movement
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Extra resources for Selected aspects of fractional Brownian motion
B tH u B tH /u2Œ0;t ; indeed, these two processes are centered, Gaussian and have the same covariance. 14) Since B H has continuous paths, we have R 1 H tH BH u du 0 Bu e ! M1H R1 H H t Bu du e 0 Moreover, as t ! s. ˇ R1 H H t B ˇ u2Œ0;1 ˇ u du e 0 By dominated convergence, the desired conclusion follows by letting t ! 14). 3. 15) p t Proof. t / D 3t H log t Clog t , t > 0. t / 3H t log 1CH as t ! 1, one can choose M > 0 large enough so that x W ŒM; 1/ ! M / is a strictly decreasing bijection.
2. It is divided into two steps. Step 1. Assume for the time being that m D 1. Fix x > 0. By monotone convergence and continuity, we have Ä E sup Xk2 n ! 11) as well, Ã Â Â sup P sup Xu 6 x 6 P kD1;:::;2n 6e x2 2 2 > 2x. If supu2Œ0;1 Xu 6 Ä E u2Œ0;1 6 u2Œ0;1 n sup kD1;:::;2n Xk2 n x: Ä Xk2 E n Ã sup kD1;:::;2n Xk2 n 6 x : By letting x ! s. (since X has continuous paths), we get a contradiction. Hence m < 1. Step 2. 2 is concluded. 3) to any value of H . We follow Molchan  and Aurzada .
S/; t 2 Œ0; T ; 0 admits a unique solution x W Œ0; T ! R belonging to C ˛ . ˛Cˇ 1/ 2 ˛ 6 1 and 00 27 j1 º 6 M . Fix ˛ 2 . 11) < 1. Consider the Banach space Ba˛ WD ¹x W Œ0; ! 0/ D a and jxj˛ 6 1º equipped with the norm j j˛ . x/ 2 In other words, the ball is invariant by the operator T . On the other hand, for any x; y 2 Ba˛ and s; t 2 Œ0; , we have Ba˛ . t / . s//d ˇˇ . s/ 0 0 . t //d . t //d . y/j˛ 6 jx yj˛ 6 jx yj˛ ˛ jgjˇ M 1 C C˛;ˇ C 2C˛;ˇ ˛ 1 ˛ M jgjˇ 3 C C˛;ˇ 6 jx yj˛ : 2 Applying the fixed point argument to T W Ba˛ !
Selected aspects of fractional Brownian motion by Ivan Nourdin