By Ciprian Tudor
Self-similar techniques are stochastic strategies which are invariant in distribution less than appropriate time scaling, and are an issue intensively studied within the previous few a long time. This e-book offers the fundamental homes of those techniques and specializes in the research in their version utilizing stochastic research. whereas self-similar tactics, and particularly fractional Brownian movement, were mentioned in different books, a few new periods have lately emerged within the clinical literature. a few of them are extensions of fractional Brownian movement (bifractional Brownian movement, subtractional Brownian movement, Hermite processes), whereas others are options to the partial differential equations pushed by means of fractional noises.
In this monograph the writer discusses the fundamental homes of those new periods of self-similar procedures and their interrelationship. whilst a brand new technique (based on stochastic calculus, particularly Malliavin calculus) to learning the habit of the differences of self-similar techniques has been constructed during the last decade. This paintings surveys those fresh strategies and findings on restrict theorems and Malliavin calculus.
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Extra resources for Analysis of Variations for Self-similar Processes: A Stochastic Calculus Approach
The integral with respect to db involves logarithms and it cannot lead to the covariance of the bifractional Brownian motion. 4 The Solution to the Fractional-White Heat Equation t+s = (2π)− 2 αH Cd d 43 − d2 +1 a 2H −2 (t + s) − a da 0 t−s − a 2H −2 (t − s) − a − d2 +1 da − d2 +1 da 0 (d) + R1 (t, s) where (d) R1 (t, s) s = (2π)− 2 αH Cd d a 2H −2 (t + s) − a 0 s − a 2H −2 (t − s) + a − d2 +1 da 0 − t a 2H −2 a − (t − s) − d2 +1 t+s da − t−s a 2H −2 (t + s) − a − d2 +1 da . 26) At this point, we perform the change of variable a → t+s − d2 +1 a 2H −2 (t + s) − a a t+s 1 da = (t + s)2H − 2 d 0 and we obtain a 2H −2 (1 − a)− 2 +1 da d 0 = β 2H − 1, − and in the same way, with the change of variable a → t−s − d2 +1 a 2H −2 (t − s) − a d d + 2 (t + s)2H − 2 2 a t−s , 1 da = (t − s)2H − 2 d 0 we obtain a 2H −2 (1 − a)− 2 +1 da d 0 = β 2H − 1, − d d + 2 (t − s)2H − 2 .
15). Actually, we will use the following transfer formula (see ). 2). 7). See also Sect. 3 in the next chapter. g. ). 15). 5). 1 The process (U (t, x))t∈[0,T ],x∈Rd exists and satisfies sup t∈[0,T ],x∈Rd E U (t, x)2 < +∞ if and only if d < 4H . 4 The Solution to the Fractional-White Heat Equation 37 and the last integral is finite if and only if 2H > d2 . 5 This implies that, in contrast to the white-noise case, we are allowed to consider the spatial dimension d to be 1, 2 or 3. Suppose that s, t ∈ [0, T ] and let R(t, s) = E U (t, x)U (s, x) where x ∈ Rd is fixed.
Then prove that for large n r(a, a + n) = 2−K 2H K(2H K − 1)n2(H K−1) + H K(K − 1) (a + 1)2H − a 2H n2(H K−1)+(1−2H ) + · · · . Deduce that for every a ∈ N we have r(a, a + n) = ∞ if 2H K > 1 r(a, a + n) < ∞ if 2H K ≤ 1. 25 (See ) Let 0 < H < 1 and define XtH = ∞ 3 1 − e−θt θ 2 −H dWθ 0 where (Wθ )θ≥0 is a Wiener process. Let B H be a fBm independent from W . Prove that: 1. If H < 1 2 the process StH = − H (2H − 1) H X + BtH 2Γ (2 − 2H ) t is a sub-fBm. 2. If H > 12 the process StH = H (2H − 1) H X + BtH 2Γ (2 − 2H ) t is a sub-fBm.
Analysis of Variations for Self-similar Processes: A Stochastic Calculus Approach by Ciprian Tudor