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 booklet provides the elemental houses of those approaches and specializes in the research in their version utilizing stochastic research. whereas self-similar techniques, and particularly fractional Brownian movement, were mentioned in different books, a few new periods have lately emerged within the medical literature. a few of them are extensions of fractional Brownian movement (bifractional Brownian movement, subtractional Brownian movement, Hermite processes), whereas others are suggestions to the partial differential equations pushed by way of fractional noises.
In this monograph the writer discusses the elemental homes of those new sessions of self-similar methods and their interrelationship. even as a brand new technique (based on stochastic calculus, particularly Malliavin calculus) to learning the habit of the diversities of self-similar tactics has been built during the last decade. This paintings surveys those fresh options 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
1). Similarly, the function R Y is a covariance. If d = 1, we have Cd = 2 and 1 R(t, s) = 2αH (2π)− 2 β 2H − 1, 3 2 1 1 (1) (t + s) 2 − (t − s) 2 + R1 (t, s) with √ (1) 2πR1 (t, s) = 2αH s 1 1 (s − a)2H −2 (t + a) 2 − (t − a) 2 da 0 s − 2αH 1 (s − a) 2 (t + a)2H −2 + (t − a)2H −2 da 0 s =H 1 1 (s − a)2H −1 (t + a)− 2 + (t − a)− 2 da 0 s − 2αH 1 (s − a) 2 (t + a)2H −2 + (t − a)2H −2 da 0 where we used integration by parts in the first integral. In the case d = 3 we have the following. 4 Assume d = 3.
38) The solution exists when the above integral is well-defined. As for the heat equation, it depends on the dimension d and on the spatial covariance of the noise. For example, when the noise is white both in time and in space the solution exists if and only if d = 1. 56 2 Solutions to the Linear Stochastic Heat and Wave Equation The necessary and sufficient condition for the existence of the solution follows from . 35) admits a unique mild solution (u(t, x))t∈[0,T ],x∈Rd if and only if 1 μ(dξ ) < ∞.
6 The process U is self-similar (with respect to t) of order H − d4 . 20). Indeed, for every c > 0, ct R(ct, cs) = αH (2π)−d/2 0 cs |u − v|2H −2 (ct + cs) − (u + v) − d2 dvdu 0 = c2H − 2 R(t, s) d by the change of variables u˜ = uc , v˜ = vc . 18) with respect to the variable t. We will give sharp upper and lower bounds for the L2 -norm of this increment. We will assume in the sequel that T = 1. Concretely, we prove the following result. 2 There exists two strictly positive constants C1 , C2 such that for any t, s ∈ [0, 1] and for any x ∈ Rd C1 |t − s|2H − 2 ≤ E U (t, x) − U (s, x) ≤ C2 |t − s|2H − 2 .
Analysis of Variations for Self-similar Processes: A Stochastic Calculus Approach by Ciprian Tudor