By Ciprian Tudor
Self-similar strategies are stochastic strategies which are invariant in distribution lower than compatible time scaling, and are a subject matter intensively studied within the previous couple of a long time. This booklet offers the elemental homes of those strategies and makes a speciality of the learn in their edition utilizing stochastic research. whereas self-similar approaches, 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 strategies to the partial differential equations pushed via fractional noises.
In this monograph the writer discusses the fundamental houses of those new periods of self-similar strategies and their interrelationship. even as a brand new method (based on stochastic calculus, in particular Malliavin calculus) to learning the habit of the diversities of self-similar procedures has been constructed over the past decade. This paintings surveys those contemporary innovations and findings on restrict theorems and Malliavin calculus.
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Extra info for Analysis of Variations for Self-similar Processes: A Stochastic Calculus Approach
The lines of the below proof will explain why the case d = 2 has to be excluded. 7 Suppose d = 2. The covariance function R(t, s) can be decomposed as follows R(t, s) = (2π)− 2 αH Cd β 2H − 1, − d d +2 2 (t + s)2H − 2 − (t − s)2H − 2 d d (d) + R1 (t, s) where Cd = 2 2−d , β(x, y) 1 x−1 y−1 (1 − t) dt and 0 t R1 (t, s) = (2π)− 2 αH Cd is the Beta function defined for x, y > 0 by β(x, y) = s d (d) s − daa 2H −2 (t + s) − a − d2 +1 − (t − s) + a − d2 +1 0 da(s − a)− 2 +1 (t − a)2H −2 + (t + a)2H −2 .
20) 0 Proof The following holds t R(t, s) = (2π)−d αH s 0 dudv|u − v|2H −2 0 t = αH (2π)−d/2 0 s 1 Rd e− 2 |ξ | 2 (t+s−u−v) |u − v|2H −2 (t + s) − (u + v) − d2 dvdu. 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.
1, for every s, t t R(t, s) = Eu(t, x)u(s, y) = αH (2π)−d 0 × Rd s dudv|u − v|2H −2 0 dξ |ξ |−α e− (t−u)|ξ |2 2 e− (s−v)|ξ |2 2 . Let c > 0. By the change of variables u˜ = uc , v˜ = vc in the integral dudv and then √ by the change of variables ξ˜ = cξ in the integral dξ we get R(ct, cs) = c2H − d−α 2 R(t, s). 18), that is, 2 E u(t, x) − u(s, y) . 52 2 Solutions to the Linear Stochastic Heat and Wave Equation We will make the following assumption: μ(ξ ) ∼ c|ξ |−α dξ, with 0 < α < d. 33) This means that for every function h such that the below integrals are finite, there exists two strictly positive constants c and c such that c Rd h(ξ )|ξ |−α dξ ≤ Rd h(ξ )μ(dξ ) ≤ c Rd h(ξ )|ξ |−α dξ.
Analysis of Variations for Self-similar Processes: A Stochastic Calculus Approach by Ciprian Tudor