Appendix A Self-similar Processes with Self-similarity: Basic Properties The first paper to give a rigorous mathematical analysis of self-similar processes is [106]. There are several monographs (see e.g. [75, 160]) that the reader can con- sult in order to build a more complete picture of self-similarity and related topics. A.1 One-Parameter Self-similar Processes Let us define the concept of self-similarity. Definition A.1 A stochastic process (Xt )t≥0 is called self-similar if there exists a H real number H>0 such that for any c>0 the processes (Xct )t≥0 and (c Xt )t≥0 have the same finite dimensional distributions. Remark A.1 A self-similar process satisfies X0 = 0 almost surely. Definition A.2 A stochastic process (Xt )t≥0 is said to be with stationary increments if for any h>0 the distribution of the process (Xt+h − Xh)t≥0 does not depend on h. The self-similar processes with stationary increments all have the same covari- ance. Theorem A.1 Let (Xt )t≥0 be a non-trivial H -self-similar process with stationary 2 ∞ increments such that EX1 < . Then 1 2 2H 2H 2H EXt Xs = EX t + s −|t − s| . 2 1 C.A. Tudor, Analysis of Variations for Self-similar Processes, 251 Probability and Its Applications, DOI 10.1007/978-3-319-00936-0, © Springer International Publishing Switzerland 2013 252 A Self-similar Processes with Self-similarity: Basic Properties Proof Let s ≤ t. Writing 1 2 2 2 Xt Xs = X + X − (Xt − Xs) 2 t s we get 1 2 2 2 EXt Xs = X + X − EX − 2 t s t s 1 = EX2 t2H + s2H −|t − s|2H . 2 1 Functions of the moments of a self-similar process with stationary increments can yield information concerning the self-similarity index. Proposition A.1 Let (Xt )t≥0 be a non-trivial H -self-similar process with station- ary increments. Then (i) If E|X1| < ∞, then 0 <H≤ 1. (ii) If E|X1| < ∞ and H = 1 then Xt = tX1 a.s. | |α ∞ ≤ 1 (iii) If E X1 < for some α 1 then H<α . Proposition A.2 Let (Xt )t≥0 be a non-trivial H -self-similar process with station- ary increments such that EX2 < ∞. Define, for any integer n ≥ 1 1 r(n) = E X1(Xn+1 − Xn) . = 1 →∞ Then, if H 2 , as n ∼ − 2H −2 2 r(n) H(2H 1)n EX1. Proof From Proposition A.1, 1 r(n) = EX2 (n + 1)2H + (n − 1)2H − 2n2H 2 1 and it suffices to study the asymptotic behavior of the sequence on the right-hand side above when n →∞. = 1 = ≥ Remark A.2 If H 2 then r(n) 0 for any n 1. Definition A.3 We say that a process X exhibits long-range dependence (or it is a long-memory process) if rn =∞ n≥0 where r(n)= E(X − X )(X + − X ).Otherwise,if 1 0 n 1 n rn < ∞ n≥0 we say that X is a short-memory process. A.2 Multiparameter Self-similar Processes 253 From Proposition A.2 and Definition A.3 we conclude that if (Xt )t≥0 is a non- 2 ∞ trivial H -self-similar process with stationary increments and with EX1 < then 1 ≤ 1 X is with long-range dependence for H>2 and with short-memory if H 2 . A.2 Multiparameter Self-similar Processes We first introduce the concept of self-similarity and stationary increments for two- parameters processes. 2 Definition A.4 A two-parameter stochastic process (Xs,t)(s,t)∈T , T ⊂ R ,isself- similar with order (α, β) if for any h, k > 0 the process (Xs,t)(s,t)∈T defined as α β Xs,t := h k X s t ,(s,t)∈ T h , k has the same finite dimensional distributions as the process X. 2 Definition A.5 A process (Xs,t)(s,t)∈I with I ⊂ R is said to be stationary if for every integer n ≥ 1 and (si,tj ) ∈ I , i, j = 1,...,n, the distribution of the random vector (Xs+s1,t+t1 ,Xs+s2,t+t2 ,...,Xs+sn,t+tn ) does not depends on (s, t), where s,t ≥ 0,(s+ si,t + ti) ∈ I , i = 1,...,n. We will say that a two-parameter stochastic process (Xs,t)(s,t)∈R2 has stationary increments if for every h, k > 0 the process − − + (Xt+h,s+k Xt,s+k Xt+h,s Xt,s)(s,t)∈R2 is stationary. So, a two-parameter stochastic process has stationary increments if its rectan- gular increments are stationary. The concept can be extended to multiparameter stochastic processes. d Definition A.6 A stochastic process (Xt)t∈T , where T ⊂ R , is called self-similar with self-similarity order α = (α1,...,αd )>0 if for any h = (h1,...,hd )>0the ˆ stochastic process (Xt)t∈T given by ˆ α α1 αd = = ··· t t Xt h X t h1 hd X 1 ,..., d h h1 hd has the same law as the process X. Let us recall the notion of the increment of a d-parameter process X on a rect- d angle [s, t]⊂R , s = (s1,...,sd ), t = (t1,...,td ), with s ≤ t. This increment is denoted by X[s,t] and it is given by d− ri X[s,t] = (−1) i Xs+r·(t−s). (A.1) r∈{0,1}d 254 A Self-similar Processes with Self-similarity: Basic Properties When d = 1 one obtains X[s,t] = Xt − Xs while for d = 2 one gets X[s,t] = − − + Xt1,t2 Xt1,s2 Xs1,t2 Xs1,s2 . d Definition A.7 A process (Xt, t ∈ R ) has stationary increments if for every h > 0, d d d h ∈ R the stochastic processes (X[0,t], t ∈ R ) and (X[h,h+t], t ∈ R ) have the same finite dimensional distributions. Appendix B The Kolmogorov Continuity Theorem This result is used to obtain the continuity of sample paths of stochastic processes. Theorem B.1 Consider a stochastic process (Xt )t∈T where T ⊂ R is a compact set. Suppose that there exist constants p,C > 0 and β>1 such that for every s,t ∈ T p β E|Xt − Xs| ≤ C|t − s| . ˜ β−1 Then X has a continuous modification X. Moreover for every 0 <γ < p | ˜ − ˜ | p Xt Xs ∞ E sup γ < . s,t∈T ;s=t |t − s| In particular X admits a modification which is Hölder continuous of any order ∈ β−1 α (0, p ). There exists a two-parameter version of the Kolmogorov continuity theorem (see e.g. [12]). Theorem B.2 Let (Xs,t)s,t∈T be a two-parameter process, vanishing on the axis, with T a compact subset of R. Suppose that there exist constants C,p > 0 and x,y > 1 such that p x y E|Xs+h,t+k − Xs+h,t − Xs,t+k + Xs,t| ≤ Ch k for every h, k > 0 and for every s,t ∈ T such that s + h, t + k ∈ T . Then X admits a continuous modification X˜ . Moreover X˜ has Hölder continuous paths of any orders ∈ x−1 ∈ y−1 ∈ x (0, p ), y (0, p ) in the following sense: for every ω Ω, there exists a Cω > 0 such that for every a s,t,s ,t ∈ T Xs,t(ω) − Xs,t (ω) − Xs ,t (ω) + Xs ,t (ω) ≤ Cω t − t s − s . C.A. Tudor, Analysis of Variations for Self-similar Processes, 255 Probability and Its Applications, DOI 10.1007/978-3-319-00936-0, © Springer International Publishing Switzerland 2013 Appendix C Multiple Wiener Integrals and Malliavin Derivatives Here we describe the elements from the Malliavin calculus that we need in the monograph. As mentioned, we give only a flavor of the Malliavin calculus, the basic tools necessary to follow the exposition in Part II of the monograph. Consider a real separable Hilbert space H and (B(ϕ), ϕ ∈ H) an isonormal Gaussian process on a probability space (Ω, A,P), which is a centered Gaussian family of random variables such that E(B(ϕ)B(ψ)) = ϕ,ψ H. Denote by In the multiple stochastic integrals with respect to B (see [136]). This In is actually an isometry between the Hilbert space H n(symmetric tensor product) equipped with the scaled norm 1 √ ·H⊗n and the Wiener chaos of order n which is defined as the closed linear n! span of the random variables Hn(B(ϕ)) where ϕ ∈ H, ϕH = 1 and Hn is the Hermite polynomial of degree n ≥ 1 (−1)n x2 dn x2 Hn(x) = exp exp − ,x∈ R. n! 2 dxn 2 The isometry of multiple integrals can be written as: for m, n positive integers, ˜ E In(f )Im(g) = n! f,g˜ H⊗n if m = n, (C.1) E In(f )Im(g) = 0ifm = n. We also have ˜ In(f ) = In(f) ˜ ˜ where f denotes the symmetrization of f defined by f(x1,...,xn) = 1 f(x ,...,x ). n! σ ∈Sn σ(1) σ(n) We recall that any square integrable random variable which is measurable with respect to the σ -algebra generated by B can be expanded into an orthogonal sum of multiple stochastic integrals F = In(fn) (C.2) n≥0 n where fn ∈ H are (uniquely determined) symmetric functions and I0(f0) = E[F ]. C.A. Tudor, Analysis of Variations for Self-similar Processes, 257 Probability and Its Applications, DOI 10.1007/978-3-319-00936-0, © Springer International Publishing Switzerland 2013 258 C Multiple Wiener Integrals and Malliavin Derivatives Let L be the Ornstein-Uhlenbeck operator LF =− nIn(fn) n≥0 ∞ 2 2 ∞ where F isgivenby(C.2) such that n=1 n n fn H⊗n < .