General conditions of weak convergence of discrete-time multiplicative scheme to asset price with memory

Mišura, Julija S.; Ralchenko, Kostiantyn; Shklyar, S. V. Article General conditions of weak convergence of discrete-time multiplicative scheme to asset price with memory Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Mišura, Julija S.; Ralchenko, Kostiantyn; Shklyar, S. V. (2020) : General conditions of weak convergence of discrete-time multiplicative scheme to asset price with memory, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 8, Iss. 1, pp. 1-29, https://doi.org/10.3390/risks8010011 This Version is available at: https://hdl.handle.net/10419/257966 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory Yuliya Mishura *, Kostiantyn Ralchenko and Sergiy Shklyar Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska 64, Kyiv 01601, Ukraine; [email protected] (K.R.); [email protected] (S.S.) *Correspondence: [email protected] Received: 25 December 2019; Accepted: 27 January 2020; Published: 30 January 2020   Abstract: We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space D[ 0, T] . Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process. Keywords: weak convergence; fractional Brownian motion; asset price model with memory; binary market model; Cholesky decomposition 1. Introduction The question of approximating prices in financial markets with continuous time using prices in markets with discrete time goes back to the approximation of Black–Scholes prices with prices changing in discrete time. For an initial acquaintance with the subject, we recommend a book, Föllmer and Schied (2011) , that starts with the central limit theorem for approximation of the Black–Scholes model by the Cox–Ross–Rubinstein model. However, this area of research is immeasurably wider, since there are many more market models. Of course, they are functioning in discrete time, but their analytical research is easier to carry out in continuous time. Therefore, we need various theorems on the convergence of random sequences to random processes with continuous time, and it is also desirable to produce the convergence of some connected functionals. For example, functional limit theorems make it possible to go to a limit in stochastic integrals and stochastic differential equations. Such equations are widely used for modeling in physics, biology, finance and other fields. Concerning finance, functional limit theorems allow us to investigate how the convergence of stock prices affects the convergence of option prices. The last of these questions is widely considered in many papers; we mention now only Hubalek and Schachermayer (1998) . Concerning the weak limit theorems for financial market we mention the book Prigent (2003) . Diffusion approximation of financial markets was described, in particular, in the papers Mishura (2015a,2015b,2015c) , see references therein. All the above-mentioned works relate to the case when the ultimate stochastic process and the corresponding market model are Markov, that is, they have no memory. However, the presence of memory in financial markets has already been so convincingly recorded that for many years models have been studied that could well model this memory, and the question of Risks 2020,8, 11; doi:10.3390/risks8010011 www.mdpi.com/journal/risks Risks 2020,8, 11 2 of 29 approximation of non-Markov asset prices and other components of financial market processes by the discrete-time random sequences is also studied. As regards purely theoretical results on functional limit theorems in which the limit process is not Markov, we cite the papers Davydov (1970), where the limit process is stationary, and Gorodetskii (1977), where the limit process is semi-stable and Gaussian. In turn, with regard to memory modeling, considering the processes with shortor long-range dependence, it is easiest to use fractional Brownian motion that is a Gaussian self-similar process with stationary correlated increments. There are two approaches to the problem: To model prices themselves using processes with memory, in particular, to consider the models involving fBm, or to concentrate the model’s memory in stochastic volatility. The first approach has the peculiarity that an ultimate market with memory allows arbitrage, while prelimit markets can be arbitrage-free. The existence of arbitrage was first established in the paper Rogers (1997) and discussed in detail in the book Mishura (2008) . However, such an approach has the right to exist, if only because regardless of possible financial applications, it is reasonable to prove functional limit theorems in which the limit process is a fractional Brownian motion or some related process. For the first time, a discrete approximation of fractional Brownian motion by a binomial market model was considered in the paper Sottinen (2001) , and a fairly thorough analysis of the number of arbitrage points in such a market was made in the paper Cordero et al. (2016). However, even fractional Black–Scholes model can be approximated by various discrete-time sequences, and the purpose of this article is to formulate and illustrate by examples the functional limit theorem and its multiplicative version, in which both the prelimit sequence of processes and the limiting process are quite general, but simple to consider. Moreover, the fractional binomial market considered by Sottinen (2001) is a special case of our model. Thus, the main objectives of this article and its novelty are as follows. To start with, we consider an additive stochastic sequence that is based on the sequence of iid random variables and has the coefficients that allow for this stochastic sequence to be dependent on the past. For such a sequence, we formulate the conditions of the weak convergence to some limit process in terms of coefficients and the characteristic function of any basic random variable. These conditions are stated in Theorem 1. This theorem is of course a special case of general functional limit theorems, but it has the advantages that it is formulated in terms of coefficients, that the coefficients are such that they immediately show the dependence on the past, and that the limit process in it is not required to have any special properties with respect to distribution, self-similarity etc. However, then, in order to apply our general theorem to more practical situations, in Theorem 2we adapt the general conditions to the case where the limit process is Gaussian. Then we go to the multiplicative scheme in order to get the almost surely positive limit process that can model the asset price on the financial market. So, we assume that all multipliers in the prelimit multiplicative scheme are positive, and this imposes additional restrictions on the coefficients, and in addition, we consider only Bernoulli basic random variables. The next goal is to apply these general results to the case, where the limit processes in the additive scheme are fractional Brownian motion (fBm) and Riemann–Liouville fBm. In the case of the limit fBm we consider the prelimit processes that are constructed regarding to Cholesky decomposition of the covariance function of fBm. The result concerning fBm is new in the sense that nobody before considered the multiplicative scheme with exponent of fBm in the limit, and the result concerning Riemann–Liouville fBm is new in the sense that nobody before considered Riemann–Liouville fBm itself and in exponent as the limiting process. In both cases we were lucky in the sense that such coefficients are suitable also for the multiplicative scheme. Our proofs require deep study of the properties of the Cholesky decomposition for the covariance matrix of fBm. It turns out that all elements of the upper-triangular matrix in this decomposition are positive and moreover, the rows of this matrix are increasing. We also suppose that the columns of this matrix are decreasing. This conjecture is confirmed by numeric results, however its proof remains an interesting open problem. For the moment, we can only prove the uniform upper bound for the elements in each column, which is sufficient for our purposes. As for stochastic volatility with memory, it is not considered in the present paper, but we can refer the reader to Gatheral et al. (2018) and Bezborodov et al. (2019), among many others. Risks 2020,8, 11 3 of 29 The paper is organized as follows. In Section 2we establish sufficient conditions for the weak convergence of continuous-time random walks of rather general form to some limit in the space D[ 0, T] . The case of Gaussian limit is studied in more detail. Multiplicative version of this result is also obtained. Sections 3and 5are devoted to two particular examples of the general scheme investigated in Section 2. In Section 3we consider a discrete process that converges to fractional Brownian motion. This example is based on Cholesky decomposition of the covariance matrix of fractional Brownian motion. In Section 4we investigate possible perturbations of the coefficients in the scheme, studied in Section 3. Moreover, Section 4contains a numerical example, which illustrates the results of Section 3; moreover, we discuss there some our conjectures and open problems. Section 5is devoted to another example, where the limit process is a so-called Riemann–Liouville fractional Brownian motion. In Appendix Awe establish auxiliary results concerning the Cholesky decomposition of the covariance matrix of a stationary Gaussian process. In particular, we explore the connection between this decomposition and time series prediction problems. 2. General Conditions of Weak Convergence Let T> 0, (Ω , F , F= (Ft)t∈[0,T] ,P ) be a stochastic basis, i.e., a complete probability space (Ω,F,P)with a filtration Fsatisfying standard assumptions. 2.1. Convergence of Sums For any N≥ 1 consider the uniform partition tN k=kT/N, 0 ≤k≤N of [ 0, T] . Let {ξi,i≥1} be a sequence of iid random variables with E[ξi]=0 and Eξ2 i=1. Assume we are given for each N≥ 1 a triangular array of real numbers ncN j,k, 1 ≤j≤k≤No . Define a stochastic process XN(tN k) = k ∑ j=1 cN j,kξj,k=0, 1, . . . , N; let XN(t) = XN(tN k−1) for t∈[tN k−1 , tN k) , k= 1, . . . , N . Because of the dependence of the coefficients cN j,k on k, the increments of XNmay depend on the past, and the dependence may be strong. Let us first establish general conditions of weak convergence of the sequence XN,N≥1 in terms of coefficients cN j,k and characteristic function ϕ(λ) = E eiλξ1 of the underlying noise. We use the Skorokhod topology in the space of càdlàg functions D([ 0, T]) . A detailed discussion of the selection of topology can be found in D([0, T]) (Billingsley 1999, Chapter 13). Theorem 1. Assume that the following assumptions hold: (A1) There exists a stochastic process {X(t),t∈[0, T]} such that XN→X , N→∞ , in the sense of finite-dimensional distributions, that is for any l ≥1, 0=t0<t1<t2<··· <tl≤T,and λ1, . . . , λl∈R, l ∏ n=1 [Ntn/T] ∏ p=[Ntn−1/T]+1 ϕl ∑ m=n λmcN p,[Ntm/T]→E"exp (i l ∑ n=1 λnX(tn))#,N→∞. (1) (A2) There exist positive constants K and αsuch that for all integer N ≥1and 0≤j<k≤N j ∑ n=1cN n,k−cN n,j2+ k ∑ n=j+1cN n,k2≤Kk−j N1+α . (2) Then the weak convergence of measures holds: XN⇒X, N →∞in D([0, T]). Risks 2020,8, 11 4 of 29 Proof. First, note that E"exp (i l ∑ m=1 λmXN(tm))#=E"exp (i l ∑ m=1 λm [Ntm/T] ∑ p=1 ξpcN p,[Ntm/T])# =E exp    i l ∑ m=1 λm m ∑ n=1 [Ntn/T] ∑ p=[Ntn−1/T]+1 ξpcN p,[Ntm/T]    =E exp    i l ∑ n=1 [Ntn/T] ∑ p=[Ntn−1/T]+1 ξp l ∑ m=n λmcN p,[Ntm/T]    = l ∏ n=1 [Ntn]/T ∏ p=[Ntn−1/T]+1 ϕl ∑ m=n λmcN p,[Ntm/T]. (3) Therefore, the convergence of finite-dimensional distributions is equivalent to Condition (1). In order to prove the weak convergence, it suffices to establish the tightness of the sequence XN,N≥1. To start with, let us mention that for 0 ≤t1<t2≤T, EXN(t2)−XN(t1)2 =E  [Nt1/T] ∑ n=1cN n,[Nt2/T]−cN n,[Nt1/T]ξn!2 +E    [Nt2/T] ∑ n=[Nt1/T]+1 cN n,[Nt2/T]ξn  2   = [Nt1/T] ∑ n=1cN n,[Nt2/T]−cN n,[Nt1/T]2+ [Nt2/T] ∑ n=[Nt1/T]+1cN n,[Nt2/T]2=:AN(t1,t2). (4) Further, let us prove that there exists C>0 such that AN(t1,t2)AN(t2,t3)≤C(t3−t1)2+2α(5) for all N≥1 and for all 0 ≤t1<t2<t3≤T. We consider two cases. Case 1: t3−t1<T/N .In this case we have that [Nt3/T]−[Nt1/T]<N(t3−t1)/T+ 1 < 2, which means that [Nt3/T]−[Nt1/T]≤ 1. This implies that at least one of the following equalities is true: [Nt1/T] = [Nt2/T] or [Nt2/T] = [Nt3/T] . If [Nt1/T] = [Nt2/T] , then AN(t1 , t2) = 0 and Inequality (5) holds for any C> 0. Similarly, it holds in the case [Nt2/T]=[Nt3/T] , because AN(t2,t3) = 0. Case 2: t3−t1≥T/N. In this case [Nt2/T]−[Nt1/T] N≤t2−t1 T+1 N≤2(t3−t1) T. Then Condition (2) implies that AN(t1,t2)≤K[Nt2/T]−[Nt1/T] N1+α ≤K2 T1+α (t3−t1)1+α. The same bound holds for AN(t2,t3). Therefore, we have AN(t1,t2)AN(t2,t3)≤K22 T2+2α (t3−t1)2+2α, that is (5) holds with C=K2(2/T)2+2α. Risks 2020,8, 11 5 of 29 Thus, (5) is proved in both cases. Now using Inequalities (4) and (5), we may write EhXN(t2)−XN(t1)XN(t3)−XN(t2)i ≤EXN(t2)−XN(t1) 2EXN(t3)−XN(t2) 21/2 =AN(t1,t2)AN(t2,t3)1/2 ≤C1/2(t3−t1)1+α. Consequently, the sequence of processes is tight (Billingsley 1999, Theorem 13.5). Hence, the statement follows. If X is a Gaussian process, we can formulate the sufficient conditions for the convergence of finite-dimensional distributions in terms of the covariance function. Theorem 2. Assume that there exists a stochastic process X={X(t),t∈[0, 1]} such that the following conditions hold: (C1) X is Gaussian and centered. (C2) For all t,s∈[0, 1], [Nt]∧[Ns] ∑ j=1 cN j,[Nt]cN j,[Ns]→covX(t),X(s),as N →∞. (C3) max 1≤j≤k≤NcN j,k→0, as N →∞. Then the finite-dimensional distributions of XNconverge to those of X as N →∞. Proof. Let us consider the characteristic function of l-dimensional distribution: E"exp (i l ∑ m=1 λmXN(tm))#, 0 =t0<t1<t2<··· <tl≤1, λ1, . . . , λl∈R. According to (3), ZN:= l ∑ m=1 λmXN(tm) = [Ntl] ∑ p=1 αN,pξp, where for 1 ≤n≤l,[Ntn−1] + 1≤p≤[Ntn],N≥1, we define αN,p:= l ∑ m=n λmcN p,[Ntm]. For every N≥ 1, the random variables ηN,p:=αN,pξp , p≥ 1, are independent. We will apply Lindeberg’s CLT (see Billingsley 1995, Theorem 27.2) for the scheme of series nηN,1, . . . , ηN,[Ntl]o . Let us calculate the variance: Risks 2020,8, 11 6 of 29 σ2 N:=Var [Ntl] ∑ p=1 ηN,p!= [Ntl] ∑ p=1 α2 N,p= l ∑ n=1 [Ntn] ∑ p=[Ntn−1]+1 l ∑ m=n λmcN p,[Ntm]!2 = l ∑ n=1 [Ntn] ∑ p=[Ntn−1]+1 l ∑ m=n l ∑ q=n λmλqcN p,[Ntm]cN p,[Ntq] = l ∑ m=1 l ∑ q=1 λmλq m∧q ∑ n=1 [Ntn] ∑ p=[Ntn−1]+1 cN p,[Ntm]cN p,[Ntq] = l ∑ m=1 l ∑ q=1 λmλq [Ntm]∧[Ntq] ∑ p=1 cN p,[Ntm]cN p,[Ntq]. Hence, by the assumption (C2), we have σ2 N→ l ∑ m=1 l ∑ q=1 λmλqcovX(tm),X(tq),N→∞. (6) Now we are ready to verify Lindeberg’s condition. We have for any ε>0 LN(ε):=1 σ2 N [Ntl] ∑ p=1 Ehη2 N,p 1 {|ηN,p|≥εσN}i=1 σ2 N [Ntl] ∑ p=1 α2 N,pE  ξ2 p 1 (|ξp|≥εσN |αN,p|)  . We can estimate αN,pfor [Ntn−1] + 1≤p≤[Ntn]as follows αN,p= l ∑ m=n λmcN p,[Ntm]≤max 1≤j≤k≤NcN j,k l ∑ m=1|λm|. Therefore, σN αN,p≥σN max1≤j≤k≤NcN j,k∑l m=1|λm|=:aN. Note that due to (C3) and (6), aN→+∞,N→∞. Hence, LN(ε)≤1 σ2 N [Ntl] ∑ p=1 α2 N,pEhξ2 p 1 {|ξp|≥εaN}i=1 σ2 N [Ntl] ∑ p=1 α2 N,p!Ehξ2 1 1 {|ξ1|≥εaN}i, because ξp,p≥1, are iid. Since σ2 N=∑[Ntl] p=1α2 N,p, we see that LN(ε)≤Ehξ2 1 1 {|ξ1|≥εaN}i→0, N→∞, by the dominated convergence theorem. According to Lindeberg’s CLT, ZN→N(0, σ2), where σ2=lim N→∞σ2 N= l ∑ m=1 l ∑ q=1 λmλqcovX(tm),X(tq), Risks 2020,8, 11 7 of 29 see (6). Thus, E"exp (i l ∑ m=1 λmXN(tm))#=E[exp {iZN}]→exp −σ2 2 =exp (−1 2 l ∑ m=1 l ∑ q=1 λmλqcovX(tm),X(tq))=E"exp (i l ∑ m=1 λmX(tm))#. 2.2. Multiplicative Scheme Consider now a multiplicative counterpart of the process considered above. Namely, let nbN j,k, 1 ≤j≤k≤Nobe a triangular array of real numbers. Define SN(t) = [Nt/T] ∏ k=1 (1+ZN k),t∈[0, T], where ZN k= k ∑ j=1 bN j,kξj,k=1, . . . , N. To assure that the values of SN are positive, we assume that {ξn,n≥1} are iid Rademacher variables, i.e., P(ξn=1) = P(ξn=−1) = 1/2, and that bN j,ksatisfy the following assumption: k ∑ j=1bN j,k<1, k=1, . . . , N. Our aim is to investigate the weak convergence of SN to some positive process S . It is more convenient to work with logarithms, i.e. to consider log SN(t) = [Nt/T] ∑ k=1 log(1+ZN k),t∈[0, T]. We will need a uniform version of the above boundedness assumption: (B1) sup N≥1,k=1,...,N k ∑ j=1bN j,k<1. We will also need the following assumption (B2) N ∑ k=1 k ∑ j=1bN j,k2→0, N→∞. Theorem 3. Assume (B1) and (B2). Let also the assumptions (A1) and (A2) hold for cN j,k= k ∑ i=j bN j,i, 1 ≤j≤k≤N, with some process X. Then SN(t),t∈[0, T]converges in D([0, T]) to nS(t) = eX(t),t∈[0, T]o. Proof. Let supN≥1,k=1,...,N∑k j=1bN j,k=a∈(0, 1). By the Taylor formula, for x∈(−a,a), log(1+x) = x−θ(x)x2, Risks 2020,8, 11 8 of 29 where 0 <θ(x)≤C(a). Therefore, log SN(t) = [Nt/T] ∑ k=1 log(1+ZN k) = [Nt/T] ∑ k=1 ZN k− [Nt/T] ∑ k=1 θ(ZN k)ZN k2=:XN 1(t) + XN 2(t). Write XN 1(t) = [Nt/T] ∑ k=1 k ∑ j=1 bN j,kξj= [Nt/T] ∑ j=1 ξj [Nt/T] ∑ k=j bN j,k= [Nt/T] ∑ j=1 cN j,kξj. By Theorem 1,XN 1⇒Xin D([0, T]). Further, E"sup t∈[0,T]XN 2(t)#≤C(a) N ∑ k=1 EZN k2=C(a) N ∑ k=1 k ∑ j=1bN j,k 2→0, N→∞. Therefore, supt∈[0,T]XN 2(t) P → 0, N→∞ , so XN 2⇒ 0, N→∞ , in D([ 0, T]) . By Slutsky’s theorem (see, e.g., Grimmett and Stirzaker 2001, p. 318), we get log SN⇒X , N→∞ , in D([ 0, T]) , whence the claim follows. Remark 1. The statement of Theorem 3remains valid, if we replace Rademacher random variables by any other sequence {ξn,n≥1} of iid random variables such that E [ξn] = 0,E [ξ2 n] = 1, and |ξn|≤ 1for all n≥ 1. The latter condition along with the assumption (B1) ensures that ZN k>− 1for all 1 ≤k≤N , and consequently, the values of SNare positive. 3. Fractional Brownian Motion as a Limit Process and Prelimit Coefficients Taken from Cholesky Decomposition of its Covariance Function Let H∈(1 2 , 1 ) , T= 1. Let BH=BH t,t∈[0, 1] be a fractional Brownian motion, i.e., a centered Gaussian process with covariance function R(s,t) = 1 2s2H+t2H−|t−s|2H. (7) For N≥1 we define the triangular array ndj,k, 1 ≤j≤k≤Noby the following relation: p∧r ∑ j=1 dj,pdj,r=R(p,r). (8) It is known that such sequence ndj,k, 1 ≤j≤k≤No exists and it is unique, since (8) is the Cholesky decomposition of positively definite matrix (the covariance matrix of fBm). Define cN j,k=dj,k NH. (9) Theorem 4. Let {ξi,i≥1} be a sequence of iid random variables with E [ξi]= 0and E ξ2 i= 1. Assume that ncN j,k, 1 ≤j≤k≤Noare defined by (9)and XN(t) = [Nt] ∑ j=1 cN j,[Nt]ξj,t∈[0, 1]. Then XN→BH, N →∞, weakly in D([0, 1]). Risks 2020,8, 11 15 of 29 The function z(t , s) is the kernel of the following Molchan–Golosov representation of the fractional Brownian motion as an integral with respect to Wiener process W ={Wt,t≥0}: BH=Zt 0z(t,s)dWs. Therefore the covariance function of BHequals R(t,s) = Zt∧s 0z(t,u)z(s,u)du, and we obtain from (31)that k ∑ j=1bN j,k2≤Rk−1 N,k−1 N+Rk N,k N−2Rk−1 N,k N=1 N2H. (32) Conditions (B2) and (B1) are derived from the bound (32) similarly to the derivation of Lemmas 6and 7 from Equality (29). 4. Possible Perturbations of the Coefficients in Cholesky Decomposition. Numerical Example and Discussion of Open Problems 4.1. Possible Perturbations of the Coefficients in Cholesky Decomposition We now discuss the question how it is possible to perturb the coefficients (8) and (9) in the pre-limit sequence so that the convergence to fractional Brownian motion is preserved. In this sense, we estimate the rate of convergence of the perturbations to zero, sufficient to preserve the convergence. Theorem 6. 1. Let the coefficients ncN j,k, 1 ≤j≤k≤No and the random variables {ξi,i≥1} be the same as in Theorem 4. Consider the perturbed coefficients ˜ cN j,k=cN j,k+εN j, where a sequence {εN j, 1 ≤j≤N}satisfies the following conditions: (i) There exist positive constants C and αsuch that k ∑ n=j+1εN n2≤Ck−j N1+α for all 0≤j<k≤N; (ii) N ∑ j=1εN j2→0, as N →∞. Then the processes XN(t) = [Nt] ∑ j=1 ˜ cN j,[Nt]ξj,t∈[0, 1], converge to BH, as N →∞, weakly in D([0, 1]). 2. Assume, additionally, that the following assumption holds: (iii) There exists N0>0such that for all N ≥N0, εN j<1−j1/2 NH, Risks 2020,8, 11 16 of 29 and ξj,j≥1are iid Rademacher random variables. Let ˜ bN j,k=   ˜ cN j,k−˜ cN j,k−1,j<k, ˜ cN j,j,j=k. Then the sequence of stochastic processes (SN(t) = [Nt] ∏ k=1 1+ k ∑ j=1 ˜ bN j,kξj!,t∈[0, 1],N≥N0) converge in D[0, 1]to nS(t) = eBH(t),t∈[0, 1]o. Proof. 1. Let us prove that the conditions (A2),(C2) and (C3) remain valid, if we replace the coefficients cN j,kby ˜ cN j,k. Applying (i) and Lemma 1, we get j ∑ n=1˜ cN n,k−˜ cN n,j2+ k ∑ n=j+1˜ cN n,k2= j ∑ n=1cN n,k−cN n,j2+ k ∑ n=j+1cN n,k+εN n2 ≤2 j ∑ n=1cN n,k−cN n,j2+2 k ∑ n=j+1cN n,k2+2 k ∑ n=j+1εN n2 ≤2k−j N2H +2Ck−j N1+α , whence (A2) follows. Further, for 0 ≤s≤t≤1, we have [Nt]∧[Ns] ∑ j=1 ˜ cN j,[Nt]˜ cN j,[Ns]= [Ns] ∑ j=1cN j,[Nt]+εN jcN j,[Ns]+εN j = [Ns] ∑ j=1 cN j,[Nt]cN j,[Ns]+ [Ns] ∑ j=1εN j2+ [Ns] ∑ j=1 εN jcN j,[Ns]+ [Ns] ∑ j=1 εN jcN j,[Nt]. (33) The first term in the right-hand side of (33) converges to R(t , s) by (25) , as N→∞ . The second term is bounded by the sum ∑N j=1εN j2 , which tends to zero according to assumption (ii). Using the Cauchy–Schwarz and (7), we can bound the third term as follows:  [Ns] ∑ j=1 εN jcN j,[Ns]≤v u u t [Ns] ∑ j=1εN j2[Ns] ∑ j=1cN j,[Ns]2≤v u u t N ∑ j=1εN j2R[Ns] N,[Ns] N→0, as N→∞, by (ii), since R[Ns] N,[Ns] N=[Ns] N2H→s2H, as N→∞. Similarly, for the forth term we have  [Ns] ∑ j=1 εN jcN j,[Nt]≤v u u t [Ns] ∑ j=1εN j2[Ns] ∑ j=1cN j,[Nt]2≤v u u t N ∑ j=1εN j2[Nt] ∑ j=1cN j,[Nt]2→0, as N→∞. Thus, ∑[Nt]∧[Ns] j=1˜ cN j,[Nt]˜ cN j,[Ns]→R(t,s), as N→∞, i. e. the assumption (C2) is verified. Risks 2020,8, 11 17 of 29 Finally, the assumption (C3) also holds true, since max 1≤j≤k≤N˜ cN j,k≤max 1≤j≤k≤NcN j,k+max 1≤j≤NεN j, max1≤j≤k≤NcN j,k→0, N→∞, by (26), and max1≤j≤NεN j≤r∑N j=1εN j2→0, N→∞, by (ii). 2. Now let us verify the conditions (B1) and(B2) for the convergence of the corresponding multiplicative scheme. Note that ˜ bN j,k=   ˜ cN j,k−˜ cN j,k−1,j<k, ˜ cN j,j,j=k.=   cN j,k−cN j,k−1,j<k, ˜ cN j,j+εN j,j=k.=   bN j,k,j<k, bN j,j+εN j,j=k.(34) Therefore, k ∑ j=1˜ bN j,k2= k−1 ∑ j=1bN j,k2+bN k,k+εN k2≤2 k ∑ j=1bN j,k2+2εN k2. Consequently, using (ii) and Lemma 6we get N ∑ k=1 k ∑ j=1˜ bN j,k2≤2 N ∑ k=1 k ∑ j=1bN j,k2+2 N ∑ k=1εN k2→0, as N→∞, i. e., (B2) holds. Applying successively (34), (30), and (iii), we get k ∑ j=1˜ bN j,k≤ k ∑ j=1bN j,k+εN k≤k1/2 NH+εN k<1, for all N≥N0 and all 1 ≤k≤N . Hence, the assumption (B1) is also satisfied, and the convergence of the multiplicative scheme follows from Theorem 3. Example 1. The following sequences of εN jsatisfy the conditions (i)–(iii) of Theorem 6: •εN j=1 Nβ,β>1 2, •εN j=jγ Nβ,γ≥0,β−γ>1 2. 4.2. Numerical Example and Discussion of Open Problems First, let us illustrate the results of Section 3with a numerical example. Take H= 0.75, N= 10. In this case the covariance matrix of fractional Brownian motion nR(j,k) = cov(BH j,BH k),j,k=1, . . . , 10o equals                   1.00000 1.41421 1.68386 1.90192 2.09017 2.25830 2.41166 2.55358 2.68629 2.81139 1.41421 2.82843 3.51229 4.00000 4.40631 4.76268 5.08417 5.37945 5.65408 5.91189 1.68386 3.51229 5.19615 6.09808 6.77403 7.34847 7.85821 8.32161 8.74961 9.14933 1.90192 4.00000 6.09808 8.00000 9.09017 9.93426 10.6621 11.3137 11.9098 12.4629 2.09017 4.40631 6.77403 9.09017 11.1803 12.4386 13.4361 14.3058 15.0902 15.8114 2.25830 4.76268 7.34847 9.93426 12.4386 14.6969 16.1086 17.248 18.2504 19.1599 2.41166 5.08417 7.85821 10.6621 13.4361 16.1086 18.5203 20.0738 21.3459 22.4734 2.55358 5.37945 8.32161 11.3137 14.3058 17.248 20.0738 22.6274 24.3137 25.7109 2.68629 5.65408 8.74961 11.9098 15.0902 18.2504 21.3459 24.3137 27.0000 28.8114 2.81139 5.91189 9.14933 12.4629 15.8114 19.1599 22.4734 25.7109 28.8114 31.6228                   Risks 2020,8, 11 18 of 29 Then the upper-triangular matrix of its Cholesky decomposition is given by                   1 1.41421 1.68386 1.90192 2.09017 2.2583 2.41166 2.55358 2.68629 2.81139 0 0.91018 1.24256 1.43958 1.59349 1.7238 1.83873 1.94263 2.03816 2.12704 0 0 0.90378 1.22457 1.41016 1.55336 1.67362 1.77909 1.87406 1.96107 0 0 0 0.90040 1.21504 1.39427 1.53132 1.6457 1.74555 1.83513 0 0 0 0 0.89857 1.20967 1.38511 1.51842 1.62917 1.72551 0 0 0 0 0 0.89741 1.20618 1.37909 1.50985 1.61811 0 0 0 0 0 0 0.89661 1.20373 1.37483 1.50375 0 0 0 0 0 0 0 0.89603 1.20192 1.37165 0 0 0 0 0 0 0 0 0.89558 1.20053 0 0 0 0 0 0 0 0 0 0.89523                   We see that this example confirms the results of Lemmas 3and 4. In particular, all elements of this matrix are non-negative and its rows are increasing. Moreover, all diagonal elements are less or equal than 1, and are bigger than √2−22H−1= 0.76537. Furthermore, the bound (23) also holds. Indeed, for H= 0.75, CH=H·22−2H √2−22H−1= 1.38582. For example, for r= 10 we have that max1≤j≤10 dj,10 = 2.81139 is less than CH·102H−1=4.38235. We suppose that the columns of the above matrix are decreasing. More precisely, our conjecture can be formulated as follows. Conjecture A1. For all r ≥1, d1,r>d2,r>··· >dr,r. However, the proof of this fact is an open problem. Further, the corresponding covariance matrix of the increments process BH k−BH k−1 , k= 1, . . . , 10, is equal to                   1.00000 0.41421 0.26965 0.21806 0.18825 0.16813 0.15336 0.14192 0.13271 0.12510 0.41421 1.00000 0.41421 0.26965 0.21806 0.18825 0.16813 0.15336 0.14192 0.13271 0.26965 0.41421 1.00000 0.41421 0.26965 0.21806 0.18825 0.16813 0.15336 0.14192 0.21806 0.26965 0.41421 1.00000 0.41421 0.26965 0.21806 0.18825 0.16813 0.15336 0.18825 0.21806 0.26965 0.41421 1.00000 0.41421 0.26965 0.21806 0.18825 0.16813 0.16813 0.18825 0.21806 0.26965 0.41421 1.00000 0.41421 0.26965 0.21806 0.18825 0.15336 0.16813 0.18825 0.21806 0.26965 0.41421 1.00000 0.41421 0.26965 0.21806 0.14192 0.15336 0.16813 0.18825 0.21806 0.26965 0.41421 1.00000 0.41421 0.26965 0.13271 0.14192 0.15336 0.16813 0.18825 0.21806 0.26965 0.41421 1.00000 0.41421 0.12510 0.13271 0.14192 0.15336 0.16813 0.18825 0.21806 0.26965 0.41421 1.00000                   and the upper-triangular matrix of the corresponding Cholesky decomposition has the following form:                   1 0.41421 0.26965 0.21806 0.18825 0.16813 0.15336 0.14192 0.13271 0.12510 0 0.91018 0.33238 0.19702 0.15391 0.13031 0.11493 0.10391 0.09553 0.08888 0 0 0.90378 0.32080 0.18559 0.14319 0.12027 0.10547 0.09496 0.08702 0 0 0 0.90040 0.31464 0.17923 0.13705 0.11438 0.09985 0.08958 0 0 0 0 0.89857 0.31109 0.17545 0.13331 0.11075 0.09634 0 0 0 0 0 0.89741 0.30877 0.17291 0.13077 0.10825 0 0 0 0 0 0 0.89661 0.30712 0.17110 0.12892 0 0 0 0 0 0 0 0.89603 0.30590 0.16973 0 0 0 0 0 0 0 0 0.89558 0.30495 0 0 0 0 0 0 0 0 0 0.89523                   Risks 2020,8, 11 19 of 29 We observe that the values along all diagonals of this matrix decrease. These numerical results allows us to formulate the following conjecture. Conjecture A2. For all 1≤j≤k, `j,k> `j+1,k+1. Remark 5. 1. For the moment, we can proof only non-strict inequality for the case j=k , i.e., the monotonicity along the main diagonal, see Remark A2 below. 2. Conjecture A2 implies Conjecture A1. This becomes clear, if we rewrite the relation (20) between dj,k and `j,kas follows dj,k=`j,j+`j,j+1+`j,j+2+···+`j,k, 1 ≤j≤k. 3. In Appendix A.3 below we formulate Conjecture A3, which is a sufficient condition for Conjecture A2. 5. Riemann–Liouville Fractional Brownian Motion as a Limit Process Let H∈(1 2, 1),T=1. Let us define ZH(t) = Zt 0(t−u)H−1/2 dWu,t∈[0, 1], (35) where W={Wu,u∈[0, 1]} is a Wiener process. The process X is known as Riemann–Liouville fractional Brownian motion or type II fractional Brownian motion, see, e.g., Davidson and Hashimzade (2009); Marinucci and Robinson (1999). Define bN j,k=H−1 2 NH(k−j+1)H−3/2, 1 ≤j≤k≤N. (36) cN j,k= k ∑ i=j bN j,i=H−1 2 NH k ∑ i=j (i−j+1)H−3/2 =H−1 2 NH k−j+1 ∑ l=1 lH−3/2, 1 ≤j≤k≤N. (37) Theorem 7. Let {ξi,i≥1} be a sequence of iid random variables with E [ξi]= 0and E ξ2 i= 1. Assume that ncN j,k, 1 ≤j≤k≤Noare defined by (37)and XN(t) = [Nt] ∑ j=1 cN j,[Nt]ξj,t∈[0, 1]. Then XN⇒ZH, N →∞weakly in D([0, 1]). First, let us prove the convergence of finite-dimensional distributions. Lemma 8. Under assumptions of Theorem 7, the finite-dimensional distributions of XN converge to those of ZH. Proof. In order to prove the lemma, we will verify the conditions of Theorem 2. Let us start with proving that the covariance function of XN converges to the covariance function of ZH as N→∞ . Indeed, for s≤twe have [Ns] ∑ j=1 cN j,[Nt]cN j,[Ns]=(H−1 2)2 N2H [Ns] ∑ j=1 [Nt]−j+1 ∑ l=1 lH−3/2 [Ns]−j+1 ∑ k=1 kH−3/2. Risks 2020,8, 11 20 of 29 By the Euler–Maclaurin formula, N ∑ l=1 lH−3/2 =NH−1/2 H−1 2 +O(1), as N→∞. (38) Therefore [Ns] ∑ j=1 cN j,[Nt]cN j,[Ns]∼1 N2H [Ns] ∑ j=1 ([Nt]−j+1)H−1/2([Ns]−j+1)H−1/2 =[Ns]2H N2H [Ns] ∑ j=1[Nt] [Ns]−j−1 [Ns]H−1/2 1−j−1 [Ns]H−1/2 1 [Ns] →s2HZ1 0t s−yH−1/2 (1−y)H−1/2 dy =Zs 0(t−u)H−1/2 (s−u)H−1/2 du =EZt 0(t−u)H−1/2 dWuZs 0(s−u)H−1/2 dWu=covZH(t),ZH(s), hence, the condition (C2) is satisfied. Note that by (37) and (38), max 1≤j≤k≤NcN j,k=H−1 2 NHmax 1≤j≤k≤N k−j+1 ∑ l=1 lH−3/2 =H−1 2 NH N ∑ l=1 lH−3/2 =ON−1/2,N→∞, (39) and the condition (C3) also holds. Thus the assumptions of Theorem 2are satisfied. This concludes the proof. Now let us verify the tightness condition (A2). Lemma 9. Let the numbers ncN j,k, 1 ≤j≤k≤no be defined by (37) . Then there exists a constant C> 0such that for all 1≤j<k≤N, j ∑ n=1cN n,k−cN n,j2+ k ∑ n=j+1cN n,k2≤Ck−j NH+1/2 . (40) Proof. We estimate each of the sums in the left-hand side of (40). Using (37), we get j ∑ n=1cN n,k−cN n,j2=(H−1 2)2 N2H j ∑ n=1 k−n+1 ∑ l=j−n+2 lH−3/2!2 =(H−1 2)2 N2H j ∑ i=1 i+k−j ∑ l=i+1 lH−3/2!2 . The inner sum can be bounded as follows i+k−j ∑ l=i+1 lH−3/2 ≤iH 2−3 4 i+k−j ∑ l=i+1 lH 2−3 4≤iH 2−3 4 i+k−j ∑ l=i+1Zl l−1xH 2−3 4dx =iH 2−3 4Zi+k−j ixH 2−3 4dx =iH 2−3 4(i+k−j)H 2+1 4−iH 2+1 4 H 2+1 4≤iH 2−3 4(k−j)H 2+1 4 H 2+1 4 , where we used the inequality xβ−yβ≤(x−y)βfor x>y>0 and 0 <β≤1. Then j ∑ n=1cN n,k−cN n,j2≤(H−1 2)2 (H 2+1 4)2 (k−j)H+1 2 N2H j ∑ i=1 iH−3 2. Risks 2020,8, 11 21 of 29 Since iH−3/2 ≤xH−3/2 for x∈[i−1, i], we see that j ∑ i=1 iH−3/2 ≤ j ∑ i=1Zi i−1xH−3/2 dx =Zj 0xH−3/2 dx =jH−1/2 H−1 2 . (41) Consequently, we have j ∑ n=1cN n,k−cN n,j2≤H−1 2 (H 2+1 4)2j NH−1 2k−j NH+1 2≤H−1 2 (H 2+1 4)2k−j NH+1 2. (42) Now we estimate the second sum in (40). By the change of variables m=k−n+1, we get k ∑ n=j+1cN n,k2=(H−1 2)2 N2H k ∑ n=j+1 k−n+1 ∑ l=1 lH−3/2!2 =(H−1 2)2 N2H k−j ∑ m=1 m ∑ l=1 lH−3/2!2 . We bound the inner sum using (41) and estimate m2H−1≤(k−j)2H−1. We obtain k ∑ n=j+1cN n,k2≤1 N2H k−j ∑ m=1 m2H−1≤(k−j)2H N2H=k−j NH−1 2k−j NH+1 2≤k−j NH+1 2. (43) Combining (42) and (43), we conclude the proof. Now let us verify the conditions of Theorem 3. We start with condition (B1). Lemma 10. Let nbN j,k, 1 ≤j≤k≤no be a triangular array of non-negative numbers defined by (36) . Then for all N ≥1and for all k =1, . . . , N, k ∑ j=1 bN j,k≤1 √2. (44) Proof. It follows from (36) and (41) that k ∑ j=1 bN j,k=H−1 2 NH k ∑ j=1 (k−j+1)H−3/2 =H−1 2 NH k ∑ l=1 lH−3/2 ≤kH−1/2 NH. (45) If k≥2, then kH−1/2 NH=k NH1 √k≤1 √2and (44) is valid. In the remaining case k=1 we have k ∑ j=1 bN j,k=bN 1,1 =H−1 2 NH≤H−1 2<1 2<1 √2, and (44) also holds. Now let us verify the condition (B2). Lemma 11. The numbers nbN j,k, 1 ≤j≤k≤nodefined by (36)satisfy the condition (B2). Risks 2020,8, 11 22 of 29 Proof. We have N ∑ k=1 k ∑ j=1bN j,k2=H−1 22 N2H N ∑ k=1 k ∑ j=1 (k−j+1)2H−3=H−1 22 N2H N ∑ k=1 k ∑ l=1 l2H−3 =H−1 22 N2H N ∑ l=1 N ∑ k=l l2H−3=H−1 22 N2H N ∑ l=1 l2H−3(N−l+1) ≤H−1 22 N2H−1 N ∑ l=1 l2H−3→0, as N→∞, since H∈(1/2, 1)and ∑N l=1l2H−3<∞. Thus, we have proved that all assumptions of Theorem 3are satisfied. As a consequence, we obtain the following result. Theorem 8. Assume that nbN j,k, 1 ≤j≤k≤no is a triangular array of real numbers defined by (36) , ξj,j≥1 are iid Rademacher random variables, the process ZH is given by (35) . Then the sequence of stochastic processes (SN(t) = [Nt] ∏ k=1 1+ k ∑ j=1 bN j,kξj!,t∈[0, 1]) converge in D[0, 1]to nS(t) = eZH(t),t∈[0, 1]o. Author Contributions: All authors have read and agree to the published version of the manuscript. Conceptualization, Y.M.; investigation, Y.M., K.R. and S.S.; writing–original draft preparation, Y.M., K.R. and S.S.; writing–review and editing,Y.M., K.R. and S.S. Funding: Y.M. and K.R. acknowledge that the present research is carried through within the frame and support of the ToppForsk project nr. 274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. Monotonicity Along the Diagonal of a Triangular Matrix in the Cholesky Decomposition of a Positive Definite Toeplitz Matrix In this appendix we establish the connection between the prediction of a stationary Gaussian process and the Cholesky decomposition of its covariance matrix. It turns out that the positivity of the coefficients of predictor implies the monotonicity along the diagonals of a triangular matrix in the Cholesky decomposition. Let X={Xn , n∈Z} be a centered stationary Gaussian discrete-time process with the known autocovariance function: γ(k) = cov(Xn+k,Xn) = E[Xn+kXn]. Assume that all finite-dimensional distributions of X are non-degenerate multivariate normal distributions. In other words, we assume that for all n∈N , the symmetric covariance matrix of n subsequent values of X, Γn=cov      X1 X2 . . . Xn       =     γ(0)γ(1). . . γ(n−1) γ(1)γ(0). . . γ(n−2) .................. γ(n−1)γ(n−2). . . γ(0)      , (A1) Risks 2020,8, 11 23 of 29 is non-degenerate. Appendix A.1. Prediction of a Stationary Stochastic Process Let us construct the predictor of Xn by the observations X1 , . . . , Xm . Since the joint distribution X1 , . . . , Xm , Xn is Gaussian, we see that the conditional distribution Xn|(X1 , . . . , Xm) is also Gaussian. (Anderson 2003, Section 2.5). The parameters of this distribution are given by E[Xn|X1, . . . , Xm] = cov  Xn,   X1 . . . Xm       cov   X1 . . . Xm      −1   X1 . . . Xm    =γ(n−1). . . γ(n−m)Γ−1 m   X1 . . . Xm   , Var[Xn|X1, . . . , Xm] =Var(Xn)−cov  Xn,   X1 . . . Xm       cov   X1 . . . Xm      −1 cov     X1 . . . Xm   ,Xn   =γ(0)−γ(n−1). . . γ(n−m)Γ−1 m   γ(n−1) . . . γ(n−m)   . Denote    αn,m,1 . . . αn,m,m   =Γ−1 m   γ(n−1) . . . γ(n−m)   , (A2) σ2 n,m=γ(0)−γ(n−1). . . γ(n−m)Γ−1 m   γ(n−1) . . . γ(n−m)   . Then Xn|(X1, . . . , Xm)∼ N m ∑ k=1 αn,m,kXk,σ2 n,m!. (A3) The following theorem claims that if the coefficients of the one-step-ahead predictor are positive, then the coefficients of the multi-step-ahead predictor are also positive. Theorem A1. Assume that the inequality αm+1,m,k> 0holds for all 1 ≤k≤m . Then for all 1 ≤k≤m<n , αn,m,k>0, (A4) αn,m,k>αn+1,m+1,k+1. (A5) Proof. For a discrete-time stochastic process, the coefficients of the multi-step-ahead predictor are evaluated by the following recursive formula: αn+1,m,k=αn+1,n,k+ n ∑ i=m+1 αn+1,n,iαi,m,k, 1 ≤k≤m≤n. (A6) Risks 2020,8, 11 24 of 29 (Here, by convention, ∑n i=n+1. . . =0.) Fix m and k , 1 ≤k≤m . By assumption, αm+1,m,k> 0. According to (A6) , if αi,m,k> 0 for all i=m+ 1, . . . , n , then αn+1,m,k> 0. Hence, by induction, αn+1,m,k> 0 for all n≥m+ 1. Inequality (A4) is proved. Now let us establish Inequality (A5) . Again, we prove it by induction. We use the following recursive formula from the Levinson–Durbin algorithm (Shumway and Stoffer 2017;Subba Rao 2018): αm+2,m+1,k+1=αm+1,m,k−αm+2,m+1,1αm+1,m,m+1−k, 1 ≤k≤m. (A7) It implies that αm+2,m+1,k+1<αm+1,m,k, 1 ≤k≤m. The induction hypothesis is the following: For all mand k, 1 ≤k≤m, and for all j=1, . . . , J αm+j+1,m+1,k+1<αm+j,m,k. Hence, we need to prove the inequality αm+J+2,m+1,k+1<αm+J+1,m,k, 1 ≤k≤m. (A8) Using (A6), we obtain αm+J+2,m+1,k+1=αm+J+2,m+J+1,k+1+ m+J+1 ∑ i=m+2 αm+J+2,m+J+1,iαi,m+1,k+1 =αm+J+2,m+J+1,k+1+ m+J ∑ i=m+1 αm+J+2,m+J+1,i+1αi+1,m+1,k+1 <αm+J+1,m+J,k+ m+J ∑ i=m+1 αm+J+1,m+J,iαi,m,k=αm+J+1,m,k. Thus, Inequality (A8) is proved. Hence, αm+j+1,m+1,k+1<αm+j,m,k for all m and k , 1 ≤k≤m , and for all j∈N. Equivalently, Inequality (A5) holds for all n,mand ksuch that 1 ≤k≤m<n. Appendix A.2. Cholesky Decomposition of the Covariance Matrix Fix N . Recall that, by assumption, the covariance matrix ΓN of the random vector (X1 , . . . , XN)> is non-degenerate. In this case the matrix ΓNcan be uniquely represented as a product ΓN=LNL> N=     `1,1 0 . . . 0 `1,2 `2,2 . . . 0 ........... `1,N`2,N. . . `N,N           `1,1 `1,2 . . . `1,N 0`2,2 . . . `2,N ........... 0 0 . . . `N,N      , (A9) where LN is a lower-triangular matrix with positive diagonal elements. This representation is called Cholesky decomposition. Remark A1. The numbers `k,ndo not depend on N: `k,n(N1) = `k,n(N2)for all k ≤n≤min(N1,N2), where `k,n(N)denotes an element of the matrix LNin Decomposition (A9)for a certain value of N.