Optimal design, financial and risk modelling with stochastic processes having semicontinuous covariances
aa r X i v : . [ s t a t . M E ] D ec Optimal design, financial and risk modelling withstochastic processes having semicontinuous covariances
M. Stehl´ık a,1, ∗ , Ch. Helpersdorfer a , P. Hermann a a Department of Applied Statistics, Johannes Kepler University Linz, Altenbergerstraße69, 4040 Linz, Austria b Institute of Statistics, University of Valpara´ıso, Blanco 951, Valpara´ıso, Region deValpara´ıso, Chile
Abstract
A.N. Kolmogorov proposed several problems on stochastic processes, whichhas been rarely addressed later on. One of the open problems are stochasticprocesses with discontinuous covariance function. For example, semicontin-uous covariance functions have been used in regression and kriging by manyauthors in statistics recently. In this paper we introduce purely topologicallydefined regularity conditions on covariance kernels which are still applicablefor increasing and infill domain asymptotics for regression problems, krigingand finance. These conditions are related to semicontinuous maps of Orn-stein Uhlenbeck (OU) processes. Beside this new regularity conditions relaxthe continuity of covariance function by consideration of semicontinuous co-variance. We provide several novel applications of the introduced class foroptimal design of random fields, random walks in finance and probabilitiesof ruins related to shocks, e.g. by earthquakes. In particular we construct arandom walk model with semicontinuous covariance.
Keywords:
Optimal design, semicontinuous covariance, correlated process,strong convergence, topology
Dedication:
This paper discusses processes with jumps in correlations,issue dedicated to Andrey N. Kolmogorov. ∗ Corresponding author
Email addresses: [email protected] (M. Stehl´ık), [email protected] (P. Hermann)
Preprint submitted to Information Sciences August 29, 2018 . Introduction
Mostly used dependence structure in regression problems is covariance.[22] states in his book: “One of the fundamental assumptions, the knowl-edge of the covariance function, is in most cases almost unrealistic. It seemsto be artificial, that the first moment E ( Y ( x )) is assumed to be unknownwhereas the more complicated second one is assumed to be known...”. Re-cently, discontinuous covariance functions have been used in both regressionand kriging by many authors. For instance, it was observed that in some sit-uations it can be useful to use semicontinuous covariance functions instead ofcontinuous ones. One example is an application of input deformations withBrownian motion filters for discontinuous regression (see [10]). For a treat-ment of discontinuous nature of kriging interpolation see e.g. [19]. Anotherwidely spread use of semicontinuous covariances follows from usage of nuggeteffect, see [6]. In computer experiment literature, typically used covariancefunctions vary from analytically smooth Gaussian to one-sided differentiableexponentially decaying (see [30]). However, less smooth covariance have notbeen studied from statistical perspective.But how erratic may a covariance function be? If we would like to con-sider the pragmatical point of view, then any practically relevant discontin-uous covariance function should be measurable. Then using the result of [7],measurable covariance function C admits decomposition C = C + C , where C is a continuous covariance and C vanishes Lebesgue almost everywhere.In the latter we assume (without loss of generality) domination by a Lebesguemeasure, which covers most of practical applications. However, it is essentialto recall the assumption of measurability of C (see [7]). [7] has also proventhat if C is isotropic and positive definite on R m , m > C is continuousexcept perhaps at d = 0 . However, for m = 1 the latter is not true anymore,as an example we may take C ( q ) = 1 for all rational numbers q ∈ Q and0 otherwise, which is positive definite, isotropic and discontinuous on Q. Inthis paper we consider the more complicated, but very practical, case d = 1(e.g. time).Beside the above discussed problem, the path restriction from continu-ity of covariance is well recognized (at least from [3] to the best knowledgeof author, even though Kolmogorov mentioned this problem already). [3]has proven that for a stationary Gaussian process with a continuous corre-lation function, assuming real values, one of the following alternatives holds:either, with probability one, the sample functions are continuous or, with2robability one, they are unbounded in every finite interval. But this is toorestrictive for many practical statistical problems. As a remedy, we suggestthe semicontinuity of covariance functions as a good and practical substitutefor a continuity. More precisely, semicontinuity is a more appropriate vari-ant of a “continuity” framework, which can still justify increasing domainasymptotics under mild regularities on space. We provide weak conditionson covariance functions which lead to feasible results for increasing domainasymptotics. We represent a class of processes obtained by specific semicon-tinuous maps of covariance functions of stationary Ornstein-Uhlenbeck (OU)processes. Several examples aim at convincing the reader that this way ofapproach to regularity conditions for covariance functions opens a direct wayto relax continuity conditions in order to benefit statistical science. Let us consider for the sake of simplicity the isotropic stationary process(see e.g. [6]) Y ( x ) = θ + ε ( x )with the design points x , ..., x n taken from a compact design space X .For the sake of simplicity we consider X = [ a, b ] . The mean parameter E ( Y ( x )) := θ ∈ Θ is unknown, the variance-covariance structure C r ( d ) de-pends on another unknown parameter r ∈ Ω and d i is the distance betweentwo particular design points, x i and x i +1 . Parametric spaces Θ and Ω are opensets with respect to standard topology. Let us assume E ( Y ( s + h ) − Y ( s )) =0 and define 2 γ ( h ) = var ( Y ( s + h ) − Y ( s ))(equation make sense only when right side depends only on h ). If this is thecase, we will say that the process is intrinsically stationary , the function 2 γ ( h )is then called variogram and γ ( h ) is called semivariogram (for more detailssee [6]). Let us briefly introduce the most common isotropic semivariograms,for further discussion see e.g. [6]. Now consider the three basic isotropicmodels: linear, spherical and exponential. • Linear model valid in R n , n ≥ ,γ ( d ) = (cid:26) τ + σ d, for d > Spherical model valid in R , R , R γ ( d ) = τ + σ ( d r − ( dr ) ) , for 0 < d ≤ r , τ + σ , for r < d ,0 otherwise. (2) • Exponential model valid in R n , n ≥ ,γ ( d ) = (cid:26) τ + σ (1 − exp( − rd )) , for d > Definition 1. (abc class)
We assume the class of positive definite functions C r ( d ) : Ω × R + → R such thata) C r (0) = 1 , C r ( d ) ≥ for all r ∈ Ω and < d < + ∞ , b) for all r mapping d → C r ( d ) is semicontinuous, almost everywhereconvex and decreasing on (0 , + ∞ ) c) lim d → + ∞ C r ( d ) = 0 . Remark 1. i) First, notice that conditions abc in Definition 1 define a classof valid covariance functions for continuous C r ( d ) (due to the celebrated cri-terion of [26]). However, not every semi-continuous, but almost everywhereconvex and non-increasing map C r ( d ) defines a valid correlation structure in (0 , + ∞ ) n , since positive definiteness may be violated. Therefore we have writ-ten “positive definite” explicitly in Definition 1. However, it is easy to checkthat for n = 2 abc define always a valid correlation, since C r ( d ) is decreasing.For higher n one can use the fact that C r ( d ) has maximally countable manypoints of discontinuity, which will be well separated by a sufficiently regularsupport topology (which may allow us to make a modified proof from [26]). Amore detailed classification of such classes is out of scope of this paper andwill be a valuable direction for further research. ii) Relaxation of condition b) from (typically assumed) continuity to semi-continuity is worth some words. First, from practical point of view, many ovariance structures recently used in theory and practice have been indeeddiscontinuous and still semicontinuous. A survey to semicontinuous func-tions can be found in [23]. To illustrate semicontinuous (but not continuous)covariance functions (i.e. covariance function satisfying abc) let us considerfor appropriately chosen c and D the covariance defined at R + C ( d ) = σ , for d = 0 , cσ , for < d ≤ D , , else (4) which is the prolongation of the covariance given by c ∈ [0 , , D, d ∈ [0 , in[20]. Figure 1 contains several trajectories for different values of parameters c and σ of random walk Y t = P ti =1 X i based on Gaussian process X i withcovariance (4). For this computation we fixed D = 2 , d = 0 . . The impactof the parameters σ and c should be visible within one plot. Therefore thecomparative line (black) for the path of Y t was computed for c = 0 . and σ = 1 . It is clear that increasing c yields in a shift in direction of the originas well as the effect of a decrease of σ . − Path of Yt for different values of c and sigman = 100, d = 0.1, c = 0.9, D = 2 t Y t c = 0.1, sigma = 1c = 0.5, sigma = 1c = 0.2, sigma = 1c = 0.1, sigma = 0.5c = 0.1, sigma = 0.2 Figure 1: Paths for different values of parameters c and σ on the basis of equation (4) The next theorem provides the representation of covariance functions whichare semicontinuous maps of covariance of Ornstein Uhlenbeck process.
Theorem 1. (Representation Theorem)
Let C be abc. Then C r ( d ) = σ exp( − ψ r ( d )) , where ψ r : [0 , + ∞ ) → R ∪ { + ∞} is: semicontinuous, non decreasing, lim d → + ∞ ψ r ( d ) = + ∞ . Proof. (Representation Theorem) 5e have ψ r ( d ) = − log (cid:18) C r ( d ) σ (cid:19) , (5)where σ = C r (0) . Thus semicontinuity follows from (5), since it is a composi-tion of a continuous and semicontinuous function. Non-decreasingness followsfrom (5), since it is a composition of a monotonous and non-increasing func-tion. We have lim d → + ∞ ψ r ( d ) = − lim d → + ∞ log C r ( d ) = − log (0) = + ∞ . iii) Notice that assumptions abc are fulfilled with many covariance struc-tures (e.g. power exponential correlation family or Mat´ern class, see e.g. [30]). iv)
Let C , C ∈ abc then for any α, β > αC + βC ∈ abc. Let C ∈ abc , then for any α ∈ N also C α ∈ abc. Here N denotes the setof all natural numbers. The paper is organized as follows: in section 2 we study properties ofabc class of covariance functions in more details. Also a discussion on thetopological convergences preserving the regularity conditions abc is givenfor different choices of the target space regularities. Pointwise convergentsequences of abc-covariance in metric space is proven to be convergent to theabc-covariance. Strong convergence, a purely topologically defined uniformconvergence for non-uniform spaces is found to be an appropriate candidatefor preserving conditions abc when target space is endowed with generaltopology. Section 3 considers optimal design for regression problem withcorrelated errors for abc covariance function. In particular, the lower andupper bound for M θ ( d ) for the class of processes with covariances satisfyingabc is given and the monotonicity of functions d → LB ( d ) , M θ ( d ) , U B ( d )is proven. Optimality of equidistant design is proven for parameter θ andcovariance functions from abc class. We also provide conditions for existenceof admissible optimal design for parameter r in abc class. Several examplesillustrate increasing domain asymptotics for various covariance functions. Insection 4 we provide illustrations of introduced stochastic processes and theirapplications to finance. The first application is forecasting of stock marketswith implications to Efficient Market Hypothesis. The second applicationis to model the probability of ruin with semicontinuous covariance, withpotential importance regarding catastrophes such as earthquakes or rain-storms. To maintain the continuity of explanation, the technicalities andproofs are put into the Appendix. 6 . abc class Here we provide results on topological convergences preserving regularityconditions abc. We prove that the pointwise convergence is sufficient for thelocally separable metric domain and metric target space (usually both are R with Euclidean topology). However, there are situations where it is moreappropriate to endow the target space R by a different topology: for this casewe give regularity conditions on the topology under which abc is preservedby strong convergence. We show that the minimal regularity of the targetspace is to be a regular and fully normal topological space.We have found strong convergence (as introduced by [16]) to be the mostrelevant for our setup. It is defined purely topologically and it is a topolog-ical uniform convergence for non-uniform spaces as the analogue of uniformconvergence for metric and uniform spaces which preserve the continuity ofthe functions. Strong convergence can be defined by the convergent netsand has many nice properties, e.g. preserving the fixed point property (FPP)(see [15]). The strong convergence is the appropriate one, since the uniformconvergence in a non compact space does not preserve FPP (see [15]). Insuch a context strong convergence is the weakest one with FPP: pointwiseconvergence does not have such a property and convergences based on graphtopology, fine and open cover are “too strong”, i.e. FPP is also preservedby the graph topology, but the graph convergence implies the strong con-vergence. Theorem 2 shows that properties abc are preserved by pointwiseconvergence for the metric target space. Theorem 2.
Let X n be a sequence of isotropic random fields with covariancekernels K n defined on a locally separable metric space mapping to the metricspace satisfying conditions abc. Let K n be uniformly convergent to K, then K also satisfies abc. Remark 2.
Notice, that if we have a continuous covariance, we will need tohave a uniform convergence to preserve continuity.
Now let us consider the general situation, i.e. general topology is endowedon the target space. Theorem 3 gives the preservation of abc for continuouscovariances.
Theorem 3.
Let { X γ } , γ ∈ Γ be a net of isotropic random fields with contin-uous covariance kernels K γ mapping to the regular topological space satisfyingconditions abc. Let K γ be strongly convergent to K, then K also satisfies abc. Lemma 1.
Let X be a topological space and Y be a regular space. Let { f γ : X → Y, γ ∈ Γ } be a net of functions semicontinuous at x ∈ X thatconverges strongly to a function f . Then f is semicontinuous at x . Example 2 in [16] shows, that the regularity of target space Y cannot beomitted. Now we need a Lemma providing a more general version of inter-change of limits theorem for a net of not necessarily continuous covariancefunctions satisfying abc to justify that limit also satisfies c). Lemma 2.
Let Z be a topological space, Y is a fully normal, T -space and ∅ 6 = X ⊆ Z. Let a ∈ Z be an accumulation point of a set X. Let net f γ (: X → Y ) s → f and ∀ γ ∈ Γ exists lim x → a f γ ( x ) := A γ ∈ Y. Then a net { A γ } is convergent and the limits interchange is valid, i.e. lim x → a lim γ ∈ Γ f γ ( x ) = lim γ ∈ Γ lim x → a f γ ( x )Now we are ready to formulate the general theorem for preserving abcclass. Theorem 4.
Let { X γ } , γ ∈ Γ be a net of isotropic random fields with co-variance kernels K γ mapping to the regular and fully normal topological spacesatisfying conditions abc. Let K γ be strongly convergent to K. Then K alsosatisfies abc.
3. Optimal Design for regression with correlated errors from abcclass
The determination of optimal designs for models with correlated errors issubstantially more difficult and for this reason not so well developed. Herewe concentrate on Ornstein-Uhlenbeck processes. For the influential papersthere is a pioneering work of [11], who considered the weighted least squareestimate, but considered mainly equidistant designs. Optimal design forestimation of parameters of OU processes has been studied in [14], optimaldesigns for prediction of OU sheets in [2]. We can find applications of variouscriteria of design optimality for second-order spatial models in the literature.8ince in our setup the information matrix is scalar, maximizing of M θ leadsto the optimal design in the sense of D, E, A or G optimality. Theoreticaljustifications for using the Fisher information for D -optimal designing undercorrelation can be found in [1, 25, 37]. In our setup, Fisher informationmatrix for trend parameter θ is defined as M θ ( n ) = 1 T C − ( r ) 1 , (6)where n denotes the number of design points and D is the vector of distances.According to the results of [25] the Fisher information matrix on r has theform M r ( n ) := 12 tr (cid:26) C − ( n, r ) ∂C ( n, r ) ∂r C − ( n, r ) ∂C ( n, r ) ∂r (cid:27) . (7)Notice, that Ornstein-Uhlenbeck process is a special case of class abcfor ψ r ( d ) = rd. For Ornstein Uhlenbeck processes all distance gradientsof M θ increase with same speed, since FIM has the form M θ ( n, D ) = 1 + P n − i =1 exp( rd i ) − rd i )+1 (see [14] or [36]). Thus the optimal design is equidistant atany fixed compact design space.The FIM for both parameters is M r M θ and collapses for the OU processfor two point designs, nevertheless it gives a design with a finite distance for n > . This is proven in the following:
Lemma 3.
The distance of neighboring points in the optimal design for es-timation of parameters ( r, θ ) is collapsing for n = 2 and it is equidistant for n > . This section considers how the nominal level of Fisher information of thedesign without nugget would be affected by a nugget. We consider class ofabc covariances of the form:
Cov ( x s , x t ) = (cid:26) . . . s = tc exp − r ( | t − s | ) . . . s = t, (8)where 0 < c ≤ c = 1 we receive a standard OUcovariance without nugget, however, c < τ = 1 − c. Let M θ,c , M r,c denote the Fisher information for trend θ and covariance parameter r , respectively. We define effectiveness in the following form M θ,c /M θ, , M r,c /M r, . (9)9e call (9) effectiveness, since it is similar to efficiency, where the ratio ofFisher information for design and optimal design is computed. Figure 2 illus-trates the behavior of effectiveness (9) at the design space X = [0 , d = y − x and fix the parameters r = 1 and θ = 1. It is clear that decrease of covariance have decreasing effecton effectiveness . M θ ( n )Hereafter, we introduce the lower and upper bounds for M θ and studytheir properties, together with properties of M θ . Let us consider a lower andupper bound for M θ ( n, D ) of the forms LB ( n, D ) := n inf x x T C − xx T x .U B ( n, D ) := n sup x x T C − xx T x . It is easy to see that LB ( n, D ) ≤ M θ ( n, D ) ≤ U B ( n, D ) . The followingtheorem holds.
Theorem 5. i) Let C r ( d ) be a covariance structure satisfying abc. Then forany design { x, x + d , x + d + d , ..., x + d + ... + d n − } and for any subsetof distances d j , j = 1 , ..., n − . a) the lower bound function ( d i , ..., d i m ) → LB ( n, D ) is nondecreasing in D . The upper bound function ( d i , ..., d i m ) → U B ( n, D ) is nondecreasing in D . Fisher information ( d i , ..., d i m ) → M θ ( n, D ) is nondecreasing in D . b) Especially, for any equidistant design ( ∀ i : d i = d ) functions d → LB ( n, D ) , d → U B ( n, D ) and d → M θ ( n, D ) are nondecreasing. ii) Denote by a ( n, n − the ratio M θ ( n, D ) /M θ ( n − , D ) . Then lim ∀ i : d i → + ∞ a ( n, n −
1) = nn − . iii) equidistant design is optimal for θ in abc on every compact designspace X, more precisely for X = [0 , any point ( d , ..., d n − ) of a set ⊗ n − i =1 ψ − r ( L/ ( n − such that d i ≥ , P d i ≤ is a set of optimal inter-distances iv) for a stationary OU there does not exist an admissible design forparameter r (i.e. optimal design for r is collapsing). However, in abc class notnecessarily e.g. nugget effect can bring a regularization and thus admissibledesigns may exist. fficiency Level for M Θ @ d D Linear Model M Θ @ d D(cid:144) M Θ @ d D c = (a) Effectiveness for θ of linearmodel (1), comparison with onejump Efficiency Level for M r @ d D Linear Model M r @ d D(cid:144) M r @ d D c = (b) Effectiveness for r of linearmodel (1), comparison with onejump Efficiency Level for M Θ @ d D Spherical Model M Θ @ d D(cid:144) M Θ @ d D c = (c) Effectiveness for θ of spheri-cal model (2), comparison with onejump Efficiency Level for M r @ d D Spherical Model M r @ d D(cid:144) M r @ d D c = (d) Effectiveness for r of spheri-cal model (2), comparison with onejump Efficiency Level for M Θ @ d D M Θ @ d D(cid:144) M Θ @ d D c = (e) Effectiveness for θ of exponen-tial model (3), comparison with onejump Efficiency Level for M r @ d D M r @ d D(cid:144) M r @ d D c (f) Effectiveness for r of exponen-tial model (3), comparison with onejumpFigure 2: Effectiveness for linear model shown in (a), (b); the spherical model in (c), (d)and the exponential model in (e), (f). Remark 3.
Notice that i) of Theorem 5 shows that the interval over whichobservations are to be made should be extended as far as possible. This issupporting the idea of increasing domain asymptotics. Also notice, that both esults i, ii) of Theorem 5 generalize the findings of [11] and [14].Particularly, to illustrate result i,b) let us consider the equidistant n-pointdesign for parameter θ of Ornstein Uhlenbeck process. The covariance matrixis Toeplitz with entries c | i − j | , c = exp( − rd ) . Then we know that C − is tridi-agonal and that (1 − c ) C − has the entry − c in every upper and sub-diagonalposition and has main diagonal entries , c , ..., c , (see [13], Example13, page 409). Thus we get M θ ( d ) = − n + ne rd e rd (see also Lemma 1 in [14])and so M θ ( n, D ) is an increasing function of distance d. The formula for Fisher information M r ( n ) on correlation parameter r has been recently derived (see [36] and [21]): M r ( n ) = P n − i =1 d i ( e rdi +1)( e rdi − . Thereexists no admissible design for r. Example 1.
Stationary Ornstein Uhlenbeck process with the nugget
Let us have Y ( x i ) = f ( x i , ϑ ) + e ( x i ) , i = 1 , , x , x ∈ X, C r ( d ) = e − rd , d := | x − x | and only covariance parameter r is the parameter of interest. Then we knowthat the maximal Fisher information is obtained for d = 0 (Collapsing effect,[14]). To avoid such ’inconvenient’ behavior we decrease the non-diagonalelements by multiplying with factor α, < α < . By this we include thenugget effect (micro-scale variation effect) of the form γ ( d, r ) = (cid:26) for d = 0 , − α + α (1 − exp( − rd )) otherwise. (10) If γ ( d ) → − α > , as d → , then − α has been called the nugget effect by [18]. This is because it is believed that microscale variation is causing adiscontinuity at the origin. Then we obtain M r = α d exp( − dr )( α exp( − dr ) + 1)(1 − α exp( − dr )) . [34] have proven that the distance d of the optimal design is an increasingfunction of nugget effect − α. The nugget effect makes C r discontinuous, C r (0) = 1 − α and C r ( d ) = α exp( − rd ) for d > . Thus C r is the memberof abc class for α < / . We see that in coherence with Theorem 5 iv) theoptimal design is not collapsing. This issue intrinsically relates to ”twin-points” design (see [5] and [4]). emark 4. Loewner optimality:
The most gratifying criterion refers toLoewner comparison of information matrices. It goes hand in hand withestimation problems, testing hypothesis and general parametric model build-ing (see Sections 3.4-3.10 in [27]). Notice that Theorem 5 says that underthe regularity conditions abc the Loewner comparison of two information ma-trices amounts to comparing their distance vectors.
Example 2.
Let us consider the two point optimal design (i.e. design max-imizing M θ ). Then for the class of decreasing covariances (corresponding tothe increasing variograms) we obtain the optimal design with the maximalinter-point distance. More formally, let { x, z } be the two point design incompact design space X ⊆ R k and let us assume increasing semivariogram γ. Then M θ = − γ ( d ) and the design { x, z, || x − z || = diamX } is optimal.The information gained by the optimal design has the form − γ ( diamX ) . Herewe remind that many semivariograms are increasing, e.g. linear, spherical,exponential, Gaussian, rational quadratic among others. There exist also nonmonotonous semivariograms, e.g. wave variogram.
Example 3.
To illustrate Theorem 5 iii , let us consider design space X =[0 , , discontinuous covariance function C ( d ) = exp( − rd ) for d < / and otherwise and the two point design for the sake of simplicity. Then M θ isincreasing for d < / and stands constant otherwise. Therefore both designs { , / } and { , } are optimal (not only equidistant design is optimal). Example 4.
Let us digress to the simple linear regression Y ( x ) = ϑ + ϑ x + ǫ ( x ) , with modified N¨ather covariance structure, studied in [32]. Therewe consider a modification of Example 6.4 discussed in [22] having the designspace X = [ − , and a covariance functionC ( d ) = (cid:26) σ (1 − dr ) for d < r, otherwise. (11) Notice that for r < covariance function C ( d ) is not differentiable with re-spect to parameter r, however, both one sided derivatives exist. The processwith such a correlation can be thought of as a model for function required tohave one sided first order derivatives (see [31]). The classic Fisher infor-mation assumes the differentiability with respect to the parameter. Still, theFisher information can be well defined over some open set.Notice, that for r > , the modified N¨ather covariance structure (11) con-stitutes on [ − , a linear semivariogram structure with γ ( d ) = dr . Let us ssume now, that only the intercept is the parameter of interest and covari-ance parameter r > is fixed. Then design {− , } is uniformly optimalbecause only correlated observations are possible. We have M θ = r r − ( x n − x ) for design { x , x , ..., x n } , − ≤ x < x ... < x n − < x n ≤ and informationgained by the optimal design has the form max M θ = r/ ( r − . So max M θ decreases with the positive correlation. As far as C ( d ) decreases with thedistance, the optimal distance is maximal. The same concept occurs also inthe case of observations, when both slope and intercept are estimated (see[32]). [22] has shown that since the covariance function can be representedas a linear function of responses, a uniformly optimal design is available forestimating ( ϑ , ϑ ) , which concentrates on the points {− , , } . It is clear that optimal designs for both parameters θ and r in any givencompact interval are in some sense trade-offs between collapsing and equidis-tant design, since the optimal strategy for the estimation of the trend pa-rameter θ conflicts with the one for estimating the correlation parameter r. This may led to compromises like geometric progressive designs (GPD, usede.g. by [36] for the case of OU process) or compound designs (see e.g. [21]).
4. Illustrations and Applications to Finance
If one deals with stochastic models in finance, one will naturally encounterthe Efficient Market Hypothesis (EMH) which forms the basis for many finan-cial market models of modern portfolio management ([8]). The core state-ment of the EMH is: nobody who is using the available information canachieve permanently above-average returns when dealing on financial mar-kets. At the same time it is supposed that the actors of financial markets actabsolutely rationally. The EMH distinguishes between three versions as theweak-form, semi-strong-form and strong form EMH, respectively. For moredetails see [9] among others.In all three forms of EMH, actors on financial markets have the sameinformation available, so one cannot achieve above-average returns perma-nently (see [9]). If the financial markets act efficiently as the EMH claims, thefinancial crisis, which began in 2007 and lasts till today, cannot be explainedeasily. Can the forming and bursting of financial bubbles be explained by theuse of insider information or by irrational decisions of the actors, or shouldeven the EMH be questioned?[17] carried out an investigation to the weak-form EMH of stock markets.Aggregated stock exchange indices of 50 states formed the database. From14.1.1995 until 31.12.2005, the pitches were collected on a daily basis. Theexamined 11 years led therefore to 2,870 values per country. Afterwardsthese 2,870 values were divided into rolling windows. These rolling windowsrepresent a part of the time row, including 200 sample points. They arecontinuously shifted by one day. The 2,670 samples per country were checkedby a hypothesis test if they correspond to a random-walk. To evaluate theefficiency of a stock exchange index, the proportion of samples which didnot correspond to a random-walk by a confidence level of 5 %, was taken.The lower the proportion of a country is, the more its stock exchange indexcorresponds to an efficient market. From the results of [17] one can supposethat efficiency as stated in the EMH is not given on stock markets - at leaston a temporarily prospect. Here we develop an alternative random walkmodel with semicontinuous covariance. Such model is based on an Ornstein-Uhlenbeck process whose covariance structure will be modified.
This section presents a model based on the abc covariance structure forhigh frequency data, which is simple in its implementation. The influences ofthe model parameters on the time series will be analyzed by simulations withthe software package R ([28]). We adjust the Ornstein-Uhlenbeck process asfollows. For t ∈ R Y ( t ) = Z t x ( s ) ds, for t ≥ , (12) x = ( x (0) , . . . , x ( t )) ∼ N (0 , Σ) , with Cov ( x ( s ) , x ( t )) = e −| t − s | (13)The vector x represents realizations of a stationary Ornstein-Uhlenbeckprocess and so Y ( t ) is a so called Integrated-Ornstein-Uhlenbeck process.For the following simulations, the discrete parameter space is in the interval[1 , t is defined as 1. Hence,the process changes to its empirical version: Y t = t X i =1 x i , for t ∈ T, (14) x = ( x , x , . . . , x t ) ∼ N (0 , Σ) , with Cov ( x s , x t ) = e −| t − s | (15)15igure 3 shows the theoretical autocorrelation function of x i and a pathof the stochastic process Y t . . . . . . . Lag A C F (a) Theoretical Autocorrelation Func-tion − t Y t (b) Path of Y t Figure 3: (a) Theoretical autocorrelation function of the x i and (b) a path of Y t . The introduced model can be adopted only with limitations for a spe-cific data. Therefore the covariance structure is extended to abc with twoparameters c and r . The vector x changes to: x = ( x , x , . . . , x t ) ∼ N (0 , Σ) , with Cov ( x s , x t ) = (cid:26) . . . s = tc exp − r ( | t − s | ) . . . s = t (16)With r > c ∈ (0 , c < r , whereas theplots (c) and (d) those of c with the aid of the theoretical autocorrelationfunction of x i and a path of Y t .Through the parameter c a jump at Lag 1 of the theoretical autocorrela-tion function is given. To improve the adjustment of the model on the data,further jumps at different Lags are enabled. The problem at this point is,that the covariance matrix has to be positive semi-definite. The model is ex-panded, so that up to four jumps in the theoretical autocorrelation function16
20 40 60 80 100 . . . . . . Lag A C F r = 0.01r = 0.1 r = 0.5r = 1 r = 5r = 10 (a) Theoretical autocorrelationfunction (fixed c , varying r ) t Y t r = 0.01r = 0.1 r = 0.5r = 1 r = 5r = 10 (b) Path of Y t (fixed c , varying r ) . . . . . . Lag A C F c = 1c = 0.8 c = 0.6c = 0.4 c = 0.2c = 0.1 (c) Theoretical autocorrelationfunction (fixed r , varying c ) t Y t c = 1c = 0.8 c = 0.6c = 0.4 c = 0.2c = 0.1 (d) Path of Y t (fixed r , varying c )Figure 4: (a) Theoretical autocorrelation function of the x i and a path of Y t by varying c with r = 0 .
07 (a) and (b). The case of fixed c = 1 and varying r is plotted in (c) and (d). can occur, in order gain information about the influences of the jumps onthe process Y t : x = ( x , x , . . . , x t ) ∼ N (0 , Σ) , with Cov ( x s , x t ) = . . . s = t . exp ( − r | t − s | ) . . . < | t − s | < . exp ( − r | t − s | ) . . . ≤ | t − s | < . exp ( − r | t − s | ) . . . ≤ | t − s | < . exp ( − r | t − s | ) . . . ≤ | t − s | < ∞ (17)17raphical representations of the influences are given in Figure 5, wherethe number of jumps is increased by steps. . . . . . . Lag A C F
4. Sprung bei Lag 88 mit c=0,53. Sprung bei Lag 73 mit c=0,62. Sprung bei Lag 30 mit c=0,71. Sprung bei Lag 1 mit c=0,8kein Sprung (a) Theoretical autocorrelation function t Y t
4. Sprung bei Lag 88 mit c=0,53. Sprung bei Lag 73 mit c=0,62. Sprung bei Lag 30 mit c=0,71. Sprung bei Lag 1 mit c=0,8kein Sprung (b) Path of Y t with up to four jumpsFigure 5: (a) Theoretical autocorrelation function of the x i and (b) a path of Y t with upto four jumps with r = 0 , Through the parameters r and c the model is completely specified. To fitthis model to a specific data, the parameters r and c have to be estimated. For the next simulation setup covariance structure was chosen as powerexponential:
Cov ( x s , x t ) = (cid:26) . . . s = texp − ( | t − s | p ) . . . s = t, (18)for different values of p = 1 , . In order to give a graphical overviewof the impacts of the parameters on the covariance structure and as a con-sequence on the trajectory, the paths of Y t are plotted for different values ofthe parameters. Therefore the known covariance structure is extended to be Cov ( x s , x t ) = (cid:26) . . . s = tc exp − r ( | t − s | ) p . . . s = t. (19)It is now possible to check for the impacts of every parameter on the pathof Y t , ceteris paribus. The visualization of the differences can be seen in18igure 6. Main differences can be observed for a value of r = 0 .
35 wherethe path is shifted upwards. The highest difference in the other directioncan be observed for a value of c = 0 .
2, however, changes in the parametersfrom r = 1 , c = 1 and p = 1 can lead to changes to both directions in thetrajectory. Path of Yt for different values of c,r and p t Y t c = 1, r = 1, p = 1c = 1, r = 0.35, p = 1c = 1, r = 0.8, p = 1c = 1, r = 1, p = 2 c = 1, r = 1, p = 10c = 0.2, r = 1, p = 1c = 0.8, r = 1, p = 1 Figure 6: Paths for different values of parameters c,r and p
As proven by [7], if C is isotropic and positive definite on R m , m > C is continuous except perhaps at the origin d = 0 (nugget effect). However,for m = 1 the latter is not true anymore and here we illustrate differencebetween both cases, i.e. A) Nugget effect (discontinuity at the origin d = 0) B) several jumps , i.e. covariance function of equation (17)Figure 7 shows simulated differences between both cases, A and B for theprocess X i itself and its related random walk Y t . It can be seen that for smallvalues of r ( < .
1) the differences between both cases are obvious. The effectof scaling parameter r is obvious, since the differences between A and B arenegligible (i.e. differences around 9 × − ) for e.g. r = 1 . Up to this point, the empirical autocorrelation function, which can be es-timated from simulated data, was disregarded. By plotting both theoreticaland empirical autocorrelation functions in one plot, it can be seen that thetrue correlation between the data is underestimated by the empirical auto-correlation function. Top row of Figure 8 does not show differences in theempirical covariance function visible with regard to having continuous or no19 ifferences between 1 and 4 jumps for different r t Y t − − − − − r = 0.01r = 0.025r = 0.05r = 0.1 (a) Difference between cases A and B forvarious r − Differences in Paths of Yt regarding jumps in covariance for r = 0.025 t Y t no jump1 jump2 jumps3 jumps4 jumps (b) Comparison between cases A and B:random walks for r = 0 . . Figure 7: Comparison between cases A and B continuous covariance matrix. If the parameter space T is changed to theinterval [1; 6 , Differences between the theoretical and the empirical autocorrelationfunctions can be used to analyze whether data arises from a process withor without jumps in the covariance. Hence, the differences are calculated atevery lag and its absolute value is summed up. This sum is called sum of20
20 40 60 80 100 . . . . . . Lag A C F . . . . . . . . . . . . Lag A C F . . . . . . . . . . . . Lag A C F . . . . . . . . . . . . Lag A C F . . . . . . Figure 8: Theoretical and empirical autocorrelation function of the x i with continuouscovariance matrix (left plot) and with 2 jumps in the covariance matrix with r = 0 .
01 (topright plot). Extending the parameter space to the interval [1; 6 , residuals T and defined as T = n − X L =0 | ρ ( L ) − ˆ ρ ( L ) | . (20)Calculating the sum of residuals from data of Figure 8 leads to a value of63.4 for the data with continuous covariance matrix (top left) and 48.96 forthe data with two jumps in the covariance matrix (top right). Data from plotsin the second row lead to sums of residuals of 352.62 and 271.54, respectively.The sum of residuals is smaller if there are jumps in the covariance matrix.Therefore, we analyzed if there is a general difference in the parameter spaceof [1; 100]. To determine whether the sum of residuals is smaller when thereare jumps in the covariance matrix, various samples, with fixed r and sample21ize, were generated. The jump points and jump heights c are varied in a way,that the covariance matrix remains positive semi-definite. Table 1 shows theresults and confirm the assumption. cc c = 0 . s
30 40 50 60 70 80 90 99sum of residuals 46.78 47.48 48.12 48.71 49.23 49.70 50.13 50.493 jumps, 1-2. as above and 3. jump at Lag s with c = 0.6 s
73 75 85 98sum of residuals 45.63 45.73 46.18 46.704 jumps, 1-3. jumps as above, 4. jump at Lag 88 (c = 0.5)sum of residuals: 45.18
Table 1: r = 0.01, n = 100. Comparison with respect to sum of residuals of 1 to 4 jumpsin the correlation structure given parameters of c and s . The smaller the jump height c , the more decreases the sum of residuals.Additionally, the lag, i.e. when the jump takes place, influences the sum ofresiduals. As it is impossible to simulate every combination from r , numberof jumps, jump height and jump point, following restrictions are made toperform hypothesis testing for continuous covariance matrix: there is onlyone jump at Lag 1 in the covariance matrix. In addition r and n are fixed.Assuming, that c arises from an unknown distribution function F c , 10,000different paths are simulated and their sum of residuals is calculated. c is inthis case a random number from F c . The sums of residuals are illustratedwith a histogram and scatter plot afterwards. Figure 9 shows the results forvarious distribution functions F c .When using Gamma or Poisson distribution values of c greater than onewere neglected. The fact, that c arises from a truncated distribution function,was disregarded. Figure 9 shows, that if the distribution function F c is known,there is a correlation to the distribution function of the sum of residuals. Thescatter plots show a linear correlation between the jump heights c and thesum of residuals. The core statement from Figure 9 is that if we know the22 istogramm der Summe der Residuen Summe der Residuen D i c h t e
10 20 30 40 50 60 . . . . . (a) U(0,1) Scatterplot der Summe der Residuen und c c S u mm e de r R e s i duen (b) U(0,1) Histogramm der Summe der Residuen
Summe der Residuen D i c h t e
10 20 30 40 50 60 . . . . . (c) Gamma(2,1) Scatterplot der Summe der Residuen und c c S u mm e de r R e s i duen (d) Gamma(2,1) Histogramm der Summe der Residuen
Summe der Residuen D i c h t e
10 20 30 40 50 60 . . . . (e) Poisson(5)/10 Scatterplot der Summe der Residuen und c c S u mm e de r R e s i duen (f) Poisson(5)/10 Histogramm der Summe der Residuen
Summe der Residuen D i c h t e
10 20 30 40 50 . . . . (g) Bin(20,0.5)/20 Scatterplot der Summe der Residuen und c c S u mm e de r R e s i duen (h) Bin(20,0.5)/20Figure 9: Sum of residuals with r = 0 . n = 100 and c distributed (a)(b) Uniform( c ∼ U (0; 1)), (c)(d) Gamma ( c ∼ Γ(2; 1)), (e)(f) Poisson ( c ∼ P (5)10 ) and (g)(h) Binomial( c ∼ Bin (20;0 , ). distribution function F c , then the sum of residuals T can be used as ahypothesis test for a continuous covariance matrix.In the following, it is shown how the histogram of the sum of residualsdevelops when there are two or three jumps in the correlation matrix. Again, r , n and the jump points are fixed. To simplify the problem, it is assumedthat c arises from a Uniform distribution. According to the definition: c is in the interval [0;1], c must be in the interval [0; c ] and c in the interval[0; c ], respectively. The simulations are illustrated in Figure 10.Contrary to the case, where covariance matrix consisted of one jump, itis impossible to derive the distribution function of T when there are two orthree jumps. Nevertheless, a correlation between the jump height and thesum of residuals can be assumed through the scatter plot. A simulation experiment was conducted in order to show how forecastcan be performed. We consider that data from Figure 11 shows the stockprice of the last 90 days of a company (black line). The aim is to predict thestock price of the next ten days.Taking first differences leads to the x i . Normally, x i are used to estimatethe parameters r and c and to test whether the covariance matrix is contin-23 istogramm der Summe der Residuen Summe der Residuen D i c h t e . . . . . c + c S u mm e de r R e s i duen Histogramm der Summe der Residuen
Summe der Residuen D i c h t e . . . . . Scatterplot der Summe der Residuen und c + c + c c + c + c S u mm e de r R e s i duen Figure 10: Sum of residuals with r = 0 . n = 100 with 2 jumps ( c ∼ U (0; 1); c ∼ U (0; c )) (top row) and 3 jumps ( c ∼ U (0; 1); c ∼ U (0; c ); c ∼ U (0; c )) (bottommrow). uous or not. Since this is a fictitious example and the data is simulated, theparameters are known. There are two jumps in the covariance matrix with r = 0 .
1. The first jump is at lag 1 with c = 0 . c = 0 .
8. So the covariance matrix is discontinuous in thisexample.The forecast for the future stock price is determined by dynamic simu-lation. First, the correlated x i are transformed into standard normal dis-tributed random variables by z i = A − · ( x , ..., x ) ′ such that AA ′ = Σ . The matrix A corresponds to the lower triangular matrix of the Cholesky-Decomposition. For one random prediction, the vector z will be completedby 10 standard normal random numbers and back transformed to correlatedrandom variables by x ∗ i = A ∗ · ( z , ..., z , z ∗ , ..., z ∗ ) ′ . The covariance matrixhas to be used in the right dimension. Summing up the vector cumulatively,leads to the original time series including the ten predicted values. If thisprocedure is repeated 10,000 times, it is possible to create a (1 − α )- confi-24
20 40 60 80 100 − − Aktienkurs t Y t − − − − − − (a) 95% confidence interval for the fore-cast of the stock priceFigure 11: Fictitious stock price of the last 90 days of a company with 95% confidenceinterval for its forecast dence interval like illustrated in Figure 11 (blue as forecast and red dashedlines as confidence bounds). As soon as insurance companies are unable to pay the claims for damagesof the assured person, one defines this as ruin of the insurance company. Forprediction of probability of ruin, Cram´er-Lundberg-model or an alternativeapproach by [35] can be used. We refer to [35] for more details on the lattermodel.To be able to calculate probabilities of ruin with the presented model, aprocess is formulated, which indicates the available capital of an insurancecompany. In order to the collective risk model, the process is called surplusprocess U t . For t ∈ N and U (0) = u : U t = u + t X i =1 x i , for t > , (21) x = ( x , x , . . . , x t ) ∼ N (0 , Σ) , with Cov ( x s , x t ) = c e − r | t − s | , (22)25here u stands for the initial surplus of the insurer at t = 0 and x i standsfor the profit or loss of the insurance company at point i . An insurer generatesa profit, if it gets more money by rates than it has to spend at point i . Inthe other way around the insurer has a loss of money while considering, thatthe insurer provides competitive rates, E ( x i ) = 0. The probability of ruin ψ ( u, t ) is calculated in the same way like [35] showed in their paper: ψ ( u, t ) = 1 − P r ( U j ≥ j = 0 , , , ..., t ) (23)A simulated experiment is conducted in order to show how the forecastis done. Figure 12(a) shows the surplus process U t of the last 90 weeks of aninsurance company. Prozess der freien Reserve t U t (a) Fictitious surplus process U t of thelast 90 weeks of an insurance company Histogramm der Ruinzeiten t D i c h t e
90 100 110 120 130 140 . . . . . (b) Predicted probability of ruinFigure 12: Fictitious surplus process U t of the last 90 weeks of an insurance company andits predicted probability of ruin The prediction of the probability of ruin is done the same way like theforecast of stock markets. For simulating the data the values u = 4 and r = 0 . c = 0 . c = 0 . u , leads to a predicted path. If this procedure isrepeated 10,000 times, the points at which U t gets negative can be illustratedwith a histogram. Such a histogram is shown in Figure 12(b) and illustratesthe predicted probability of ruin ψ ( u, t ). The value at the end of the predic-tion interval expresses the probability that the insurance company will nothave a ruin in the next 50 weeks. To be able to compare the results, thedata from Figure 12 are analyzed as they if were uncorrelated. This corre-sponds to the application of the Cram´er-Lundberg-Model. The histogram ofthe simulated points of ruin is shown in Figure 13 and illustrates that theprobability of ruin is underestimated at the beginning compared to Figure12(b). Histogramm der Ruinzeiten t D i c h t e
90 100 110 120 130 140 . . . . . . (a) Predicted probability of ruin withuncorrelated data
90 100 110 120 130 140 . . . . . . Quotienten der prognistizierten Ruinwahrscheinlichkeiten t Q uo t i en t (b) Quotients of the predicted probabil-ities of ruinFigure 13: Predicted probability and quotients of predicted probabilites of ruin with un-correlated data The quotients Q of the predicted probabilities of ruin are shown in Figure13(b) and calculated via Q = ψ ( u, t ) k ψ ( u, t ) ua (24)
5. Conclusions
Recently, semicontinuous covariance functions have been used by manyauthors. However, an appropriate discussion on the regularity conditions and27tatistical properties is up to the best knowledge of the authors still missing.As we have shown in the paper covariance function satisfying conditions abcstill possesses some important features of the continuous covariance, justi-fying the increasing domain asymptotics, e.g. equidistant optimal design fortrend parameter. Moreover, it may have advantage of new desirable features,e.g. possible existence of admissible designs for correlation parameter. Alsoin class abc compromise designs seems to be appropriate, like GPD. An-other possibility is to construct compound designs. As has been discussed in[19], kriging is providing discontinuous surfaces even for smooth covariancefunctions. Therefore a natural idea may appear to employ semicontinuouscovariance functions from class abc, which may be more flexible for suchmodeling.Due to the fact that the hypothesis of efficient markets of [8] is doubtedby results of [17], a new random walk model for finance data was developedin this paper. For this reason a stationary Ornstein-Uhlenbeck process wasused and its covariance matrix modified. This modification enables to adaptthe process as good as possible to real data and moreover discontinuous co-variance structure. Parameter r affects the strength of dependency and c permits jumps in covariance structure. It was shown that both parametershave big impact on resulting paths of the processes. Generally valid formula-tions were not possible to be done, because the necessary property of positivesemi-finiteness of the covariance matrix was not given for all combinationsof r and c .In practical applications it is hard to prove the jumps in covariance struc-ture. This could be difficult especially in small samples, however it does notmean that no jumps are existing in the covariance. This is why a test-statisticwas suggested to check for a discontinuous covariance matrix. Distributionof the sum of the residuals with one jump, known jump discontinuity andfixed r only enables assumptions about the continuity if distribution func-tion of jump height c is known. More jumps lead to higher complexity anda solution of this problem needs further research in this area.
6. Acknowledgements
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Proof.
Lemma 1Let us have n = 2 , i.e. we have two point design x , x . Then optimaldesign for parameters ( θ, r ) is collapsing (see [14]), i.e. M r M θ attains itsmaximum at x = x . Let us have n > . Then we have ∂F∂d i < . Therefore also the direc-tional derivative is negative in all canonical directions with the start at thebeginning of coordinate system. Therefore M r M θ attains its maximum at d = d = ... = d n − , P d i = 1 , so the optimal design is equidistant. Proof.
Theorem 1Pointwise convergence of maps defined on a locally separable metric spacemapping to a metric space preserves the semicontinuity of maps (see [24]).Thus K ≥ d → + ∞ C ( d ) = lim d → + ∞ lim n → + ∞ K n ( d,
0) =lim n → + ∞ lim d → + ∞ K n ( d,
0) = 0 . Proof.
Lemma 2Condition a) is satisfied, since strong convergence implies the pointwiseconvergence when target space is regular in topological space (see [16]) andpointwise convergence preserves inequalities.Condition b) is satisfied since strong convergence preserves continuitywhen target space is a regular topological space (see [16]).Condition c) is satisfied, since the regularity of target space guarantiesthat strong convergence preserves continuity and implies pointwise conver-gence. Using these facts we finally getlim d → + ∞ C ( d ) = lim d → + ∞ lim γ ∈ Γ K γ ( d,
0) = lim γ ∈ Γ lim d → + ∞ K γ ( d,
0) = 0 . Proof.
Theorem 3Condition a) follows from Theorem 5.Condition b) is satisfied because of Lemma 2 and regularity of targetspace.Condition c) is satisfied because of Lemma 3, regularity and fully nor-mality of target space.
Proof.
Theorem 4First, let us recall the Frobenius theorem (see [29], p.46). An irreduciblepositive matrix A always has a positive characteristic value λ ( A ) which is32 simple root of the characteristic equation and not smaller than the moduliof other characteristic values. Moreover, if A ≥ B ≥ λ ( A ) ≥ λ ( B ) . Now let + ∞ > d > d ≥ . Then C i,j ( d , r ) ≤ C i,j ( d , r ) for all i, j =1 , .., n and thus C ( d , r ) ≥ C ( d , r ) ≥ . Employing the Frobenius theoremwe have λ ( C ( d , r )) ≥ λ ( C ( d , r )) . Our matrix is symmetric and real, thuswe have λ min ( C − ( d , r )) ≤ λ min ( C − ( d , r )) , where λ min ( A ) denotes the minimaleigenvalue of matrix A. Now, M θ ( n ) = 1 T C − ( d, r ) 1 ≥ n inf x x T C − ( d,r ) xx T x = λ min ( C − ( d, r )) andthus we have proven that for an equidistant design the lower bound function d → n inf x x T C − ( d,r ) xx T x is non decreasing. Similarly we can prove the rest of1). Now let C ( d ) be decreasing. We will show that M θ is increasing. Noticethat λ max ( C − ) = ρ ( C − ) is decreasing with d, more precisely if 0 ≤ A ≤ B then ρ ( A ) ≤ ρ ( B ) and if 0 ≤ A < B and A + B is irreducible, then ρ ( A ) < ρ ( B ) (see [12]). Let us have + ∞ > d > d ≥ . Then C ( d ) < C ( d )and we have ρ ( C − ( d )) < ρ ( C − ( d )) . Let ǫ = ρ ( C − ( d )) − ρ ( C − ( d ))2 , then wehave (see Lemma 5.6.10 in [13]) such a matrix norm || . || ⋆ that ρ ( C − ( d )) ≤|| C − ( d ) || ⋆ < ρ ( C − ( d )) + ǫ < ρ ( C − ( d )) ≤ || C − ( d ) || ⋆ . Here we use thefact that ρ ( A ) = inf {|| A || , || . || is a matrix norm } . Thus we have || C − ( d ) || ⋆ < || C − ( d ) || ⋆ and norms || . || ⋆ and l-1 norm || . || = P i,j | A i,j | are equivalentand M θ ( d ) = || C − ( d ) || . Thus we have M θ ( d ) < M θ ( d ) . To prove ii) let us consider the open set U of all covariance matrices C r ( d )with bounded inverse in a Banach space of real matrices n × n. Then theidentity I ( n ) = lim ∀ i : d i → + ∞ C r ( d ) ∈ U and map C ( n ) → C ( n ) − is smooth.This implies a ( n, n − ∞ ) = lim ∀ i : d i → + ∞ T C ( n ) − r ( d )11 T C ( n − − r ( d )1 = 1 T I ( n )11 T I ( n − nn − . (cid:3) To prove iii) let us consider representation C ( d ) = exp( − ψ r ( d )) . Processconsidered in distances ξ i = ψ r ( d i ) is Ornstein Uhlenbeck and for such aprocess equidistant design is optimal (see [14]) and all neighboring point dis-tances ξ i increase with same speed. ψ r ( d ) is nondecreasing function, thereforeall neighboring point distances d i of original process increase with same speed.Therefore also for the original process (with covariance C ( d )) is equidistantdesign optimal.iv) Optimal design for parameter r in stationary Ornstein Uhlenbeckprocess is collapsing (see [14] and [36]). Process considered in distances33 i = ψ r ( d i ) is Ornstein Uhlenbeck. ψ r ( dd