# Explicit Solutions of the Extended Skorokhod Problems in Affine Transformations of Time-Dependent Strata

Authors: Marek Slaby
Journal: Journal of Probability and Statistics
 (2021)

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## Abstract

The goal of this paper is to expand the explicit formula for the solutions of the Extended Skorokhod Problem developed earlier for a special class of constraining domains inℝn with orthogonal reflection fields. We examine how affine transformations convert solutions of the Extended Skorokhod Problem into solutions of the new problem for the transformed constraining system. We obtain an explicit formula for the solutions of the Extended Skorokhod Problem for any ℝn- valued càdlàg function with the constraining set that changes in time and the reflection field naturally defined by any basis. The evolving constraining set is a region sandwiched between two graphs in the coordinate system generating the reflection field. We discuss the Lipschitz properties of the extended Skorokhod map and derive Lipschitz constants in special cases of constraining sets of this type.


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## Body

## 1. Introduction

The Skorokhod Problem (SP) has its origin in [1], Skorokhod’s 1961 paper Stochastic equations for diffusions in a bounded region. It was extended by Tanaka in [2] to simple multidimensional cases and to more general cases by Harrison and Reiman in [3] and by Lions and Sznitman in [4].Since then, it has been studied extensively in many directions. The Skorokhod map (SM), a part of its solution, together with its slightly more general version, the extended Skorokhod map (ESP), constitute a very important tool in studying processes whose values are restricted to some domain by a constraining force acting along directions specified for each point on the boundary. They found the most significant applications in studying reflected diffusion processes and queueing networks. Two of the most recent directions are particularly relevant to this work: the search for an explicit formula for the solutions and the introduction of time-dependent barriers. In the one-dimensional case, the first explicit formulas were developed in [5] for the solutions of the SP and in [6] for the solutions of the ESP. In addition, the results in [6] allowed also for the constraining interval to change in time. Somewhat different explicit formulas were developed independently by the author in [7] for the solutions of the SP and in [8] for ESP. In both papers, time-dependent boundaries were considered.Theℝn-valued case however is obviously much more complicated. In [9], the author managed to derive an explicit formula for solutions of the ESP for a special class of time-dependent constraining domains called strata with orthogonal reflection fields. It is those results that we intend to generalize now onto a much larger class of constraining domains and more general reflection fields.In [9], we have introduced a special kind of a constraining set called a stratum. We now introduce its generalization. Let E=e1,e2,…,en be the standard orthonormal basis and V=v1,v2,…,vn be any basis of ℝn. We will use x1,x2,…,xnV to represent the vector x=x1v1+x2v2+⋯+xnvn in terms of its coordinates with respect to V. The subscript will be omitted when V is the standard orthonormal basis.Definition 1.
A closed setG in ℝn will be called a quasistratum if there is a basis V such that(1)G=x1,x2,…,xnV:xi∈ai,bifori=1,2,…,n−1,xn∈Ax1,…,xn−1,Bx1,…,xn−1,where ai≤bi for i=1,2,…,n−1 and A,B are two real-valued continuous functions defined on a1,b1×⋯×an−1,bn−1 such that Ax≤Bx for every x. For short, we will write(2)G=SVa1,b1×⋯×an−1,bn−1,A,B.
The superscript will be omitted whenV is a standard orthonormal basis. In the special case when A and B are constant functions, G will be called a quasiblock.
In two dimensions, a quasiblock is simply a parallelogram, and in three dimensions, it is a parallelepiped. In general, a quasiblock inℝn is a parallelotope. This perhaps not quite popular name was introduced by H.S.M. Coxeter in [10]. Alternatively, a quasiblock in ℝn can be described as an n-dimensional parallelepiped.
Note that, in the special case, whenV is an orthonormal basis, the quasistratum becomes a stratum and a quasiblock becomes a block in the sense of Definition 2.1 in [9]. By TV, we will denote the unique linear transformation TV:ℝn⟶ℝn mapping the standard orthonormal basis onto V, such that TVei=vi for i=1,2,…,n. Then,(3)SVa1,b1×⋯×an−1,bn−1,A,B=TVG,where G=Sa1,b1×⋯×an−1,bn−1,A,B. Note that TV can be represented by a matrix whose columns are v1,v2,…,vn. Any invertible affine transformation of ℝn can be represented as a composition of a translation with TV for some basis V.Definition 2.
A family of pairsGt,dt⋅:t≥0 will be called an evolving quasistratum constraining system if there is a basis V such that Gt=SVαt1,βt1×⋯×αtn−1,βtn−1,At,Bt,Gt:t≥0 is càdlàg with respect to the Hausdorff distance between constraining sets, and dt satisfies the following conditions.
For anyx=x1,x2,…,xnV on the boundary of Gt,(4)dtx=dtVx=∑i∈ItVrivi−∑i∈JtVrivi:ri≥0,fori∈ItVx∪JtVx,where(5)ItVx=i:1≤i<nandxi=αtiori=nandxn=Ax1,x2,…,xn−1,JtVx=i:1≤i<nandxi=βtiori=nandxn=Bx1,x2,…,xn−1.
Finally,dtx=0 for any x in the interior of Gt.
In the special case whenGt is a quasiblock for every t≥0, the evolving quasistratum constraining system will be called an evolving quasiblock constraining system.
Throughout this paper,D0,∞,ℝn will denote the space of ℝn-valued right continuous functions with left limits, the so-called càdlàg functions, defined on 0,∞. The subspace of D0,∞,ℝn consisting of functions taking values in G at t=0 will be denoted by DG0,∞,ℝn.Remark 1.
LetGt,dt⋅:t≥0 be an orthogonal evolving stratum constraining system with Gt=Sαt1,βt1×⋯×αtn−1,βtn−1,At,Bt, let V be any basis, and let dtVx=TVdtTV−1x. Then, TVGt,dtV:t>0 is a quasistratum constraining system. In particular,(6)TVGt=SVαt1,βt1×⋯×αtn−1,βtn−1,At,Bt,(7)dtVx=∑i∈ItVrivi−∑i∈JtVrivi:ri≥0,fori∈ItVx∪JtVx.Proof.
Equation (6) follows immediately from (3). Applying (4) to both bases V and E leads to TV∘dtx=dtV∘TVx, which implies (7).
Given a time-dependent constraining systemGt,dt⋅:t≥0 in ℝn and an ℝn-valued càdlàg function ψ such that ψ0∈G0, the Extended Skorokhod Problem (ESP) is to find a pair of functions ϕ,η, both from D0,∞,ℝn such that ϕ=ψ+η, and for every t≥0, the following conditions are satisfied:(8)ϕt∈Gt,(9)ηt−ηs∈co¯∪u∈s,tduϕu,for everys∈0,t,(10)ηt−ηt−∈dtϕt,where ηt− is the limit from the left of η at t and co¯A is the closure of the convex hull of the set A. The function ϕ is called the extended Skorokhod map (ESM).
The ESP was introduced in [11] and is a generalization of the SP originally introduced in [1] in the real-valued case and further developed in the multidimensional case in [2]. The technical difference is that, in the Skorokhod Problem, the constraining function η is required to have finite variation. The discussions in this paper are restricted to the ESP, however, corresponding modified versions of all results could be formulated for the SP as well. The SP in time-dependent domains was first studied by Constantini et al. in [12]. Further work in time-dependent intervals was continued by Burdzy et al. in [13], by Burdzy et al. in [6], and by Slominski and Wojciechowski in [14]. More general multidimensional time-dependent domains were studied by Nyström and Önskog in [15].
The explicit formula was first derived in [5] for the solutions of the SP on 0,a. Explicit formulas for the solutions of ESP on the time-dependent interval in ℝ were developed by Burdzy et al. in [6] and independently by the author in [8]. In [9], the author further extended the explicit formulas onto the solutions of ESP on strata in ℝn with orthogonal constraining fields.
The SM and the ESM are natural tools used to study reflected deterministic and stochastic processes. In particular, a lot of research was done in the area of reflected diffusions, where the solutions of the SP or ESP are applied. Some of the recent papers include [16–19].
Significant contributions on Lipschitz continuity properties of the SM came from Dupuis and Ishii in [20] and Dupuis and Ramanan in [21, 22].
Very useful in working with ESP are projections onto the constraining domains. In the one-dimensional case, we define(11)πa,b=a,ifx≤a,x,ifa≤x≤b,b,ifx≥b.
We can use it then to define a projection on a quasiblock inℝn. If G=SVa1,b1×⋯×an,bn, then πGx=πa1,b1x1,πa2,b2x2,…,πan,bnxnV. To define a projection on a quasistratum, we need to extend first functions A and B onto ℝn−1. An extension of A:a1,b1×⋯×an−1,bn−1⟶ℝ onto ℝn−1, denoted by A¯ will be defined by(12)A¯tx1,x2,…,xn−1=Atπαt1,βt1x1,παt2,βt2x2,…,παtn−1,βtn−1xn−1.
Similarly, we defineB¯, the extension of B onto ℝn−1. Finally, we define the projection onto a quasistratum G=SVa1,b1×⋯×an−1,bn−1,A,B by(13)πGx=πa1,b1x1,…,πan−1,bn−1xn−1,πA¯x1,…,xn−1,B¯x1,…,xn−1xnV.
The solution of ESP forψ is particularly easy to find when ψ is a piecewise constant function with a finite number of jumps.Example 1.
LetV=v1,v2,…,vn be any basis of ℝn, let 0=t0<t1<t2<⋯<tm−1<tm=∞, and let Gt=SVαt1,βt1×⋯×αtn,βtn, where αtk=aik and βtk=bik when ti≤t<ti+1 for i=0,1,…,m−1, and k=1,2,…,n.
Then, for any simple functionψ=∑i=0m−1uiIti,ti+1 with u0∈G0, the ESM with respect to Gt,dtV can be constructed as follows:(14)ϕt=πGtψt=u0,for0≤t<t1,ϕt=πGtϕti−1+ψti−ψti−1,forti≤t<ti+1,i=1,2,…,m−1.
The projections and the recursive construction shown above are very well known and easily verifiable. This technique has been used in [5, 9, 11, 15, 18, 20, 21, 23].
In [9], we obtained an explicit formula for the solution of the ESP on an evolving orthogonal stratum constraining system. In Theorem 2.1 of [9], we have shown that if Gt,dt⋅:t≥0 is an orthogonal evolving stratum constraining system with Gt=Sαt1,βt1×⋯αtn−1,βtn−1,At,Bt, then for any ψ∈DG00,∞,ℝn the solution ϕ,η of the evolving ESP on Gt,dt⋅ is given by(15)η=−Ξα1,β1ψ1,−Ξα2,β2ψ2,…,−Ξαn,βnψn,ϕ=ψ+η,where(16)αtn=Atψ1t−Ξαt1,βt1,ψ2t−Ξαt2,βt2,…,ψn−1t−Ξαtn−1,βtn−1,(17)βtn=Btψ1t−Ξαt1,βt1,ψ2t−Ξαt2,βt2,…,ψn−1t−Ξαtn−1,βtn−1.
In the above, for everyi=1,2,…,n,(18)Ξαi,βiψit=Iτβi≤ταiIτβi,∞Hαi,βiψit+Iταi<τβiIταi,∞Lαi,βiψit,where IA is an indicator function of A, τα=inft>0:αt>ψt, τβ=inft>0:ψt>βt,(19)Hα,βψt=sup0≤s≤tψs−βs∧infs≤r≤tψr−αr,Lα,βψt=inf0≤s≤tψs−αs∨sups≤r≤tψr−βr.
It is our main goal here to extend the explicit formula onto much more general constraining domains by dropping the orthogonality condition.


## 2. Transformations of Constraining Systems

We now examine how the solutions of an ESP are affected by affine mappings ofℝn.Proposition 1.
LetGt,dt⋅:t≥0 be an orthogonal evolving stratum constraining system, let T:ℝn⟶ℝn be an invertible affine transformation, and let T0=T−T0 be its linear transformation component. For any ψ∈DG00,∞,ℝn, if ϕ,η is a solution of ESP for ψ with respect to Gt,dt⋅:t≥0, then Tϕ,T0η is the unique solution of ESP for Tψ with respect to TGt,dtV⋅:t≥0, where V=T0E and E is the standard orthonormal basis.Proof.
Letψ∈DG00,∞,ℝn, let ϕ,η be the solution of ESP for ψ with respect to Gt,dt⋅:t≥0, and let T be an affine transformation. Since ϕt∈Gt, it follows immediately that Tϕt∈TGt for every t≥0 and so (8) holds.
Let0≤s≤t. Since ηt−ηs∈co¯∪u∈s,tduϕu, we have(20)T0ηt−T0ηs=T0ηt−ηs∈T0co¯∪u∈s,tduϕu=co¯T0∪u∈s,tduϕu=co¯∪u∈s,tT0duϕu=co¯∪u∈s,tduVTϕu,where V=T0E, by Remark 1. Thus, (9) holds for every s∈0,t.
Finally, fort>0, we have(21)T0ηt−T0ηt−=T0ηt−ηt−∈T0dtϕt=T0dtϕt=dtVTϕt.
Thus,Tϕ,T0η is a solution of ESP for Tψ. Suppose there is another solution ϕ˜,η˜ of ESP for Tψ with respect to TGt,dtV⋅:t≥0. Then, T−1ϕ˜,T0−1η˜ is a solution of ESP for ψ with respect to Gt,dt⋅:t≥0.
Indeed, we will show thatT−1ϕ˜,T0−1η˜ satisfies conditions (8)–(10).
SinceTψ=ϕ˜+η˜, we have T−1ϕ˜+T0−1η˜=T0−1ϕ˜−T0+T0−1η˜=T0−1ϕ˜+η˜−T0=T−1ϕ˜+η˜=T−1Tψ=ψ.
Sinceϕ˜t∈TGt, it follows that T−1ϕ˜t∈Gt for every t≥0 and so (8) holds.
Sinceη˜t−η˜s∈co¯∪u∈s,tduVϕ˜ufor everys∈0,t, we have (22)T0−1η˜t−T0−1η˜s=T0−1η˜t−η˜s∈T0−1co¯∪u∈s,tduVϕ˜u=co¯∪u∈s,tT0−1duVϕ˜u=co¯∪u∈s,tduT−1ϕ˜u,and so (9) holds.
Finally, sinceη˜t−η˜t−∈dtVϕt, we get that T0−1η˜t−T0−1η˜t−=T0−1η˜t−η˜t−∈T0−1dtVϕ˜t=dtT−1ϕ˜t, which shows that (10) holds as well. By Theorem 2.1 of [9], the solution to the evolving ESP with respect to Gt,dt⋅:t≥0 is unique. Therefore, T−1ϕ˜,T0−1η˜=ϕ,η, which means that Tϕ,T0η=ϕ˜,η˜ and so the solution Tϕ,T0η is unique.
The above result suggests that, through the use of affine transformations, the orthogonal evolving constraining systems can generate a much larger class of constraining systems. Moreover, the affine transformation provides the link between the solutions of ESP with respect to the image constraining system and the solutions of ESP with respect to the original orthogonal constraining system.Definition 3.
A time-dependent constraining systemG˜t,d˜t⋅:t≥0 in ℝn is generated by an orthogonal constraining system, if there is an orthogonal evolving stratum constraining system Gt,dt⋅:t≥0 and an affine mapping T:ℝn⟶ℝn such that, for every ψ∈DG00,∞,ℝn, if ϕ,η is the solution of ESP for ψ with respect to Gt,dt⋅:t≥0, then T∘ϕ,T0∘η is the solution of ESP for T∘ψ with respect to TGt,T0dtT0−1⋅:T≥0. Such a mapping will be referred to as preserving the solutions of ESP.
The next results will show that every quasistratum constraining system is generated by an orthogonal constraining system.Proposition 2.
LetG˜t,d˜t:t≥0 be an evolving quasistratum constraining system in ℝn, let V be the associated basis as described in Definition 2, and let TV:ℝn⟶ℝn be the linear mapping such that TVei=vi for i=1,2,…,n. Then, G˜t,d˜t:t≥0 is generated by an orthogonal evolving stratum constraining system and TV is preserving the solutions of the ESP.Proof.
By Definition2, there are functions αi≤βi,i=1,2,…,n−1, and A≤B such that, for every t≥0, G˜t=GtV=SVαt1,βt1×⋯×αtn−1,βtn−1,At,Bt and d˜tx=dtVx as defined in (4).
LetGt=Sαt1,βt1×⋯×αtn−1,βtn−1,At,Bt and let dt=dtE be the associated orthogonal constraining field. Then, G˜t=TVGt and dtVx=TVdtTV−1x for every x, as seen in Remark 1.


## 3. Explicit Solutions of ESP

The relationship between the solutions of the ESP on an evolving orthogonal constraining systemGt,dt and the corresponding solutions of the ESP on a constraining system G˜t,d˜t generated by it as described in Propositions 1 and 2 permits now an extension of the explicit formulas (15).Theorem 1.
LetG˜,d˜ be an evolving quasistratum constraining system in ℝn; that is, there is a basis V=v1,v1,…,vn, such that G˜t=SVαt1,βt1×⋯×αtn−1,βtn−1,At,Bt and d˜t=dtV, for every t≥0. Then, for any ψ˜∈DG˜00,∞,ℝn, the evolving ESP on G˜,d˜ has the unique solution ϕ˜,η˜ given by(23)η˜=TV−Ξα1,β1ψ1,−Ξα2,β2ψ2,…,−Ξαn,βnψn,ϕ˜=ψ˜+η˜,where TV:ℝn⟶ℝn is the linear transformation defined by TVei=vi for every i=1,2,…,n,(24)ψt=ψ1t,ψ2t,…,ψnt=TV−1ψ˜1t,ψ˜2t,…,ψ˜ntV=TV−1ψ˜t,(25)αtn=Atψ1t−Ξαt1,βt1,ψ2t−Ξαt2,βt2,…,ψn−1t−Ξαtn−1,βtn−1,(26)βtn=Btψ1t−Ξαt1,βt1,ψ2t−Ξαt2,βt2,…,ψn−1t−Ξαtn−1,βtn−1.
In the above, for everyi=1,2,…,n,(27)Ξαi,βiψit=Iτβi≤ταiIτβi,∞Hαi,βiψit+Iταi<τβiIταi,∞Lαi,βiψit,where ταi=inft>0|αit−ψit>0, τβi=inft>0|ψit−βit>0, while(28)Hα,βψt=sup0≤s≤tψs−βs∧infs≤r≤tψr−αr,Lα,βψt=inf0≤s≤tψs−αs∨sups≤r≤tψr−βr.Proof.
LetG˜,d˜ be an evolving quasistratum constraining system in ℝn. By Proposition 2, there is a basis V of ℝn such that G˜t=TVGt and d˜tx=dtVx=TVdtTV−1x, where Gt,dt is an orthogonal stratum constraining system having a representation Sαt1,βt1×⋯×αtn−1,βtn−1,At,Bt. Let ψt=ψ1t,ψ2t,…,ψnt=TV−1ψ˜t. Then, ψ∈DG00,∞,ℝn and by Theorem 2.1 in [9], as shown in (15), the ESP for ψ on G,d has a unique solution ϕ,η given by(29)η=−Ξα1,β1ψ1,−Ξα2,β2ψ2,…,−Ξαn,βnψn,ϕ=ψ+η,where αn,βn are defined by (16), (17), and Ξαi,βi is defined by (18) for i=1,2,…,n. Hence, by Proposition 1, TVϕ,TVη is a solution of ESP for ψ˜ on G˜,d˜. The uniqueness of ϕ˜,η˜ follows immediately from the uniqueness of ϕ,η.


## 4. Lipschitz Properties

Lipschitz continuity is the most desirable property of the ESM. Unfortunately, establishing the best Lipschitz conditions tends to get very technical. In this paper, the discussion of this property will be limited to quasiblocks under two different norms.Given any invertible linear operatorA:ℝn⟶ℝn and the associated basis V=v1,v1,…,vn, where vi=Aei,i=1,2,…,n, we define a V norm on ℝn by(30)xV=A−1x.Note thatx1v1+x2v2+⋯+xnvnV=x1e1+x2e2+⋯+xnen. This norm induces a corresponding norm on D0,∞,ℝn defined by(31)ψV=supt≥0ψtV,forψ∈D0,∞,ℝn.Some easily obtainable Lipschitz conditions for solutions of the ESP on evolving quasistratum and quasiblock constraining systems are inherited from the evolving orthogonal stratum or block constraining systems via the linear transformation.Proposition 3.
LetG˜,d˜ be an evolving quasiblock constraining system in ℝn that can be represented as an image of an orthogonal evolving block constraining system via an invertible linear transformation TV, where V is the image of the orthonormal basis through TV. If ϕ˜1,η˜1 and ϕ˜2,η˜2 are the solutions of the ESP for ψ˜1 and ψ˜2 with respect to G˜,d˜, then the following Lipschitz conditions hold:(32)η˜1−η˜2V≤nψ˜1−ψ˜2V,(33)ϕ˜1−ϕ˜2V≤1+nψ˜1−ψ˜2V.Proof.
LetG˜,d˜ be an evolving quasiblock constraining system in ℝn. Then, by Proposition 2, there is a basis V and an evolving block constraining system G,d whose image through TV is G˜,d˜. Let ψ1=TV−1∘ψ˜1 and ψ2=TV−1∘ψ˜2. Then, by Proposition 1, the solutions of the ESP for ψ1 and ψ2 with respect to G,d are ϕ1,η1 and ϕ2,η2, where ϕi=TV−1∘ϕ˜i, ηi=TV−1∘η˜i for i=1,2. By Proposition 3.1 in [9],(34)η1−η2≤nψ1−ψ2.
Therefore,(35)η˜1−η˜2V=supt≥0η˜1t−η˜2tV=supt≥0TVη1t−TVη2tV=supt≥0TVη1t−η2tV=supt≥0η1t−η2t=η1−η2≤nψ1−ψ2=nsupt≥0ψ1t−ψ2t=nsupt≥0‖TVψ1t−ψ2tV=nsupt≥0TVψ1t−TVψ2tV=nsupt≥0ψ˜1t−ψ˜2tV=nψ˜1−ψ˜2V,and so (32) is proven. The Lipschitz condition (33) follows now immediately via the triangle inequality:(36)ϕ˜1−ϕ˜2V=ψ˜1−ψ˜2+η˜1−η˜2V≤ψ˜1−ψ˜2V+η˜1−η˜2V=1+nψ˜1−ψ˜2V.
The Lipschitz constants in Proposition3 are tight as the Lipschitz constant in Proposition 3.1 in [9] is tight.
More useful for applications of the ESM is Lipschitz continuity in the standard norm onD0,∞,ℝn generated by the Euclidean norm on ℝn.
We shall useT to denote the norm of T as a linear operator. Thus,(37)T=supx≤1Tx=infK:Tx≤Kxfor allx.Remark 2.
LetT:ℝn⟶ℝn be a linear transformation such that Tei=vi,i=1,2,…,n. Then, for every x=x1,x2,…,xn∈ℝn,(38)∑i=1nxivi≤Tx.Proof.
(39)∑i=1nxivi=∑i=1nxiTei=Tx≤Tx.
One easy way to obtain a Lipschitz constant for the ESM is to use the operator norms ofTV and TV−1.Proposition 4.
LetG˜,d˜ be an evolving quasiblock constraining system in ℝn that can be represented as an image of an orthogonal block constraining system via an invertible linear transformation T and let V be the image of the orthonormal basis through T. If ϕ˜1,η˜1 and ϕ˜2,η˜2 are the solutions of the ESP for ψ˜1 and ψ˜2 with respect to G˜,d˜, then the following Lipschitz conditions hold:(40)η˜1−η˜2≤n⋅T⋅T−1⋅ψ˜1−ψ˜2,(41)ϕ˜1−ϕ˜2≤1+n⋅T⋅T−1ψ˜1−ψ˜2.Proof.
Letψi=T−1ψ˜i, ϕi=T−1ϕ˜i, and ηi=T−1η˜i for i=1,2. For every t≥0, by Proposition 3.1 in [9],(42)η˜1t−η˜2t=Tη1t−Tη2t=Tη1t−η2t≤T⋅η1t−η2t≤T⋅n⋅sup0≤t≤∞ψ1t−ψ2t=T⋅n⋅sup0≤t≤∞T−1ψ˜1t−ψ˜2t≤n⋅T⋅T−1sup0≤t≤∞ψ˜1t−ψ˜2t=n⋅T⋅T−1⋅ψ˜1−ψ˜2,which ends the proof of (40). Now, we obtain (41) by triangle inequality. Indeed, for any t≥0,(43)ϕ˜1t−ϕ˜2t≤η˜1t−η˜2t+ψ˜1t−ψ˜2t≤n⋅T⋅T−1⋅ψ˜1−ψ˜2+ψ˜1−ψ˜2≤1+n⋅T⋅T−1ψ˜1−ψ˜2,ending the proof of (41).
The Lipschitz constants in Proposition4 are not tight as will be seen in Example 3. Unfortunately, the author’s attempts to derive the best constants seem to produce inadvertently very technical results. The Lipschitz constant derived for the ESM with an orthogonal evolving stratum constraining system in Theorem 4.1 of [9] was tight but it was not simple and rather technical to derive. That is why the results in this section are limited to quasiblock restraining systems. Still, we are going to improve the results of Proposition 4 and produce smaller constants.
We shall need some technical inequalities involving vectors and angles between vectors. We will use∠x,y to denote the smallest angle between vectors x and y. Thus, if ∠x,y=θ, then 0≤θ≤π.Lemma 1.
Ifx,y∈ℝn and ∠x,y=θ, where 0<θ<π, then(44)x−y≤tanθ2⋅x+y.
Furthermore, ifx=y, then(45)x−y=tanθ2⋅x+y.Proof.
Because the two independent vectors span a two-dimensional space, it is enough to prove (44) in ℝ2. We can also assume without loss of generality that x=r,0 for some r>0 and y=cosθ,sinθ, where 0<θ<π. Then, x+y2=r+cosθ,sinθ2=r2+2rcosθ+1 and x−y2=r−cosθ,−sinθ2=r2−2rcosθ+1. We define a function g:0,∞⟶0,∞ by(46)gr=x−y2x+y2=r2−2rcosθ+1r2+2rcosθ+1.
A quick analysis of its derivative(47)g′r=4cosθr2−1r2+2rcosθ+12,shows that the maximum value of gr occurs when r=1 and g1=tan2θ/2. Since 0<θ<π, tanθ/2>0, and therefore,(48)maxx−yx+y:∠x,y=θ=g1=tanθ2.
We need to introduce now some necessary notations. LetΛ be the set of all permutations λ:1,2,…,n⟶1,2,…,n. Given any sequence of n independent vectors v1,v2,…,vn and any λ∈Λ, we define(49)CkλV=tan12∠∑i=1kvλi,∑i=k+1nvλi,fork=1,2,…,n−1,CkλV=1,fork=n.
Finally, we set(50)CV=max1≤k≤nmaxλ∈ΛCkλV.Lemma 2.
For any sequencev1,v2,…,vn of independent vectors in ℝn,(51)max∑i=1ntivi:−1≤ti≤1,i=1,2,…,n=CV∑i=1nvi.Proof.
Letg:−1,1n⟶0,∞ be a function defined by gt1,t2,…,tn=∑i=1ntivi. Then, g is a convex nonnegative function and it attains maximum value at some extreme point of −1,1n; that is, there are n values ɛ1,ɛ2,…,ɛn such that ɛi∈−1,1 for i=1,2,…,n and(52)maxgt1,t2,…,tn:ti∈−1,1,i=1,2,…,n=gɛ1,ɛ2,…,ɛn.
Letk be the number of coordinates of ɛ1,ɛ2,…,ɛn that are positive and let λ be a permutation in Λ such that ɛλ1=ɛλ2=⋯=ɛλk=1 while ɛλi=−1 for i=k+1,…,n. Then, by Lemma 1,(53)∑i=1nɛλivλi=∑i=1kvλi−∑i=k+1nvλi≤CkλV∑i=1nvλi≤CV∑i=1nvi.Lemma 3.
Letx,y be two independent vectors in ℝn and let ∠x,y=θ. Then,(54)maxx,y≤1sinθx+y.Proof.
We can assume without loss of generality thatn=2. Consider the parallelogram with vertices at 0,x,y and x+y. In any parallelogram, the distance between the parallel lines through the opposite sides cannot exceed the length of each diagonal. The distance from x to the line through 0 and y is xsinθ and so it cannot exceed x+y. Similarly, the distance from y to the line through 0 and x is ysinθ and it cannot exceed x+y. Therefore, x≤x+y/sinθ and y≤x+y/sinθ.
Given a sequence ofn independent vectors V=v1,v2,…,vn, let Vj denote the linear subspace spanned by vectors V\vj, let Kj=sin∠vj,Vj, and let KV=min1≤j≤nKj. It will be also convenient to use the following notations: v¯=v/v and V¯=v¯1,v¯2,…,v¯n. It is important to notice two things. First,(55)KV≤Kj=sin∠vj,Vj≤sin∠vj,∑i=1nvi−vjfor everyj=1,2,…,n.
Second,KV depends only on the directions of vectors in V and not on their magnitudes, that is,(56)KV=KV¯.Lemma 4.
Letv1,v2,…,vn be independent vectors in ℝn. Then, for every j=1,2,…,n,(57)vj≤1KV¯∑i=1nvi.Proof.
Letj∈1,2,…,n. Applying Lemma 3 to vectors vj and ∑i=1nvi−vj and using (55) and (56), we can see that(58)vj≤1Kjvj+∑i=1nvi−vj≤1KV¯∑i=1nvi.Theorem 2.
LetG˜,d˜ be an evolving quasiblock constraining system in ℝn that can be represented as an image of an orthogonal evolving block constraining system via an invertible linear transformation T and let V be the image of the orthonormal basis through T. If ϕ˜1,η˜1 and ϕ˜2,η˜2 are the solutions of the ESP for ψ˜1 and ψ˜2 with respect to G˜,d˜, then the following Lipschitz conditions hold(59)η˜1−η˜2≤LV⋅ψ˜1−ψ˜2,(60)ϕ˜1−ϕ˜2≤1+LVψ˜1−ψ˜2,where(61)LV=CV¯KV¯∑i=1nv¯i.Proof.
For anyt≥0,(62)η˜1t−η˜2t=∑i=1nη1it−η2itvi=∑i=1nη1it−η2itvi⋅vivi,where, by (3.2) of [9], for every i=1,2,…,n,(63)η1it−η2it⋅vi≤max1≤i≤nmax0≤s≤tψ1is−ψ2is⋅vi.
Thus, by Lemma2,(64)η˜1t−η˜2t≤max1≤i≤nmax0≤s≤tψ1is−ψ2is⋅vi⋅CV¯⋅∑i=1nvivi=max0≤s≤tmax1≤i≤nψ1is−ψ2is⋅vi⋅CV¯⋅∑i=1nv¯i.
By Lemma4, for every 0≤s≤t,(65)max1≤i≤nψ1is−ψ2is⋅vi≤1KV¯∑i=1nψ1is−ψ2isvi=1KV¯ψ˜1s−ψ˜2s.
Therefore, for everyt≥0,(66)η˜1t−η˜2t≤max0≤s≤tCV¯KV¯∑i=1nv¯i⋅ψ˜1s−ψ˜2s=CV¯KV¯∑i=1nv¯i⋅ψ˜1−ψ˜2,which ends the proof of (59).
To prove (60), we use (59) and the triangle inequality:(67)ϕ˜1−ϕ˜2=ψ˜1−ψ˜2+η˜1−η˜2≤ψ˜1−ψ˜2+η˜1−η˜2≤1+CV¯KV¯∑i=1nv¯iψ˜1−ψ˜2.Remark 3.
The Lipschitz constants of Theorem2 depend only on the angles defining the shape of the quasiblock and not on its size.Remark 4.
In the special case of an orthogonal block constraining system, all the relevant angles are right angles and thereforeCV¯=1 and KV¯=1. Thus, the Lipschitz constant in (59) becomes ∑i=1nv¯i=n matching the result of Proposition 3.1 in [9].
The following example will show that the Lipschitz constantLV in (59) is tight in ℝ2. Essentially, it will demonstrate that for any quasiblock constraining system in ℝ2, there are functions ψ1 and ψ2 such that η1−η2=LVψ1−ψ2.Example 2.
LetG˜,d˜ be an arbitrary nonevolving quasiblock constraining system in ℝ2. Then, G˜ is a parallelogram. Let α be the obtuse angle in G˜ and let α¯=π−α. We can assume without the loss of generality that G˜,d˜ is generated by vectors v1=ae1 and v2=bcosαe1+sinαe2. More specifically, we assume that G˜ has vertices at 0, v1, v2, and v1+v2. Then, CV¯=maxtanα/2,1=tanα/2=cotα¯/2,KV¯=sinα=sinα¯=2sinα¯/2cosα¯/2, and v¯1+v¯2=2cosα/2=2sinα¯/2 and so LV=cscα¯/2.
Letψ˜1=−rcotαv¯1I0,1+cscαv¯1I1,∞ and ψ˜2=−rcscαv¯2I0,1+cotαv¯2I1,∞, where r<min−atanα,bcotα/2. Then, using projections as in Example 1, we can evaluate ϕ˜1 and ϕ˜2.
Sincer<−atanα, we have that ψ˜10=−rcotαv¯1∈G˜ and so ϕ˜10=πG˜ψ˜10=ψ˜10=−rcotαv¯1. On the other hand, ϕ˜20=πG˜ψ˜20 = 0. Therefore, η˜10=ϕ˜10−ψ˜10 = 0 and η˜20=ϕ˜20−ψ˜20=rcscαv¯2.
Sinceϕ˜10+ψ˜11−ψ˜10=ψ˜11=−rcscαv¯1∉G˜, ϕ˜11=πG˜−rcscαv¯1 = 0 and η˜11=−ψ˜11=rcscαv¯1. Since r<bcotα/2, we have that ϕ˜20+ψ˜21−ψ˜20=rcscαv¯2−cotαv¯2=rtanα/2v¯2∈G˜, ϕ˜21=πG˜rtanα/2v¯2=rtanα/2v¯2, and η˜21=ϕ˜21−ψ˜21=rcscαv¯2.
Now,η˜10−η˜20=rcscαv¯2=rcscα and η˜11−η˜21=rcscαv¯1−rcscαv¯2=rtanα/2,−1=rsecα/2=rcscα¯/2. Thus, η˜1−η˜2=rmaxcscα,cscα¯/2=rcscα¯/2. On the other hand, ψ˜10−ψ˜20=−rcotαv¯1+rcscαv¯2=re2=r and ψ˜11−ψ˜21=−rcscαv¯1+rcotαv¯2=r−sinα,cosα=r. Therefore, ψ˜1−ψ˜2=r and so η˜1−η˜2=cscα¯/2ψ˜1−ψ˜2.
In other words, in this case,η˜1−η˜2=LVψ˜1−ψ˜2. Because G˜ represents an arbitrary quasiblock constraining system in ℝ2, LV is a tight Lipschitz constant in ℝ2.
The next example will demonstrate that the Lipschitz constant in Theorem2 is not only smaller than the one in Proposition 4 but also exhibits other desirable qualities.Example 3.
We consider a simple example of a quasiblock constraining system inℝ2, where G˜=x1+1/2x2,3/2x2:0≤x1≤1,0≤x2≤1. In other words, G˜,d˜ is generated by vectors v1=e1 and v2=1/2e1+3/2e2 or by a linear transformation:(68)T=112032.
Then,∠v¯1,v¯2=π/3, C1λV=tanπ/6=3/3, C2λV=1 for any λ and so CV=1; KV¯=sinπ/3=3/2 and v¯1+v¯2=3/2,3/2=3. Thus, the Lipschitz constant given in (59) is LV=3/3/2=2.
We can compare it to the Lipschitz constant in (40) of Proposition 4. Since(69)T=maxx≤1Tx=max0≤α≤2πTcosα,sinα=max0≤α≤2πcosα+12sinα,32sinα,it is enough to maximize the function(70)gα=cosα+12sinα2+32sinα2=sinαcosα+1.
Sincemaxgα:0≤α≤2π=gπ/4=3/2, we get T=6/2.
The inverse ofT is defined by(71)T−1x1,x2=x1−33x2,233x2.
A similar analysis to the above shows that(72)T−1=max0≤α≤2πT−1cosα,sinα=max0≤α≤2πcosα−33sinα,233sinα=cos2π3−33sin2π3,233sin2π3=2.
Hence, the Lipschitz constant described in (40) is n⋅T⋅T−1=6.
Thus, for this particular quasiblock constraining system, Theorem2 provides a better Lipschitz constant than Proposition 4. Moreover, as pointed out in Remark 3, the constant in Proposition 4 depends also on the size of vectors in V. In fact, let us examine what happens when v1 changes. We consider now a quasiblock constraining system generated by V=re1,1/2e1+3/2e2, where r>0. This time(73)Tx=Tx1,x2=rx1+12x2,32x2,T−1x=T−1x1,x2=1rx1−33rx2,233x2.
Therefore,(74)T≥T1,0=r,0=r,T−1≥T−10,1=0,233=233,and so the Lipschitz constant described in (40) of Proposition 4 is n⋅T⋅T−1≥2⋅r⋅23/3>r. In other words, this Lipschitz constant increases without bounds as the magnitude of v1 increases. On the other hand, the Lipschitz constant given in (59) remains unchanged.



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