Research Article | Open Access
Marek Slaby, "Explicit Solutions of the Extended Skorokhod Problems in Affine Transformations of Time-Dependent Strata", Journal of Probability and Statistics, vol. 2021, Article ID 9992546, 10 pages, 2021. https://doi.org/10.1155/2021/9992546
Explicit Solutions of the Extended Skorokhod Problems in Affine Transformations of Time-Dependent Strata
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 in 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 - 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.
The Skorokhod Problem (SP) has its origin in , Skorokhod’s 1961 paper Stochastic equations for diffusions in a bounded region. It was extended by Tanaka in  to simple multidimensional cases and to more general cases by Harrison and Reiman in  and by Lions and Sznitman in .
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  for the solutions of the SP and in  for the solutions of the ESP. In addition, the results in  allowed also for the constraining interval to change in time. Somewhat different explicit formulas were developed independently by the author in  for the solutions of the SP and in  for ESP. In both papers, time-dependent boundaries were considered.
The -valued case however is obviously much more complicated. In , 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 , we have introduced a special kind of a constraining set called a stratum. We now introduce its generalization. Let be the standard orthonormal basis and be any basis of . We will use to represent the vector in terms of its coordinates with respect to . The subscript will be omitted when is the standard orthonormal basis.
Definition 1. A closed set in will be called a quasistratum if there is a basis such thatwhere for and are two real-valued continuous functions defined on such that for every . For short, we will writeThe superscript will be omitted when is a standard orthonormal basis. In the special case when and are constant functions, 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 in is a parallelotope. This perhaps not quite popular name was introduced by H.S.M. Coxeter in . Alternatively, a quasiblock in can be described as an -dimensional parallelepiped.
Note that, in the special case, when is an orthonormal basis, the quasistratum becomes a stratum and a quasiblock becomes a block in the sense of Definition 2.1 in . By , we will denote the unique linear transformation mapping the standard orthonormal basis onto , such that for . Then,where . Note that can be represented by a matrix whose columns are . Any invertible affine transformation of can be represented as a composition of a translation with for some basis .
Definition 2. A family of pairs will be called an evolving quasistratum constraining system if there is a basis such that is càdlàg with respect to the Hausdorff distance between constraining sets, and satisfies the following conditions.
For any on the boundary of ,whereFinally, for any in the interior of .
In the special case when is a quasiblock for every , the evolving quasistratum constraining system will be called an evolving quasiblock constraining system.
Throughout this paper, will denote the space of -valued right continuous functions with left limits, the so-called càdlàg functions, defined on . The subspace of consisting of functions taking values in at will be denoted by .
Remark 1. Let be an orthogonal evolving stratum constraining system with , let be any basis, and let . Then, is a quasistratum constraining system. In particular,
Proof. Equation (6) follows immediately from (3). Applying (4) to both bases and leads to , which implies (7).
Given a time-dependent constraining system in and an -valued càdlàg function such that , the Extended Skorokhod Problem (ESP) is to find a pair of functions , both from such that , and for every , the following conditions are satisfied:where is the limit from the left of at and is the closure of the convex hull of the set . The function is called the extended Skorokhod map (ESM).
The ESP was introduced in  and is a generalization of the SP originally introduced in  in the real-valued case and further developed in the multidimensional case in . The technical difference is that, in the Skorokhod Problem, the constraining function 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 . Further work in time-dependent intervals was continued by Burdzy et al. in , by Burdzy et al. in , and by Slominski and Wojciechowski in . More general multidimensional time-dependent domains were studied by Nyström and Önskog in .
The explicit formula was first derived in  for the solutions of the SP on . Explicit formulas for the solutions of ESP on the time-dependent interval in were developed by Burdzy et al. in  and independently by the author in . In , the author further extended the explicit formulas onto the solutions of ESP on strata in 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  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 defineWe can use it then to define a projection on a quasiblock in . If , then . To define a projection on a quasistratum, we need to extend first functions and onto . An extension of onto , denoted by will be defined bySimilarly, we define , the extension of onto . Finally, we define the projection onto a quasistratum byThe solution of ESP for is particularly easy to find when is a piecewise constant function with a finite number of jumps.
Example 1. Let be any basis of , let , and let , where and when for , and .
Then, for any simple function with , the ESM with respect to can be constructed as follows: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 , we obtained an explicit formula for the solution of the ESP on an evolving orthogonal stratum constraining system. In Theorem 2.1 of , we have shown that if is an orthogonal evolving stratum constraining system with , then for any the solution of the evolving ESP on is given bywhereIn the above, for every ,where is an indicator function of , , ,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 of .
Proposition 1. Let be an orthogonal evolving stratum constraining system, let be an invertible affine transformation, and let be its linear transformation component. For any , if is a solution of ESP for with respect to , then is the unique solution of ESP for with respect to , where and is the standard orthonormal basis.
Proof. Let , let be the solution of ESP for with respect to , and let be an affine transformation. Since , it follows immediately that for every and so (8) holds.
Let . Since , we havewhere , by Remark 1. Thus, (9) holds for every .
Finally, for , we haveThus, is a solution of ESP for . Suppose there is another solution of ESP for with respect to . Then, is a solution of ESP for with respect to .
Indeed, we will show that satisfies conditions (8)–(10).
Since , we have .
Since , it follows that for every and so (8) holds.
Since , we have and so (9) holds.
Finally, since , we get that , which shows that (10) holds as well. By Theorem 2.1 of , the solution to the evolving ESP with respect to is unique. Therefore, , which means that and so the solution 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 system in is generated by an orthogonal constraining system, if there is an orthogonal evolving stratum constraining system and an affine mapping such that, for every , if is the solution of ESP for with respect to , then is the solution of ESP for with respect to . 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. Let be an evolving quasistratum constraining system in , let be the associated basis as described in Definition 2, and let be the linear mapping such that for . Then, is generated by an orthogonal evolving stratum constraining system and is preserving the solutions of the ESP.
3. Explicit Solutions of ESP
The relationship between the solutions of the ESP on an evolving orthogonal constraining system and the corresponding solutions of the ESP on a constraining system generated by it as described in Propositions 1 and 2 permits now an extension of the explicit formulas (15).
Theorem 1. Let be an evolving quasistratum constraining system in ; that is, there is a basis , such that and , for every . Then, for any , the evolving ESP on has the unique solution given bywhere is the linear transformation defined by for every ,In the above, for every ,where , , while
Proof. Let be an evolving quasistratum constraining system in . By Proposition 2, there is a basis of such that and , where is an orthogonal stratum constraining system having a representation . Let . Then, and by Theorem 2.1 in , as shown in (15), the ESP for on has a unique solution given bywhere are defined by (16), (17), and is defined by (18) for . Hence, by Proposition 1, is a solution of ESP for on . The uniqueness of follows immediately from the uniqueness of .
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 operator and the associated basis , where , we define a norm on by
Note that . This norm induces a corresponding norm on defined by
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. Let be an evolving quasiblock constraining system in that can be represented as an image of an orthogonal evolving block constraining system via an invertible linear transformation , where is the image of the orthonormal basis through . If and are the solutions of the ESP for and with respect to , then the following Lipschitz conditions hold:
Proof. Let be an evolving quasiblock constraining system in . Then, by Proposition 2, there is a basis and an evolving block constraining system whose image through is . Let and . Then, by Proposition 1, the solutions of the ESP for and with respect to are and , where , for . By Proposition 3.1 in ,Therefore,and so (32) is proven. The Lipschitz condition (33) follows now immediately via the triangle inequality:The Lipschitz constants in Proposition 3 are tight as the Lipschitz constant in Proposition 3.1 in  is tight.
More useful for applications of the ESM is Lipschitz continuity in the standard norm on generated by the Euclidean norm on .
We shall use to denote the norm of as a linear operator. Thus,
Remark 2. Let be a linear transformation such that . Then, for every ,
Proof. One easy way to obtain a Lipschitz constant for the ESM is to use the operator norms of and .
Proposition 4. Let be an evolving quasiblock constraining system in that can be represented as an image of an orthogonal block constraining system via an invertible linear transformation and let be the image of the orthonormal basis through . If and are the solutions of the ESP for and with respect to , then the following Lipschitz conditions hold:
Proof. Let , , and for . For every , by Proposition 3.1 in ,which ends the proof of (40). Now, we obtain (41) by triangle inequality. Indeed, for any ,ending the proof of (41).
The Lipschitz constants in Proposition 4 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  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 to denote the smallest angle between vectors and . Thus, if , then .
Lemma 1. If and , where , thenFurthermore, if , then
Proof. Because the two independent vectors span a two-dimensional space, it is enough to prove (44) in . We can also assume without loss of generality that for some and , where . Then, and . We define a function byA quick analysis of its derivativeshows that the maximum value of occurs when and . Since , , and therefore,We need to introduce now some necessary notations. Let be the set of all permutations