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 
 
 
 
 ℝ
 
 
 n
 
 
 
 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 
 
 
 
 ℝ
 
 
 n
 
 
 
 - 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.


Skorokhod Map
The mapping ψ → φ from D [0, ∞) to D [0, ∞) is called the Skorokhod map on [α, β] for ψ. The original Skorokhod map was introduced by Skorokhod in 1961 as a tool for solving stochastic differential equations on the half-line R + with a reflecting boundary condition at 0. In other words in the original Skorohod's paper α = 0 and β = ∞. The explicit representation for the original Skorokhod map is p. 4 of 40 The concept of Skorokhod map with time dependent boundaries have been studied by K. Burdzy, K. Kang and K. Ramanan in The Skorokhod problem in a time dependent interval that appeared in Stochastic Processes and their Applications in 2009. They considered a general case, where both the lower and the upper boundaries are time dependent. They also developed an explicit formula for the Skorokhod map with such boundaries. In addition, their analysis includes a more relaxed version of the Skorokhod map called the extended Skorokhod map.

Definition 1 (Extended Skorokhod problem in R)
Let α, β ∈ D[0, ∞) be such that α ≤ β, and let ψ ∈ D[0, ∞). A pair of real valued càdlàg functions (φ, η), is said to be a solution of the extended Skorokhod problem (ESP) on [α, β] for ψ, if the following three properties are satisfied: iii. For every t ≥ 0 Solution of the extended Skorokhod Problem in R Let α, β ∈ D[0, ∞) be such that α ≤ β, and let ψ ∈ D[0, ∞). The evolving ESP for ψ on [α, β] has the unique solution (φ, η) given by η = −Ξ α,β (ψ) and φ = ψ + η, where p. 9 of 40 A stratum and a block in R n A closed set G in R n will be called a stratum if it admits the following representation where a i ≤ b i for i = 1, 2, . . . , n − 1 and A, B are two real valued continuous functions on [a 1 , b 1 ] × . . . × [a n−1 , b n−1 ] such that A(x ) ≤ B(x ) for every x . In such case we will shortly write In the special case when A and B are constant functions G will be called a block. In other words a block is a cross product of n intervals.

Hausdorff distance
We are interested in the constraining domains in R n that change with time and so we shall need to introduce the convergence for sets. This will be defined in the sense of the Housdorff metric. For any two sets G 1 , G 2 ⊂ R n their Hausdorff distance is defined by where d (x , G) = inf y ∈G x − y and where · is the Euclidean norm on R n . It is well known that the set of all non-empty compact subsets of R n forms a complete metric space with d H .
p. 11 of 40 Definition 2 (A càdlàg family of strata) A family {G t : t ≥ 0} of closed subsets of R n will be called càdlàg if the function t → G t is càdlàg with respect to the Hausdorff metric d H .
To represent càdlàg family of strata we shall use the following notation Orthogonal evolving stratum constraining system A family of pairs {(G t , d t (·)) : t ≥ 0} will be called an orthogonal evolving stratum constraining system if G t is a stratum for every Orthogonal evolving block constraining system In the special case when G t is a block for every t the orthogonal evolving stratum constraining system will be called an orthogonal evolving block constraining system.
p. 14 of 40 Definition 3 (Solution of ESP on an orthogonal evolving stratum constraining system) Given an orthogonal evolving stratum constraining system ) if the following conditions hold for every t ≥ 0: Explicit formula for the solutions of ESP on an orthogonal evolving stratum constraining system p. 17 of 40

Projections in R n
In the vector valued case we will need similar projections onto blocks and strata. Given Finally, the projection on a stratum p. 18 of 40 Example 1 (ESM for a simple function) Consider the ESP for a function ψ ∈ S ([0, ∞)], R n ) with an orthogonal evolving stratum constraining system (G t , d t (·)) such that where 0 = t 0 < t 1 < t 2 < ... < t m < ∞ and t m+1 = ∞. Then, the corresponding ESM is the function φ such that for t ∈ [t k , t k+1 ), k = 0, 1, ..., m Expansion from strata to quasistrata We are ready now to expand our explicit formula from strata to a much bigger class of constraining domains that will be called quasistrata.
Let E = {e 1 , e 2 , . . . , e n } be the standard orthonormal basis and V = {v 1 , v 2 , . . . , v n } be any basis of R n . We will use (x 1 , x 2 , . . . , x n ) V to represent the vector x = x 1 v 1 + x 2 v 2 + . . . + x n v n in terms of its coordinates with respect to V . The subscript will be omitted when V is the standard orthonormal basis.
p. 20 of 40 Quasistratum A closed set G in R n will be called a quasi-stratum if there is a basis V such that where a i ≤ b i for i = 1, 2, . . . , n − 1 and A, B are two real valued continuous functions defined on [a 1 , b 1 ] × . . . × [a n−1 , b n−1 ] such that A(x) ≤ B(x) for every x. For short, we will write The superscript will be omitted when V is a standard orthonormal basis. In the special case when A and B are constant functions G will be called a quasi-block.

Quasiblock
In two dimensions a quasi-block is simply a parallelogram, in three dimensions it is a parallelepiped. In general, a quasi-block in R n is a parallelotope. This perhaps not quite popular name was introduced by H.S.M. Coxeter in his 1973 book Regular politopes. Alternatively, a quasi-block in R n can be described as an n-dimensional parallelepiped.

Quasistratum as a linear transformation of a stratum
Note that in the special case, when V is an orthonormal basis the quasi-stratum becomes a stratum and a quasi-block becomes a block. By T V we will denote the unique linear transformation T V : R n → R n mapping the standard orthonormal basis onto V , i.e. such that T V (e i ) = v i for i = 1, 2, . . . , n. Then B] . Note that T V can be represented by a matrix whose columns are v 1 , v 2 , . . . , v n . Any invertible affine transformation of R n can be represented as a composition of a translation with T V for some basis V .
p. 23 of 40 Evolving quasi-stratum constraining system A family of pairs {(G t , d t (·)) : t ≥ 0} will be called an evolving quasi-stratum constraining system if there is a basis V such that is càdlàg with respect to the Hausdorff distance between constraining sets, and d t satisfies the following conditions. For any where I V t (x) = i : 1 ≤ i < n and x i = α i t or i = n and x n = A x 1 , x 2 , ..., x n−1 , and J V t (x) = i : 1 ≤ i < n and x i = β i t or i = n and x n = B x 1 , x 2 , ..., x n−1 .
Finally, d t (x) = 0 for any x in the interior of G t .

p. 24 of 40
Evolving quasi-block constraining system In the special case when G t is a quasi-block for every t ≥ 0, the evolving quasi-stratum constraining system will be called an evolving quasi-block constraining system.

p. 25 of 40
Proposition 1 (How solutions of ESP are affected by affine mappings of R n ) Let {(G t , d t (·)) : t ≥ 0} be an orthogonal evolving stratum constraining system, let T : R n −→ R n be an invertible affine transformation and let T 0 = T − T (0) be its linear transformation component. For any ) is a solution of ESP for ψ with respect to {(G t , d t (·)) : t ≥ 0}, then (T φ, T 0 η) is the unique solution of ESP for T ψ with respect to (TG t , d V t (·)) : t ≥ 0 , where V = T 0 (E ) and E is the standard orthonormal basis.

p. 27 of 40
Idea: expanding the results for the orthogonal constraining systems via affine transformations The above result suggests that through the use of affine transformations the orthogonal evolving constraining systems can generate 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. p. 28 of 40 Constraining system generated by an orthogonal constraining system A time-dependent constraining system (G t ,d t (·)) : t ≥ 0 in R n is generated by an orthogonal constraining system, if there is an orthogonal evolving stratum constraining system {(G t , d t (·)) : t ≥ 0} and an affine mapping T : : t ≥ 0 . Such a mapping will be referred to as preserving the solutions of ESP.
Proposition 2 (Every quasi-stratum constraining system is generated by an orthogonal constraining system) Let G t ,d t : t ≥ 0 be an evolving quasi-stratum constraining system in R n , let V be the associated basis as described in the definition, and let T V : R n −→ R n be the linear mapping such that T V (e i ) = v i for i = 1, 2, . . . , n. Then G t ,d t : t ≥ 0 is generated by an orthogonal evolving stratum constraining system and T V is preserving the solutions of the ESP.
p. 30 of 40 Theorem 1 (Explicit solutions of ESP on quasi-stratum constraining systems) Page 1 of 2 Let G ,d be an evolving quasi-stratum constraining system in for every t ≥ 0. Then for anyψ ∈ DG 0 ([0, ∞) , R n ) the evolving ESP on G ,d has the unique solution φ ,η given bỹ η = T V −Ξ α 1 ,β 1 (ψ 1 ), −Ξ α 2 ,β 2 (ψ 2 ), ..., −Ξ α n ,β n (ψ n ) andφ =ψ+η, where T V : R n −→ R n is the linear transformation defined by T V (e i ) = v i for every i = 1, 2, . . . , n, Theorem 1 (Explicit solutions of ESP on quasi-stratum constraining systems) Page 2 of 2 In the above, for every i = 1, 2, . . . , n,  Continuity and Lipschitz Conditions -constant K V Given a sequence of n independent vectors V = {v 1 , v 2 , . . . , v n }, let V j denote the linear subspace spanned by vectors V \ {v j }, let K j = sin ∠ (v j , V j ) and let K V = min 1≤j≤n K j . 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, for every j = 1, 2, . . . , n Second, K V depends only on the directions of vectors in V and not on their magnitudes, i.e.
p. 34 of 40 Theorem 2 (Continuity and Lipschitz Conditions) Let G ,d be an evolving quasi-block constraining system in R n 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 forψ 1 andψ 2 with respect to G ,d then the following Lipschitz conditions hold η 1 −η 2 ≤ L V · ψ 1 −ψ 2 ,