Research Article | Open Access
Marek Slaby, "An Explicit Representation of the Extended Skorokhod Map with Two Time-Dependent Boundaries", Journal of Probability and Statistics, vol. 2010, Article ID 846320, 18 pages, 2010. https://doi.org/10.1155/2010/846320
An Explicit Representation of the Extended Skorokhod Map with Two Time-Dependent Boundaries
We consider the extended Skorokhod problem for real-valued càdlàg functions with the constraining interval , where and change in time as values of two càdlàg functions. We find an explicit form of the solution and discuss its continuity properties with respect to the uniform, and , metrics on the space of càdlàg functions. We develop a useful technique of extending known results for the Skorokhod maps onto the larger class of extended Skorokhod maps.
The Skorokhod problem (SP) was introduced originally in  as a tool for solving stochastic differential equations in a domain with one fixed reflective boundary. Given a function , a solution of the Skorokhod problem on is a pair of functions such that is nondecreasing, and . The mapping is called a Skorokhod map (SM) on and is well defined for all càdlàg functions .
The Skorokhod problem has been studied since  in more general settings. Chaleyat-Maural et al. studied in  a Skorokhod map constraining functions to . A multidimensional version of the SP was introduced by Tanaka in . Over the years, numerous applications were found for the SM particularly in queueing theory. In , Kruk et al. provided an explicit formula and studied the properties of the two-sided Skorokhod map constraining the function to remain in the interval , where is a positive constant. More recently attempts were made towards relaxing the rigidity of the constrains. Burdzy et al. in  found an explicit representation for the so called extended Skorokhod map (ESM), which is a relaxed version of the SM. [The constraining interval in that paper varies with time.] Another explicit representation for the SM with two time-dependent boundaries, different from the representation in  and based on the approach in , was developed by the author in . In addition, a number of properties of the SM were studied in  including its continuity and Lipschitz conditions.
In this paper we obtain an alternative form of the explicit formula for the ESM with two time-dependent boundaries developed in  that is simpler to understand and potentially more useful for applications and generalizations to higher dimensions. We develop methods of extending certain properties of the SM onto the ESM and use them to analyze continuity properties of the ESM.
Throughout the paper, will denote real-valued càdlàg functions on , and will denote càdlàg functions on taking values in and in , respectively. A function is càdlàg if it is right-continuous and has finite left limits at every . Similarly, we will use and to denote subspaces of consisting of nondecreasing functions and functions with bounded variation on every finite interval, respectively. We will use to denote the effective domain of , that is, .
On the space of the càdlàg functions we will consider the topology of the uniform convergence and the topology of the uniform convergence on compact sets. For every , let and . Let be a sequence of functions in or in . We say that converges to uniformly on compact sets if for every . Equivalently, we could say that converges to uniformly on compact sets if on for large enough and converges to uniformly on for every .
Definition 1.1 (extended Skorokhod problem). Let be such that , and let . A pair of real-valued càdlàg functions, , is said to be a solution of the extended Skorokhod problem (ESP) on for if the following three properties are satisfied:(i)for every ; (ii)for every (iii)for every
Traditionally in (iii) is defined as zero so that has a jump at 0, whenever . The map defined by is called the extended Skorokhod map (ESM) on . In the traditional Skorokhod problem, conditions (ii) and (iii) are replaced by a stronger condition.
Definition 1.2 (Skorokhod problem). Let be such that . Given , a pair of functions is said to be a solution of the Skorokhod problem on for if the following two properties are satisfied:(i)for every (ii) and have the decomposition , where
Burdzy et al. have shown in Theorem 2.6 of  that for any and such that , there is a well-defined ESM and it is represented by where is given by They obtained their result first for simple functions and then extended it by the limiting process. In Section 2 we will develop an alternative version of this formula.
It is easy to see that if is a solution of the SP on for , then it is also a solution of the ESP on for . Conversely, it is shown in Proposition 2.3 of  that a solution of the ESP solves also the corresponding SP whenever . Furthermore, Corollary 2.4 of  shows that if and is a solution of the ESP on for , then . Therefore we can identify the ESM with the SM in this special case.
Remark 1.3. If , then .
2. Alternative Explicit Formula for the Two-Sided Extended Skorokhod Map with Time-Dependent Boundaries
We will make the explicit formula (1.4) more user friendly by developing a new expression for the constraining term that is easier to understand and shows more promise for possible extensions to higher dimensions than (1.5). Given and such that , we introduce two pairs of times Note that Also note that the four times depend on . When necessary we will indicate it by using full notation such as or .
Remark 2.1. Let , and , If and pointwise, then
Proof. Let . By (2.2) there is such that . Let . There is such that for . Hence Thus . Since is arbitrary, we conclude that . By similar argument we can show that .
Remark 2.2. Let and be such that . For any , there are three possibilities Similarly, in terms of and , the following three cases are possible: Clearly, for every if and only if , and for every if and only if .
Remark 2.3. It follows from the definition of the ESM that for every and Similarly, we obtain that for every and .
Under the assumption that , we define two increasing sequences of times and similar to the sequences used in . If , we set and ; for , we set If , we set , we define for all by (2.9), and we define for by (2.8).
It is easy to see that unless one of the times equals , at which point all the following times are also . Also note that the time sequences depend on , and . Finally, as in Proposition 2.1 of , if , then The following observations follow immediately from the definition of .
Remark 2.4. If or if and , then and for every If , then and for every ,
Similarly, by definition of , we make the following conclusions.
Remark 2.5. If , then and for every If , then and for every ,
The following result establishes a straight-forward representation for the constraining term of the ESM similar to the representation of Theorem 2.2 of .
Theorem 2.6. Let be such that , let and let be defined by (1.5).
If , then If , then
We precede the proof with two technical lemmas. The first one examines on .
Lemma 2.7. Under the assumptions of Theorem 2.6, for every and for every ,
The next lamma examines on .
Lemma 2.8. Under the assumptions of Theorem 2.6, if or if and , then for every ,
Proof. Let be a nonnegative integer, and let . We can write , where and are defined by (2.24) and (2.25), respectively, and We first show the upper bound, By (2.12) Thus we have shown that . To show the opposite inequality for , it suffices to show that Let be arbitrary and , and let be such that Then, by (2.15), Therefore, by (2.19) and (2.38), Since (2.39) holds for every , the proof of (2.36) is complete for . To complete the proof for in the case of , it suffices to show that In this case , and so . Also since and . Therefore which ends the proof.
Proof of Theorem 2.6. Let . If , then by Remark 2.3. If for some , then (2.21) holds by Lemma 2.7, and if for some , then (2.21) holds by Lemma 2.8.
Similarly, when , then and (2.22) holds for by Remark 2.3, for , by Lemma 2.7, and for , by Lemma 2.8.
We introduce two functions It is easy to verify that the following relationship holds:
Corollary 2.9. Let , be such that . Then for every ,
Proof. If and , then both sides of (2.45) are zero.
Suppose that . Then . If , then by Remark 2.3.
Let for some . We have shown in the proof of Lemma 2.7 that Therefore (2.45) holds by (2.19) and (2.21).
Consider now for some . We have shown in the proof of Lemma 2.8 that Hence, again, we conclude (2.45) from (2.21).
Suppose now that and set , and . Then and so we can apply the already proven case of (2.45) to and . We obtain, by (2.44), By Remark and Remark in , , and so the proof of (2.45) is complete.
Corollary 2.10. Let , be such that . Then for every ,
Proof. By Remark 2.2, there are three possible cases. If , then , and (2.49) holds trivially.
Consider the case when . If , then, as in Remark 2.2, , and (2.49) holds.
If , then by (2.3), , and so, by (2.10), there is such that or . If , then, as in the proof of Lemma 2.7, , and therefore, by (2.46), .
If , then by (2.31) and (2.47), we have again.
In the case of we can apply the already proven case of (2.49) to , , and , as in the proof of Corollary 2.9, to complete the proof.
In the following final result, we extend the representation of (2.49) to a general case, thus producing an alternative representation formula for the ESM with two time-dependent reflective boundaries.
Theorem 2.11. Let , be such that . Then for every and every ,
Proof. Let for every , and define and . Then , and . By Corollary 2.10, for every and every ,
To complete the proof of (2.50), it suffices to show that the right-hand side of (2.51) converges to the right-hand side of (2.50) uniformly on compact sets. By Proposition 2.5 in , we could then conclude that . In fact we will show the uniform convergence of the right-hand sides.
It is easy to see that for every Hence, by Remark 2.1,
To show the convergence of we consider a mapping defined by It is easy to see that is continuous in the uniform metric. In fact, it can be shown that Since , we get that uniformly. Similarly, uniformly.
We consider four possible cases: , and . If , then by (2.52), for every , and so the right-hand sides of both (2.51) and (2.50) are zero.
Suppose that , then also for almost all . We show first that converges to uniformly. If has a jump at , then and so also for sufficiently large . Thus for large enough we have that converges uniformly to . If is continuous at , then and so . Because is right-continuous, . Hence Thus we have established that and therefore where the convergence is uniform.
By a similar argument we show convergence in the case of . We consider now the final case of . Because , it follows by (2.7), that . Analogously we can show that . Hence, for every , By (2.53) and (2.59), we can find an increasing sequence of positive integers such that Let Then it follows from (2.60) that Therefore, as in (2.58), Similarly, Both limits, in (2.64) and in (2.65), are in fact the uniform limits. Since, by Proposition 2.5 in , and , the limits in (2.64) and (2.65) must be the same, that is, Therefore uniformly, and so (2.50) holds again.
3. Continuity Properties of the ESM in Metrics on
In  we have established a number of continuity properties of the Skorokhod map under the assumptions that and . We are going to extend some of these properties onto the ESM. This will be done in two steps. First we will allow , , and secondly we will let .
Remark 3.1. Let be the solution of the SP [resp., ESP] for on for some and . Consider another pair of constrains and . If , then is also the solution of the SP [resp., ESP] for on
Instead of checking that the proofs of all the continuity properties in  are valid when the constraining functions are allowed to take infinite values, we develop in the next lemma a convenient tool for expanding such properties to this more general case.
Lemma 3.2. Let and be such that . For any there is a nonincreasing sequence and a nondecreasing sequence such that the following conditions hold:(i)for every ;(ii)for every ; (iii)for every , there is such that for all and all (iv) for every .
Proof. Let with , let , and let . For every , we define
It is easy to verify that for every ,
Inequalities in (3.3) show the monotonic properties of and and prove statement (i) as well.
To prove (ii) we note that, by (3.4), for and so (ii) immediately follows.
For , define and set . Let and let . If , then . If , then . Similarly, we show that if and if , which completes the proof of (iii).
Finally, (iv) follows from (i) and Remark 3.1.
The limiting process used in the proof of Theorem 2.11 provides a useful technique of extending properties of the Skorokhod map with separated constraining boundaries onto the general ESM. In the following proposition we set a formal method that will allow us to replace the assumption of by a weaker assumption of .
Definition 3.3. Consider , a family of mappings from to indexed by a set of pairs , where . We will say that is closed if the following condition is satisfied: for any sequence in if , and