A Dirichlet process characterization of a class of reflected diffusions
aa r X i v : . [ m a t h . P R ] O c t The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2010
A DIRICHLET PROCESS CHARACTERIZATION OFA CLASS OF REFLECTED DIFFUSIONS
By Weining Kang and Kavita Ramanan Carnegie Mellon University and Carnegie Mellon University
For a class of stochastic differential equations with reflection forwhich a certain L p continuity condition holds with p >
1, it is shownthat any weak solution that is a strong Markov process can be decom-posed into the sum of a local martingale and a continuous, adaptedprocess of zero p -variation. When p = 2, this implies that the re-flected diffusion is a Dirichlet process. Two examples are providedto motivate such a characterization. The first example is a class ofmultidimensional reflected diffusions in polyhedral conical domainsthat arise as approximations of certain stochastic networks, and thesecond example is a family of two-dimensional reflected diffusions incurved domains. In both cases, the reflected diffusions are shown tobe Dirichlet processes, but not semimartingales.
1. Introduction.
Background and motivation.
This work identifies fairly general suf-ficient conditions under which a reflected diffusion can be decomposed as thesum of a continuous local martingale and a continuous adapted process ofzero p -variation, for some p greater than one. As motivation for such a char-acterization, two examples of classes of reflected diffusions are considered.The first example consists of a large class of multidimensional, obliquely re-flected diffusions in polyhedral domains that arise in applications. Reflecteddiffusions in this class are shown not to be semimartingales, but to belongto the class of so-called Dirichlet processes. Dirichlet processes are processesthat can be expressed (uniquely) as the sum of a local martingale and acontinuous process that has zero quadratic variation, and thus correspond Received June 2008. Supported in part by NSF Grants DMS-0406191, DMS-0405343, CMMI-0728064.
AMS 2000 subject classifications.
Primary 60G17, 60J55; secondary 60J65.
Key words and phrases.
Reflected Brownian motion, reflected diffusions, rough paths,Dirichlet processes, zero energy, semimartingales, Skorokhod problem, Skorokhod map,extended Skorokhod problem, generalized processor sharing, diffusion approximations.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2010, Vol. 38, No. 3, 1062–1105. This reprint differs from the original inpagination and typographic detail. 1
W. KANG AND K. RAMANAN to the case when p = 2 in the decomposition mentioned above. The secondexample consists of a class of two-dimensional reflected diffusions in curved“valley-shaped” domains that were first considered by Burdzy and Toby in[3]. Once again, the reflected diffusion is shown to admit a decomposition ofthe type mentioned above, but in this case the magnitude of p depends, ina sense made precise in the sequel, on the curvature of the domain.Processes that admit a decomposition of the type mentioned above areclearly an extension of the class of continuous semimartingales. As is wellknown, semimartingales form an important class of processes for stochas-tic integration, they are stable under C transformations and admit an Itˆochange-of-variable formula. However, there are many natural operations thatlead out of the class of semimartingales and motivate the consideration ofDirichlet processes. For example, C functionals of Brownian motion, certainfunctionals of symmetric Markov processes associated with Dirichlet forms[17], and Lipschitz functionals of a broad class of semimartingale reflecteddiffusions in bounded domains [26, 27], are all Dirichlet processes that arein general not semimartingales. Moreover, Dirichlet processes exhibit manynice properties analogous to semimartingales. They admit a natural, Doob–Meyer-type decomposition [5], they are stable under C transformations (seeProposition 11 of [28] and also [1]) and there are extensions of stochasticcalculus and Itˆo’s formula that apply to Dirichlet processes (see [12, 14] andChapter 4 of [28]) or, more generally, to processes that admit a decompo-sition as the sum of a local martingale and a continuous, adapted processof bounded p -variation, for p ∈ (1 ,
2) [1]. Furthermore, the theory of roughpaths (see, e.g., [16] or [21]) applies to processes whose paths have bounded p -variation for an arbitrary p ∈ [1 , ∞ ).The theory of reflected diffusions is most well-developed for semimartin-gale or symmetric reflected diffusions. In particular, the Skorokhod problemapproach to the study of reflected diffusions [8, 22, 29] is automatically lim-ited to semimartingales, while the Dirichlet form approach is best suitedto analyze symmetric diffusions (see, e.g., [4, 17]). However, using the sub-martingale formulation of Stroock and Varadhan [30] or the extended Sko-rokhod problem [22], it is possible to construct reflected diffusions that areneither semimartingales nor symmetric processes [2, 3, 23, 24, 31]. This leadsnaturally to the question of determining when these reflected diffusions aresemimartingales and, if they are not semimartingales, whether they belongto some other tractable class of processes such as Dirichlet processes. Therehas been a substantial body of work that shows that, under certain con-ditions on the domain and reflection directions (namely, the completely- S condition and generalizations of it), the associated reflected diffusion is asemimartingale [22, 32]. In contrast, it has been a longstanding open prob-lem (see Section 4(iii) of [32]) to develop a theory for multidimensionalreflected diffusions for which this condition fails to hold (some results in two EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES dimensions can be found in [2, 3, 31]). As shown in [23, 24], such reflected dif-fusions arise naturally as approximations of a so-called generalized processorsharing model used in telecommunication networks. Thus, the developmentof such a theory is also of interest from the perspective of applications.The first main result of this work (Theorem 3.1) shows that multidi-mensional reflected diffusions that belong to a slight generalization of thefamily of reflected diffusions obtained as approximations in [23, 24] fail to besemimartingales. In two dimensions and for the case of reflected Brownianmotion, this result follows from Theorem 5 of [31] (also see [2] for an alter-native proof of this result). However, the analysis in [31] uses constructionsin polar coordinates that appear not to be easily generalizable to higher di-mensions. We follow a different approach, which is independent of dimensionand which allows us to establish the result for uniformly elliptic reflected dif-fusions, with possibly state-dependent diffusion coefficients, rather than justreflected Brownian motion.The next main result (Theorem 3.5) shows that reflected diffusions thatbelong to a broad class admit a decomposition as the sum of a local mar-tingale and a process of zero p -variation, for some p >
1. This class consistsof weak solutions to stochastic differential equations with reflection thatare Markov processes and have locally bounded drift and dispersion coeffi-cients and satisfy a certain L p continuity requirement (see Assumption 2).This continuity requirement is satisfied, for example, when the associatedextended Skorokhod map is H¨older continuous, but also holds under muchweaker conditions that do not even require that the (extended) Skorokhodmap be well-defined (see Remark 2.4). When the extended Skorokhod mapis well-defined and Lipschitz continuous, this implies that the associated re-flected diffusion is a Dirichlet process. Using the latter result, it is shownin Corollary 3.6 that the nonsemimartingale reflected diffusions consideredin Theorem 3.1 are Dirichlet processes. Our next result concerns the classof reflected Brownian motions introduced in [3], which were shown in [2]not to be semimartingales. In Corollary 3.7, Theorem 3.5 is applied to showthat even in cusplike domains, the associated reflected Brownian motionsare Dirichlet processes, thus partially resolving an open question raised in[3].The paper is organized as follows. Some common notation used through-out the paper is first summarized in Section 1.2. The class of stochasticdifferential equations with reflection under consideration, and the relatedmotivating examples, are introduced in Section 2. Section 3 contains a rig-orous statement of the main results; the proof of Theorem 3.1 is presentedin Section 4, while the proofs of Theorem 3.5 and Corollary 3.6 are given inSection 5. Some elementary results required in the proofs are relegated tothe Appendix. W. KANG AND K. RAMANAN
Notation.
As usual, R + or [0 , ∞ ) denote the space of all nonnega-tive reals, and N denotes the space of all positive integers. Given two realnumbers a and b , a ∧ b and a ∨ b denote the minimum and maximum, respec-tively, of a and b . For each positive integer J ≥ R J denotes J -dimensionalEuclidean space and the nonnegative orthant in this space is denoted by R J + = { x ∈ R J : x i ≥ i = 1 , . . . , J } . The Euclidean norm of x ∈ R J is de-noted by | x | and the inner product of x, y ∈ R J is denoted by h x, y i . Thevectors ( e , e , . . . , e J ) represent the usual orthonormal basis for R J , with e i being the i th coordinate vector. Given a vector u ∈ R J , u T denotes itstranspose, with analogous notation for matrices For x, y ∈ R J and a closedset A ⊂ R J , d ( x, y ) denotes the Euclidean distance between x and y , and d ( x, A ) = inf y ∈ A d ( x, y ) denotes the distance between x and the set A . Foreach r ≥ N r ( A ) = { x ∈ R J : d ( x, A ) ≤ r } . The unit sphere in R J is repre-sented by S (0). Given a set A ⊂ R J , A ◦ denotes its interior, A its closureand ∂A its boundary.The space of continuous functions on [0 , ∞ ) that take values in R J isdenoted by C [0 , ∞ ), and, given a set G ⊂ R J , C G [0 , ∞ ) denotes the subset offunctions f in C [0 , ∞ ) such that f (0) ∈ G . The spaces C [0 , ∞ ) and C G [0 , ∞ )are assumed to be equipped with the topology of uniform convergence oncompact sets. Given f ∈ C [0 , ∞ ) and T ∈ [0 , ∞ ), Var [0 ,T ] f denotes the R + ∪{∞} -valued number that equals the variation of f on [0 , T ]. Also, given areal-valued function f on [0 , ∞ ), its oscillation is defined by Osc ( f ; [ s, t ]) = sup s ≤ u ≤ u ≤ t | f ( u ) − f ( u ) | ; 0 ≤ s ≤ t < ∞ . For each A ∈ R J , I A ( · ) denotes the indicator function of the set A , whichtakes the value 1 on A and 0 on the complement of A .Given two random variables U ( i ) defined on a probability space (Ω ( i ) , F ( i ) , P ( i ) ) and taking values in a common Polish space S , i = 1 ,
2, the notation U (1) ( d ) = U (2) will be used to imply that the random variables are equal indistribution. Given a sequence of S -valued random variables { U ( n ) , n ∈ N } and U , with U ( n ) defined on (Ω ( n ) , F ( n ) , P ( n ) ) and U defined on (Ω , F , P ), U ( n ) ⇒ U is used to denote weak convergence of the sequence U ( n ) to U .Also, if the sequence of random variables are all defined on the same prob-ability space (Ω , F , P ), the notation U ( n ) ( P ) →
2. The class of reflected diffusions.
The class of stochastic differentialequations with reflection under study are introduced in Section 2.1, and thebasic assumptions are stated in Section 2.2. Some useful ramifications of theassumptions and a motivating example are then presented in Section 2.3.
EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Stochastic differential equations with reflection.
The so-called ex-tended Skorokhod problem (ESP), introduced in [22], is a convenient toolfor the pathwise construction of reflected diffusions. The data associatedwith an ESP is the closure G of an open, connected domain in R J and aset-valued mapping d ( · ) defined on G such that d ( x ) = { } for x ∈ G ◦ , d ( x )is a nonempty, closed and convex cone in R J with vertex at the origin forevery x ∈ ∂G and the graph of d ( · ) is closed. Roughly speaking, given a con-tinuous path ψ , the ESP associated with ( G, d ( · )) produces a constrainedversion φ of ψ that is restricted to live within the domain G by adding to ita “constraining term” η whose increments over any interval lie in the closureof the convex hull of the union of the allowable directions d ( x ) at the points x visited by φ during this interval. We now state the rigorous definition ofthe ESP. (In [22], the ESP was formulated more generally for c`adl`ag paths,but the formulation below will suffice for our purposes since we consideronly continuous processes.) Definition 2.1 (Extended Skorokhod problem). Suppose (
G, d ( · )) and ψ ∈ C G [0 , ∞ ) are given. Then ( φ, η ) ∈ C G [0 , ∞ ) × C [0 , ∞ ) are said to solve theESP for ψ if φ (0) = ψ (0), and if for all t ∈ [0 , ∞ ), the following propertieshold:(1) φ ( t ) = ψ ( t ) + η ( t );(2) φ ( t ) ∈ G ;(3) for every s ∈ [0 , t ) η ( t ) − η ( s ) ∈ co (cid:20) [ u ∈ ( s,t ] d ( φ ( u )) (cid:21) , (2.1)where co[ A ] represents the closure of the convex hull generated by theset A .If ( φ, η ) is the unique solution to the ESP for ψ , then we write φ = ¯Γ( ψ ),and refer to ¯Γ as the extended Skorokhod map (ESM).If a unique solution to the ESP exists for all ψ ∈ C G [0 , ∞ ), then the as-sociated ESM ¯Γ is said to be well-defined on C G [0 , ∞ ). In this case, it iseasily verified (see Lemma A.1) that if φ = ¯Γ( ψ ), then for any s ∈ [0 , ∞ ), φ s = ¯Γ( ψ s ), where for t ∈ [0 , ∞ ), ψ s ( t ) . = φ ( s ) + ψ ( s + t ) − ψ ( s ) , φ s ( t ) . = φ ( s + t ) . (2.2)Moreover, a well-defined ESM is said to be Lipschitz continuous on C G [0 , ∞ )if for every T < ∞ , there exists K T < ∞ such that, given ψ ( i ) ∈ C G [0 , ∞ )with corresponding solution ( φ ( i ) , η ( i ) ) to the ESP, for i = 1 ,
2, we havesup s ∈ [0 ,T ] | φ (1) ( s ) − φ (2) ( s ) | ≤ K T sup s ∈ [0 ,T ] | ψ (1) ( s ) − ψ (2) ( s ) | . (2.3) W. KANG AND K. RAMANAN
The ESP is a generalization of the so-called Skorokhod Problem (SP)introduced in [29]. Unlike the SP, the ESP does not require that the con-straining term η have finite variation on bounded intervals (compare Def-initions 1.1 and 1.2 of [22]). The ESP can be used to define solutions tostochastic differential equations with reflection (SDERs) associated with agiven pair ( G, d ( · )) and drift and dispersion coefficients b : R J R J and σ : R J R J × R N . Definition 2.2.
Given (
G, d ( · )), b ( · ) and σ ( · ), the triple ( Z t , B t ) , (Ω , F , P ) , {F t } is said to be a weak solution to the associated SDER if and only if:(1) {F t } is a filtration on the probability space (Ω , F , P ) that satisfies theusual conditions;(2) { B t , F t } is an N -dimensional Brownian motion.(3) P ( R t | b ( Z ( s )) | ds + R t | σ ( Z ( s )) | ds < ∞ ) = 1 ∀ t ∈ [0 , ∞ ).(4) { Z t , F t } is a J -dimensional, adapted process such that P -a.s., ( Z, Y )solves the ESP for X , where Y . = Z − X and X ( t ) = Z (0) + Z t b ( Z ( s )) ds + Z t σ ( Z ( s )) dB ( s ) ∀ t ∈ [0 , ∞ ) . (2.4)(5) The set { t : Z ( t ) ∈ ∂G } has P -a.s. zero Lebesgue measure. In other words, P -a.s., Z ∞ I ∂G ( Z ( s )) ds = 0 . (2.5)This is similar to the usual definition for weak solutions for SDEs (see, e.g.,Definitions 3.1 and 3.2 of [19]), except that property 4 is modified to definereflection and property 5 captures the notion of “instantaneous” reflection(see, e.g., pages 87–88 of [15]). A strong solution can also be defined in ananalogous fashion. Definition 2.3.
Given a probability space (Ω , F , P ) and an N -dimensionalBrownian motion B on (Ω , F , P ), Z is said to be a strong solution tothe SDER associated with ( G, d ( · )), b ( · ), σ ( · ) and initial condition ξ if P ( Z (0) = ξ ) = 1 and properties 3–5 of Definition 2.2 hold with {F t } equal tothe completed and augmented filtration generated by the Brownian motion B .For a precise construction of the filtration {F t } referred to in Definition2.3, see (2.3) of [19]. In what follows, given the constraining process Y inproperty 4 of Definition 2.2, the quantity L will denote the associated totalvariation measure: in other words, for 0 ≤ s ≤ t < ∞ , we define L ( s, t ) . = Var ( s,t ] Y and L ( t ) . = L (0 , t ] . (2.6) EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Observe that the process L in the second definition in (2.6) is {F t } -adaptedand takes values in the extended nonnegative reals, R + .2.2. Main assumptions.
We now introduce certain basic assumptions on(
G, d ( · )), b ( · ) and σ ( · ) that will be used in this work. In Section 2.3, weprovide a concrete motivating example of a family of SDERs that arisein applications which satisfies all the stated assumptions. In Section 2.4, weprovide another example of a class of SDERs that satisfy these assumptions.The latter class, which consists of two-dimensional reflected diffusions incurved domains, was first studied by Burdzy and Toby in [3].The first assumption concerns existence of solutions. General conditionson G and d ( · ) under which this assumption is satisfied can be found inLemma 2.6, Theorem 3.3 and Theorem 4.3 of [22]. Assumption 1.
There exists a weak solution ( Z t , B t ) , (Ω , F , P ) , {F t } tothe SDER associated with ( G, d ( · )) , b ( · ) and σ ( · ) such that { Z t , F t ; t ≥ } isa Markov process under P . Next, we impose a kind of L p -continuity condition on the ESM. Assumption 2.
There exist p > , q ≥ and K T < ∞ , T ∈ (0 , ∞ ) , suchthat the weak solution Z to the SDER satisfies, for every ≤ s ≤ t ≤ T , E [ | Y ( t ) − Y ( s ) | p |F s ] ≤ K T E h sup u ∈ [ s,t ] | X ( u ) − X ( s ) | q |F s i , (2.7) where X is the process defined by (2.4) and Y . = Z − X . Remark 2.4.
Assumption 2 holds under rather mild conditions on theESP—for example, when the following oscillation inequality is satisfied forany solution ( φ, η ) to the ESP for a given ψ : for every 0 ≤ s ≤ t < ∞ , thereexists C s,t < ∞ such that Osc ( φ, [ s, t ]) ≤ C s,t Osc ( ψ, [ s, t ]) . In this case, since (
Z, Y ) solve the ESP for X , we have for 0 ≤ s ≤ t ≤ T , | Y ( t ) − Y ( s ) | ≤ Osc ( Y, [ s, t ]) ≤ C s,t Osc ( X, [ s, t ]) ≤ C T sup u ∈ [ s,t ] | X ( u ) − X ( s ) | , where C T = max ≤ s ≤ t ≤ T C s,t < ∞ , and so Assumption 2 holds with p = q = 2and K T = 4 C T . The oscillation inequality can be shown to hold in manysituations of interest (see, e.g., Lemma 2.1 of [32]). If the ESM associatedwith ( G, d ( · )) is well-defined and Lipschitz continuous on C G [0 , ∞ ), thenthe oscillation inequality is also automatically satisfied, and so Assumption2 again holds with p = q = 2. Furthermore, it is easy to see that if the W. KANG AND K. RAMANAN
ESM is well-defined and H¨older continuous on C G [0 , ∞ ) with some exponent α ∈ (0 , p ≥ /α and q = αp . An exampleof such an ESM is provided in Section 2.4 (see also Section 5.3 and, inparticular, Remark 5.5). Assumption 3.
The coefficients b and σ are locally bounded, that is,they are bounded on every compact subset of G . A motivating example and ramifications of the assumptions.
Wenow describe a family of multi-dimensional ESPs (
G, d ( · )) that arise in ap-plications. Fix J ∈ N , J ≥
2. The J -dimensional ESPs in this family havedomain G = R J + and a constraint vector field d ( · ) that is parametrized by a“weight” vector α = ( α , . . . , α J ) with α i > i = 1 , . . . , J , and P Ji =1 α i = 1.Associated with each weight vector α is the ESP ( R J + , d ( · )), where for x ∈ ∂G = ∂ R J + , d ( x ) . = (cid:26) X i : x i =0 β i d i : β i ≥ (cid:27) with ( d i ) j . = ( − α j − α i , for j = i ,1 , for j = i ,for i, j = 1 , . . . , J . Reflected diffusions associated with this family were shownin [23, 24] to arise as heavy traffic approximations of the so-called generalizedprocessor sharing (GPS) model in communication networks (see also [7] and[9]). Indeed, the characterization of this class of reflected diffusions servesas one of the motivations for this work.Next, we introduce a family of SDERs that is a slight generalization ofthe class of GPS ESPs. Definition 2.5.
We will say (
G, d ( · )) , b ( · ) and σ ( · ) define a Class A SDER if they satisfy the following conditions:(1) The ESM associated with the ESP (
G, d ( · )) is well-defined and, for every T < ∞ , is Lipschitz continuous (with constant K T < ∞ ) on C G [0 , T ].(2) G is a closed convex cone with vertex at the origin, V = { } and thereexists ~ v ∈ G such that h ~ v , d i = 0 for all d ∈ d ( x ) ∩ S (0) , x ∈ ∂G \ { } ;(3) There exists a constant ˜ K < ∞ such that for all x, y ∈ G , | σ ( x ) − σ ( y ) | + | b ( x ) − b ( y ) | ≤ ˜ K | x − y | and | σ ( x ) | ≤ ˜ K, | b ( x ) | ≤ ˜ K (1 + | x | ) . EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES (4) The covariance function a : G → R J × R J defined by a ( · ) = σ T ( · ) σ ( · ) isuniformly elliptic, that is, there exists λ > u T a ( x ) u ≥ λ | u | for all u ∈ R J , x ∈ G. (2.8)We expect that the conditions in property 3 can be relaxed to a localLipschitz and linear growth condition on both b and σ , and the main resultcan still be proved by using localization along with the current arguments.However, to keep the notation simple, we impose the slightly stronger as-sumptions above. Remark 2.6.
ESPs in the GPS family defined above were shown tosatisfy properties 1 and 2 (the latter with ~ v = e + · · · + e J ) of Definition 2.5in Theorem 3.6 and Lemma 3.4 of [22], respectively.In Theorem 2.7, we summarize some consequences of Assumptions 1–3,and also show that Class A SDERs satisfy these assumptions. The proofessentially follows from Theorem 4.3 of [22] and Proposition 4.1 of [18]. Thefollowing set, V . = { x ∈ ∂G : there exists d ∈ S (0) such that { d, − d } ⊂ d ( x ) } , (2.9)was shown in [22] to play an important role in the analysis. Theorem 2.7.
Suppose ( G, d ( · )) , b ( · ) and σ ( · ) satisfy Assumptions 1and 2, and let ( Z t , B t ) , (Ω , F , P ) , {F t } be a weak solution to the associatedSDER. Then Z is an F t -semimartingale on [0 , T V ) , where T V . = inf { t ≥ Z ( t ) ∈ V} , (2.10) and P -a.s., Z admits the decomposition Z ( · ) = Z (0) + M ( · ) + A ( · ) , (2.11) where for t ∈ [0 , T V ) , M ( t ) . = Z t σ ( Z ( s )) · dB ( s ) , A ( t ) . = Z t b ( Z ( s )) ds + Y ( t ) , (2.12) and Y has finite variation on [0 , t ] and satisfies Y ( t ) = Z t γ ( s ) dL ( s ) , (2.13) where L is given by (2.6) and γ ( s ) ∈ d ( Z ( s )) , dL -a.e. s ∈ [0 , t ] . Moreover, if ( G, d ( · )) , b ( · ) and σ ( · ) satisfy properties 1 and 3 of Definition 2.5, then theyalso satisfy Assumption 1, Assumption 2 (with p = q = 2 ) and Assumption3. In this case, { Z t , F t } is in fact the pathwise unique strong solution tothe SDER, is a strong Markov process and has E [ | Z ( t ) | ] < ∞ for every t ∈ (0 , ∞ ) if E [ | Z (0 | ] < ∞ . W. KANG AND K. RAMANAN
Proof.
Let X be the process defined by (2.4). Then X is clearly asemimartingale and property 4 of Definition 2.2 shows that P -a.s., ( Z, Z − X )satisfy the ESP for X . Moreover, Theorem 2.9 of [22] shows that Y = Z − X has P -a.s. finite variation on any closed sub-interval of [0 , T V ). This showsthat Z is an F t -semimartingale on [0 , T V ) with the decomposition given in(2.11)–(2.13), and thus establishes the first assertion of the theorem.Next, suppose ( G, d ( · )), b ( · ) and σ ( · ) satisfy properties 1 and 3 of Defini-tion 2.5. Then property 3 of Definition 2.5 implies Assumption 3 is satisfied.In addition, by Remark 2.4, property 1 ensures that Assumption 2 holdswith p = q ≥
2. Moreover, Theorem 4.3 of [22] and Proposition 4.1 of [18]show that, in fact, the associated SDER admits a pathwise unique strongsolution Z , which is also a strong Markov process. Thus, Assumption 1 isalso satisfied. Hence, we have shown that Assumptions 1–3 hold. The lastassertion of the theorem can be established using standard techniques, by amodification of the proof in Theorem 4.3 of [22], in the same manner as thisresult is proved for strong solutions to SDEs, and so we omit the details ofthe proof. (cid:3) We conclude this section by stating a consequence of property 2 of Def-inition 2.5 that will be useful in the sequel. Let Γ denote the (extended)Skorokhod map associated with the 1-dimensional (extended) Skorokhodproblem with G = R + and d (0) = R + , d ( x ) = 0 if x >
0. It is well known(see, e.g., [29] or Lemma 3.6.14 of [19]) that Γ is well-defined on C R + [0 , ∞ ),and in fact has the explicit formΓ ( ψ )( t ) = ψ ( t ) + sup s ∈ [0 ,t ] [ − ψ ( s )] ∨ . (2.14) Lemma 2.8.
Suppose that ( G, d ( · )) satisfies property 2 of Definition 2.5.If ( φ, η ) solves the associated ESP for ψ ∈ C G [0 , ∞ ) , then h φ, ~ v i = Γ ( h ψ, ~ v i ) . The proof of this lemma is exactly analogous to the proof of Corollary 3.5of [22], and is thus omitted.2.4.
Another motivating example.
We now describe a family of two-dimensional reflecting Brownian motions (henceforth abbreviated to RBMs)in “valley-shaped” domains with vertex at the origin and horizontal direc-tions of reflection. This family of reflected diffusions, which was first studiedin [3], is parameterized by two continuous real-valued functions L and R defined on [0 , ∞ ), with L (0) = R (0) = 0 and L ( y ) < R ( y ) for all y >
0. Theassociated domain G is then given by G . = { ( x, y ) ∈ R : y ≥ , L ( y ) ≤ x ≤ R ( y ) } . EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Let ∂ G . = { ( x, y ) ∈ ∂G \ (0 ,
0) : x = L ( y ) } and, likewise, let ∂ G . = { ( x, y ) ∈ ∂G \ (0 ,
0) : x = R ( y ) } . Then the reflection vector field is defined by d ( x, y ) = (1 , , ( x, y ) ∈ ∂ G ,( − , , ( x, y ) ∈ ∂ G , { v : v ≥ } , ( x, y ) = (0 , , ,
0) to ensure that the Brownian motion can be constrained within thedomain. To conform with the general structure of ESPs, at (0 ,
0) we in factdefine d ( · ) to be the convex cone (which, in this case, equals a half-space)generated by the three directions (1 , − ,
0) and (0 , V = { } for this ESM and, when L and R are linear functions, this reflected diffusionis a special case of the Class A SDER’s introduced in the last section.It was shown in Theorem 1 of [3] (see also Section 4.3 of [2]) that theESM ¯Γ corresponding to (
G, d ( · )) is well-defined and thus, when B is astandard two-dimensional Brownian motion, Z = ¯Γ( B ) is a well-defined re-flected Brownian motion starting at (0 ,
0) and is also a Markov process (seeTheorem 2 of [3]). In Proposition 4.13 of [2], RBMs in this family were shownnot to be semimartingales. As an application of the results of this paper, weshow that when L and R are sufficiently regular, Z nevertheless admits auseful decomposition (see Corollary 3.7).
3. Statement of main results.
Theorem 2.7 shows that if V = ∅ then Z is a semimartingale. In fact, it was shown in Theorem 1.3 of [22] that when V = ∅ , the ESM coincides with the SM. The main focus of this work is to un-derstand the behavior of reflected diffusions Z associated with ESPs ( G, d ( · ))for which V 6 = ∅ , with the GPS family being a representative example. In[22], it was shown that for the GPS family of ESPs, Z is a semimartingaleuntil the first time it hits the origin. However, the first result of the presentpaper (Theorem 3.1) shows that Z is not a semimartingale on [0 , ∞ ). Theorem 3.1.
Suppose ( G, d ( · )) , b ( · ) and σ ( · ) describe a Class A SDER.Then the unique pathwise solution Z to the associated SDER is not a semi-martingale. The proof of Theorem 3.1 is given in Section 4.3. As mentioned in Section1, for the special case when G is a convex wedge in R and the directions ofconstraint on the two faces are constant and point at each other, b ≡ σ is the identity matrix (i.e., Z is a reflected Brownian motion), this resultfollows from Theorem 5 of [31] (with the parameters α = 1 and the wedgeangle π less than 180 ◦ therein). The fact that, when J = 2, the reflected W. KANG AND K. RAMANAN
Brownian motion Z defined here is the same as the reflected Brownian mo-tion defined via the submartingale formulation in [31] follows from Theorem1.4(2) of [22]. This two-dimensional result can also be viewed as a specialcase of Proposition 4.13 of [2]. However, the proofs in [2] and [31] do notseem to extend easily to higher dimensions. In this paper, we take a differ-ent approach that is applicable in arbitrary dimensions and to more generaldiffusions, in particular providing a different proof of the two-dimensionalresult mentioned above.As is well known, when a process is a semimartingale, C functionalsof the process can be characterized using Itˆo’s formula. Theorem 3.1 canthus be viewed as a somewhat negative result since it suggests that Class A reflected diffusions and, in particular, reflected diffusions associated with theGPS family that arise in applications, may not possess desirable properties.However, we show in Corollary 3.6 that these diffusions are indeed tractableby establishing that they belong to the class of Dirichlet processes (in thesense of F¨ollmer). This follows as a special case of a more general result,which is stated below as Theorem 3.5.In order to state this result, we first recall the definitions of zero p -variation processes and Dirichlet processes (see, e.g., Theorem 2 of [13]). Definition 3.2.
For p >
0, a continuous process A is of zero p -variationif and only if for any T > X t i ∈ π n | A ( t i ) − A ( t i − ) | p ( P ) → { π n } of partitions of [0 , T ] with ∆( π n ) . = max t i ∈ π n ( t i +1 − t i ) → n → ∞ . If the process A satisfies (3.1) with p = 2, then A is saidto be of zero energy. Definition 3.3.
The stochastic process Z is said to be a Dirichlet pro-cess if the following decomposition holds: Z = M + A, (3.2)where M is an F t -adapted local martingale and A is a continuous, F t -adapted, zero energy process with A (0) = 0.Note that this is weaker than the original definition of a Dirichlet processgiven by F¨ollmer [13], which requires that M and A in the decomposition(3.2) be square integrable and that A satisfy E [ P t i ∈ π n | A t i − A t i − | ] → π n ) →
0, rather than satisfy (3.1) with p = 2. However, our definition canbe viewed as a localized version and coincides with Definition 2.4 of [5] (seealso Definition 12 of [28]). EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Remark 3.4.
The decomposition of a Dirichlet process Z , into a localmartingale and a zero energy process starting at 0, is unique. For any p > π n of [0 , T ], X t i ∈ π n | A ( t i +1 ) − A ( t i ) | p ≤ max t i ∈ π n | A ( t i +1 ) − A ( t i ) | p − Var [0 ,T ] ( A ) . Therefore, it follows that if A is continuous and of finite variation, then it isalso of zero p -variation, for all p >
1. In particular, this shows that the classof Dirichlet processes generalizes the class of continuous semimartingales.
Theorem 3.5.
Suppose ( G, d ( · )) , b ( · ) and σ ( · ) satisfy Assumptions 1and 3, let Z be an associated weak solution that satisfies Assumption 2 forsome p > , and let Y = Z − X , where X is defined by (2.4). Then Y haszero p -variation. As an immediate consequence of Theorem 3.5, Definition 3.3 and Theorem2.7, we have the following result.
Corollary 3.6.
Suppose ( G, d ( · )) , b ( · ) and σ ( · ) satisfy Assumptions1 and 3, and also Assumption 2 with p = 2 . Then the associated reflecteddiffusion is a Dirichlet process. In particular, reflected diffusions associatedwith Class A SDERs are Dirichlet processes.
The next consequence of Theorem 3.5 concerns the class of reflected dif-fusions described in Section 2.4.
Corollary 3.7.
Suppose that L and R are two continuous functionson [0 , y ] given by L ( y ) = − c L y α L , R ( y ) = c R y α R , y ∈ [0 , ∞ ) , (3.3) for some α L , α R , c L , c R ∈ (0 , ∞ ) , and let α = min( α L , α R ) . If α ≥ , then theassociated two-dimensional reflected diffusion Z described in Section 2.4 isa Dirichlet process, that is, admits the decomposition Z = B + A , where A is a process with zero quadratic variation. It was shown in [2] that, for every α > Z is not a semimartingale. Incontrast, Corollary 3.7 establishes a positive result in this direction, showingthat even when the domain has a cusp-like shape (i.e., corresponding to α > L and R arelinear (i.e., when α L = α R = 1) the domain is wedge-shaped and the reflecteddiffusion Z is associated with a Class A SDER. In this case, Corollary 3.7 W. KANG AND K. RAMANAN follows from Corollary 3.6. The proof of Corollary 3.7 in the general caseis given in Section 5.3. It is natural to expect that the reflected diffusionwould also be a Dirichlet process when α <
1, since this corresponds to nicer“flatter” domains. However, this does not directly follow from the simpleproof of Corollary 3.7 given in Section 5.3 (see Remark 5.4).
4. Reflected diffusions associated with Class A SDERs.
Throughout thissection, we will assume that (
G, d ( · )), b ( · ) and σ ( · ) describe a Class A SDER.Let B be an N -dimensional Brownian motion on a given probability space(Ω , F , P ), let {F t } be the right-continuous augmentation of the filtration gen-erated by B (see Definition (2.3) given in [19]). Also, let Z be the pathwiseunique strong solution to the associated SDER (which exists by Theorem2.7), let X be defined by (2.4), let Y . = Z − X and let L be the total varia-tion process of Y as defined in (2.6). We use E to denote expectation withrespect to P and, for z ∈ G , let P z (resp., E z ) denote the probability (resp.,expectation) conditioned on Z (0) = z .This section is devoted to the proof of Theorem 3.1. The key step is toshow that the constraining process Y in the extended Skorokhod decompo-sition for Z has P -a.s. infinite variation. More precisely, let ~ v be the vectorthat satisfies property 3 of Definition 2.5 and, for any given ε ≥
0, considerthe hyperplane H ε . = { x ∈ R d : h ~ v , x i = ε } ∩ G, (4.1)and let τ ε . = inf { t ≥ Z ( t ) ∈ H ε } . (4.2)We now state the key result in the proof of Theorem 3.1. Theorem 4.1.
There exists
T < ∞ such that P ( L ( T ) = ∞ ) > . A somewhat subtle point to note is that Theorem 4.1 does not immedi-ately establish the fact that Z is not a semimartingale because we do notknow a priori that, if Z were a semimartingale, then its Doob decomposi-tion must be of the form Z = M + A given in (2.11) and (2.12). However,in Section 4.3 (see Proposition 4.12) we establish that this is indeed thecase, thus obtaining Theorem 3.1 from Theorem 4.1. First, in Section 4.1,we establish Theorem 4.1 for the case when b ≡
0. The proof for the generalcase is obtained from this result via a Girsanov transformation in Section4.2.
EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES The zero drift case.
Throughout this section, we assume b ≡ Proposition 4.2. If b ≡ , then we have E [ e − L ( τ ) ] = 0 , (4.3) and hence, L ( τ ) = ∞ , P -a.s. (4.4)When combined with Lemma 4.11, which shows that P ( τ < ∞ ) = 1when b ≡
0, Proposition 4.2 yields Theorem 4.1. The proof of Proposition 4.2is given in Section 4.1.3. The proof relies on an upper bound for E [ e − L ( τ ) ],which is obtained in Section 4.1.1, and some weak convergence results, whichare established in Section 4.1.2.4.1.1. An upper bound.
To begin with, we use the strong Markov prop-erty of Z to obtain an upper bound on E [ e − L ( τ ) ]. Recall the definition of τ given in (4.2) with ε = 0, noting that H = { } because G is a closedconvex cone with vertex at 0. Moreover, for ε >
0, we recursively define twosequences of random times { τ εn } n ∈ N and { α εn } n ∈ N as follows: α ε . = 0 and for n ∈ N , τ εn . = inf { t ≥ α εn − : Z ( t ) ∈ H ε } , (4.5) α εn . = inf { t ≥ τ εn : Z ( t ) ∈ H } . Since Z is continuous and H ε and H are closed, it is clear that τ , τ εn and α εn are F t -stopping times. For conciseness, we will often denote τ ε simplyby τ ε , since this is consistent with the notation of τ ε given in (4.2). Lemma 4.3.
For every ε ∈ (0 , , E [ e − L ( τ ) ] ≤ E [ P Z ( τ ε ) ( τ ≥ τ )] E [ P Z ( τ ε ) ( τ ≥ τ )] + E [ E Z ( τ ε ) [(1 − e − L ( τ ) ) I { τ <τ } ]] . Proof.
From the elementary inequality L ( τ ) ≥ ∞ X n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) , it immediately follows that E [ e − L ( τ ) ] ≤ E [ e − P ∞ n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) ] . (4.6) W. KANG AND K. RAMANAN
For n ≥ α εn ≥ α ε and τ εn ≥ α ε . Hence, on the set { α ε ≥ τ } , we have α εn ∧ τ = τ εn ∧ τ = τ for every n ≥
2. Therefore, the right-hand side of(4.6) can be decomposed as E [ e − P ∞ n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) ] = E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε ≥ τ } ]+ E [ e − P ∞ n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) I { α ε <τ } ] . Conditioning on F α ε , using the fact that I { α ε <τ } , L ( α ε ∧ τ ) and L ( τ ε ∧ τ )are F α ε -measurable, the strong Markov property of Z and the fact that Z ( α ε ) = 0, last term above can be rewritten as E [ e − P ∞ n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) I { α ε <τ } ]= E [ E [ e − P ∞ n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) I { α ε <τ } |F α ε ]]= E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε <τ } E [ e − P ∞ n =2 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) |F α ε ]]= E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε <τ } E Z ( α ε ) [ e − P ∞ n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) ]]= E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε <τ } ] E [ e − P ∞ n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) ] . Combining the last two assertions and rearranging terms, we obtain E [ e − P ∞ n =1 ( L ( α εn ∧ τ ) − L ( τ εn ∧ τ )) ] = E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε ≥ τ } ]1 − E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε <τ } ] . Together with (4.6), this yields the inequality E [ e − L ( τ ) ] ≤ E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε ≥ τ } ]1 − E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε <τ } ] . (4.7)We now show that the upper bound stated in the lemma follows from(4.7). Due to the nonnegativity of L ( α ε ∧ τ ) − L ( τ ε ∧ τ ) and the strongMarkov property of Z , we have E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε ≥ τ } ] ≤ E [ I { α ε ≥ τ } ]= E [ E [ I { α ε ≥ τ } |F τ ε ]](4.8) = E [ P Z ( τ ε ) ( τ ≥ τ )] , where recall that τ = inf { t ≥ Z ( t ) ∈ H } . Similarly, once again condition-ing on F τ ε and using the strong Markov property of Z , we obtain E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε <τ } ]= E [ E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε <τ } |F τ ε ]]= E [ E Z ( τ ε ) [ e − L ( τ ∧ τ ) I { τ <τ } ]] . EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Therefore,1 − E [ e − ( L ( α ε ∧ τ ) − L ( τ ε ∧ τ )) I { α ε <τ } ]= E [1 − E Z ( τ ε ) [ e − L ( τ ∧ τ ) I { τ <τ } ]](4.9) = E [ P Z ( τ ε ) ( τ ≥ τ )] + E [ E Z ( τ ε ) [(1 − e − L ( τ ) ) I { τ <τ } ]] . The lemma follows from (4.7), (4.8) and (4.9). (cid:3)
Next, we establish an elementary lemma that holds when the drift is zero.Recall the vector ~ v of property 2 of Definition 2.5. Lemma 4.4.
When b ≡ , the process h Z, ~ v i is an F t -martingale on [0 , τ ] and for every ε > , P -a.s., P Z ( τ ε ) ( τ ≥ τ ) = ε. (4.10) Proof.
First, note that H = { } = V by property 2 of Definition 2.5and so T V defined in (2.10) coincides with τ . From Lemma 2.8 and thecontinuity of the sample paths of Y , it follows that for t ∈ [0 , τ ], h Y ( t ) , ~ v i = 0and so P -a.s., h Z ( t ) , ~ v i = h Z (0) , ~ v i + ˜ M , t ∈ [0 , τ ] , (4.11)where ˜ M . = h R · σ ( Z ( s )) · dB ( s ) , ~ v i is an F t martingale on [0 , τ ] since σ isuniformly bounded by property 3 of Definition 2.5. This establishes the firstassertion of the lemma.The quadratic variation h ˜ M i of ˜ M is given by h ˜ M i ( t ) = Z t ~ v T a ( Z ( s )) ~ v ds, t ∈ [0 , ∞ ) , where a . = σ T σ . By property 4 of Definition 2.5, a ( · ) is uniformly elliptic.Therefore, P -a.s., h ˜ M i is strictly increasing and h ˜ M i ∞ . = lim t →∞ h ˜ M i ( t ) = ∞ . For t ∈ [0 , ∞ ), let T ( t ) . = inf { s ≥ h ˜ M i ( s ) > t } , G t . = F T ( t ) , ˜ B ( t ) . = ˜ M ( T ( t )) . Then { ˜ B t , G t } t ≥ is a standard one-dimensional Brownian motion (see, e.g.,Theorem 4.6 on page 174 of [19]). Define ˜ τ ε . = inf { t ≥ B ( t ) = ε } . By (4.11),we have P -a.s., P Z ( τ ε ) ( τ ≥ τ ) = P (˜ τ ≥ ˜ τ | ˜ B (0) = ε ) = ε, where the latter follows from well-known properties of Brownian motion.This proves (4.10). (cid:3) W. KANG AND K. RAMANAN
Remark 4.5.
From Lemmas 4.3 and 4.4, we conclude that for every ε > E [ e − L ( τ ) ] ≤ εε + E [ E Z ( τ ε ) [(1 − e − L ( τ ε ) ) I { τ <τ } ]] . Thus, in order to establish (4.3), it suffices to show that for some sequence { ε k } k ∈ N such that ε k → k → ∞ ,lim inf k →∞ ε k E [ E Z ( τ εk ) [(1 − e − L ( τ εk ) ) I { τ <τ } ]] = ∞ . This is established in Section 4.1.3 using scaling arguments. Since Z is areflected diffusion (rather than just a reflected Brownian motion), the scalingarguments are more involved and rely on some weak convergence results thatare established in Section 4.1.2. The reader may prefer to skip forward tothe proof of Proposition 4.2 in Section 4.1.3 and refer back to the results inSection 4.1.2 when required.4.1.2. A weak convergence result.
Recall that we have assumed that thedrift b ≡
0. Now, let { ε k } k ∈ N and { x k } k ∈ N be sequences such that ε k → k → ∞ and x k ∈ H ε k for k ∈ N . For each k ∈ N , let Z ( k ) be the pathwiseunique solution to the associated SDER with initial condition x k , and let X ( k ) , Y ( k ) and L ( k ) be the associated processes as defined in Definition 2.2and (2.6). For k ∈ N , consider the scaled process B k ( t ) . = B ( ε k t ) ε k , t ∈ [0 , ∞ ) , which is a standard Brownian motion due to Brownian scaling. Similarly,define A k ( t ) . = A ( k ) ( ε k t ) ε k , A = X, Y, Z, L, (4.12)and let F kt . = F ε k t for t ∈ [0 , ∞ ). Clearly, the processes Z k , B k , Y k and L k are {F kt } -adapted and L k ( t ) = Var [0 ,t ] Y k for every t ≥
0. For ( r, R ) ∈ (0 , ∞ ) such that r < R , let θ kr,R . = inf { t ≥ h Z k ( t ) , ~ v i / ∈ ( r, R ) } , k ∈ N . (4.13)This section contains two main results. Roughly speaking, the first re-sult (Lemma 4.7) shows that for the question under consideration, we canin effect replace the state-dependent diffusion coefficient σ ( · ) by σ (0). Thisproperty is then used in Corollary 4.8 to provide bounds on the total varia-tion sequence L k ( θ kr,R ), as ε k →
0. First, we observe that there exists a simpleequivalence between ( X k , Z k , Y k ) and another triplet of processes that willbe easier to work with. EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Remark 4.6.
For notational conciseness, we define the scaled diffusioncoefficient σ k ( x ) . = σ ( ε k x ) , x ∈ R J , k ∈ N . By the definition of Z ( k ) and the scaling (4.12), it then follows that X k ( t ) = x k ε k + 1 ε k Z ε k t σ ( Z ( k ) ( s )) dB ( s ) = x k ε k + Z t σ k ( Z k ( s )) dB k ( s ) , where the last equality holds by the time-change theorem for stochasticintegrals (see Proposition 1.4 in Chapter V of [25]). This implies Z k is astrong solution to the SDER associated with ( G, d ( · )), b ≡ σ k and theBrownian motion { B k ( t ) , F kt } t ≥ defined on (Ω , F , P ), with initial condition x k /ε k . If σ satisfies properties 3 and 4 of Definition 2.5 then so does σ k , andthus ( G, d ( · )), b ≡ σ k also describe a Class A SDER. Therefore, byTheorem 2.7 there exists a pathwise unique solution ˜ Z k to the associatedSDER for the Brownian motion { B t , F t } with initial condition x k /ε k . Let˜ X k and ˜ Y k be the processes associated with ˜ Z k , defined in the usual manneras follows: ˜ X k ( t ) = x k ε k + Z t σ k ( ˜ Z k ( s )) dB ( s ) , t ∈ [0 , ∞ ) , (4.14)and ˜ Y k = ˜ Z k − ˜ X k . From the fact that solutions to Class A SDERs areunique in law by Theorem 2.7, it then follows that( X k , Z k , Y k ) ( d ) = ( ˜ X k , ˜ Z k , ˜ Y k ) , (4.15)where recall that ( d ) = indicates equality in distribution. Lemma 4.7.
Given x ∗ ∈ R J + , let ( Z, Y ) satisfy the ESP pathwise for X . = x ∗ + σ (0) B, (4.16) and let θ r,R . = inf { t ≥ h Z ( t ) , ~ v i / ∈ ( r, R ) } . (4.17) Suppose b ≡ and x k /ε k → x ∗ as k → ∞ . Then the following propertieshold: (1) As k → ∞ , E h sup t ∈ [0 ,T ] | ˜ Z k ( t ) − Z ( t ) | i → and ( X k , Z k , Y k ) ⇒ ( X, Z, Y ) ; W. KANG AND K. RAMANAN (2)
For all but countably many pairs ( r, R ) ∈ (0 , ∞ ) such that r < R , as k → ∞ , we have max i =1 ,...,J sup s ∈ [0 ,θ kr,R ] Y ki ( s ) ⇒ max i =1 ,...,J sup s ∈ [0 ,θ r,R ] Y i ( s ) . Proof.
Note that since x k /ε k ∈ H for every k ∈ N and H is closed, wemust have x ∗ ∈ H . We first prove property 1. Let ˜ X k , ˜ Z k and ˜ Y k be as inRemark 4.6. Then, by (4.15), it clearly suffices to show that ( ˜ X k , ˜ Z k , ˜ Y k ) ⇒ ( X, Z, Y ). From (4.14) and (4.16), it follows that for t ∈ [0 , ∞ ), | ˜ X k ( t ) − X ( t ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) x k ε k − x ∗ + Z t ( σ k ( ˜ Z k ( s )) − σ (0)) dB ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) x k ε k − x ∗ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z t ( σ k ( Z ( s )) − σ (0)) dB ( s ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z t (cid:18) σ k ( ˜ Z k ( s )) − σ k ( Z ( s )) (cid:19) dB ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . Using the fact that ( a + b + c ) ≤ a + b + c ) for all a, b, c ∈ R and takingthe supremum over t ∈ [0 , T ] and then expectations of both sides, we obtain E h sup t ∈ [0 ,T ] | ˜ X k ( t ) − X ( t ) | i ≤ (cid:12)(cid:12)(cid:12)(cid:12) x k ε k − x ∗ (cid:12)(cid:12)(cid:12)(cid:12) + 3 E (cid:20) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ( σ k ( Z ( s )) − σ (0)) dB ( s ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) + 3 E (cid:20) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ( σ k ( ˜ Z k ( s )) − σ k ( Z ( s ))) dB ( s ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) . Since σ is uniformly bounded, the stochastic integrals on the right-hand sideare martingales. By applying the Burkholder–Davis–Gundy (BDG) inequal-ity, the Lipschitz condition on σ , the definition of σ k and Fubini’s theorem,we obtain E (cid:20) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ( σ k ( ˜ Z k ( s )) − σ k ( Z ( s ))) dB ( s ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) ≤ C E (cid:20)Z T | σ k ( ˜ Z k ( s )) − σ k ( Z ( s )) | ds (cid:21) ≤ C ˜ K ε k E (cid:20)Z T | ˜ Z k ( s ) − Z ( s ) | ds (cid:21) ≤ C ˜ K ε k Z T E h sup u ∈ [0 ,s ] | ˜ Z k ( u ) − Z ( u ) | i ds, EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES where C < ∞ is the universal constant in the BDG inequality. Using similararguments, we also see that E (cid:20) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ( σ k ( Z ( s )) − σ (0)) dB ( s ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) ≤ C ˜ K ε k Z T E h sup u ∈ [0 ,s ] | Z ( u ) | i ds ≤ C ˜ K ε k T E h sup t ∈ [0 ,T ] | Z ( t ) | i . Combining the last three displays, and setting ˜ C T . = 3 C ˜ K (1 ∨ T ) < ∞ , wehave E h sup t ∈ [0 ,T ] | ˜ X k ( t ) − X ( t ) | i ≤ ˜ C T ε k Z T E h sup u ∈ [0 ,s ] | ˜ Z k ( u ) − Z ( u ) | i ds (4.19) + R k ( T ) , where R k ( T ) . = 3 (cid:12)(cid:12)(cid:12)(cid:12) x k ε k − x ∗ (cid:12)(cid:12)(cid:12)(cid:12) + ˜ C T ε k E h sup t ∈ [0 ,T ] | Z ( t ) | i . By the assumed Lipschitz continuity of Γ, E h sup t ∈ [0 ,T ] | Z ( t ) | i ≤ K T E h sup t ∈ [0 ,T ] | x ∗ + σ (0) B ( t ) | i ≤ K T | x ∗ | + 2 K T | σ (0) | E h sup t ∈ [0 ,T ] | B ( t ) | i < ∞ . Since x k /ε k → x ∗ and ε k → k → ∞ , it follows thatlim k →∞ R k ( T ) = 0 . (4.20)On the other hand, combining the inequality in (4.19) with the Lipschitzcontinuity of the map ¯Γ, we obtain E h sup t ∈ [0 ,T ] | ˜ Z k ( t ) − Z ( t ) | i ≤ K T R k ( T ) + K T ˜ C T ε k Z T E h sup u ∈ [0 ,s ] | ˜ Z k ( u ) − Z ( u ) | i ds. An application of Gronwall’s lemma then shows that E h sup t ∈ [0 ,T ] | ˜ Z k ( t ) − Z ( t ) | i ≤ K T R k ( T ) e K T ˜ C T ε k , W. KANG AND K. RAMANAN which converges to zero as k → ∞ due to (4.20) and the fact that ε k → k → ∞ . This proves (4.18). In turn, substituting the last inequality backinto (4.19) and, again using (4.20) and the fact that ε k →
0, we also obtain E h sup t ∈ [0 ,T ] | ˜ X k ( t ) − X ( t ) | i → k → ∞ , which implies ˜ X k ⇒ X . Since the mapping from ˜ X k ( ˜ X k , ˜ Z k , ˜ Y k ) is con-tinuous, by the continuous mapping theorem it follows that ( ˜ X k , ˜ Z k , ˜ Y k ) ⇒ ( X, Z, Y ) and the first property of the lemma is established.We now turn to the proof of the second property. By the first property,we know that ( Z k , Y k ) ⇒ ( Z, Y ) as k → ∞ . This immediately implies thatfor all but countably main pairs ( r, R ) ∈ (0 , ∞ ) such that r < R , we have,as k → ∞ ,( Z k ( · ∧ θ kr,R ) , Y k ( · ∧ θ kr,R ) , θ kr,R ) ⇒ ( Z ( · ∧ θ r,R ) , Y ( · ∧ θ r,R ) , θ r,R ) . (For an argument that justifies this implication, see, e.g., the proof of The-orem 4.1 on page 354 of [11].) Using the continuity of the map ( f, g, t ) max i =1 ,...,J sup s ∈ [0 ,t ] g i ( s ) from C [0 , ∞ ) × C [0 , ∞ ) × R + to R + , an applicationof the continuous mapping theorem yields the second property. (cid:3) Corollary 4.8.
Suppose b ≡ and x k /ε k → x ∗ as k → ∞ . Then foreach pair ( r, R ) ∈ (0 , ∞ ) such that r < R , the following properties hold: (1) P (sup k ∈ N L k ( θ kr,R ) < ∞ ) = 1 . (2) ε k L k ( θ kr,R ) ⇒ . (3) P ( L ( θ r,R ) < ∞ ) = 1 and if r < h x, ~ v i < R , P ( L ( θ r,R ) > > . Proof. If h x ∗ , ~ v i < r or h x ∗ , ~ v i > R , then θ r,R = 0 and θ kr,R = 0 for all k sufficiently large. In this case, properties (1)–(3) hold trivially. Hence, forthe rest of the proof, we assume that r ≤ h x ∗ , ~ v i ≤ R .We start by proving property 1. Let ˜ X k , ˜ Z k and ˜ Y k be defined as inRemark 4.6, and let ˜ L k be defined as in (2.6), but with Y replaced by ˜ Y k .By (4.15), it follows that ( L k , θ kr,R ) and ( ˜ L k , ˜ θ kr,R ) have the same distributionfor each k ∈ N , where ˜ θ kr,R is defined in the obvious way:˜ θ kr,R . = inf { t ≥ h ˜ Z k ( t ) , ~ v i / ∈ ( r, R ) } . We now argue that P (˜ θ kr,R < ∞ ) = 1. Indeed, for k ∈ N such that x k /ε k / ∈ ( r, R ) this holds trivially. On the other hand, if ˜ Z k (0) = x k /ε k ∈ ( r, R ) thenthis follows because Lemma 2.8 and the uniform ellipticity condition showthat, on (0 , ˜ θ r,R ), h ˜ Z k ( t ) , ~ v i = h ˜ X k ( t ) , ~ v i is a continuous martingale whose EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES quadratic variation is strictly bounded away from zero. Thus, h ˜ Z k , ~ v i is P -a.s. unbounded, and hence ˜ θ kr,R is P -a.s. finite. Therefore, to prove property1, it suffices to show that P (cid:16) sup k ∈ N ˜ L k (˜ θ kr,R ∧ T ) < ∞ (cid:17) = 1 , T > . Fix T ∈ (0 , ∞ ). Since r >
0, there exists δ > h y, ~ v i < r for all y with | y | ≤ δ . Let ˜ κ kδ . = inf { t ≥ | ˜ Z k ( t ) | ≤ δ } . Then θ kr,R ≤ ˜ κ kδ for all k ∈ N .Let ˜ C k . = sup t ∈ [0 ,T ] | ˜ Z k ( t ) | ∨ | ˜ X k ( t ) | . By property 1 of Lemma 4.7, it follows that ( ˜ X k , ˜ Z k ) ⇒ ( X, Z ) as k →∞ . Using the continuity of the map ( f, g ) sup s ∈ [0 ,T ] | f ( s ) | ∨ | g ( s ) | from C [0 , ∞ ) × C [0 , ∞ ) to R + , an application of the continuous mapping theoremyields ˜ C k ⇒ C as k → ∞ , where C . = sup t ∈ [0 ,T ] | Z ( t ) | ∨ | X ( t ) | . Also, due tothe Lipschitz continuity of the ESM ¯Γ and (4.16), P -a.s., we havesup t ∈ [0 ,T ] | Z ( t ) | ≤ K T sup t ∈ [0 ,T ] | X ( t ) | ≤ K T (cid:16) | x ∗ | + | σ (0) | sup s ∈ [0 ,T ] | B ( s ) | (cid:17) < ∞ , and hence P -a.s., C < ∞ . It then follows that P (sup k ∈ N ˜ C k < ∞ ) = 1. More-over, V = { } and for each ω ∈ Ω, ( Z ( · , ω ) , Y ( · , ω )) solves the ESP for X ( · , ω ).Therefore, it follows from Lemma 2.8 of [22] that there exist ρ >
0, indepen-dent of k , a finite set I = { , . . . , I } and a collection of open sets {O i , i ∈ I } of R J and associated vectors { v i ∈ S (0) , i ∈ I } that satisfy the following twoproperties:(1) [ { x ∈ G : | x | ≤ C } \ N δ/ (0) ◦ ] ⊂ [ S i ∈ I O i ].(2) If y ∈ { x ∈ G : | x | ≤ C } ∩ N ρ ( O i ) for some i ∈ I then h d, v i i ≥ ρ for every d ∈ d ( y ) with | d | = 1.Moreover, as in the proof of Theorem 2.9 of [22], for each ω ∈ Ω, we can definea sequence { ( T m ( ω ) , i m ( ω )) , m = 0 , , . . . } defined recursively as follows. Let T ( ω ) . = 0 and let i ( ω ) ∈ I be such that Z (0 , ω ) = x ∗ ∈ O i ( ω ) . Note thatbecause x ∗ ∈ ( r, R ) implies | x | > δ > δ/
2, such an i exists by property (1)above. Next, for each m = 0 , , . . . , whenever T m ( ω ) < κ δ/ ( ω ) . = inf { t ≥ Z ( t, ω ) ∈ N δ/ (0) } , define T m +1 ( ω ) . = inf { t > T m ( ω ) : Z ( t, ω ) / ∈ N ρ/ ( O i m ( ω ) ) ◦ or Z ( t, ω ) ∈ N δ/ (0) } . If T m +1 ( ω ) < T ∧ κ δ/ ( ω ), choose i m +1 ( ω ) ∈ I such that Z ( T m +1 ( ω ) , ω ) ∈O i m +1 ( ω ) . Note that such an i m +1 ( ω ) exists by property (1) above. Let N ( ω ) < ∞ be the smallest integer such that T N ( ω ) ( ω ) ≥ T ∧ κ δ/ ( ω ) and re-define T N ( ω ) ( ω ) = T ∧ κ δ/ ( ω ). (Note that N ( ω ) and { ( T m ( ω ) , i m ( ω )) , m = W. KANG AND K. RAMANAN , , . . . } are constructed in the same way as M and { T m , m ∈ N } in Theorem2.9 of [22], except that we replace ρ and δ by ρ/ δ/
2, respectively.)Since, as shown in Lemma 4.7, ( X k , Z k , Y k ) ⇒ ( X, Z, Y ) as k → ∞ and( X, Z, Y ) has continuous paths, by invoking the Skorokhod representationtheorem, we may assume without loss of generality that there exists ˜Ωwith P ( ˜Ω) = 1 such that for every ω ∈ ˜Ω, ( X k ( ω ) , Z k ( ω ) , Y k ( ω )) → ( X ( ω ) ,Z ( ω ) , Y ( ω )) uniformly on [0 , T ] as k → ∞ . Let ¯ k < ∞ be such that for all k > ¯ k , sup t ∈ [0 ,T ] | Z k ( t, ω ) − Z ( t, ω ) | < ( ρ ∧ δ ) /
4. Then Z k ( · , ω ) will stay in N ρ ( O i m ( ω ) ) during the interval [ T m ( ω ) , T m +1 ( ω )). Exactly as in the proofof Lemma 2.9 of [22] (note that the argument there only requires that φ ( t ) ∈ N ρ ( O k m − ) for t ∈ [ T m − , T m )), we can then argue that ˜ L k ( T ∧ τ kδ ( ω ) , ω ) ≤ (4 ˜ C k ( ω ) N ( ω )) /ρ for ω ∈ ˜Ω. Together with the fact that P (sup k ∈ N ˜ C k < ∞ ) =1 and N ( ω ) < ∞ for each ω ∈ Ω, this shows that P ( ˜ L k (˜ τ kδ ∧ T ) < ∞ ) = 1.Since ˜ L k (˜ θ kr,R ∧ T ) ≤ ˜ L k (˜ τ kδ ∧ T ), we then have P (sup k ∈ N ˜ L k (˜ θ kr,R ∧ T ) < ∞ ) =1. This completes the proof of property 1.Property 2 follows directly from property 1 and the fact that ε k → k → ∞ . In addition, by Theorem 2.7 it follows that Z is a semimartingaleon [0 , T V ), with Y being the bounded variation term in the decomposition.The first assertion of property 3 is thus a direct consequence of the fact that θ r,R < T V . For the second assertion of property 3, notice that with positiveprobability, the Brownian motion X = x ∗ + σ (0) B will exit G before it hitsone of the two levels H r or H R . Since Z lies in G and Z = X + Y , thisimplies that, with positive probability, Y is not identically zero in the interval[0 , θ r,R ). This, in turn, implies that L ( θ r,R ) is strictly positive with positiveprobability. Thus, the second assertion of property 3 is also established, andthe proof of the corollary is complete. (cid:3) A scaling argument.
Since the equality E [ e − L ( τ ) ] = 0 impliesthat P -a.s., L ( τ ) = ∞ , in order to prove Proposition 4.2 it suffices to estab-lish the former equality. In turn, by Remark 4.5, this equality holds if thereexists a sequence { ε k } k ∈ N such that ε k → k → ∞ , andlim inf k →∞ ε k E [ E Z ( τ εk ) [(1 − e − L ( τ ) ) I { τ <τ } ]] = ∞ . (4.21)We will show that (4.21) holds by using the strong Markov property andscaling arguments. First, we need to introduce some additional notation. Fix ε >
0. Let Λ ε denote the following union of hyperplanes:Λ ε . = [ n ∈ Z H n ε . (4.22)For x ∈ Λ ε , let N ε ( x ) denote the pair of hyperplanes in Λ ε that are adjacentto the hyperplane on which x lies. In other words, let N ε ( x ) . = H n − ε ∪ H n +1 ε , x ∈ H n ε , n ∈ Z . (4.23) EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES For future reference, note that for y ∈ R J + and x ∈ H n ε , n ∈ Z , yε ∈ N (cid:18) xε (cid:19) ⇒ y ∈ N ε ( x ) . (4.24)Let { β εn } n ∈ N be the sequence of random times defined recursively by β ε . = 0and for n ∈ N , β εn . = inf { t ≥ β εn − : Z ( t ) ∈ N ε ( Z ( β εn − )) } . (4.25)It is easy to see that { β εn } n ∈ N defines a sequence of stopping times (forcompleteness, a proof is provided in Lemma B.1).Observe that L is nondecreasing and for x ∈ H ε , P x -a.s., β εn ≤ τ for every n ∈ N . Now Z ( τ ε ) = 0 because ε >
0. Hence, for every n ∈ N , E Z ( τ ε ) [(1 − e − L ( τ ) ) I { τ <τ } ] ≥ E Z ( τ ε ) [(1 − e − L ( β εn ) ) I { τ <τ } ] . (4.26)Using the elementary identity1 − e − L ( β εn ) = 1 − e − L ( β εn − ) + e − L ( β εn − ) (1 − e − ( L ( β εn ) − L ( β εn − )) ) , conditioning on F β εn − , and invoking the strong Markov property of Z , theright-hand side of (4.26) can be expanded as E Z ( τ ε ) [(1 − e − L ( β εn ) ) I { τ <τ } ]= E Z ( τ ε ) [(1 − e − L ( β εn − ) ) I { τ <τ } ]+ E Z ( τ ε ) [ E Z ( τ ε ) [ e − L ( β εn − ) (1 − e − ( L ( β εn ) − L ( β εn − )) ) I { τ <τ } |F β εn − ]]= E Z ( τ ε ) [(1 − e − L ( β εn − ) ) I { τ <τ } ]+ E Z ( τ ε ) [ e − L ( β εn − ) E Z ( β εn − ) [(1 − e − L ( β ε ) ) I { τ <τ } ]] . Observing that the first term on the right-hand side is identical to the termon the left-hand side, except for a shift down in the index n , we can iteratethis procedure and use the relation L ( β ε ) = L (0) = 0 to conclude that forany n ∈ N , E Z ( τ ε ) [(1 − e − L ( β εn ) ) I { τ <τ } ](4.27) = n X m =1 E Z ( τ ε ) [ e − L ( β εm − ) E Z ( β εm − ) [(1 − e − L ( β ε ) ) I { τ <τ } ]] . Let { ε k } k ∈ N and { x k } k ∈ N be sequences such that x k ∈ H ε k for k ∈ N ,and ε k → k → ∞ . Since H is compact and x k /ε k ∈ H for every k ∈ N , we can assume without loss of generality (by choosing an appropriatesubsequence, if necessary) that there exists x ∗ ∈ H such that x k /ε k → x ∗ ,as k → ∞ . We now show that, when ε is replaced by ε k , each term in the W. KANG AND K. RAMANAN sum on the right-hand side of (4.27), is O ( ε k ) (as k → ∞ ), with a constantthat is independent of m . This proof relies on the estimates obtained in thenext two lemmas. In both lemmas, Z ( k ) , Y ( k ) , L ( k ) , Z k , Y k and L k denotethe processes defined at the beginning of Section 4.1.2, and for ε >
0, let β ε ( k ) , . = 0 and for n ∈ N , β ε ( k ) ,n . = inf { t ≥ β ε ( k ) ,n − : Z ( k ) ( t ) ∈ N ε ( Z ( k ) ( β ε ( k ) ,n − )) } , (4.28)and, likewise, let ζ k, . = 0 and for n ∈ N , define ζ k,n . = inf { t ≥ ζ k,n − : Z k ( t ) ∈ N ( Z k ( ζ k,n − )) } . (4.29)Note that these sequences of stopping times are defined in a manner anal-ogous to the sequence { β εn } n ∈ N defined in (4.25), except that Z is replacedby Z ( k ) and Z k , respectively. Moreover, these definitions, together with thescaling relations (4.12) and (4.24), yield the following equivalence relation ε k ζ k,n = β ε k ( k ) ,n , k, n ∈ N , (4.30) Lemma 4.9.
Suppose b ≡ . Then there exists C > such that lim inf k →∞ ε k inf x ∈ H εk E x [(1 − e − L ( β εk ) ) I { τ <τ } ] ≥ C. (4.31) Proof.
Since the law of ( Z ( k ) , Y ( k ) , L ( k ) ) under P is the same as the lawof ( Z, Y, L ) under P x k , we havelim inf k →∞ ε k E x k [(1 − e − L ( β εk ) ) I { τ <τ } ](4.32) = lim inf k →∞ ε k E [(1 − e − L ( k ) ( β εk ( k ) , ) ) I { τ k ) <τ k ) } ] , where τ ε ( k ) and τ k,ε are defined as follows: τ ε ( k ) . = inf { t ≥ Z ( k ) ( t ) ∈ H ε } , (4.33) τ k,ε . = inf { t ≥ Z k ( t ) ∈ H ε/ε k } , and recall the definition of β ε ( k ) , given in (4.28). Assume, without loss ofgenerality, that k is large enough so that ε k <
1. Then, for each x ≥ f x ( ε ) = 1 − e − εx , we inferthat for x ≥
0, there exists ε ∗ k = ε ∗ k ( x ) ∈ (0 , ε k ) such that1 − e − ε k x ε k = xe − ε ∗ k x ≥ xe − x . EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Using the above inequality along with the equalities L ( k ) ( β ε k ( k ) , ) = ε k L k ( ζ k, ), ε k τ k, = τ k ) and ε k τ k, = τ k ) , which hold due to the scaling relations (4.12)and (4.24), we have for all k sufficiently large,1 ε k E [(1 − e − L ( k ) ( β εk ( k ) , ) ) I { τ k ) <τ k ) } ] = E (cid:20)(cid:18) − e − ε k L k ( ζ k, ) ε k (cid:19) I { τ k, <τ k, } (cid:21) ≥ E [ L k ( β k, ) e − L k ( β k, ) I { τ k, <τ k, } ] . Comparing this with (4.31) and (4.32), it is clear that to prove the lemmait suffices to show that there exists ˜
C > k →∞ E [ L k ( ζ k, ) e − L k ( ζ k, ) I { τ k, <τ k, } ] ≥ ˜ C. (4.34)Let X = x ∗ + σ (0) B , where B is standard Brownian motion, and let ( Z, Y )satisfy the ESP for X , as in Lemma 4.7. Then, since x k /ε k → x ∗ ∈ H , byLemma 4.7(2) it follows that there exist r ∈ (1 / ,
1) and R ∈ (1 ,
2) such thatas k → ∞ , max i =1 ,...,J sup s ∈ [0 ,θ kr,R ] Y ki ( s ) ⇒ max i =1 ,...,J sup s ∈ [0 ,θ r,R ] Y i ( s ) , where recall the definitions of θ kr,R and θ r,R given in (4.13) and (4.17), re-spectively. By the Portmanteau theorem, this implies that P (cid:16) max i =1 ,...,J sup t ∈ [0 ,θ r,R ] Y i ( t ) > δ (cid:17) ≤ lim inf k →∞ P (cid:16) max i =1 ,...,J sup t ∈ [0 ,θ kr,R ] Y ki ( t ) > δ (cid:17) ≤ lim inf k →∞ P ( L k ( θ kr,R ) > δ ) . Together with the fact that property 3 of Corollary 4.8 implies that thereexists δ > P (cid:16) max i =1 ,...,J sup t ∈ [0 ,θ r,R ] Y i ( t ) > δ (cid:17) > δ, and the inequality ζ k, ≥ θ kr,R for all k , it follows that there exists K < ∞ such that P ( L k ( ζ k, ) > δ ) ≥ δ, k ≥ K. (4.35)Next, choose r ′ ∈ (0 , /
2) and R ′ ∈ (2 , ∞ ) and note that ζ k, ≤ θ kr ′ ,R ′ because Z k (0) ∈ H and N ( Z k (0)) = H / ∪ H . Hence, property 1 of Corollary 4.8implies that there exists c < ∞ such thatsup k ∈ N P ( L k ( θ kr ′ ,R ′ ) < c ) ≥ P (cid:16) sup k ∈ N L k ( θ kr ′ ,R ′ ) < c (cid:17) ≥ − δ . (4.36) W. KANG AND K. RAMANAN
On the other hand, since P Z ( τ εk ) ( τ ≥ τ ) = ε k by Lemma 4.4 and ε k → k → ∞ , we havelim k →∞ P ( τ k, < τ k, ) = lim k →∞ P ( τ k ) < τ k ) ) = lim k →∞ (1 − ε k ) = 1 . Hence, by choosing
K < ∞ larger if necessary, we can assume that P ( τ k, < τ k, ) ≥ − δ , k ≥ K. (4.37)Now, define the set S k . = { τ k, < τ k, , e − L k ( ζ k, ) ≥ e − c , L k ( ζ k, ) > δ } . Then (4.35), (4.36) and (4.37), together show that for k ≥ K , P ( S k ) ≥ δ/ k ≥ K , E [ L k ( ζ k, ) e − L k ( ζ k, ) I { τ k, <τ k, } ] ≥ E [ L k ( ζ k, ) e − L k ( ζ k, ) I { τ k, <τ k, } I S k ] ≥ δe − c δ , and so (4.34) holds with ˜ C = δ e − c /
2. This completes the proof of the lemma. (cid:3)
Lemma 4.10.
Suppose b ≡ . For every n ∈ N , lim k →∞ sup x ∈ H εk E x [1 − e − L ( β εkn ) ] = 0 . Proof.
Fix n ∈ N . We prove the lemma using an argument by con-tradiction. Suppose that there exists δ > { ε k } k ∈ N , such that ε k ↓ k → ∞ and for every k ∈ N ,sup x ∈ H εk E x [1 − e − L ( β εkn ) ] ≥ δ . For each k ∈ N , let x k ∈ H ε k be such that E x k [1 − e − L ( β εkn ) ] ≥ δ . (4.38)Since, the law of ( Z ( k ) , Y ( k ) , L ( k ) ) under P is the same as the law of ( Z, Y, L )under P x k , (4.38) is equivalent to the inequality E [1 − e − L ( k ) ( β εk ( k ) ,n ) ] ≥ δ . (4.39)The scaling relations in (4.12) and (4.24) show that E [1 − e − L ( k ) ( β εk ( k ) ,n ) ] = E [1 − e − ε k L k ( β k, n ) ] . (4.40) EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Moreover, since Z k (0) = x k /ε k ∈ H , it follows that h Z k ( t ) , ~ v i ∈ [2 − n , n ]for t ∈ [0 , ζ k,n ], Therefore, there exist 0 < r < − n and R > n such that ζ k,n ≤ θ kr,R , where θ kr,R is defined in (4.13). As a result, we conclude that E [1 − e − ε k L k ( ζ k,n ) ] ≤ E [1 − e − ε k L k ( θ kr,R ) ] → k → ∞ , where the last limit holds due to the weak convergence ε k L k ( θ kr,R ) ⇒ x − e − x is a bounded con-tinuous function. When combined with (4.40), this contradicts (4.38) andthus proves the lemma. (cid:3) We are now in a position to complete the proof of Proposition 4.2.
Proof of Proposition 4.2.
First, observe that by Lemma 4.9, thereexists
C >
K < ∞ such that for all k ≥ K , the relationinf x ∈ H εk E x [(1 − e − L ( β εk ) ) I { τ <τ } ] ≥ C ε k is satisfied. Together with the fact that Z ( τ ε k ) ∈ H ε k and, for any x ∈ H ε k , P x -a.s., h Z ( β ε k n − ) , ~ v i ≤ n − ε k , (4.41)implies that for all k large enough so that ε k < − ( n − ε and for m =1 , . . . , n , E Z ( τ εk ) [ e − L ( β εkm − ) E Z ( β εkm − ) [(1 − e − L ( β εk ) ) I { τ <τ } ]](4.42) ≥ C E Z ( τ εk ) [ e − L ( β εkm − ) h Z ( β ε k m − ) , ~ v i ] . When combined with (4.26) and (4.27), this shows that E [ E Z ( τ εk ) [(1 − e − L ( τ ) ) I { τ <τ } ]](4.43) ≥ C n X m =1 E Z ( τ εk ) [ e − L ( β εkm − ) h Z ( β ε k m − ) , ~ v i ] . Each summand on the right-hand side can be rewritten in the more conve-nient form E Z ( τ εk ) [ e − L ( β εkm − ) h Z ( β ε k m − ) , ~ v i ]= E Z ( τ εk ) [ h Z ( β ε k m − ) , ~ v i ] − E Z ( τ εk ) [(1 − e − L ( β εkm − ) ) h Z ( β ε k m − ) , ~ v i ] . W. KANG AND K. RAMANAN
Since b ≡
0, Lemma 4.4 and the uniform bound (4.41) show that h Z, ~ v i is amartingale on [0 , β ε k n ]. In addition, because β ε k m − ≤ β ε k n and h Z ( τ ε k ) , ~ v i = ε k ,it follows that E [ E Z ( τ εk ) [ h Z ( β ε k m − ) , ~ v i ]] = E [ ε k ] = ε k . Furthermore, by (4.41), Lemma 4.10 and the bounded convergence theorem,we have for any n ∈ N and m = 1 , . . . , n ,lim sup k →∞ ε k E [ E Z ( τ εk ) [(1 − e − L ( β εkm − ) ) h Z ( β ε k m − ) , ~ v i ]] ≤ n − lim k →∞ E h sup x ∈ H εk E x [1 − e − L ( β εkm − ) ] i = 0 . Combining the last three assertions, we see that for every n ∈ N and m =1 , . . . , n , lim inf k →∞ ε k E [ E Z ( τ εk ) [ e − L ( β εkm − ) h Z ( β ε k m − ) , ~ v i ]] = 1 . Together with (4.43), this shows that for every n ∈ N ,lim inf k →∞ ε k E [ E Z ( τ εk ) [(1 − e − L ( τ ) ) I { τ <τ } ]] ≥ nC . Taking the limit as n → ∞ , we obtain (4.21), thus completing the proof ofthe proposition. (cid:3) The general drift case.
In this section, we establish Theorem 4.1.Specifically, we use a Girsanov transformation to generalize the case of zerodrift, established in Proposition 4.2, to arbitrary Lipschitz drifts with lineargrowth, as specified in property 3 of Definition 2.5. As before, let Z be theunique strong solution to the Class A SDER, which exists by Theorem 2.7,and let τ be the first hitting time to H , as defined in (4.2). We begin witha simple lemma that shows that τ is finite with positive P probability. Lemma 4.11.
We have P ( τ < ∞ ) > . (4.44) Moreover, if inf x : h x,~ v i≤ h b ( x ) , ~ v i ≥ , then P ( τ < ∞ ) = 1 . (4.45) Proof.
Recall the definition of X and M given in (2.4) and (2.12). ByTheorem 2.7, we know that P -a.s., Z satisfies the ESP for X . Hence, byLemma 2.8 it follows that b Z = Γ ( b X ), where Γ is the 1-dimensional Sko-rokhod map and, for H = Z, M, X , we define b H . = h H, ~ v i . Let T ( t ) . = inf { s ≥ EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES h c M i s > t } . Then, due to the uniform ellipticity of a , T is strictly increasingand, since c M is a continuous martingale, c M ( T ( · )) is a 1-dimensional Brow-nian motion. In turn, this implies b Z is a one-dimensional reflected Brownianmotion with drift Z t h b ( Z ( T ( s ))) , ~ v i dT ( s ) = Z t h b ( Z ( T ( s ))) , ~ v i ~ v T a ( Z ( s )) ~ v ds. Since h b ( x ) , ~ v i /~ v T a ( x ) ~ v is continuous on G , there exists κ ∈ ( −∞ , ∞ ) suchthat h b ( x ) , ~ v i ~ v T a ( x ) ~ v > κ for all x ∈ G, h x, ~ v i ≤ . Consider the process ˜ X defined by ˜ X ( t ) . = κt + M ( T ( t )) for t ∈ [0 , ∞ ), and let˜ Z . = Γ ( ˜ X ) be a one-dimensional reflected Brownian motion with constantdrift κ . Then b X ( T ( t )) − b X ( T ( s )) ≥ ˜ X ( t ) − ˜ X ( s ) for every 0 ≤ s ≤ t , and sothe comparison principle for Γ (see, e.g., equation (4.1) in Lemma 4.1 of[20]) shows that b Z ( T ( t )) ≥ ˜ Z ( t ) for every t ∈ [0 , b τ ], where b τ . = inf { t > b Z ( T ( t )) = 1 } . Since T ( b τ ) = τ , it follows that P ( b Z ( T ( t ) ∧ τ ) ≥ ˜ Z ( t ∧ b τ ) for all t ≥
0) = 1 . Since T is strictly increasing, we have τ = ∞ if and only if b τ = ∞ . There-fore, on the set { τ = ∞} , we must have˜ Z ( t ) ≤ b Z ( T ( t )) < t ∈ [0 , ∞ ) . However, ˜ Z will hit 1 with positive P probability, and in fact will hit 1 P -a.s. if κ ≥ Z ( T ( · )). This implies both (4.44) and (4.45), and so the proof of the lemmais complete. (cid:3) Proof of Theorem 4.1.
The uniform ellipticity of a ( · ) ensures that a − ( · ) exists. Let µ . = − σ T a − b , note that µ T µ = b T ab , and define D ( t ) . = exp (cid:26)Z t µ ( Z ( s )) dB ( s ) − Z t b T ( Z ( s )) a ( Z ( s )) b ( Z ( s )) ds (cid:27) (4.46)for t ∈ [0 , ∞ ). Property 3 of Definition 2.5 guarantees that µ has at mostlinear growth, and so, as is well-known, { D ( t ) , F t } is a martingale (see, e.g.,Corollary 5.16 of [19]).Fix T < ∞ . Define a new probability measure Q on (Ω , F , {F T } ) bysetting Q ( A ) = E [ D ( T ) I A ] for A ∈ F T . W. KANG AND K. RAMANAN
Define ˜ B ( t ) . = B ( t ) + Z t σ T ( Z ( s )) a − ( Z ( s )) b ( s ) ds, t ∈ [0 , T ] . By Girsanov’s theorem (see Theorem 5.1 of [19]), under Q , { ˜ B t , F t } t ∈ [0 ,T ] is a Brownian motion and Z ( t ) = Z t σ ( Z ( s )) d ˜ B ( s ) + Y ( t ) , t ∈ [0 , T ] , where ( Z, Y ) satisfy the ESP pathwise for Z − Y . Since, under Q , Z is thesolution to a Class A SDER with no drift, by Proposition 4.2, it follows that Q ( L ( τ ) < ∞ , τ ≤ T ) = 0 . Since P ≪ Q [with d P /d Q = D − ( T ) on F T ], this implies P ( L ( τ ) < ∞ , τ ≤ T ) = 0 . Since
T < ∞ is arbitrary, sending T → ∞ (along a countable sequence), weconclude that P ( L ( τ ) < ∞ , τ < ∞ ) = 0 . However, P ( τ < ∞ ) > P ( L ( τ ) = ∞ , τ < ∞ ) >
0, which in turn implies that there exists
T < ∞ such that P ( L ( T ) = ∞ ) >
0, which proves Theorem 4.1. In addition, note that if inf x ∈ G : h x,~ v i≤ h b ( x ) , ~ v i ≥
0, then P ( τ < ∞ ) = 1 and so we in fact have P ( L ( τ ) = ∞ ) = 1. (cid:3) The semimartingale property for Z . Recall from Theorem 2.7 thatthe process Z has the decomposition Z = M + A , where M = Z · σ ( Z ( s )) dB ( s ) , A = Z · b ( Z ( s )) ds + Y, (4.47)and Y is the constraining term associated with the ESP. M is clearly a (local)martingale, but Theorem 4.1 shows that Y is not P -a.s. of finite variationon bounded intervals. However, as mentioned earlier, Theorem 4.1 does notimmediately imply that Z is not a semimartingale because we do not knowa priori that the above decomposition must be the Doob decomposition of Z if it were a semimartingale. In Proposition 4.12 below, we show that thelatter statement is indeed true, thus showing that Z is not a semimartingale. Proposition 4.12. If Z were a semimartingale, then its Doob decom-position must be Z = M + A . EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Proof.
Suppose that Z is a semimartingale, and let its (unique) Doobdecomposition take the form Z = ˜ M + ˜ A, where ˜ M is an {F t } -adapted continuous local martingale and ˜ A is an {F t } -adapted continuous, process with P -a.s. finite variation on bounded intervals.Fix R < ∞ and let θ R . = inf { t ≥ | M ( t ) | ≥ R } . For each ε >
0, define twosequences of stopping times { τ εn } n ∈ N and { ξ εn } n ∈ N as follows: ξ ε . = 0 and for n ∈ N , let τ εn . = inf { t ≥ ξ εn − : Z ( t ) ∈ H ε } ∧ θ R ,ξ εn . = inf { t ≥ τ εn : Z ( t ) ∈ H ε/ } ∧ θ R . (For notational conciseness, we have suppressed the dependence of thesestopping times on R .) By uniqueness of the Doob decomposition, clearly Z ( · ∧ ξ εn ) − Z ( · ∧ τ εn ) is an {F t } -adapted semimartingale, with Doob decom-position Z ( t ∧ ξ εn ) − Z ( t ∧ τ εn ) = ˜ M ( t ∧ ξ εn ) − ˜ M ( t ∧ τ εn ) + ˜ A ( t ∧ ξ εn ) − ˜ A ( t ∧ τ εn )On the other hand, due to the identity Z = M + A = ˜ M + ˜ A , we also have Z ( t ∧ ξ εn ) − Z ( t ∧ τ εn ) = M ( t ∧ ξ εn ) − M ( t ∧ τ εn ) + A ( t ∧ ξ εn ) − A ( t ∧ τ εn ) . Since M is an {F t } -adapted continuous (local) martingale, and M is uni-formly bounded on [0 , θ R ], the stopped processes M ( · ∧ ξ εn ) and M ( · ∧ τ εn )are {F t } -adapted continuous martingales. Hence, M ( · ∧ ξ εn ) − M ( · ∧ τ εn ) isalso an {F t } -adapted continuous martingale. Moreover, Theorem 2.7 impliesthat Y ( · ∧ ξ εn ) − Y ( · ∧ τ εn ) has P -a.s. finite variation on each bounded timeinterval. Since A = Y + R · b ( Z ( s )) ds , A ( · ∧ ξ εn ) − A ( · ∧ τ εn ) also has P -a.s.finite variation on each bounded time interval. By uniqueness of the Doobdecomposition, we conclude that for every ε > t ∈ [0 , ∞ ), M ( t ∧ ξ εn ) − M ( t ∧ τ εn ) = ˜ M ( t ∧ ξ εn ) − ˜ M ( t ∧ τ εn ) . Summing over n ∈ N on both sides of the last equation, we obtain ∞ X n =1 ( M ( t ∧ ξ εn ) − M ( t ∧ τ εn )) = ∞ X n =1 ( ˜ M ( t ∧ ξ εn ) − ˜ M ( t ∧ τ εn )) . (4.48)On the other hand, because P -a.s., M (0) = 0 and ξ εn → θ R as n → ∞ , wecan write M ( t ∧ θ R ) as a telescopic sum: M ( t ∧ θ R ) = ∞ X n =1 ( M ( t ∧ ξ εn ) − M ( t ∧ ξ εn − )) , t ∈ [0 , ∞ ) . W. KANG AND K. RAMANAN
Next, observe that M ( t ∧ θ R ) − ∞ X n =1 ( M ( t ∧ ξ εn ) − M ( t ∧ τ εn ))= ∞ X n =1 ( M ( t ∧ τ εn ) − M ( t ∧ ξ εn − ))= Z t ∞ X n =1 I ( ξ εn − ,τ εn ] ( s ) dM ( s )= Z t ∞ X n =1 I ( ξ εn − ,τ εn ] ( s ) I [0 ,ε ] ( h ~ v , Z ( s ) i ) dM ( s ) , where the last equality holds because h ~ v , Z ( s ) i ≤ ε for s ∈ ( ξ εn − , τ εn ]. Whencombined with Doob’s maximal martingale inequality, this yields E " sup s ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ( s ∧ θ R ) − ∞ X n =1 ( M ( s ∧ ξ εn ) − M ( s ∧ τ εn )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M ( t ∧ θ R ) − ∞ X n =1 ( M ( t ∧ ξ εn ) − M ( t ∧ τ εn )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 4 E "(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t ∞ X n =1 I ( ξ εn − ,τ εn ] ( s ) I [0 ,ε ] ( h ~ v , Z ( s ) i ) dM ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E (cid:20)Z t I [0 ,ε ] ( h ~ v , Z ( s ) i ) | a ( Z ( s )) | ds (cid:21) . By Assumption 3, a is bounded on the set { x : h ~ v , x i ≤ ε } . Hence, an appli-cation of the bounded convergence theorem shows thatlim ε → E (cid:20)Z t I {h ~ v ,Z ( s ) i≤ ε } | a ( Z ( s )) | ds (cid:21) = | a (0) | E (cid:20)Z t I {h ~ v ,Z ( s ) i =0 } ds (cid:21) = 0 , where the last equality is a consequence of the fact that h ~ v , Z i is a uniformlyelliptic one-dimensional reflected diffusion (see Lemma 2.8) and consequentlyspends zero Lebesgue time at the origin (see, e.g., page 90 of [15]).An exactly analogous argument, with ˜ θ R . = inf { t ≥ | ˜ M | ( t ) ≥ R } and˜ ξ εn , ˜ τ εn defined in a fashion analogous to ξ εn , τ εn , but with θ R replaced by ˜ θ R ,shows thatlim ε → E " sup s ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ M ( s ∧ ˜ θ R ) − ∞ X n =1 ( ˜ M ( s ∧ ˜ ξ εn ) − ˜ M ( s ∧ ˜ τ εn )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES ≤ lim ε → J J X i =1 E (cid:20)Z t I [0 ,ε ] ( h ~ v , Z ( s ) i ) d h ˜ M i i ( s ) (cid:21) = 4 J J X i =1 E (cid:20)Z t I { } ( h ~ v , Z ( s ) i ) d h ˜ M i i ( s ) (cid:21) = 4 J J X i =1 E (cid:20)Z t I { } ( Z i ( s )) d h ˜ M i i ( s ) (cid:21) , where the last equality uses the property that Z i ( s ) = 0 for every i = 1 , . . . , J if and only if h ~ v , Z ( s ) i = 0 (see property 2 of Definition 2.5). Due to theassumption that ˜ Z i is a semimartingale with decomposition ˜ M i + ˜ A i , theoccupation times formula for continuous semimartingales (see, e.g., Corol-lary 1.6 in Chapter VI of [25]) and the fact that the set { x : x i = 0 } has zeroLebesgue measure, we have, P -a.s., for i = 1 , . . . , J , Z t I { } ( Z i ( s )) d h ˜ M i i ( s ) = Z t I { } ( Z i ( s )) d h Z i i ( s ) = 0 . Combining the last four displays with (4.48), we conclude that M ( t ∧ θ R ) =˜ M ( t ∧ ˜ θ R ), P -a.s., for every t ≥
0. This in turn implies that θ R = ˜ θ R P -a.s.Sending R → ∞ and invoking the continuity of both M and ˜ M , we concludethat M = ˜ M P -a.s. In turn, this implies A = ˜ A , thus completing the proofof the theorem. (cid:3) The proof of Theorem 3.1 is now a simple consequence of Theorem 4.1and Proposition 4.12.
Proof of Theorem 3.1. If Z were a semimartingale under P , then byProposition 4.12, Z = M + A is the Doob decomposition for Z . In particular,this implies that P ( L ( T ) < ∞ ) = 1 for every T ∈ [0 , ∞ ), where recall that L ( T ) = Var [0 ,T ] Y . However, this contradicts the assertion of Theorem 4.1that there exists T < ∞ such that P ( L ( T ) = ∞ ) >
0. Thus, we concludethat Z is not a semimartingale. (cid:3) Remark 4.13.
It is natural to expect that similar, but somewhat moreinvolved, arguments could be used to show that the semimartingale propertyfails to hold for a more general class of reflected diffusions in the nonnega-tive orthant, in particular those that arise as approximations of generalizedprocessor sharing networks (rather than just a single station, as consideredin [23, 24]). Such diffusions would satisfy properties 1, 2 and 4 of Definition2.5 but would have more complicated V -sets (see [10] for a description ofthe ESP associated with such a network). This is a subject for future work. W. KANG AND K. RAMANAN
5. Dirichlet process characterization.
This section is devoted to the proofof Theorem 3.5. Specifically, here we only assume that (
G, d ( · )), b ( · ) and σ ( · )satisfy Assumptions 1 and 3, and let ( Z t , B t ) , (Ω , F , P ) , {F t } be a Markovprocess that is a weak solution to the associated SDER that satisfies As-sumption 2 for some constants p > , q ≥ K T < ∞ , T ∈ (0 , ∞ ). Asusual, let X be as defined in (2.4), and let Y = Z − X , so that we can write Z ( t ) = Z (0) + Z t b ( Z ( s )) ds + Z t σ ( Z ( s )) dB ( s ) + Y ( t ) , t ∈ [0 , ∞ ) . Note that R · b ( Z ( s )) ds is a process of bounded variation, and therefore ofbounded p -variation for any p > P -a.s., Y has zero p -variation.In Section 5.1, we first show that it suffices to establish a localized version(5.3) of the zero p -variation condition on Y . This is then used to proveTheorem 3.5 in Section 5.2.5.1. Localization.
Fix
T >
0, let { π n , n ≥ } be a sequence of partitionsof [0 , T ] such that ∆( π n ) → n → ∞ . As mentioned above, to proveTheorem 3.5 we need to establish the following result: X t i ∈ π n | Y ( t i ) − Y ( t i − ) | p ( P ) → π n ) → ∞ . (5.1)For each m ∈ (0 , ∞ ), let ζ m . = inf { t > | Z ( t ) | ≥ m } . (5.2)It is easy to see that P -a.s., ζ m → ∞ as m → ∞ . We now show that thelocalized version, (5.3) below, is equivalent to (5.1). Lemma 5.1.
The result (5.1) holds if and only if for each m ∈ (0 , ∞ ) , X t i ∈ π n | Y ( t i ∧ ζ m ) − Y ( t i − ∧ ζ m ) | p ( P ) → as ∆( π n ) → . (5.3) Proof.
First, assume (5.3) holds for every m ∈ (0 , ∞ ). Then, for every m ∈ (0 , ∞ ) and δ > P (cid:18) X t i ∈ π n | Y ( t i ) − Y ( t i − ) | p ≥ δ (cid:19) ≤ P (cid:18) X t i ∈ π n | Y ( t i ) − Y ( t i − ) | p ≥ δ, ζ m > T (cid:19) + P ( ζ m ≤ T ) EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES = P (cid:18) X t i ∈ π n | Y ( t i ∧ ζ m ) − Y ( t i − ∧ ζ m ) | p ≥ δ, ζ m > T (cid:19) + P ( ζ m ≤ T ) ≤ P (cid:18) X t i ∈ π n | Y ( t i ∧ ζ m ) − Y ( t i − ∧ ζ m ) | p ≥ δ (cid:19) + P ( ζ m ≤ T ) . Taking limits as ∆( π n ) →
0, the first term on the right-hand side vanishesdue to (5.3). Next, sending m → ∞ , and using the fact that ζ m → ∞ P -a.s.,the second term also vanishes, and so we obtain (5.1). This proves the “if”part of the result.In order to prove the converse result, suppose (5.1) holds. Let θ mn . =sup { t i ∈ π n : t i ≤ ζ m } , where θ mn . = T if the latter set is empty. Then X t i ∈ π n | Y ( t i ∧ ζ m ) − Y ( t i − ∧ ζ m ) | p ≤ X t i ∈ π n | Y ( t i ) − Y ( t i − ) | p + | Y ( ζ m ∧ T ) − Y ( θ mn ) | p . Taking limits as ∆( π n ) → P -a.s the last term vanishes since | ζ m ∧ T − θ mn | ≤ ∆( π n ) and Y is continuous. Therefore, (5.3) follows from (5.1). (cid:3) The decomposition result.
For each ε >
0, recursively define two se-quences of stopping times { τ εn } n ∈ N and { ξ εn } n ∈ N as follows: ξ ε . = 0 and for n ∈ N , τ εn . = inf { t ≥ ξ εn − : d ( Z ( t ) , V ) = ε } , (5.4) ξ εn . = inf { t ≥ τ εn : d ( Z ( t ) , V ) = ε/ } . For each ε >
0, we have the decomposition X t i ∈ π n | Y ( t i ) − Y ( t i − ) | p = X t i ∈ π n ∞ X k =1 | Y ( t i ) − Y ( t i − ) | p I ( τ εk ,ξ εk ) ( t i − )+ X t i ∈ π n ∞ X k =0 | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ] ( t i − ) . Therefore, for any given δ >
0, we have P (cid:18) X t i ∈ π n | Y ( t i ) − Y ( t i − ) | p > δ (cid:19) ≤ P X t i ∈ π n ∞ X k =1 | Y ( t i ) − Y ( t i − ) | p I [ τ εk ,ξ εk ) ( t i − ) > δ ! (5.5) + P X t i ∈ π n ∞ X k =0 | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ) ( t i − ) > δ ! . W. KANG AND K. RAMANAN
Under additional uniform boundedness assumptions on b and σ , the proofof (5.1) is essentially a consequence of the following two lemmas, whichprovide estimates on the two terms on the right-hand side of (5.5). Lemma 5.2.
Suppose b and σ are uniformly bounded. Then, for each ε > , lim ∆(Π n ) → P X t i ∈ π n ∞ X k =1 | Y ( t i ) − Y ( t i − ) | p I [ τ εk ,ξ εk ) ( t i − ) > δ ! = 0 . (5.6) Proof.
Fix ε > n ∈ N , and letΩ εn . = (cid:26) Z ( t ) / ∈ V , ∀ t ∈ [ k ∈ N : ξ εk ≤ T [ ξ εk , ξ εk + ∆( π n )] (cid:27) . Also, define N ε . = inf { k ≥ τ εk > T or ξ εk > T } . Observe that P -a.s., N ε < ∞ since Z has continuous sample paths and there-fore crosses the levels { z ∈ G : d ( z, V ) = ε } and { z ∈ G : d ( z, V ) = ε/ } at mosta finite number of times in the interval [0 , T ]. The continuity of Z also impliesthat for each ε > P (Ω εn ) → π n ) → . (5.7)On the set Ω εn , we have X t i ∈ π n ∞ X k =1 | Y ( t i ) − Y ( t i − ) | p I [ τ εk ,ξ εk ) ( t i − ) ≤ max t i ∈ π n | Y ( t i ) − Y ( t i − ) | p − X t i ∈ π n ∞ X k =1 L ( t i − , t i ] I [ τ εk ,ξ εk ) ( t i − )(5.8) = max t i ∈ π n | Y ( t i ) − Y ( t i − ) | p − X t i ∈ π n ∞ X k =1 L ( t i − , t i ] I [ τ εk ,ξ εk ) ( t i − ) ≤ max t i ∈ π n | Y ( t i ) − Y ( t i − ) | p − ∞ X k =1 L ( τ εk ∧ T, ( ξ εk + ∆( π n )) ∧ T ] . By definition, P -a.s. ( Z, Y ) satisfy the ESP for X . Therefore, by LemmaA.1, P -a.s., for each k ∈ N , ( Z ( τ εk ∧ T + · ) , Y ( τ εk ∧ T + · ) − Y ( τ εk ∧ T )) solvethe ESP for Z ( τ εk ∧ T ) + X ( τ εk ∧ T + · ) − X ( τ εk ∧ T ). On Ω εn , Z is away from V on [ τ εk ∧ T, ( ξ εk + ∆( π n )) ∧ T ] for each k ≥
1, and hence by Theorem 2.9 of
EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES [22] it follows that L ( τ εk ∧ T, ( ξ εk + ∆( π n )) ∧ T ] < ∞ . Together with the factthat P -a.s. N ε < ∞ , this implies that ∞ X k =1 L ( τ εk ∧ T, ( ξ εk + ∆( π n )) ∧ T ] < ∞ P -a.s. on Ω εn . On the other hand, since Y is continuous on [0 , T ] and p >
1, we havemax t i ∈ π n | Y ( t i ) − Y ( t i − ) | p − → π n ) → . Combining the above two displays with (5.7), we conclude that for every δ >
0, as ∆( π n ) → P max t i ∈ π n | Y ( t i ) − Y ( t i − ) | p − ∞ X k =1 L ( τ εk ∧ T, ( ξ εk + ∆( π n )) ∧ T ] > δ ! → . Together with (5.8), this shows that (5.6) holds and completes the proof ofthe lemma. (cid:3)
In the next lemma, q ≥ Lemma 5.3.
Suppose b and σ are uniformly bounded. Then there existsa finite constant C < ∞ such that for each ε > , lim △ ( π n ) → P X t i ∈ π n ∞ X k =0 | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ) ( t i − ) > δ ! (5.9) ≤ Cδ E "Z T ∞ X k =0 I [ ξ εk ,τ εk +1 ] ( t ) dt , if q = 2 , , if q > . Proof.
Fix ε >
0. Then by Markov’s inequality [whose application isjustified by (5.12) when q >
2, and by (5.13) when q = 2] and the monotoneconvergence theorem P X t i ∈ π n ∞ X k =0 | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ) ( t i − ) > δ ! (5.10) ≤ δ X t i ∈ π n ∞ X k =0 E [ | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ) ( t i − )] . Recall that a = σ T σ , and let ¯ C > | b | , | σ | and | a | .By Assumption 2, the definition (2.4) of X and the elementary inequality W. KANG AND K. RAMANAN | x + y | q ≤ q ( | x | q + | y | q ), there exists K T < ∞ such that for each t i ∈ π n , E [ | Y ( t i ) − Y ( t i − ) | p |F t i − ] ≤ K T E h sup u ∈ [ t i − ,t i ] | X ( u ) − X ( t i − ) | q |F t i − i ≤ q K T E (cid:20) sup u ∈ [ t i − ,t i ] (cid:12)(cid:12)(cid:12)(cid:12) Z ut i − b ( Z ( v )) dv (cid:12)(cid:12)(cid:12)(cid:12) q + sup u ∈ [ t i − ,t i ] (cid:12)(cid:12)(cid:12)(cid:12) Z ut i − σ ( Z ( v )) dB v (cid:12)(cid:12)(cid:12)(cid:12) q (cid:12)(cid:12)(cid:12) F t i − (cid:21) ≤ q K T E (cid:20) ¯ C q ( t i − t i − ) q + (cid:18) qq − (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) Z t i t i − σ ( Z ( v )) dB v (cid:12)(cid:12)(cid:12)(cid:12) q (cid:12)(cid:12)(cid:12) F t i − (cid:21) ≤ q K T ¯ C q ( t i − t i − ) q + 2 q K T (cid:18) qq − (cid:19) q ˜ K E (cid:20)(cid:18)Z t i t i − | a ( Z ( v )) | dv (cid:19) q/ (cid:12)(cid:12)(cid:12) F t i − (cid:21) , where the third inequality holds due to the uniform bound on b ( · ), theMarkov property of Z and Doob’s maximal martingale inequality, while thefourth inequality follows, with ˜ K < ∞ a universal constant, by an applicationof the martingale moment inequality, which is justified since the uniformboundedness on a ensures that the stochastic integral is a martingale.Define ˜ C . = 2 q K T [ ¯ C q ∨ ( q q ¯ C q/ ˜ K/ ( q − q )]. Using the bound on a , thelast inequality shows that for each t i ∈ π n , E [ | Y ( t i ) − Y ( t i − ) | p |F t i − ] ≤ ˜ C [( t i − t i − ) q + ( t i − t i − ) q/ ] . (5.11)We now consider two cases. If q >
2, it follows from (5.11) that, for allsufficiently large n such that ∆( π n ) < E [ | Y ( t i ) − Y ( t i − ) | p |F t i − ] ≤ C ∆( π n ) q/ − ( t i − t i − ) . Multiplying both sides of this inequality by I [ ξ εk ,τ εk +1 ) ( t i − ), which is F t i − -measurable since τ εk and ξ εk are stopping times, then taking expectations andsubsequently summing over k = 0 , , . . . , and t i ∈ π n , it follows that X t i ∈ π n ∞ X k =0 E [ | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ) ( t i − )] ≤ C ∆( π n ) q/ − T. (5.12)Since ∆( π n ) q/ − → n → ∞ , combining this with (5.10), we then obtainlim △ ( π n ) → P X t i ∈ π n ∞ X k =0 | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ) ( t i − ) > δ ! = 0 . EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES On the other hand, if q = 2, again multiplying both sides of (5.11) by I [ ξ εk ,τ εk +1 ) ( t i − ), then taking expectations, subsequently summing over k =0 , , . . . , and t i ∈ π n , and then using the monotone convergence theorem tointerchange expectation and summation, we obtain X t i ∈ π n ∞ X k =0 E [ | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ) ( t i − )] ≤ ˜ C ∆( π n ) q + E " X t i ∈ π n ( t i − t i − ) ∞ X k =0 I [ ξ εk ,τ εk +1 ) ( t i − ) < ∞ . Sending ∆( π n ) → I [ ξ εk ,τ εk +1 ) ( · ) and the definitionof the Riemann integral, we obtainlim ∆( π n ) → X t i ∈ π n ∞ X k =0 E [ | Y ( t i ) − Y ( t i − ) | p I [ ξ εk ,τ εk +1 ) ( t i − )] ≤ ˜ C E "Z T ∞ X k =0 I [ ξ εk ,τ εk +1 ) ( t ) dt . Together with (5.10), this shows that (5.9) holds with C = 2 ˜ C . (cid:3) Proof of Theorem 3.5.
Due to Lemma 5.1, using a localization ar-gument and the local boundedness of b and σ stated in Assumption 3, wecan assume without loss of generality that a, b and σ are bounded. Then,combining (5.5) with Lemmas 5.2 and 5.3, we havelim ∆( π n ) → P (cid:18) X t i ∈ π n | Y ( t i ) − Y ( t i − ) | p > δ (cid:19) ≤ Cδ E "Z T ∞ X k =0 I [ ξ εk ,τ εk +1 ] ( t ) dt , if q = 2,0 , if q > ε >
0, and so (5.1) holds for the case q >
2. If q = 2, sending ε ↓ ξ εk and τ εk , we see that the term on the right-hand side convergesto Cδ E (cid:20)Z T I V ( Z ( t )) dt (cid:21) = 0 , where the last equality follows from (2.5) and the fact that V ⊂ ∂G . Thisproves (5.1), and Theorem 3.5 then follows from the discussion at the be-ginning of Section 5. (cid:3) W. KANG AND K. RAMANAN
Proof of Corollary 3.7.
Suppose that, as in (3.3), the functions L and R on [0 , ∞ ) are given by L ( y ) = − c L y α L and R ( y ) = c R y α R forsome α L , α R , c L , c R ∈ (0 , ∞ ). As defined in Section 2.4, let ( G, d ( · )) and Γbe the associated ESP and ESM, and let Z = ( Z , Z ) be the associatedtwo-dimensional RBM: Z = Γ( B ), where B = ( B , B ) is a standard two-dimensional Brownian motion. Then Assumptions 1 and 3 are automaticallysatisfied for this family of reflected diffusions. In order to prove the corollary,it suffices to show that Assumption 2 holds. Indeed, then all the assumptionsof Theorem 3.5 are satisfied, and Corollary 3.7 follows as a consequence.We now recall the representation for Y . = Z − B that was obtained in Sec-tion 4.3 of [2]. First, note that Z is a one-dimensional RBM on [0 , ∞ ) withthe pathwise representation Z = Γ ( B ), where Γ is the one-dimensionalreflection map on [0 , ∞ ). Thus, Y = Λ ( B ), where Λ ( ψ ) . = Γ ( ψ ) − ψ isgiven explicitly byΛ ( ψ )( t ) . = sup ≤ s ≤ t [ − ψ ( s )] + , ψ ∈ C [0 , ∞ ) , t ∈ [0 , ∞ ) . (5.13)(Recall that C [0 , ∞ ) is the space of continuous functions on [0 , ∞ ), equippedwith the topology of uniform convergence on compact sets.) Since Y isa nondecreasing process, it is clearly of finite variation. Therefore, to es-tablish Assumption 2, it suffices to show that the inequality (2.7) holdswith Y replaced by Y . From Section 4.3 of [2], it follows that pathwise Z = ¯Γ ℓ,r ( B ), where ¯Γ ℓ,r is the ESM whose domain is the time-dependentinterval [ l ( · ) , r ( · )], with l ( t ) . = L ( Z ( t )) and r ( t ) . = R ( Z ( t )), for t ∈ [0 , ∞ ).A precise definition of ¯Γ ℓ,r is stated as Definition 2.2 of [2], but for thepresent purpose it suffices to note that Theorem 2.6 of [2] establishes theexplicit representation ¯Γ ℓ,r ( ψ ) = ψ − Ξ ℓ,r ( ψ ), where for ψ ∈ C [0 , ∞ ) such that ψ (0) ∈ [ ℓ (0) , r (0)], and t ∈ [0 , ∞ ),Ξ ℓ,r ( ψ )( t ) . = max (cid:16)h ∧ inf u ∈ [0 ,t ] ( ψ ( u ) − ℓ ( u )) i , (5.14) sup s ∈ [0 ,t ] h ( ψ ( s ) − r ( s )) ∧ inf u ∈ [ s,t ] ( ψ ( u ) − ℓ ( u )) i(cid:17) . Thus, we see that Y = Λ ( B , B ), where Λ is the map from C [0 , ∞ ) to C [0 , ∞ ) given by Λ : ( ψ , ψ )
7→ − Ξ L ◦ Γ ( ψ ) ,R ◦ Γ ( ψ ) ( ψ ) . From the explicit expression for Ξ ℓ,r given in (5.14), it can be easily veri-fied that the map ( ℓ, r, ψ )
7→ − Ξ ℓ,r ( ψ ) from C [0 , ∞ ) to C [0 , ∞ ) is Lipschitzcontinuous. In addition, it follows from (5.13) that the map ψ Γ ( ψ ) from C [0 , ∞ ) to itself is also Lipschitz continuous. If L and R are H¨older con-tinuous with exponent α = α L ∧ α R ∈ (0 , EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES maps ℓ = L ◦ Γ and r = R ◦ Γ are also H¨older continuous with exponent α .When combined, the above statements then imply that the map Λ is locallyH¨older continuous on C [0 , ∞ ) with exponent α , and so (2.7) holds for any p ≥ /α with, correspondingly, q = αp . On the other hand, if L and R arelocally Lipschitz continuous [i.e., if (3.3) is satisfied with α ≥ p = q = 2.Thus, the result follows in this case as well (note that, due to the localiza-tion result of Section 5.1, it suffices for the ESM to be locally Lipschitz orlocally H¨older, that is, Lipschitz continuous or H¨older continuous on pathsthat lie in a compact set on any finite time interval). Remark 5.4.
The proof above also shows that when α < Z = B + A ,where A is a process of zero p -variation for every p > /α . However, this islikely to be a sub-optimal result, since given that Z is a Dirichlet processeven when the domain is cusp-like, one would expect that Z would alsobe a Dirichlet process when the domain is flatter (corresponding to α > p -variation that vanishes for a given α , to better understand the relationshipbetween the “roughness” of the paths of Z and the curvature of the boundaryof the domain. Such questions motivate a “rough paths” analysis (see, e.g.,[21] and [16]) of reflected stochastic processes. Remark 5.5.
The above class of reflected diffusions provides one ex-ample of a situation where the ESM is locally H¨older continuous, but the(generalized) completely- S condition does not hold. However, we believe thatit should be possible to combine a localization argument of the kind used in[6] and the sufficient condition for Lipschitz continuity of the ESM obtainedin Theorem 3.3 of [22] to identify a broad class of piecewise smooth domainsand directions of reflection where the generalized completely- S conditionfails to hold, but for which the associated ESM is locally H¨older continuous.APPENDIX A: ELEMENTARY PROPERTIES OF THE ESP Lemma A.1. If ( φ, η ) is a solution to the ESP ( G, d ( · )) for ψ ∈ C G [0 , ∞ ) ,then for each ≤ s < ∞ , ( φ s , η s ) is a solution to the ESP for φ ( s )+ ψ s , where φ s ( · ) . = φ ( s + · ) , ψ s ( · ) . = ψ ( s + · ) − ψ ( s ) and η s ( · ) . = η ( s + · ) − η ( s ) . Moreover, if the ESM is well-defined and Lipschitz continuous on C G [0 , ∞ ) then for every T < ∞ , there exists ˜ K T < ∞ such that for every ≤ s < t ≤ T + s , | η ( t ) − η ( s ) | ≤ ˜ K T sup u ∈ [0 ,t − s ] | ψ ( s + u ) − ψ ( s ) | . W. KANG AND K. RAMANAN
Proof.
Fix s ∈ [0 , ∞ ) and a path ψ ∈ D G [0 , ∞ ). The first statementfollows from Lemma 2.3 of [22]. It implies that η s = ¯Γ( ψ ) − ψ , where ψ . = φ ( s ) + ψ s . On the other hand, consider the path ψ which is equal to theconstant φ ( s ) on [0 , ∞ ), that is, ψ ( u ) . = φ ( s ) for all u ∈ [0 , ∞ ). Then clearly( φ ( s ) ,
0) is the unique solution to the ESP for ψ , that is, 0 = ¯Γ( ψ )( u ) − ψ ( u ) for all u ∈ [0 , ∞ ). Using the Lipschitz continuity of the ESM, for δ ∈ [0 , T − s ] we obtain | η s ( δ ) − | ≤ sup u ∈ [0 ,δ ] | ¯Γ( ψ )( u ) − ψ ( u ) − ¯Γ( ψ )( u ) + ψ ( u ) |≤ sup u ∈ [0 ,δ ] | ¯Γ( ψ )( u ) − ¯Γ( ψ )( u ) | + sup u ∈ [0 ,δ ] | ψ ( u ) − ψ ( u ) |≤ K T sup u ∈ [0 ,δ ] | ψ s ( u ) | + sup u ∈ [0 ,δ ] | ψ s ( u ) | , where K T < ∞ is the Lipschitz constant of ¯Γ on [0 , T ]. The lemma followsby letting ˜ K T . = K T + 1 and δ = t − s . (cid:3) APPENDIX B: AUXILIARY RESULTSFor completeness, we provide the proof of the fact that the sequences oftimes defined in Section 4.1.3 are stopping times.
Lemma B.1. { β εn } n ∈ N , { β ε ( k ) ,n } n ∈ N , k ∈ N , are sequences of {F t } -stoppingtimes. Also, { β k,εn } n ∈ N , k ∈ N , are sequences of {F kt } -stopping times. Proof.
Clearly, β ε . = 0 is an {F t } -stopping time. Now, suppose β εn − isan {F t } -stopping time and note that for each ε > n ∈ N and t ∈ [0 , ∞ ), { β εn ≤ t } = [ k ∈ Z [ { β εn − ≤ t } ∩ { Z ( β εn − ) ∈ H k ε } ∩ A εk,n ( t )] , where A εk,n ( t ) . = n sup s ∈ [ β εn − ,t ] h Z ( s ) , ~ v i ≥ k +1 ε o ∪ n inf s ∈ [ β εn − ,t ] h Z ( s ) , ~ v i ≤ k − ε o . Then { β εn − ≤ t } ∈ F t because β εn − is an {F t } -stopping time. Since Z is con-tinuous we also know that { β εn − ≤ t } ∩ { Z ( β εn − ) ∈ H k ε } lies in F t . In ad-dition, the continuity of h Z, ~ v i and the fact that [2 k +1 ε, ∞ ) and ( −∞ , k − ε ]are closed show that { β εn − ≤ t } ∩ A εn,k ( t ) ∈ F t . When combined, this impliesthat { β εn ≤ t } ∈ F t or, equivalently, that β εn is an {F t } -stopping time, andthe first assertion follows by induction. The proof for the other sequences isexactly analogous. (cid:3) EFLECTED DIFFUSIONS AND DIRICHLET PROCESSES Acknowledgments.
The authors are grateful to F. Coquet for first posingthe question as to whether the generalized processor sharing reflected diffu-sion is a Dirichlet process. The second author would also like to thank S. R.S. Varadhan for his hospitality and advice during her stay at the CourantInstitute, during which part of this work was completed, and is grateful toA. S.-Sznitman for a useful discussion.REFERENCES [1]
Bertoin, J. (1989). Sur une int´egrale pour les processus `a α -variation born´ee. Ann.Probab. Burdzy, K. , Kang, W. N. and
Ramanan, K. (2009). The Skorokhod problem in atime-dependent interval.
Stochastic Process. Appl.
Burdzy, K. and
Toby, E. (1995). A Skorohod-type lemma and a decomposition ofreflected Brownian motion.
Ann. Probab. Chen, Z.-Q. (1993). On reflecting diffusion processes and Skorokhod decompositions.
Probab. Theory Related Fields Coquet, F. , Jakubowski, A. , M´emin, J. and
S lomi´nski, L. (2006). Natural decom-position of processes and weak Dirichlet processes. In
Memoriam Paul-Andr´eMeyer: S´eminaire de Probabilit´es XXXIX . Lecture Notes in Math.
Dupuis, P. and
Ishii, H. (1993). SDEs with oblique reflection on nonsmooth domains.
Ann. Probab. Dupuis, P. and
Ramanan, K. (1998). A Skorokhod problem formulation and largedeviation analysis of a processor sharing model.
Queueing Systems Theory Appl. Dupuis, P. and
Ramanan, K. (1999). Convex duality and the Skorokhod problem.I.
Probab. Theory Related Fields
Dupuis, P. and
Ramanan, K. (1999). Convex duality and the Skorokhod problem.II.
Probab. Theory Related Fields
Dupuis, P. and
Ramanan, K. (2000). A multiclass feedback queueing network witha regular Skorokhod problem.
Queueing Syst. Ethier, S. N. and
Kurtz, T. G. (1986).
Markov Processes . Wiley Series in Probabil-ity and Mathematical Statistics: Probability and Mathematical Statistics . Wiley,New York. MR838085[12]
F¨ollmer, H. (1981). Calcul d’Itˆo sans probabilit´es. In
Seminar on Probability, XV(Univ. Strasbourg, Strasbourg, 1979/1980) (French) . Lecture Notes in Math.
F¨ollmer, H. (1981). Dirichlet processes. In
Stochastic Integrals (Proc. Sympos.,Univ. Durham, Durham, 1980) . Lecture Notes in Math.
F¨ollmer, H. , Protter, P. and
Shiryayev, A. N. (1995). Quadratic covariationand an extension of Itˆo’s formula.
Bernoulli Freidlin, M. (1985).
Functional Integration and Partial Differential Equations . Annals of Mathematics Studies . Princeton Univ. Press, Princeton, NJ.MR833742[16]
Friz, P. and
Victoir, N. (2009).
Multidimensional Stochastic Processes as RoughPaths: Theory and Applications. Cambridge Studies of Advanced Mathematics .Cambridge University Press. To appear. W. KANG AND K. RAMANAN[17]
Fukushima, M. , ¯Oshima, Y. and Takeda, M. (1994).
Dirichlet Forms and Sym-metric Markov Processes . de Gruyter Studies in Mathematics . de Gruyter,Berlin. MR1303354[18] Kang, W. N. and
Ramanan, K. (2009). Stationary distributions of reflected diffu-sions in polyhedral domains. Preprint.[19]
Karatzas, I. and
Shreve, S. E. (1988).
Brownian Motion and Stochastic Calculus . Graduate Texts in Mathematics . Springer, New York. MR917065[20]
Kruk, L. , Lehoczky, J. , Ramanan, K. and
Shreve, S. (2007). An explicit formulafor the Skorokhod map on [0 , a ]. Ann. Probab. Lyons, T. J. , Caruana, M. J. and
L´evy, T. (2007). Differential equations drivenby rough paths. In
Ecole d’Et´e des probabilit´es de Saint-Flour XXXIV, 2004 ( J.Picard , ed.).
Lecture Notes in Math. . Springer, Berlin. MR2314753[22]
Ramanan, K. (2006). Reflected diffusions defined via the extended Skorokhod map.
Electron. J. Probab. (36), 934–992 (electronic). MR2261058[23] Ramanan, K. and
Reiman, M. I. (2003). Fluid and heavy traffic diffusion lim-its for a generalized processor sharing model.
Ann. Appl. Probab. Ramanan, K. and
Reiman, M. I. (2008). The heavy traffic limit of an unbalancedgeneralized processor sharing model.
Ann. Appl. Probab. Revuz, D. and
Yor, M. (1999).
Continuous Martingales and Brownian Motion , 3rded.
Grundlehren der Mathematischen Wissenschaften [Fundamental Principlesof Mathematical Sciences] . Springer, Berlin. MR1725357[26]
Rozkosz, A. (2003). On a decomposition of symmetric diffusions with reflectingboundary conditions.
Stochastic Process. Appl.
Rozkosz, A. and
S lomi´nski, L. (2000). Diffusion processes coresponding to uni-formly elliptic divergence form operators with reflecting boundary conditions.
Studia Math.
Russo, F. and
Vallois, P. (2007). Elements of stochastic calculus via regularization.In
S´eminaire de Probabilit´es XL . Lecture Notes in Math.
Skorokhod, A. V. (1961). Stochastic equations for diffusions in a bounded region,
Theor. of Prob. and Appl. Stroock, D. W. and
Varadhan, S. R. S. (1971). Diffusion processes with boundaryconditions.
Comm. Pure Appl. Math. Williams, R. J. (1985). Reflected Brownian motion in a wedge: Semimartingaleproperty.
Probab. Theory Related Fields Williams, R. J. (1995). Semimartingale reflecting Brownian motions in the orthant.In