Comparison Results for Reflected Jump-diffusions in the Orthant with Variable Reflection Directions and Stability Applications

We consider reflected jump-diffusions in the orthant n + with time-and state-dependent drift, diffusion and jump-amplitude coefficients. Directions of reflection upon hitting boundary faces are also allow to depend on time and state. Pathwise comparison results for this class of processes are provided, as well as absolute continuity properties for their associated regulator processes responsible of keeping the respective diffusions in the orthant. An important role is played by the boundary property in that regulators do not charge times spent by the reflected diffusion at the intersection of two or more boundary faces. The comparison results are then applied to provide an ergodicity condition for the state-dependent reflection directions case.


Introduction
Reflecting stochastic differential equations have found a wide variety of applications over the last decades.They play an increasingly important role in several disciplines such as economics and operations research, where they serve to model portfolio and consumption processes, option pricing and subsidy phenomena in interdependent economies, among others (see for example [10; 15; 21; 16] and references therein).They also play a central role in several fields of applications in the electrical engineering context, where they are used to model, in conjunction with weak convergence methods, from systems such as adaptive antenna arrays to stochastic communication networks (see for example [12; 13] and references therein).
In the context of stochastic networks, such models appear as heavy-traffic limits of complex network models, otherwise difficult to analyze, giving rise to corresponding approximations in terms of reflected diffusions.Reflections are taken into account via the Skorokhod map, and are due to the non-negativeness requirements for the buffer occupation processes in the network (see for example [26; 4] and references therein).Reflected jump-diffusions appear in this queueing application context when for example network stations are subject to service interruptions (see [11; 25] and references therein).In the same way, time-and state-dependence in the corresponding drift and diffusion coefficients, as well as directions of reflection, obey to the corresponding dependence in network traffic parameters, such as arrival and service rates, and station-to-station routing probabilities (see for example [13; 14] and references therein).
Due to the wide variety of applications of reflected diffusion models, as summarized above, the availability of comparison properties for such class of models have become of practical and theoretical importance.For example, the pathwise comparison between buffer occupation processes in queueing networks or cumulative subsidies transferred among entities in interdependent economies, both considered as the respective model parameters change, are of self-explicatory importance.Such pathwise results in general demand not only the comparison between the respective constrained jump-diffusion processes (constructed in terms of the Skorokhod map, as detailed further in the next section), but also the corresponding establishment of absolute continuity properties for the associated regulator processes (constraining the diffusions to the domain of interest, as for example the non-negative orthant).In this context, comparison results for reflected jump-diffusions in the orthant have traditionally been restricted to the case of normal reflection directions upon hitting boundary faces (see [23]).However a comparison result in the oblique reflection directions case is available in the context of the deterministic Skorokhod problem in the orthant (see [21]), the framework in which it is established makes its application to the diffusion setting only possible when the stochastic integral term driving the respective diffusion is state-independent.We will show in the paper that this requirement is essentially a boundary condition at the faces of the orthant, being able to include then a controlled state-dependency in that term over the interior of the orthant covering for example the important case of a product-form setting (see [17; 19]) in queueing network applications.A crucial role is played here by an appropriate boundary behavior characterization, and in particular by the boundary property in that regulators do not charge times spent by the reflected diffusion at the intersection of two or more boundary faces (see [18; 19]).
A direct application of the comparison results established in the paper is to provide a simple ergodicity condition for continuous reflected diffusions in the orthant with state-dependent reflection directions.Stability conditions available in the literature have been established for the constant reflection directions case (see [1]) and critically depend on the Lipschitz continuity of the corre-sponding Skorokhod map, which is not ensured in the state-dependent case.
The organization of the paper is as follows.In Section 2 we specify the setting to be considered throughout as well as some related notation.In Section 3 we establish the main comparison results of the paper.In Section 4 we apply those results to derive an ergodicity condition for the continuous case.Finally, in Section 5 we provide the corresponding proofs of the main results in Section 3.

Setting and Notation
Let n ≥ 2 be an integer1 and (Ω, , ( t ) t≥0 , ) be a stochastic basis satisfying the usual hypotheses, i.e., 0 contains all the -null sets of and the filtration ( t ) t≥0 is right continuous.Throughout the paper we consider pair of processes (X , Z) satisfying reflecting stochastic differential equations (RSDEs) in the orthant n with state-and timedependent reflection directions upon hitting boundary faces, of the form where2 • X = (X t ) t≥0 = (X 1 t , . . ., X n t ) t≥0 is an ( t ) t≥0 -adapted n + -valued c àd l à g3 semi-martingale.
with each E i being an arbitrary Polish space (e.g., with the usual Euclidian distance), δ = (δ i j ) n i, j=1 : with each N i (ds, d r i ) being an independent Poisson random measure over [0, ∞) × E i with intensity measure λ i ds ⊗ G i (d r i ), where λ i ≥ 0 and G i is a probability distribution on (E i , (E i )).Note since the Poisson measures are independent, any number of them do not "jump" simultaneously at any time (a.s.), and therefore to ensure that a jump does not take X outside the orthant we ask for each δ i j to be such that δ i j (t, x, r j ) ≥ −x i for all t ≥ 0, x = (x l ) n l=1 ∈ n + and r j ∈ E j .
• Z = (Z t ) t≥0 = (Z 1 t , . . ., Z n t ) t≥0 is a continuous ( t ) t≥0 -adapted n + -valued process with each Z i being non-decreasing and such that Z i 0 = 0 and As usual, and since the j-th column of R gives the reflection direction upon hitting the interior of the j-th face F j .= {x = (x l ) n l=1 ∈ n + : x j = 0}, we refer to R as the reflection matrix 5 and assume, without loss of generality, the normalization R ii (•, •) ≡ 1 for each i.
We now identify different sets of conditions that will be alternatively considered as assumptions in the results given in the paper.Condition 2.1.b and R are continuous.Also, b, γ, δ and R satisfy a linear growth condition and are Lipschitz continuous, both in the state variable x ∈ n + and uniformly in all the other corresponding variables, i.e., there exists K ∈ (0, ∞) such that, for all x, y ∈ n with the usual Euclidian and Frobenius norms in n and n×n , respectively.Moreover 6 , sup is continuous, for each i.
Condition 2.2.For each i, j, i = j, there exists m i j ≥ 0 such that and, with m ii .= 0 for each i and M .= (m i j ) n i, j=1 ∈ n×n , we have σ(M ) < 1 with σ(M ) denoting the spectral radius of M .Conditions 2.1 and 2.2 in particular guarantee that, given an 0 -measurable initial condition X 0 ∈ n + and b, γ, W , δ, N and R as above, the pair (X , Z) is the pathwise unique strong solution of RSDE (2.1).Indeed, this follows from [6] in the continuous case (i.e., in absence of jumps), jumps being taken then into account via standard piecewise construction arguments (see for example [13, Section 3.7, pp.134]).Whenever that is the case, we write omitting γ, W , δ and N in the notation since in all comparison results between pairs they will be the same for both 7 . 5One in general may assume each R i j as given on F j and extended to the whole orthant by setting Conditions 2.1 and 2.2 in fact guarantee the well posedness of the Skorokhod problem (SP) in the orthant with state-dependent reflection directions (see [24] for a detailed treatment of the (modified) SP in the orthant with state-dependent reflection directions), and one may then write and 8 , i.e., with (X , Z) solving the SP for U and R on an a. s. pathwise basis 9 .D([0, ∞) : G) denotes here, as usual, the space of c àd l àg functions mapping [0, ∞) into G ⊆ n .Condition 2.2 is standard in the context of RSDEs and SPs in non-smooth domains (see for example [6; 5; 21; 8]) as it guarantees that R(t, x) is completely-S, for each x ∈ n + and t ≥ 0, in that for each principal sub-matrix R(t, x) extracted from R(t, x) there always exists a non-negative vector v, of the corresponding proper dimension, such that R(t, x)v > 0. Each such R(t, x) is also non-singular.This structure, along with Condition 2.1 above and Condition 2.5 below, in particular guarantee that (see [18; 19]) for each for each i ∈ , with 1 {•} denoting as usual the corresponding indicator function.
As it will be seen in Section 5, the boundary property in equation (2.3) plays an important role in the establishment of the results in the paper.
Finally, we identify the following conditions on the coefficients of δ and γ and on the diffusion matrix a.
Condition 2.4.There exist measurable functions {η i j } n i, j=1 , mapping 2 + into , such that for each i, j Condition 2.5.The diffusion matrix a is positive definite for each x ∈ n + and t ≥ 0, i.e., i, j As mentioned before, since the Poisson jump measures {N i (ds, d r i )} n i=1 are independent, any number of them do not "jump" simultaneously at any time (a.s.).Therefore, Condition 2.3 is useful when comparing processes X and X , coming from as it ensures that for each i (X − X ) i s (X − X ) i s− ≥ 0 at any jump instant (i.e., that jumps cannot alter the order between corresponding components).In this same comparison context, Condition 2.4 will guarantee for the semi-martingale local time at level zero associated with each difference (X − X ) i to be null.It also makes each γ i j independent of the position over the corresponding i-th face F i , and hence each diagonal diffusion coefficient a ii too.Condition 2.4 is required for comparisons even in the case of normally reflected jump-diffusions in the orthant (see [23]), and it encompasses the important case of a product-form setting (see [17; 19]) in queueing network applications.
In addition to play a role in the establishment of relationship (2.3) above, Condition 2.5 also guarantees for reflections from the boundary to be instantaneous (see [18; 19]).

Main Results: Comparison Properties
We establish in this section the main results of the paper, regarding comparison properties between different pairs (X , Z) = RSDE(X 0 , b, R) and ( X , Z) = RSDE( X 0 , b, R).
The following additional notation will be used from now on in the paper, with I denoting the identity matrix in n×n .

Notation. Consider
with relationship (2.2) in Condition 2.1 holding, respectively, and define the mapping ψ = (ψ 1 , . . ., ψ n ) : [0, ∞) × n + → n (resp., ψ) by setting each ψ i (resp., ψ i ) as the net-drift including jumps in the i-th coordinate, i.e., and similarly for ψ with b in place of b.We write when each Radon-Nikodym derivative above is less than or equal to 1 a. s.
In order not to opaque the continuity in the exposition of the results, we postpone their corresponding proofs to Section 5. We begin with the case when R(•, •) ≡ I.
both under Conditions 2.1 to 2.4, respectively.Assume that Then we have The next result considers the case when R(•, •) ≡ I.In that case, a full comparison between the tuples (X , Z) and ( X , Z) is possible when R(•, •) is constant, a partial comparison being possible otherwise.As it will become clear in Section 5, the main difficulty in getting a full comparison for non-constant R(•, •) relies on the fact that the usual alternative characterization of , as being the (unique) pathwise-minimum non-decreasing continuous process satisfying (see for example [25; 26]) is in general not guaranteed for non-constant R(•, •) (see for example [21]).
both under Conditions 2.1 to 2.4, respectively.Assume that Then we have X t ≤ X t , t ≥ 0 = 1.
The following corollary is a direct consequence of Theorems 3.1 and 3.3.
both under Conditions 2.1 to 2.4, respectively.Assume that Then we have X t ≤ X t , t ≥ 0 = 1.
Finally, the next result shows that a full comparison is possible for non-constant oblique reflection directions, provided reflections upon hitting each boundary tend to bring the remaining coordinates closer to the origin.In order to compare X and X we generally require in this case an ordering at the boundaries on the drift vectors b and b, as it will become clear in Section 5, ensuring in turn a pathwise comparison between the increments of the processes Z and Z.It is in this context where the boundary property in equation (2.3) plays an important role.The result is the following.
both under Conditions 2.1 to 2.5, respectively.Assume that with R(•, •) ≤ I, and that further each pair of drift coefficients b i and b i satisfies the same ordering as ψ i and ψ i but only on the corresponding i-th face F i , i.e., that 11b i (t, x) ≤ b i (t, y), x, y ∈ F i , x ≤ y, t ≥ 0, for each i.Then we have

Stability Applications: An Ergodic Result
In this section we use the comparison results of Section 3 to establish an ergodicity criterium related to solutions of RSDEs as in equation (2.1), the main idea being to exploit those comparison properties, and the stability results in [1] for the constant reflection directions case, to provide a simple but useful ergodicity condition in the context of state-dependent directions of reflection.
For simplicity we consider the continuous case, i.e., in absence of jumps (δ(•, •, •) ≡ 0).Therefore, we consider RSDEs of the form where of course coefficients are assumed to be time-independent.Note when b and γ are constant, X in equation (4.1) reduces to a Semi-martinagle Reflecting Brownian Motion (SRBM) in the orthant with state-dependent reflection directions (see [24]).
We denote by X x the process X in equation (4.1) when starting from X 0 = x ∈ n + , whose existence and uniqueness is ensured under Conditions 2.1 and 2.2, and introduce accordingly and as usual the family of distributions { x : x ∈ n + } on the path space of continuous functions mapping [0, ∞) into n + , denoting as x expectation with respect to x .Also, for each x ∈ n + and t ≥ 0, we write P x (t, •) for the law of X x t in n + , i.e., abusing notation in the last equality 12 .(Note P x (0, •) = δ x (•), unit mass at x ∈ n + .)We will consider in this section a boundedness condition on b and γ, and a uniform non-degeneracy condition on the corresponding diffusion matrix a (= γγ T ).Moreover, the diffusion matrix a is uniformly elliptic, i.e., there exists ς ∈ (0, ∞) such that i, j a i j (x)ξ i ξ j ≥ ς ξ 2 for all x ∈ n + and ξ = (ξ l ) n l=1 ∈ n .
Conditions 2.1 and 2.2, along with the boundedness requirement in Condition 4.1, guarantee the family {X x : x ∈ n + } satisfies the Feller property 13 .Indeed, from [21, Proposition 3.2, pp.515] and using the Burkholder-Davis-Gundy inequalities [9, Theorem 26.12, pp.524], it is easy to see that there exists a constant 0 < C < ∞ such that, for all t ≥ 0 and all x, y ∈ n + , sup 0≤s≤t Also, from [6, Theorem 5.1, pp.572] we know for each 0 < T < ∞ there exists a constant 0 < C T < ∞ such that, for all 0 Gronwall's lemma then shows that sup 0≤s≤t and therefore, on invoking again (4.2), and the arbitrariness of 0 < T < ∞, we conclude for each t > 0 and all x, y ∈ n + .The Feller property of the family {X x : x ∈ n + } then follows from standard arguments (see for example [15], proof of Lemma 8.1.4,pp.133-134.).
On the other hand, the uniform ellipticity requirement in Condition 4.1, along with Conditions 2.1 and 2.2, in particular guarantee irreducibility in that the probability measure P x (t, •) and Lebesgue measure in n + are, for each x ∈ n + and t > 0, mutually absolutely continuous.We are now in position to state and prove the advertised ergodic result.
Then there exists a unique invariant distribution for the family {X x : x ∈ n + }, in that there exists a unique probability measure π on for all 15 f ∈ b ( n + ).Moreover, for each initial distribution π 0 on ( n + , ( n + )) we have for each A ∈ ( n + ), and therefore in particular we have that the measure n weakly to π as t increases to infinity, i.e., lim Before giving the proof of the theorem we make the following remark.[1], which reads in this case sup

Remark 4.3. Note the key ergodicity condition in Theorem 4.2, equation (4.3), does not coincide in the case of constant directions of reflection, say R(•) ≡ R, with the corresponding one in
, with the supremun being pulled out in equation (4.3).This is a consequence of supporting our result in an auxiliary comparison, as it will be done in the proof below.However, as mentioned at the beginning of the section, equation ( 4.3) provides a useful ergodicity condition for the case of applications with state-dependent reflection directions.
Proof.Consider for each x ∈ n + the auxiliary RSDE given by whose well-posedness is straightforwardly ensured, and write P x (t, •), t ≥ 0, for the corresponding laws.From the theorem's assumptions, the Lipschitz continuity of the Skorokhod map in this constant reflection directions case (see [26]) and [1, Theorem 2.16, pp.8], we conclude the tightness, for each M ∈ (0, ∞), of the family of probability measures which in turn ensures, since by Theorem 3.5 in Section 3 we have X x ≤ X x a. s., the corresponding tightness of the family The above tightness, along with the theorem's assumptions and the Feller structure of the family {X x : x ∈ n + }, then give the corresponding existence of an invariant distribution π (see [7]).The convergence for each initial distribution π 0 and A ∈ ( n + ), and therefore the uniqueness of π, follow then in turn from [13, Theorems 1.1.and 1.3, pp.142 and 144, resp.].That in particular the measure converges weakly to π as t increases to infinity, is a direct consequence of Portmanteau's theorem (see for example [3]).

Proofs of the Main Results
In this section we give the proofs of the main results of the paper in Section 3. To that aim we first establish the following lemma.
both under Conditions 2.1 to 2.4, respectively, and with X 0 ≤ X 0 a. s.Then for each constant N ≥ 0, index i and t ≥ 0 we have both 16 and where φ i t .= X i t − X i t , t ≥ 0, and where the ( t ) t≥0 -stopping times T N and T are defined as 17 with T ⋄ any ( t ) t≥0 -stopping time 18 , Proof.Consider an index i, fixed throughout the proof.Note since φ i is clearly a semi-martingale with the (jointly) right-continuous in y (∈ ) and continuous in t (∈ [0, ∞)) version of the local time associated to φ i , with y indicating the corresponding level, exists (see [20]).We denote it by L φ i = (L φ i (t, y)) t≥0, y∈ .In order to prove the lemma, we first verify that Indeed, denote by ([φ i , φ i ] c t ) t≥0 the path-by-path continuous part of the quadratic variation process = 0, and note that and that, from Conditions 2.1 and 2.4, Define the mapping ρ : (0, ∞) → (0, ∞) by setting Let ε > 0 and note that Now, with we have a.s.
On the other hand, by the occupation times formula of semi-martingale local times (see [20]) The claim then follows by invoking the sample path continuity of L φ i (•, 0).We now turn into proving the lemma.From Meyer-Itô's formula (see [20]) we have 19 But, from Condition 2.3 we have 0<s≤•∧T N ∧T and, since also (φ i 0 ) + = 0 and L φ i (•, 0) ≡ 0 a. s., it is therefore easy to see that for each t ≥ 0, where we have replaced X s− by X s since X is c àd l àg and Z is continuous, and similarly for X and Z.Thus, by writing s and using that and that from the definition of Z and Z we have and also, on we obtain equation (5.1).In the same way, by writing we find, by similar arguments than before, equation (5.2).The lemma is then proved.
Having established the lemma, we now give the proofs of the results in Section 3. Set * .= {1, 2, . ..} and, for each k ∈ * , index i and x = (x l ) n l=1 ∈ n + , In addition, , for each k ∈ * and index i as well.
= ∞, we have21 for each i.By letting N ր ∞, Fatou's lemma then shows that t∧T * a. s., t ≥ 0, and therefore right-continuity gives In particular, right-continuity of X (k) and X (k) and continuity of ψ (k) and ψ (k) then give T * > 0 a. s., and therefore the consideration, on {T * < ∞}, of (X (k) shows, by a direct argument by contradiction based on equation (5.7), that T * = ∞ a. s.Thus, we conclude that X (k) t ≤ X ), t ≥ s, ( i.e., with ), t ≥ s, (5.10) i.e., with the corresponding replacement of Z by Z in the right-hand-side of equation (5.9).(Note since R(•, •) ≡ I, it has been therefore correspondingly omitted in equations (5.9) and (5.10).)Then, since X (k) • ≤ X (k) • a. s., by using Meyer-Itô's formula as in the proof of Lemma 5.1, it is easy to see that, for each i, [(X ) + ] ≤ 0, t ≥ s.
with b (k) and γ (k) defined the same as before and, for each k ∈ * , x ∈ n + and t ≥ 0, with k 0 sufficiently large so as for R (k) to satisfy Condition 2.2, inherited from22 R. Also as before, we associate ψ (k) (≡ ψ) to (5.12) and ψ (k) to (5.13), and fix in what follows a k ∈ * .From Lemma 5.1, equation 23 (5.2), applied to (5.12) and (5.13) above 24 , we obtain for each i and t ≥ 0 where we set T ⋄ in the corresponding definition of T as and X (k),l t = 0 f or some l .
We claim that T ⋄ > 0 a. s.Indeed, set  Right-continuity of X (k) and X (k) , continuity of ψ, ψ (k) , R (k) , R, b and b (k) , and the facts that a. s.
Thus, again by right-continuity of X (k) and X (k) , and the fact that X (k),l > 0 for l / ∈ 0 , we conclude that for t in some vicinity [0, ξ), with ξ > 0 depending on the paths,

Finally, we write
d Z << d Z when the random measure each Z i • induces in [0, ∞) is absolutely continuous with respect to the corresponding one associated to Z i • , denote by d Z i d Z i the related Radon-Nikodym derivatives, and write d Z d Z ≤ 1 a. s.

Theorem 4 . 2 .
Consider the family {X x : x ∈ n + } as above, under Conditions 2.1, 2.2, 2.4 and 4.1.Set b i .= sup x∈ n + b i (x) for each i, and assume that there exists R = ( R i j ) n i, j=1 ∈ n×n with R ≤ I, R ii = 1 for each i and σ( R − I) < 1, and such that R(•) ≤ R and 14 , with b .

t
) f or some l ∈ 0 .