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Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the Octant

We study the variational problem that arises from consideration of large deviations for semimartingale reflected Brownian motion (SRBM) in the positive octant. Due to the difficulty of the general problem, we consider the case in which the SRBM has rotationally symmetric parameters. In this case, we are able to obtain conditions under which the optimal solutions to the variational problem are paths that are gradual (moving through faces of strictly increasing dimension) or that spiral around the boundary of the octant. Furthermore, these results allow us to provide an example for which it can be verified that a spiral path is optimal. For rotationally symmetric SRBM's, our results facilitate the simplification of computational methods for determining optimal solutions to variational problems and give insight into large deviations behavior of these processes.


Introduction and Main Results
In this paper, we analyze the variational problem associated with the large deviations principle for semimartingale reflected Brownian motion (SRBM) in the octant. The SRBM processes of interest arise from heavy traffic limits of queueing network processes. Understanding the tail asymptotics of the SRBM's can aid in computing their stationary distribution, which in turn gives insight into the behavior of the pre-limit queueing processes.
The typical analysis of large deviations for any process can often be divided into two steps: (1) proving a large deviations principle (LDP) and (2) analyzing the resulting variational problem. For particularly complex variational problems, one might further subdivide step (2) into: (2a) characterizing optimal paths and (2b) optimal path computations. Our primary interest in this paper is in step (2a), especially for SRBM's in R 3 + . To gain understanding of the difficulties of the overall investigation, we briefly review some previous results. First, with respect to step (1), an LDP for SRBM's in R d + has only been established for special cases. For a general dimension d, Majewski examined the special cases of SRBM's arising from feed-forward queueing networks [14] and SRBM's whose reflection matrix is an M -matrix (the so-called Harrison-Reiman case) [15]. Dupuis and Ramanan [7] obtained an LDP for a generalization of the Harrison-Reiman case.
It should be noted that these results still leave the LDP for d = 2 unresolved for some parameter cases (see [10] for a summary). More recently, Dai and Miyazawa [5] obtained exact asymptotics for SRBM in two dimensions using moment generating functions and techniques from complex analysis. However, the results are limited to asymptotic behavior along a ray of the quadrant. In a related follow-up paper, Dai and Miyazawa [6] provide new insights into the results of Avram et al. [1] and derive exact asymptotics for the boundary measures of SRBM in two dimensions.
The tasks outlined for step (2) are best explained by examining the case in two dimensions. In this setting, Avram et al. [1] and Harrison and Hasenbein [10] gave a complete analytical solution to the variational problem for any SRBM of interest (e.g., those possessing a stationary distribution). The analysis was carried out in a few steps. First, three general properties of optimal paths were established: convexity, scaling, and merging. Second, these properties were used to conclude that only three types of optimal paths are possible.
Finally, these path properties allow the development of a complete algebraic description of the optimal paths in two dimensions. Unfortunately, the situation in three dimensions is considerably more difficult. While the properties of convexity, scaling, and merging still apply, they are nowhere near sufficient to characterize the possible optimal paths. In order to attack the higher dimensional problem, we examine a special set of SRBM cases and develop new techniques for restricting the types of paths which must be examined.
More specifically, in this paper we investigate the variational problem associated with SRBM in the positive orthant for d = 3 in the case in which the SRBM has either rotationally symmetric or mirror symmetric data (the latter is a special case of the former). Note that neither symmetry case we analyze coincides with the much studied case of skew-symmetric SRBM's, which have tractable product form stationary distributions.
Our first contribution is to use the Bramson, Dai, and Harrison [3] stability results to derive an appealingly simple set of stability conditions for rotationally symmetric SRBM (see Theorem 9 in Section 5). However, the main contribution of the paper is to clarify the nature of optimal paths in three-dimensional variational problems and to provide new techniques to achieve this analysis. To best elucidate our contribution, we present our main result now: Theorem 1. Consider a rotationally symmetric variational problem, as given in Definition 7, arising from SRBM in R 3 + . Suppose Γ = I and θ < 0. Under Condition 1 in Section 8, there always exists an optimal path which is either (a) a gradual path (a path which moves through faces of strictly increasing dimension) or (b) a classic spiral path. This follows from Theorem 1 and the calculations in Section 10. To the best of our knowledge, this is the first time a spiral path has been shown to be optimal for this type of variational problem. Condition 1 is somewhat complicated and we discuss it in detail later. The most important restriction it imposes is that the SRBM must have reflection vectors which point "outward." We believe that this condition can be relaxed. Note that the negativity condition on the drift θ is not restrictive, since it is equivalent in our case to requiring stability of the associated SRBM. An implication of these results is that they provide the basis for tractable numerical methods for computing optimal paths in three dimensions. Complementary to our work is a recent paper by El Kharroubi et al. [8], which provides some algebraic results for paths in three dimensions. However, most of the results in [8] require a priori elimination of certain optimal path types, which we are able to provide. Important related computational methodology appeared in Majewski [15]. If one fixes the maximum number of segments in the search for an optimal path, Majewski's branch-and-bound algorithm can efficiently produce the desired path. Finally, Farlow [9] also investigates variational problems arising from rotationally symmetric SRBM. In particular, she provides evidence that spiral paths cannot be optimal unless r 1 > 1 or r 2 > 1. Furthermore, her arguments, when combined with our results, indicate that there are always spiral optimal paths in such cases.
Our hope is that the path properties we establish can be extended beyond the symmetry cases. However, it should be noted that the general case in three dimensions is already known to be fairly complex and in fact our main results do not hold for all parameter cases in three dimensions. In [7], the authors show that an optimal path to a point in the interior of the octant may have up to five linear pieces, implying that a simple characterization of paths in the general d = 3 case is non-trivial. This five-piece path is depicted in Nonetheless, we believe that deriving new properties of optimal paths for special cases is a necessary building block for solving other variational problems, and provides for a better understanding of the tail asymptotics for SRBM. This paper is structured as follows. In Section 2 we introduce the variational problem (VP) analyzed throughout the paper. This problem arises from studying large deviations of SRBM in the orthant, concepts that are described in Section 3. Section 4 introduces the symmetric cases of the SRBM and VP which are of interest to us. In Section 5, we use the framework of Bramson et al. [3] to derive the stability conditions of symmetric SRBM. The next three sections characterize the nature of (piecewise linear) optimal paths with a finite number of segments. Section 9 is devoted to paths with an infinite number of pieces and it is demonstrated that only classic spiral paths can be optimal. Finally, in Section 10 we provide an example of a spiral path that is indeed optimal.

The Variational Problem
In this section, we define the variational problem of interest in this paper. First, we give notation and definitions which follow as closely as possible to those given in Avram et al. [1].
Let d ≥ 1 be an integer and θ a constant vector in R d . Also, Γ is a d × d symmetric and strictly positive definite matrix, and R is a d × d matrix. The triple (θ, Γ, R) provides the data to variational problems and, as described later, associated reflected Brownian motion processes. Throughout the paper, all vector inequalities should be interpreted componentwise and all vectors are assumed to be column vectors. Finally, In order to more easily define the VP, we first introduce the Skorohod problem associated with the is called a path and is often denoted by x(·). We now define the Skorohod problem associated with a reflection matrix R.
Definition 1 (The Skorohod Problem). Let x be a path. An R-regulation of x is a pair of paths (z, y) ∈ y(·) is non-decreasing, y(0) = 0, When the R-regulation (y, z) of x is unique for each x ∈ C([0, ∞), R d ), the mapping . When the triple (x, y, z) is used, it is implicitly assumed that (y, z) is an R-regulation of x.
An important issue when defining the Skorohod problem is whether a solution exists for any given path x. If the reflection matrix R is completely-S, as defined below, then indeed there is a solution for every x with x(0) ≥ 0 (see Bernard and El Kharroubi [2]).
The class of P-matrices, defined below, also plays an important role in the development of SRBM theory and associated variational problems. Definition 3. A d × d matrix R is said to be a P-matrix if all of its principal minors are positive.
In addition to the issue of existence of solutions to the Skorohod problem, there is also the matter of the uniqueness of the solution, for a given path x. It is useful when defining the VP to have a notational convention which applies when solutions are not unique. To this end, we assume that if the Skorohod problem is non-unique, then ψ(x) represents a set of paths (solutions) corresponding to x. Furthermore, the expression We now define the variational problem studied in this paper.
where H d is the space of all absolutely continuous functions x(·) : [0, ∞) → R d which have square integrable derivatives on bounded intervals and have x(0) = 0.
If a given triple of paths (x, y, z) is such that the triple satisfies the Skorohod problem, z(T ) = v for some T ≥ 0, and then we will call (x, y, z) an optimal triple, for VP (5), with optimal value I(v). The function z is called an optimal path if it is the last member of an optimal triple. Such a triple (x, y, z) is also sometimes referred to as a solution to the VP (5). T is called the optimal time for such a solution.

Semi-martingale Reflected Brownian Motion
We now define the semi-martingale reflected Brownian motion (SRBM) on the positive orthant associated with the data (θ, Γ, R). Let B denotes the σ-algebra of Borel subsets of R d is called a filtered space if Ω is a set, F is a σ-field of subsets of Ω, and {F t } ≡ {F t , t ≥ 0} is an increasing family of sub-σ-fields of F, i.e., a filtration. If, in addition, P is a probability measure on (Ω, F), then (Ω, F, {F t }, P) is called a filtered probability space.
Definition 6 (SRBM). Given a probability measure ν on (R d + , B), a semi-martingale reflecting Brownian motion associated with the data (θ, Γ, R, ν) is an {F t }-adapted, d-dimensional process Z defined on some filtered probability space (Ω, F, {F t }, P ν ) such that (i) P ν -a.s., Z has continuous paths and Z(t) ∈ R d + for all t ≥ 0, Brownian motion with drift vector θ, covariance matrix Γ and X(0) has distribution ν, Recall that the parameters θ, Γ and R are called the drift vector, covariance matrix and reflection matrix of the SRBM, respectively. The results of Reiman and Williams [16] and Taylor and Williams [17] imply that the necessary and sufficient conditions for the existence the SRBM is that R is completely-S.
The measure ν on (R d + , B) is a stationary distribution for an SRBM Z if for each A ∈ B, When ν is a stationary distribution, the process Z is stationary under the probability measure P ν . In our discussion below, we are concerned only with the (unique) stationary distribution for the SRBM with data (θ, R, Γ) and therefore we drop the ν notation.

Large Deviations
The motivation for studying the variational problem introduced in Section 2 comes from the theory of large deviations. For SRBM's in R d + , we have the following statement of the large deviations principle, which has only been established for some special cases, as noted in the introduction.

Conjecture 3 (General Large Deviations Principle).
Consider an SRBM Z with data (θ, Γ, R). Suppose that R is a completely-S matrix and that there exists a probability measure P π under which Z is stationary.

Then for every measurable
and lim inf where A c and A o are respectively the closure and interior of A.
The specific connection between this LDP statement and the VP is that the function I(·) appearing above is the same function which appears in Definition 5.

Symmetric SRBM
In this paper, we study solutions to the variational problem associated with the LDP introduced in the previous section. The three-dimensional case is considerably more difficult than the two-dimensional case and thus we confine our study to SRBM's with some symmetry in the data. The special cases we study are called rotationally symmetric and mirror symmetric and are defined below. These symmetries, while restrictive, provide a considerable simplification of the analysis.
Some statements in the rest of the paper relate only to R and in this case we call R alone rotationally symmetric if and only if R has the form given in the definition above. We employ a similar convention for Γ. Notice that the rotationally symmetric Γ matrix also appears to be mirror symmetric. Since covariance matrices are by definition symmetric (in the standard matrix algebra sense), there is no sensible way to define a rotationally symmetric Γ which is not also mirror symmetric. For a rotationally symmetric Γ we have the following result, proved in the Appendix, which will be used in demonstrating optimal path properties.
Lemma 4. If Γ is rotationally symmetric, then Some readers may also be familiar with the skew-symmetry condition (see [11,12]) which is where D = diag(R) and Λ = diag(Γ). This condition is necessary and sufficient for the stationary density function of the SRBM to admit a separable, exponential form. Our notions of symmetry do not coincide in any meaningful way with the notion of skew-symmetry. It can be checked that rotationally symmetric SRBM data is also skew-symmetric if and only if r 1 + r 2 = 2ρ.
In subsequent sections, we provide results for both SRBM and the associated variational problems. Thus, for an SRBM with rotationally symmetric data we use the abbreviation RS-SRBM. Similarly, for an SRBM with mirror symmetric data we use MS-SRBM. The associated variational problems take the same data and when stating results for VPs we use the abbreviations RSVP and MSVP.

SRBM Stability Conditions
For SRBM in three dimensions Bramson et al. [3] obtained results which, in addition to previous results, give a complete characterization of existence and stability of SRBM. This characterization is summarized in Figure 2. The results of this section specialize their results for RS-SRBM. First, however, we need to define a few terms appearing in the figure.
We define the solutions to the linear complementarity problem (LCP) in dimension d as follows (see [4] for background). Vectors u, v ∈ R d comprise a solution to the LCP if Using the terminology in [3] a solution (u, v) to the LCP is called stable if v = 0 and the solution is called divergent otherwise. The existence or non-existence of a solution to the LCP must be checked in the bottom decision point in Figure 2.
Bramson et al. [3] also define various subsets of the data pairs (θ, R) which relate to the third line of decision points in Figure 2. To avoid overlapping notation we specialize their definitions now to the RS-SRBM case.
First, for a pair (θ, R), C 1 = {(θ 0 , r 1 , r 2 ) : θ 0 < 0, r 1 < 1, r 2 > 1} and Next, for a data pair (θ, R) ∈ C 1 , For general SRBM data β(θ, R) depends on θ but in the rotationally symmetric case the dependence disappears. These definitions are related to spiral piecewise linear solutions of the Skorohod problem. Section 3 in [3] should be consulted for an in-depth explanation of how these expressions arise.
We are now prepared to present a series of lemmas which lead to the main stability result of this section.
The first lemma probably appears in a textbook somewhere, but we state it here and prove it in the Appendix for completeness. For later use, note that the lemma implies that a + b + c = 0. All the results stated in this section apply to the three-dimensional case.
Lemma 5. If a reflection matrix R is non-singular and rotationally symmetric then its inverse must be of the form and (a + b + c)(1 + r 1 + r 2 ) = 1.
Since existence of an SRBM requires that R is completely-S, our first task is derive a simple condition to insure that this characterization holds.
Lemma 6. Suppose a matrix R is rotationally symmetric. Then R being completely-S is equivalent to Proof. First suppose R is completely-S and 1+r 1 +r 2 ≤ 0. We derive a contradiction. Since R is completely-S, there exists a vector u ≡ (u 1 , u 2 , u 3 ) > 0 such that Ru > 0. Summing the equations in Ru > 0 we have But if 1 + r 1 + r 2 ≤ 0, then there is no u > 0 satisfying (10), which is a contradiction. So, we have proven that completely-S implies 1 + r 1 + r 2 > 0 which is one direction of the equivalence. Now assume that 1 + r 1 + r 2 > 0. Then note that u = (1, 1, 1) satisfies Ru > 0. This implies that R is an S-matrix. We must now verify that the two-by-two principal submatrices are also S-matrices. These submatrices take the form We prove the result for S 1 only, since the argument for S 2 is completely analogous. Now, for S 1 to be an S-matrix there must exist a (u 1 , u 2 ) > 0 such that, First suppose r 1 , r 2 > 0. Then any (u 1 , u 2 ) > 0 satisfies (11) and (12). In the cases r 1 > 0, r 2 < 0 and r 1 < 0, r 2 > 0 it is clear that (11) and (12) for some positive u. In the last case, r 1 , r 2 < 0, it can be checked that (11) and (12) holding for some u is equivalent to r 1 r 2 < 1. This condition holds when r 1 and r 2 are negative because 1 + r 1 + r 2 > 0 insures r 1 , r 2 > −1. Finally, if r 1 = 0 and/or r 2 = 0 then, for example u = (1, 1) satisfies (11) and (12).
Having now dispatched with the first line in Figure 2, we present a lemma relating to the second line.
Lemma 7. Let the data (θ, Γ, R) be rotationally symmetric and let R be non-singular and completely-S.
Now we proceed to results involving the last two lines in Figure 2.
Proof. Our proof relies on the results in [3] as depicted in Figure 2. The assumptions of the lemma place us in the lower half of the figure. To further partition the proof, we divide the (r 1 , r 2 ) plane into four regions: We do not include the completely-S condition that r 1 + r 2 > −1 in this partitioning scheme because the condition is not employed directly in the algebraic arguments below. Under our assumption R −1 θ < 0, the definitions of C 1 and C 2 coincide with the Bramson et al. [3] definitions given in Section 5. Furthermore, The one point of the plane not included in the union of these regions is r 1 = r 2 = 1. The matrix R is singular in this case, which violates the assumption of the lemma. Notice that the line r 1 + r 2 = 2 bisects C 1 C 2 and that C 3 lies entirely above this line and C 4 entirely below this line.
So, for all of Case 1, r 1 + r 2 < 2 is necessary and sufficient for stability.
In this case stability of the SRBM is equivalent to the nonexistence of a divergent solution to the LCP.
Case 2a: We examine the case (r 1 , r 2 ) ∈ C 3 . Since r 1 +r 2 > 2 for all points in C 3 we need to show instability for data in this region. In the LCP, let u = (−θ 0 , 0, 0) and v = −θ 0 (0, r 1 − 1, r 2 − 1) . This is clearly a divergent solution to the LCP for any (r 1 , r 2 ) ∈ C 3 . Therefore the corresponding SRBM is never stable in this case.
Case 2b: We examine the case (r 1 , r 2 ) ∈ C 4 . Since r 1 + r 2 < 2 for all points in C 3 we need to show stability for data in this region.
Consider a solution u, v to the LCP. We show that there exist no divergent solutions for this case. If u > 0, then v = 0 and the solution is stable. If u = 0, then we must have v = θ < 0 which is not an allowable LCP solution. So if there exists a divergent solution, either one term or two terms in u = (u 1 , u 2 , u 3 ) is positive.
Suppose one term is positive and it is u 1 , which implies v 1 = 0. Then we have v 1 = θ 0 + u 1 = 0 yielding u 1 = −θ 0 . Therefore, in this case the unique a solution to the LCP must be of the form , which violates the non-negativity condition of v. Exactly analogous arguments show that neither u 2 nor u 3 can be the positive term. So, there exist no LCP solutions in which only one term in u is positive.
Next, suppose that two terms of u are positive. Again, without loss of generality, suppose u 1 > 0, u 2 > 0, and u 3 = 0, which implies v 1 = v 2 = 0. From the LCP equations we have u 1 + r 2 u 2 + θ 0 = 0 and u 2 + r 1 u 1 + θ 0 = 0. Solving these yields: If r 1 = 1, then r 2 < 1 and these equations force u 2 = 0, which contradicts our assumption on u. Similarly, we cannot have r 2 = 1. So, we now assume that both r 1 and r 2 are strictly less than 1. In this case, the solution given in (14) implies that both u 1 and u 2 are positive. Once again using the LCP equations we obtain: Note that 1 − r 1 r 2 > 0, −θ 0 > 0, and Therefore, v 3 < 0 which violates the non-negativity condition in the LCP.
We have now demonstrated that no divergent LCP solutions exist when (r 1 , r 2 ) ∈ C 4 . So, any SRBM with data in this region is stable.
Lemmas 6 through 8 then imply simple existence and stability conditions for RS-SRBM in three dimensions.
Theorem 9. Consider an SRBM in three dimensions with rotationally symmetric data (θ, Γ, R). The necessary and sufficient conditions for existence and stability of such an SRBM are θ 0 < 0 and −1 < The results in [8], which we shall make use of in later sections, require that R be a P-matrix. The next result shows that this is not a restriction in the rotationally symmetric case, given that we only study stable SRBM's. The proof is given in the Appendix.
Theorem 10. Suppose R is rotationally symmetric and r 1 +r 2 < 2. Then R being completely-S is equivalent to R being a P-matrix.

Optimal Path Preliminaries
In this section we establish some notation and review the properties pertaining to optimal paths. As much as possible we use notation which is consistent with either [1] or [8]. Many of our results rely on algebraic expressions given in [8]. First, we give expressions for the optimal costs of various types of paths.
Set I = {1, 2, . . . , d} and for K ⊂ I define the face associated with K as follows: When d = 3, if |K| = 2 then F K is an axis and if |K| = 1 then F K is a 2-dimensional face.
We define the following costs, inspired by the notation in [8].
2. (One-piece Reflected Path Cost) Let J and K be subsets of I with K ⊂ J and 0 < |K| ≤ |J| ≤ d. For points v ∈ F K and w ∈ F J , set 4. (Two-Piece Path via an Axis). Let d = 3 and K ⊂ I with |K| = 2. For i ∈ K and v ∈ F i , set Each cost above corresponds to the cost for an optimal path of a certain type, as denoted in each item in the list. These costs, and the associated paths, are the building blocks for constructing paths which are optimal in the original variational problem.
In [1], the authors established various properties of optimal paths that hold in all dimensions. The first three items in Lemma 11 restate those properties. We add a fourth property for RSVPs and MSVPs in three dimensions. These properties are frequently used to establish results in subsequent sections. The first three parts are proved in [1], the fourth result is evident using symmetry. In the Appendix, we state and prove a simple extension to the convexity property (see Lemma 24). The convexity property below implies that direct paths within a face should have constant velocity and direction. The extension shows that this also is true for reflected paths.
1. (Optimality of Linear Paths via Convexity) Let g be a convex function on R d , and x ∈ H d .
This implies that a (one-piece) linear path minimizes the unconstrained variational problem.

(Scaling) Consider a variational problem with
is the optimal triple for v andx,ŷ,ẑ is the optimal triple for kv, thenx(t) = kx(t/k),

(Symmetric Terminal Points) Consider an RSVP. If
. Furthermore, each optimal path to one of these points is rotationally symmetric translation of an optimal path to one of the other points. For an MSVP case,

Eliminating Bad Faces in RSVPs
The next result is one of the key results in the paper, since it allows us to eliminate entire categories of paths by eliminating paths whose penultimate pivot point is on a "bad face." Below, the distance between a face and a point is the standard Euclidean distance from a point to the associated face.
Theorem 12 (Bad Faces). Consider a variational problem with terminal point v ∈ int(R 3 + ) and consider an optimal triple (x, y, z) to v. Let w be the last point of z which is not in int(R 3 + ). Then (a) If the variational problem is an RSVP, then there exists an optimal path for which w is in one of the two nearest faces to v.
(b) If the variational problem is an MSVP, then there exists an optimal path for which w is in the nearest face to v. , a). Furthermore, we assume that we cannot have both a = 0 and b = 0.
First, we want to compare the optimal cost from u i to v for various values of i. Of course, by convexity (Lemma 11, part 1), the optimal path from any u i to v must be a linear path. Now, recall that Hence this portion of the cost is independent of i. So it is sufficient to analyze where the inequality is due to our assumption on v and Lemma 4. This of course implies that v, Now, let w be the last point which is not in the interior of the octant, for an optimal path with terminal point v. By convexity, the segment wv must be linear. Suppose then that w is in F 3 . If v 3 = v 2 then F 3 is the furthest face from v. Thus w must be of the form of u 3 and accordingly we take w = u 3 = (a, b, 0). Now, consider the point u 1 = (0, a, b). Note that the optimal cost from the origin to u 3 and the optimal cost from the origin to u 1 must be equal due to rotational symmetry. By the merging and convexity properties of Lemma 11, to establish (a) it suffices to show that the optimal cost from u 1 to v using a direct path is less than or equal the cost from u 3 to v via a direct path. This result was already demonstrated, as seen in (16).
At this point some discussion may be needed to see that part (a) of the theorem has been proved. Suppose We now address part (b) of the theorem. First, it can be checked that Using the calculations from the RSVP case, we havẽ By mirror symmetry, the optimal cost from the origin to u 3 and the optimal cost from the origin to u 4 must be equal.
The remainder of the proof is similar to the part (a) argument. Again, let w be the last point which is not in the interior of the octant, for an optimal path with terminal point v. Suppose first that w is in F 3 .
Unless v 1 = v 2 = v 3 (in which case the result holds trivially), then F 3 is one of the two furthest faces from v.
Recall that an MSVP is also an RSVP, so we can apply part (a) of the theorem to conclude that there must exist an optimal path to v with w ∈ F 2 . Without loss of generality, assume then that w = u 2 = (b, 0, a). By mirror symmetry, the optimal cost from the origin to u 2 and the optimal cost from the origin to u 4 must be equal. However, by (17)

Remark 1.
When v 1 = 0 and v 2 > 0, v is in the interior of F 1 . The first part of Theorem 12 holds in the following sense: For an RSVP, there exists an optimal path whose last segment does not emanate from the interior of F 3 . Furthermore, the last segment can not originate from F 2,3 although it may originate from F 1,3 .

Remark 2.
When v 1 = v 2 = 0 and v 3 > 0, v is on the axis F 1,2 . Again, the first part of Theorem 12 holds. In particular, there exists an optimal path whose last segment does not emanate from the interior of the farthest face, which is F 3 in this case.
Finally, we believe that the results of this section can be generalized to higher dimensional RSVPs and MSVPs with minor modifications to the proofs.

Further Optimal Path Characterizations
Our eventual goal is to show that optimal paths in three dimensions can be of only two types: gradual paths and classic spirals. Demonstrating this requires the establishment of a number of properties for paths with a finite or infinite number of linear segments. The results in this section are related to paths with a finite number of segments, although some of these properties are used later on to establish characterizations for paths with an infinite number of segments.
In various proofs in this section it is useful to consider paths (and the corresponding costs), which are feasible, but not necessarily optimal. Therefore, we introduce the following definition.
Note that the pair (y, z) uniquely defines x.
The next several results provide detailed characterizations of optimal paths. Unfortunately, the overall connection will not be apparent until we bring them together to prove the main results.
Then the path from v 1 to v 4 consisting of the following linear segments is strictly suboptimal: a direct segment from v 1 to v 2 , a reflected segment from v 2 to v 3 , a direct segment from v 3 to v 4 .
Consider a new path from v 1 to v 4 as follows: a direct segment from v 1 toṽ 2 , a reflected segment fromṽ 2 toṽ 3 , a direct segment fromṽ 3 to v 4 .
We show that the new path has a strictly lower cost than the original path. Notice that v 3 − v 2 =ṽ 3 −ṽ 2 so it suffices to compareĨ 0 (v 1 , v 2 ) +Ĩ 0 (v 3 , v 4 ) withĨ 0 (v 1 ,ṽ 2 ) +Ĩ 0 (ṽ 3 , v 4 ). By definitioñ It is easy to check that Thus, the newly constructed path has a strictly lower cost. If v 2 1 < v 3 1 then re-defineṽ 2 = (0, 0, v 2 . The proof of the corresponding result for this case is analogous to the first case.
This result is the key to showing that "exotic" paths which seem intuitively "bad" are indeed suboptimal.
In particular, it shows that paths which switch back and forth between two faces are not cost effective.
Analogous arguments show that the lemma holds for any pair of two-dimensional faces.
The next result is important in establishing the optimality of gradual paths. In this and later proofs, we use the standard notation e 3 = (0, 0, 1).
. Then the path from v 1 to e 3 consisting of the following linear segments is strictly suboptimal: a direct segment from v 1 to v 2 and a reflected segment from v 2 to e 3 .
Using the paths defined above we have: where the inequality is due to the fact that (x 1 , y 1 ,z 1 ) need not be optimal.
Recall thatż 1 T 1 = e 3 − v 2 . Finally, we expand the direct path costs similarly and compute: Since r 2 ≥ 0, y 1 2 ≥ 0, and v 2 1 can assumed to be positive, the last term is strictly positive, establishing the result. (If v 2 1 = 0 then the theorem holds trivially by convexity.) In general, we apply this result under the condition that r 1 , r 2 ≥ 0. By symmetry it is easily seen that the result applies to rotational variations of the paths involved in the result.
Our study of optimal path characterizations rests crucially on comparing the paths depicted in Figure 3.
For paths with both a finite number of segments and an infinite number of segments, we wish to establish that the blue path is "cheaper" than the red path. In most instances it is difficult to establish this as a general property, so we provide a sufficient condition for this blue-path-red-path condition to hold. For specific numerical instances of an RSVP, this condition is easily verified. Furthermore, a combination of numerical and analytical arguments can be used to show that the condition holds in general on R f , defined below.

Condition 1.
For an RSVP with reflection matrix R, (r 1 , r 2 ) ∈ R f and This condition is required to prove the next two results. Lemma 16 is proved in the Appendix.
Then if Condition 1 holds, the least cost two-piece path from the origin to v and from v to v is not an optimal path to v .
Proof. When v 2 ≥ v 3 , the two-piece path through v is not optimal due to Theorem 12. Next, invoking the scaling property of Lemma 11, we assume without loss of generality thatv 3 = 1.

Case 1.
Consider then the case where v 2 < v 3 , v 1 < 1, r 2 ≥ r 1 and setv = (0, v 1 , 1). We claim that the optimal two-piece path throughv is strictly better than the optimal two-piece path through v. First, Lemma 15 givesĨ 2 (v) ≥Ĩ 1 (v). In other words, the first segment of the path throughv has a lower (or equal) cost than the first segment through v .
To compare the second segments, note that Thus,Ĩ 0 (v, v ) >Ĩ 0 (v, v ) and the result is established for this case.
Case 2. Suppose next that v 2 < v 3 and v 1 ≥ 1 (with no restriction on r 1 and r 2 ). Letṽ = (0, 1, v 1 ) and consider the two-piece path to v viaṽ. Again, we show that the optimal two-piece path throughṽ is strictly better than the optimal two-piece path through v. By rotational symmetryĨ 2 (v) =Ĩ 1 (ṽ). On the As in Case 1, this impliesĨ which establishes the result for this case.
Case 3. The remaining case is when v 2 < v 3 , r 2 < r 1 , and v 1 < 1. Once again we find an alternate two-piece path to v which has a lower cost. In this case, consider the two-piece path to v via e 3 . If Condition 1 holds, then Lemma 16 indicates that for some k ∈ [0, 1], . This establishes the result for this case.
Note that Condition 1 is only needed to establish the third case. It may be possible to replace this condition by a simpler expression for special cases.
Next, we are now able to establish the result that there always exist gradual optimal paths to points on the boundary of the octant. It is important to note that the class of gradual paths do not include paths which traverse an axis and then cross the interior to a point on a two-dimensional face. Next, consider an endpoint of type F . There are three possibilities for the previous endpoint: • The endpoint is on an axis A (either one of the two axes adjoining this face, or the remaining axis) • The endpoint is on the same face SF .
• The point is on a difference face DF .
Similarly, for an endpoint of type A, there are two possibilities for the previous endpoint: • The point is on an axis A.
• The point is on the same face SF (i.e., a face adjoining the axis).
Notice that for a point of type A the previous point cannot be on the face not adjoining the axis as a consequence of the Bad Faces Theorem. With this notation, we can categorize a piecewise linear path by a finite series whose elements are in the set {SF, DF, A, O}.
For a series corresponding to a finite-piece optimal path, we infer the following rules: 1. By the convexity property of optimal paths, none of the following pairs can appear in the series: (DF, DF ), (SF, SF ), (SF, A). Without loss of generality let the terminal point be on face F 1 , denote it v 1 = (0, v 1 2 , v 1 3 ), and assume that 0 < v 1 2 ≤ v 1 3 . The endpoint before v 1 has to be a point v 2 on F 2 . So v 2 = (v 2 1 , 0, v 2 3 ) which is the DF in the series. We must have v 2 1 ≤ v 2 3 for the path to be optimal, by the assumed type of path and the Bad Faces Theorem. The next endpoint v 3 cannot be on axis F 1,3 again by the Bad Faces Theorem. Furthermore, it cannot be on axis F 1,2 due to the arguments in the proof of Lemma 17. Hence, v 3 must be in F 2,3 . Now, if the path just described is optimal, this implies that the optimal path to v 2 is from the origin to v 3 then to v 2 . Then by the scaling property, the optimal path to an arbitrary point (u 1 , 0, u 3 ) on face F 2 is of the form (F, A, O) if u 1 ≤ u 3 . By symmetry, the optimal path to an arbitrary point (0, u 2 , u 3 ) on F 1 is also of the form (F, A, O) if u 2 ≤ u 3 . However, v 1 is indeed of this form, which means we can replace the proposed optimal path of the form (F, DF, A, O) by a gradual path of the form (F, A, O).

If
Next consider the case (F, SF, DF, A, O). If the terminal point v is in say F 1 then so is the endpoint v 1 immediately preceding this point. This implies that the optimal path to v 1 is of the form (F, DF, A, O). As argued above, we can eliminate this form. All the remaining cases can be eliminated by analogous arguments that reduce the end of the series to the (F, DF, A, O) case. We conclude that any optimal path to a point

Results for Exotic Paths
This entire section is devoted to arguing that certain "exotic spirals" cannot be optimal. Depicted in the left-hand side of Figure 4 is what we call a classic spiral, a path type which has appeared in other contexts in the literature on fluid models. In Section 10 we show that such a path can indeed be the optimal solution to the variational problem we consider in the paper. For now, however, we wish to show that other types of paths, exotic spirals, cannot be better than a classic spiral. One important type of exotic spiral appears in the right-hand side of Figure 4. Eliminating this type of path from consideration is the focus of much of the next results.
Lemma 20. For any two-dimensional face, define the bisecting ray to be the ray which forms an angle of π/4 radians with the adjacent axes and whose endpoint is the origin. Consider an optimal path which the following characteristics: it contains a line segment that intersects the bisecting ray in F 1 and it contains another line segment that intersects the bisecting ray in F 2 . Then there exists an optimal path with the following characteristics: (a) The path has two segments (as defined above) which form the same angle with the bisecting rays (i.e., if the segments are rotated to lie in the same face, then they must be parallel).
(b) The path contains another segment in F 3 which intersects the bisecting ray at the same angle.
(c) The path contains an infinite number of segments.
Proof. Let v 0 v 1 ∈ F 1 and v 2 v 3 ∈ F 2 be the segments which intersect the respective bisecting rays and suppose the points are traversed by the optimal path in the order v 0 , v 1 , v 2 , and v 3 .
We prove part (a) by contradiction, assuming the segments do not form the same angles with the bisecting rays. Consider then the portion of the path from the origin to v 1 . This portion can be rotated and scaled to create an optimal path which passes through the point, call it w, where v 2 v 3 intersects the bisecting ray in F 2 . Thus, we can create a path from the origin through w to v 3 which is optimal yet has a "kink" at w.
However, this path cannot be optimal due to reflected convexity (Lemma 24). This establishes (a).
It is clear by rotation, scaling, and merging that one can form an optimal path to v 3 which intersects the bisecting ray in F 3 . Repeating the process results in the formation of an optimal path with an infinite number of such segments. This establishes (b) and (c).
We present one more lemma before giving the main result which eliminates exotic spirals from consideration. kv 1 , 0), and u = (0, ku 1 , ku 2 ). Then, for all k ∈ (0, 1) the three-piece path passing through for all x in a non-negative neighborhood of 0. For all such x, In the same non-negative neighborhood, define To establish the result, it is enough to show that f (0) < 0.
We have that So straightforward calculations yield Recalling that γ 0 > γ 1 by Lemma 4, we obtain establishing the desired result.
Theorem 22 (Elmination of Exotic Spirals). Consider an RSVP with Γ = I, θ 0 < 0, r 1 , r 2 ≥ 0 and suppose Condition 1 holds. For any optimal path to a point on the axis with a countably infinite number of segments, there exists another path, with lower or equal cost, which is of the form of the classic spiral (i.e., of the form A, A, . . . , O)).
Proof. We begin with a general principle that holds for paths with an infinite number of segments. Consider an optimal path characterization which begins with an A and contains another A at position n, elsewhere in the sequence. Then there exists an optimal path whose entire characterization must be identical to the (original) characterization starting at position n. This principle follows directly by scaling, rotation, and merging and it can be thought of as enforcing a "self-similarity" property of optimal paths. As an example, We now proceed to eliminate these two types of paths.
Part (i). We consider first a path of type (A, SDF, O). Without loss of generality, assume that the terminal point (the first A in the sequence) is e 3 and the next pivot point is v 1 ∈ int(F 2 ). Now the farthest face from v 1 can be either F 1 or F 3 (since the point is on F 2 , this cannot be the farthest face). Suppose v 1 is strictly closer to F 1 . Then by the Bad Faces Theorem, the next endpoint must be in F 1 . However, such a path can be eliminated from consideration by Lemma 14. Therefore we assume that v 1 is closer to F 3 than F 1 and the next endpoint in the path is in F 3 again by the Bad Faces Theorem. (If v 1 is equidistant to F 1 and F 3 , then the segments to F 1 and F 3 have the same costs and we choose the segment going to F 3 ).
The next point, v 3 is also in the interior of F 3 due to our assumption on the path type. Using arguments from the proof of the Bad Faces Theorem, it can be shown that v 2 must be closer to F 2 than F 1 . Next, if v 3 is closer to F 2 than F 1 then the resulting path is of "switchback" form. Such a path is suboptimal by Lemma 13. Thus, v 3 must be closer to F 1 than F 2 . Furthermore, by Lemma 20, e 3 v 1 is rotationally parallel to v 2 v 3 . Hence, after e 3 the faces containing the endpoints are in this order: F 2 , F 3 , F 3 , F 1 , F 1 , F 2 , F 2 , . . . .
By the usual arguments using rotation and scaling, all the DF segments are rotationally parallel. So, for this path type, the path is an "exotic spiral" as depicted in the right half of Figure 4. However, Lemma 21 implies that such an exotic spiral is suboptimal.
Part (ii). We now turn our attention to the other general type of exotic spiral, one with the character- segments repeated i times. By symmetry and scaling arguments, we can assume that each of these sequences is identical. Suppose that the end point v 0 is on axis F 1,2 and the SF is from v 1 on F 1 . By Lemma 14, since v 2 is on a different face than v 1 , v 1 must be closer to F 1,3 than to F 1,2 and v 2 must be in F 3 . The next point, v 3 , is either on axis F 2,3 or in F 3 . However, it must be closer to F 2,3 than to This implies however, that the total cost of any such path is infinite. Hence, the path cannot be optimal.

An Optimal Spiral Path
In Example 2 of Section 6 of El Kharoubi et al. [8], it is shown that a spiral path has a lower cost than a two-piece gradual path, for the corresponding RSVP. Here we give a related example, and using the results of previous sections, show that a spiral path is indeed optimal.
Let θ = (−1, −1, −1) , Γ = I and To establish that a spiral path is optimal, we need to undertake three steps. First, we check that Condition 1 of Section 8 holds. If so, then we know that only gradual paths or spiral paths are optimal. Second, using results from [8] we check the reflectivity characteristics of optimal paths traversing an axis. Third, to travel to a point, say e 3 , on an axis, we verify that it is less costly to traverse one of the other axes and then cross a two-dimensional face. If this is the case, then one can construct a spiral path to e 3 that is cheaper than the gradual path to e 3 (which simply travels along the axis).
The next proposition simplifies the process of checking Condition 1 and may be useful in producing other examples.
Proof. Clearly (r 1 , r 2 ) ∈ R f under the assumptions given. Recall that Condition 1 is given by For r 2 = 0 the condition reduces to (1 + r 2 1 ) 2 ≥ 0 which clearly holds for all real r 1 .
Finally, we check the spiral condition for the point e 3 . In particular either must hold. We verify the first inequality. For the parameters of our example we havẽ I 1,2 (e 3 ) = A 1,2 θ A 1,2 e 3 − A 1,2 θ, A 1,2 e 3 ≈ 0.4211.
Next, let u = (0.5, 0, 0). Theñ Therefore, for the given RSVP, the optimal path to any point on the boundary of the octant is a classic spiral optimal path.
We can more precisely characterize this optimal spiral path. In particular, the last piece connects the points ke 1 and e 3 , where k, 0 < k < 1, is the shrink factor. The optimal value of k can be calculated by defining the corresponding spiral cost as a function of k: Applying the data for this problem and setting f (k) = 0 results in the quadratic root-finding problem 1228123k 2 − 3690960k + 1626300 = 0. The appropriate root is k * ≈ 0.5363. From this we can calculate the cost of the optimal spiral asĨ

Conclusions
As mentioned in the introduction, this paper only provides a piece of the puzzle of the variational problem related to large deviations for SRBM in the orthant. We have only addressed problems with symmetric data, and even then some of the results have further restrictions on the parameters. Although we do not provide an analytical proof, in Liang [13] convincing numerical evidence indicates that Condition 1 holds whenever the SRBM is stable and r 1 , r 2 ≥ 0. Other results require that the covariance matrix Γ is the identity. The covariance condition seems more difficult to remove, since this matrix could affect the types of paths that are optimal for a given SRBM. As noted earlier, it is already know that our results cannot be generalized to arbitrary (stable) SRBM data. The example in [7] which was discussed in the introduction is particularly troubling because the reflection matrix in that example is partially rotationally symmetric (two of the three reflection vectors exhibit rotationally symmetry).
In addition, even once one has a handle on the types of paths which are optimal, computation and comparison of these path costs appear still requires considerable effort. This indicates that fully solving large deviations problems in high dimensions is likely to remain a challenge and future work in dimensions four and higher will require examination of specialized cases and increasing mathematical creativity.

Appendix
Proof of Lemma 4. It can be checked that if Γ −1 has the form of the lemma, then for a rotationally symmetric matrix Γ the nine equations in ΓΓ −1 = I are consistent. Since matrix inverses are unique, it follows immediately that Γ −1 can be written as stated in the lemma. Next, in order for ΓΓ −1 = I to hold we must have γ 0 + 2ργ 1 = 1 and Solving these equations yields which implies Note that γ 0 = 0 is not possible. To prove γ 0 > γ 1 we examine four cases.
Proof of Lemma 5. It can be checked that if R −1 has the form of the lemma, then for a rotationally symmetric matrix R the nine equations in RR −1 = I are consistent. Since matrix inverses are unique, it follows immediately that R −1 can be written as stated in the lemma. Also, from RR −1 = I, we have a + r 2 c + r 1 b = 1 b + r 2 a + r 1 c = 0 c + r 2 b + r 1 a = 0.
Summing these equations gives the claimed equality.
Proof of Theorem 10. Lemma 6 states that R being completely-S is equivalent to By definition, the conditions are necessary and sufficient for R to be a P-matrix. We prove that (21) is equivalent to (22)  1. Suppose that either r 1 = 0 or r 2 = 0. We prove the case in which r 2 = 0, the other case is analogous.
The arithmetic-geometric mean (AGM) inequality gives and thus (22) always holds. Invoking the AGM inequality again yields Recall that equality in the AGM inequality holds iff the three terms are equal. Equality of the terms in this case implies r 1 = r 2 = 1 which is not possible due to r 1 + r 2 < 2. Therefore (23) automatically holds.
Note that when r 1 < 1 this term is positive as can be seen from the right-hand side above. When r 1 ≥ 1 we have r 1 − r 2 > r 1 − 1 ≥ 0 and r 1 − r 2 ≥ 1 − r 2 > 0 and again the right-sand side above is clearly positive. This fact implies that (21) and (23) are equivalent, as argued in Case 3.
Lemma 24 (Reflected Convexity). Consider a section of a feasible triple (x, y, z) in which the path z consists of segments v 1 v 2 and v 2 v 3 , with v 1 , v 2 , v 3 ∈ F j where j ∈ {1, 2, 3}. Suppose that v 1 v 2 is a reflected segment and v 2 v 3 is direct. Then there exists a linear reflected path from v 1 to v 3 whose cost is no greater than the original path.
Proof of Lemma 16. Part (a). First, without loss of generality we set v = (v 1 , 0, 1). We prove that if Note that A 31 + A 13 ≤ 0 and (Aθ) 1 ≤ 0 in R f . So, to prove the non-negativity of the derivative, it is sufficient to show that (Aθ) 2 1 ≥
Denote a feasible triple from e 3 to v by (x(t),ỹ(t),z(t)) on [0,T ]. LetT = T * ,z(t) =zt = t(0,z 2 ,z 3 ) andỹ(t) = t(ỹ 1 , 0, 0) for someỹ 1 ≥ 0. It is clear thatz 2 = x * 2 andz 3 = x * 3 . The goal now is to determine if there exists aỹ 1 ≥ 0 such thatĨ Plugging inx(t) =z(t) − Rỹ(t), x * T * = v − v, and writingȳ =ỹ 1 T * we see that the inequality above is equivalent to So, for given problem data and points v and v if there exists aȳ ≥ 0 such that (28) is satisfied, then the desired feasible path construction can be achieved. Notice that the left-hand side (LHS) in this equation is a concave, quadratic function ofȳ. So to prove the desired inequality, it is necessary that this function has a non-negative maximum which is achieved at a non-negative value.