Localization for constrained martingale problems and optimal conditions for uniqueness of reflecting diffusions in 2-dimensional domains

We prove existence and uniqueness for semimartingale reflecting diffusions in 2-dimensional piecewise smooth domains with varying, oblique directions of reflection on each"side", under geometric, easily verifiable conditions. Our conditions are optimal in the sense that, in the case of a convex polygon with constant direction of reflection on each side, they reduce to the conditions of Dai and Williams (1996), which are necessary for existence of Reflecting Brownian Motion. Moreover our conditions allow for cusps. Our argument is based on a new localization result for constrained martingale problems which holds quite generally: as an additional example, we show that it holds for diffusions with jump boundary conditions.


Introduction
Reflecting diffusions in nonsmooth domains have been studied since the early 1980s.Despite this long history, there is no general existence and uniqueness result in the literature for curved, piecewise smooth domains or cones, not even under the restriction that the process be a semimartingale, and not even in dimension 2. This notwithstanding the fact that there are significant applications, for instance in stochastic networks (see e.g.Kang et al. (2009) or Kang and Williams (2012)).
Exhaustive results exist only for normal reflection (Tanaka (1979), Saisho (1987), Bass and Hsu (1991), Bass (1996), DeBlassie and Toby (1993), etc.), for Brownian motion in an orthant with constant direction of reflection on each face (Harrison and Reiman (1981), Reiman and Williams (1988), Taylor and Williams (1993), etc.), for Brownian motion in a 2-dimensional wedge with constant direction of reflection on each side (Varadhan and Williams (1985), Williams (1985)), for Brownian motion in a smooth cone with radially constant direction of reflection (Kwon and Williams (1991)) and for semimartingale reflecting Brownian motion in a convex polyhedral domain with constant direction of reflection on each face (Dai and Williams (1996)).In the case of a simple polyhedral domain, the assumptions of Dai and Williams (1996) are necessary for existence of a semimartingale Brownian motion (see also Reiman and Williams (1988) for the orthant case.) For a piecewise smooth domain with varying, oblique direction of reflection on each "face", the best available result is Dupuis and Ishii (1993).Unfortunately, the Dupuis and Ishii (1993) result is proved under a condition that is not easy to verify and leaves out many very natural examples.(See e.g.Remark 3.5.)In fact, the Dupuis and Ishii (1993) condition does not reduce to the assumptions of Dai and Williams (1996) in the case of a polyhedral domain.
More recently, existence and uniqueness of a semimartingale reflecting diffusion has been proved by Costantini and Kurtz (2018) in a 2-dimensional cusp with varying, oblique directions of reflection on each "side" and by Costantini and Kurtz (2022) in a d-dimensional domain with one singular point that near the singular point can be approximated by a smooth cone, with varying, oblique direction of reflection on the smooth part of the boundary.In the cusp case, even starting at the cusp, with probability one, the process never hits it again.In contrast, in the case when the domain can be approximated by a cone, the process can hit the singular point infinitely many times.Therefore the study of this case requires a new ergodic theorem for inhomogeneous subprobability transition kernels.The conditions under which the above results are proved are geometric in nature and easily verifiable.A quite general existence result for piecewise smooth domains in R d , even with cusp like points, has been obtained in Costantini and Kurtz (2019), leaving the question of uniqueness.
In dimension two, piecewise smooth domains look locally like smooth domains or like domains with one singular point.Consequently, by a localization argument, one should be able to exploit the results of Costantini and Kurtz (2018) and Costantini and Kurtz (2022) to give conditions for uniqueness of semimartingale reflecting diffusions.In this paper we carry out this program.The conditions we find (Conditions 3.1 and 3.4; see also Remark 3.3) are geometric and easy to verify and of course allow for cusps and for points where the boundary is smooth but the direction of reflection has a discontinuity.The same conditions allow to apply the results of Costantini and Kurtz (2019) to obtain existence as well.They are optimal in the sense that for a polygonal domain with constant direction of reflection on each side they reduce to the conditions of Dai and Williams (1996) (Proposition 3.7 and Remark 3.11.)The existence proof in Costantini and Kurtz (2019) makes use of the equivalence between solutions of a stochastic differential equation with reflection (SDER) and natural solutions of the corresponding constrained martingale problem (CMP), proved in the same paper.CMPs were introduced in Kurtz (1990) and Kurtz (1991)) and further studied in Kurtz and Sockbridge (2001), Costantini and Kurtz (2015) and Costantini and Kurtz (2019).Here we exploit the equivalence between SDERs and CMPs also to localize the uniqueness problem for the SDER.
In Section 2, we introduce CMPs stopped at the exit from an open set and show that, under a quite general condition, uniqueness holds for the natural solution of a CMP in a given domain if and only if it holds for the natural solution of the CMP stopped at the exit from each open set belonging to an open covering of the domain.This result holds for general CMPs in arbitrary dimension and is of independent interest.CMPs may be used to define not only reflecting diffusions, but also, for instance, diffusions with Wentzell boundary conditions and Markov processes with jump boundary conditions (see Section 7 of Costantini and Kurtz (2019).)As an example of application of our localization result to other processes besides reflecting diffusions, we show that the condition we require is typically satisfied also by diffusions with jump boundary conditions (Remark 2.10.)Since the proofs of the results of Section 2 are somewhat technical, they are postponed to Appendix A.
In Section 3, we combine the above localization results with the uniqueness results in Costantini and Kurtz (2018) and Costantini and Kurtz (2022) to obtain global uniqueness for the natural solution of the CMP corresponding to an SDER in a piecewise smooth domain in R 2 , with varying, oblique direction of reflection on each "side".As mentioned above, existence follows from Costantini and Kurtz (2019).By the equivalence between natural solutions of the CMP and solutions of the SDER, existence and uniqueness transfer to the SDER.Although most of the work of this section consists in verifying the assumptions of Costantini and Kurtz (2019), Costantini and Kurtz (2018) and Costantini and Kurtz (2022), this verification is nontrivial.In particular, if the boundary has cusps, in order to apply the results of Costantini and Kurtz (2019) one needs to use the fact that the domain admits infinitely many representations and to construct a suitable representation.
A more detailed discussion of the contents is provided at the beginning of each section.We will use the following notation.⊆ and ⊇ will denote inclusion, while ⊂ and ⊃ will denote strict inclusion.For a finite set F , |F | will denote the cardinality of F .For a metric space E, B(E) will denote the σ-algebra of Borel sets and P(E) will denote the set of probability measures on (E, B(E)); for E 0 ⊆ E, E 0 will denote the closure of E 0 .For a stochastic process Z, F Z t := σ(Z(s), s ≤ t) and F Z t + := ∩ s>t F Z s ; Finally the superscript T denotes the transpose of a matrix and B r (0) denotes a ball in R d of radius r and center the origin .

Localization for constrained martingale problems
Let E be a compact metric space, E 0 be an open subset of E, and let A ⊆ C(E) × C(E) with (1, 0) ∈ A. Let U also be a compact metric space, let Ξ be a closed subset of (E − E 0 ) × U and assume that, for every x ∈ E−E 0 , there is some u ∈ U such that (x, u) ∈ Ξ.Let B ⊆ C(E)×C(Ξ) with (1, 0) ∈ B, D := D(A) ∩ D(B) and assume D is dense in C(E).The intuition is that A is the generator for a process in E and that B determines controls that constrain the process to remain in E 0 or, more precisely, in E 0 .
Let L U be the space of Borel measures µ on for all continuous f with compact support in [0, ∞) × U.It is possible to define a metric on L U that induces the above topology and makes L U into a complete, separable metric space.Also let L Ξ be defined analogously.For any L U -valued (L Ξ -valued) random variable L, for each t ≥ 0, L([0, t] × •) is a random measure on U (Ξ).We will occasionally use the notation For a nondecreasing path l 0 ∈ D [0,∞) [0, ∞) with l 0 (0) = 0, we define where we adopt the usual convention that the infimum of the empty set is ∞.Of course, if l 0 is strictly increasing (l 0 ) −1 is just the inverse of l 0 .In addition, for every path ) such that lim t→∞ y(t) exists, we will use the notation y(∞) := lim t→∞ y(t).
The controlled martingale problem for (A, E 0 , B, Ξ), the constrained martingale problem for (A, E 0 , B, Ξ) and natural solutions of the constrained martingale problem for (A, E 0 , B, Ξ) have been introduced and studied in Kurtz (1990), Kurtz (1991), Costantini and Kurtz (2015) and Costantini and Kurtz (2019).Here, given an open subset U of E, we introduce the notions of stopped controlled martingale problem for (A, E 0 , B, Ξ; U), and of natural solution of the stopped constrained martingale problem for (A, E 0 , B, Ξ; U) and study their relations with the corresponding unstopped objects.Our main goals are COrollary 2.12 and Theorem 2.13, which correspond to Theorems 4.6.1 and 4.6.3 of Ethier and Kurtz (1986) for martingale problems.A natural solution of the stopped constrained martingale problem for (A, E 0 , B, Ξ; U) is obtained by time-changing a solution of the stopped controlled martingale problem for (A, E 0 , B, Ξ; U) (see below for precise definitions): Roughly speaking, in order to transfer the results of Section 4.6 of Ethier and Kurtz (1986) to constrained martingale problems, what we need is to be able to exchange the "stopping" and the "time-changing".
Note that the set E here corresponds to E 0 ∪ F 1 in Costantini and Kurtz (2019) and that for Lemma 2.3 below we do not need Condition 3.5 c) of Costantini and Kurtz (2019).
and Λ U 1 be a L U -valued random variable such that ) is a solution of the controlled martingale problem for (A, E 0 , B, Ξ).
Remark 2.2 Note that, in general, Y U (t), in particular Y U (0), may take values outside U.
Let (Y, λ 0 , Λ 1 ) be a solution of the controlled martingale problem for (A, E 0 , B, Ξ).Then, setting Theorem 2.3 Suppose that for every ν ∈ P(E) there exists a solution of the controlled martingale problem for (A, E 0 , B, Ξ) with initial distribution ν.
Then, for every solution of the stopped controlled martingale problem for (A, E 0 , B, Ξ; U), (Y U , λ U 0 , Λ U 1 ), there exists a solution (Y, λ 0 , Λ 1 ) of the controlled martingale problem for (A, E 0 , B, Ξ) such that, with θ defined by of the stopped controlled martingale problem, with the property that the event {θ U = ∞, lim s→∞ λ U 0 (s) < ∞} has zero probability, such that Definition 2.6 Uniqueness holds for natural solutions of the stopped constrained martingale problem for (A, E 0 , B, Ξ; U) (the constrained martingale problem for (A, E 0 , B, Ξ; U)) if any two solutions with the same initial distributions have the same distribution on In the sequel we assume the following condition on the controlled martingale problem for (A, E 0 , B, Ξ) and the open set U.
Proposition 2.9 Suppose Condition 2.7 (i) is verified.If each solution of the controlled martingale problem for (A, E 0 , B, Ξ) satisfies λ 0 (t) > 0 for all t > 0 a.s., then λ 0 is strictly increasing a.s.for each solution, and Condition 2.7 is verified for every open set U.

Proof. See Appendix A
Remark 2.10 The controlled martingale problems corresponding to reflecting diffusions will usually satisfy the assumptions of Proposition 2.9 (e.g.see Lemma 6.8 of Costantini and Kurtz (2019) where σσ T is uniformly positive definite on E 0 , b and σ are continuous and vanish outside of an open neighborhood of E 0 whose closure is included in where p is a transition function on E, p(x, •) is continuous as a function from E into P(E) and, for all x ∈ E, p(x, E 0 ) = 1.
Then the controlled martingale problem for (A, E 0 , B, Ξ) satisfies (i), (ii) and (iii) of Condition 2.7: see Section 7.1 of Costantini and Kurtz (2019), and note that, under the above assumptions, Lemma 3.1 of Costantini and Kurtz (2019) applies, so that (iii) holds with ) is a solution of the controlled martingale problem for (A, E 0 , B, Ξ), Y behaves like a diffusion with generator A till it reaches ∂E 0 ; it stays at the exit point for a unit exponential time and then it jumps into E 0 and starts behaving like a diffusion again.The corresponding natural solution of the constrained martingale problem for (A, E 0 , B, Ξ) defined in (iv) of Condition 2.7 behaves in the same way except that it jumps instantaneously.In particular both Y and X stay in E 0 for all times and Y (X) jumps at a time t if and only if If Y (0) ∈ ∂D, Y will stay at Y (0) for a unit exponential time ρ and λ 0 (t) = 0 for 0 < t ≤ ρ, therefore the assumption of Proposition 2.9 is not satisfied.However, let U be an open set of R d with smooth boundary, such that U ⊆ • E and that, denoting by Leb the surface Lebesgue measure on ∂U, Leb(∂U ∩ ∂E 0 ) = 0.Then, with θ and τ as in (iv) of Condition 2.7, the probability that In both cases λ 0 is strictly increasing in a right neighborhood ot θ, so that λ −1 0 (λ 0 (θ)) = θ.Moreover Y (θ) / ∈ U implies τ = λ 0 (θ), so that (iv) of Condition 2.7 is satisfied.Processes of this type have been considered in a variety of settings, for example Davis and Norman (1990); Shreve and Soner (1994).Semigroups corresponding to processes with nonlocal boundary conditions of this type have been considered in Arendt, Kunkel and Kunze (2016).Related models are considered in Menaldi and Robin (1985).
Theorem 2.11 Under Condition 2.7, for every natural solution X U of the stopped constrained martingale problem for (A, E 0 , B, Ξ; U), there exists a natural solution X of the constrained martingale problem for (A, E 0 , B, Ξ) such that, with τ defined by (2.10), X(• ∧ τ ) has the same distribution as X U (•).

Proof. See Appendix A
Corollary 2.12 Under Condition 2.7, if uniqueness holds for natural solutions of the constrained martingale problem for (A, E 0 , B, Ξ), then it holds for natural solutions of the stopped constrained martingale problem for (A, E 0 , B, Ξ; U).
Proof.The assertion follows immediately from Theorem 2.11.
Theorem 2.13 Suppose there exist open subsets U k ⊆ E, k = 1, 2, ..., with E = ∞ k=1 U k , such that, for each k, (A, E 0 , B, Ξ) and U k satisfy Condition 2.7 and uniqueness holds for natural solutions of the stopped, constrained martingale problem for (A, E 0 , B, Ξ; U k ).Then uniqueness holds for natural solutions of the constrained martingale problem for (A, E 0 , B, Ξ).
Proof.See Appendix A.
3 Existence and uniqueness of reflecting diffusions in a 2-dimensional, piecewise smooth domain In this section, first we formulate our assumptions on the domain where the reflecting diffusion takes values and on the directions of reflection and compare them with the assumptions of the most general previous results, namely the results of Dupuis and Ishii (1993) (Remark 3.5) and Dai and Williams (1996) (Proposition 3.7).In particular, in the case of a convex polygon with constant direction of reflection on each side, our assumptions are equivalent to those of Dai and Williams (1996), which are necessary for existence of a reflecting Brownian motion: in this sense our assumptions are optimal (Remark 3.11).
Next we prove that the two definitions of a semimartingale reflecting diffusion as a solution of a stochastic differential equation with reflection and as a natural solution of a constrained martingale problem are equivalent (Theorem 3.13) and prove existence of a reflecting diffusion (Theorem 3.14).Both these results follow immediately from the results of Section 6 of Costantini and Kurtz (2019) once one has verified that the assumptions of Section 6 of Costantini and Kurtz (2019) are satisfied (Lemma 3.12: however, in particular at a cusp point, this verification is nontrivial and requires to construct a suitable representation of the domain. Finally, we show that uniqueness holds for the constrained martingale problem stopped at the exit from a neighborhood of each corner, both when the corner is a cusp (Lemma 3.17) and when it is not (Lemma 3.16): this amounts essentially to verifying that the assumptions of Costantini and Kurtz (2018) and Costantini and Kurtz (2022), respectively, are satisfied, but, again, this is nontrivial.Corollary 2.12 is also needed here.Uniqueness for the global constrained martingale problem then follows immediately from Theorem 2.13 and transfers to the corresponding stochastic differential equation by Theorem 3.13.
We consider a domain D satisfying the following condition.
Condition 3.1 (i) D is a bounded domain that admits the representation where, for i = 1, ..., m, D i is a bounded domain defined as and The representation is minimal in the sense that, for j = 1, ..., m, where ⊂ denotes strict inclusion.
For x ∈ ∂D i , we denote by n i (x) the unit, inward normal to D i at x, i.e. n i (x) : We call a point x 0 ∈ ∂D such that |I(x 0 )| > 1 a corner and assume |I(x 0 )| = 2 at every corner.
(iii) Let x 0 be a corner and I(x 0 ) = {i, j}.
If n i (x 0 ) = −n j (x 0 ) (then we say that x 0 is a cone point), lim sup for l = i, j.
If n j (x 0 ) = −n i (x 0 ) (then we say that x 0 is a cusp point), D ∩ B r (x 0 ) is connected for all r > 0 small enough, and for some finite L.
Remark 3.2 A piecewise C 1 domain D admits infinitely many representations (3.1), and it may be that some representations verify all assumptions in Condition 3.1 and others do not.In all our results we only need that there exists a representation that verifies Condition 3.1.It may be convenient to use more than one representation with different properties (see Lemma 3.12).
Define the inward normal cone at x 0 ∈ ∂D as For I(x 0 ) = {i, j}, if x 0 is a cone point, clearly N(x 0 ) is the closed, convex cone generated by n i (x 0 ) and n j (x 0 ).If x 0 is a cusp point, by the assumption that D ∩ ∂B r (0) is connected for all r > 0 small enough, there exists one and only one unit vector τ (x 0 ) such that Remark 3.3 Let x 0 be a corner, I(x 0 ) = {i, j}, and suppose The set of possible directions of reflection on the boundary of D is defined by vector fields g i : R 2 → R 2 , i = 1, ..., m, g i of unit length on ∂D i .For x 0 ∈ ∂D, define (3.6) Condition 3.4 (i) For i = 1, ..., m, g i is a Lischitz continuous vector field such that (ii) For every x 0 ∈ ∂D, there exists a unit vector e(x 0 ) ∈ N(x 0 ) such that e(x 0 ) • g > 0, ∀g ∈ G(x 0 ) − {0}.
Remark 3.5 As mentioned in the Introduction, the best result available in the literature for a piecewise smooth domain with varying directions of reflection on each "face" is Dupuis and Ishii (1993).A very simple example that shows how the Dupuis and Ishii (1993) assumptions may not be satisfied is the following.Let D 1 be the unit ball centered at (1, 0), and let D be its intersection with the upper half plane.Of course D can be represented as , where D 2 is a bounded C 1 domain.Let n i , i = 1, 2, denote the unit, inward normal to D i , and Then, at x 0 = (0, 0) and at x 0 = (2, 0), it can be proved by contradiction that there is no convex compact set that satisfies (3.7) of Dupuis and Ishii (1993).Conditions 3.1 and 3.4 are instead satisfied.
In the case when D is a convex polygon and the direction of reflection is constant on each side, Condition 3.4 coincides with the assumptions of Dai and Williams (1996).This is an immediate consequence of the following lemma, which rephrases the assumptions of Dai and Williams (1996).The lemma holds in general for convex polyhedrons in R d . Let where n 1 , ..., n m are distinct unit vectors, b 1 , ..., b m are real numbers, and the above representation is minimal, that is, for each j = 1, ..., m, where ⊂ denotes strict inclusion.Assumption 1.1 of Dai and Williams (1996) is formulated in terms of maximal subsets of the set of indeces {1, ..., m}, defined as follows: K ⊆ {1, ..., m} is maximal if and only if K = ∅, Lemma 3.6 K ⊆ {1, ..., m} is maximal if and only if K = I(x 0 ) for some x 0 ∈ ∂D.
For K ⊂ {1, ..., m}, K is maximal if and only if for every j / ∈ K there exists x j ∈ D such that x j • n i = b i for all i ∈ K, x j • n j > b j .Then the fact that K = I(x 0 ) is maximal for every x 0 ∈ ∂D is immediate.To see that the converse holds, let K be maximal and set Proposition 3.7 Let D ⊆ R 2 be defined by (3.7) and be bounded, and let g i , i = 1, ..., m, be constant unit vectors.Then D satisfies Condition 3.1.D and g i , i = 1, ..., m, satisfy Condition 3.4 if and only if they satisfy Assumption 1.1 of Dai and Williams (1996).
Proof.Verifying that D satisfies Condition 3.1 is immediate.In particular, in this case the minimality assumption (3.8) implies that 1 ≤ |I(x 0 )| ≤ 2 for every x 0 ∈ ∂D.
In dimension 2 every polyhedron is simple (see Definition 1.4 of Dai and Williams (1996)), therefore, by Proposition 1.1 of Dai and Williams (1996), Assumption 1.1 of Dai and Williams (1996) reduces to assuming that, for each maximal K, there is a nonnegative linear combination e := i∈K η i n i such that e • g j > 0 for all j ∈ K (actually Dai and Williams (1996) requires a positive linear combination, but of course the two requirements are equivalent).Since, by Lemma 3.6, K is maximal if and only if K = I(x 0 ) for some x 0 ∈ ∂D, this is indeed Condition 3.4 (ii).As the directions of reflection g i are constant, Condition 3.4 (i) follows from (ii).
Remark 3.8 Conditions 3.1 and 3.4 allow for boundary points x 0 at which the boundary is actually smooth, but the direction of reflection has a discontinuity, i.e.
Finally, we assume that the drift b and the dispersion coefficient σ satisfy the following condition.Condition 3.9 (i) b : R 2 → R 2 and σ : R 2 → R 2×2 are Lipschitz continuous.
In most of the literature, a semimartingale reflecting diffusion is defined as a solution of a stochastic differential equation with reflection.We recall the definition below, for the convenience of the reader.Definition 3.10 Let D be a bounded domain and, for x ∈ ∂D, let G(x) be a closed, convex cone such that {(x, u) ∈ ∂D × ∂B 1 (0) : u ∈ G(x)} is closed.Let b : R 2 → R 2 and σ : R 2 → R 2×2 be bounded, measurable functions, and ν ∈ P(D).A stochastic process X is a solution of the stochastic differential equation with reflection in D with coefficients b and σ, cone of directions of reflection G, and initial distribution ν, if X(0) has distribution ν, there exist a standard Brownian motion W , a continuous, non decreasing process λ, and a process γ with measurable paths, all defined on the same probability space as X, such that , for all t ≥ 0, and the equation is satisfied a.s.. Given an initial distribution ν ∈ P(D), weak uniqueness or uniqueness in distribution holds if all solutions of (3.9) with P {X(0 A stochastic process X is a weak solution of (3.9) if there is a solution X of (3.9) such that X and X have the same distribution.
Remark 3.11 When D is a bounded, convex polyhedron in R 2 , and the direction of reflection is constant on each side, Propositions 1.1 and 1.2 of Dai and Williams (1996) prove that if there exists a semimartingale reflecting Brownian motion (i.e. a weak solution of (3.9) with b and σ constant), then Assumption 1.1 of Dai and Williams (1996) must be verified.On the other hand we have proved in Proposition 3.7 that, when specialized to this case, Condition 3.4 coincides with Assumption 1.1 of Dai and Williams (1996).In this sense Condition 3.4 is optimal.
In the following we exploit repeatedly the equivalence between the stochastic differential equation (3.9) and the constrained martingale problem for (A, D, B, Ξ), where the state space is E := D, A denotes the operator and This equivalence is proved in general dimension d in Section 6 of Costantini and Kurtz (2019) (Theorem 6.12), under quite general assumptions.In the next lemma we show that, under Conditions 3.1, 3.4 and 3.9, the assumptions of Section 6 of Costantini and Kurtz (2019) are satisfied, or more precisely, that the domain D admits a representation such that the assumptions of Section 6 of Costantini and Kurtz (2019) are verified (see Remark 3.2.)Lemma 3.12 Assume Conditions 3.1, 3.4 and 3.9.Then the domain D admits a representation such that the assumptions of Section 6 of Costantini and Kurtz (2019) are verified.
Proof.First of all note that the assumption of Section 6 of Costantini and Kurtz (2019) that the domains are simply connected is redundant: it is enough to assume that the domains are connected, as we are doing here.
If 0 is a cone point, the normal cone N(0) can be written in the form (6.3) and Conditions 6.2 a) and b) of Costantini and Kurtz (2019) are verified.Condition 3.4 (ii) implies that the matrix is a completely-S matrix.Then its transpose is also completely-S (Lemma 3 of Reiman and Williams (1988)), so that, in particular, there exists g ∈ G(0), g Costantini and Kurtz (2019) for I = {1, 2}.Since Condition 6.2 (c) of Costantini and Kurtz (2019) is clearly satisfied for I = {1} and I = {2}, it is verified for every I ⊆ I(0).Now let 0 be a cusp point and let τ = τ (0) be the vector defined in (3.4).Without loss of generality we can take (τ, n 1 ) as the basis of the coordinate system.Let r 0 > 0 be small enough that B r 0 (0) contains no other corners than 0. Then D can be represented as where ψ i is the function defining D i and χ is a smooth, nondecreasing function such that χ(t) = 0 for t ≤ 0, χ(t) = 1 for t ≥ 1.
Intuitively, we add an extra domain ∆ and replace the function ψ i , i = 1, 2, with a function ψ i that agrees with ψ i for x 1 ≥ 0, but is symmetric with respect to x 1 in a neighborhood of 0 With the addition of the extra domain ∆, the normal cone N(0) can be written in the form (6.3) of Costantini and Kurtz (2019).By defining the direction of reflection on ∂∆, γ, to be the inward normal direction, we have γ(0) = τ , so that the cone of directions of reflection at 0, G(0), does not change.However, by the symmetry of the functions ψ i , i = 1, 2, now I(0), defined by (6.9) of Costantini and Kurtz (2019) for the representation (3.12), is and Condition 6.2 (c) of Costantini and Kurtz (2019) is satisfied at 0. By iterating the above construction for each cusp point of ∂D, we obtain a representation of D that satisfies the assumptions of Section 6 of Costantini and Kurtz (2019).
Theorem 3.13 Every solution of (3.9) is a natural solution of the constrained martingale problem for (A, D, B, Ξ) defined by (3.10)- (3.11).
Conversely every natural solution of the constrained martingale problem for (A, D, B, Ξ) is a weak solution of (3.9).
Proof.By Lemma 3.12, this is just a special case of Theorem 6.12 of Costantini and Kurtz (2019).Note that a solution of (3.9) as defined in Definition 3.10 is called a weak solution in Costantini and Kurtz (2019).
Theorem 3.14 Under Conditions 3.1, 3.4 and 3.9, for every initial distribution ν ∈ P(D), there exists a strong Markov solution of (3.9) with initial distribution ν.
Proof.By Lemma 3.12, this is just a special case of Theorem 6.13 of Costantini and Kurtz (2019).
Remark 3.15 Note that the construction of the solution of (3.9) provided in Section 6 of Costantini and Kurtz (2019) (Theorem 6.7 of Costantini and Kurtz (2019) and Lemma 1.1 of Kurtz (1990)) yields also a numerical approximation of the solution.
Lemma 3.16 Let x 0 ∈ ∂D be a cone point, r 0 be small enough that ∂D ∩ B r 0 (x 0 ) contains no other corners and U := D ∩ B r 0 (x 0 ).Let A, Ξ and B be defined by (3.10)-(3.11).
Then, under Conditions 3.1, 3.4 and 3.9, uniqueness holds for natural solutions of the stopped constrained martingale problem for (A, D, B, Ξ; U).
Then, under Conditions 3.1, 3.4 and 3.9, uniqueness holds for natural solutions of the stopped constrained martingale problem for (A, D, B, Ξ; U).
Exactly the same arguments allow to construct a solution of the controlled martingale problem for (A, D, B, Ξ) with an arbitrary initial distribution ν ∈ P D and to show that λ 0 is strictly increasing for each solution of the controlled martingale problem for (A, D, B, Ξ).Hence, by Proposition 2.9, Condition 2.7 is satisfied by (A, D, B, Ξ) and U and we can conclude as in the proof of Lemma 3.16.
Theorem 3.18 Under Conditions 3.1, 3.4 and 3.9, for every initial distribution ν ∈ P(D), uniqueness in distribution holds for solutions of (3.9) with initial distribution ν.
Proof.Let A, Ξ and B be defined by (3.10)-(3.11).By Lemma 3.12, D, G, b and σ satisfy the assumptions of Section 6 of Costantini and Kurtz (2019), therefore Theorems 6.7 and Lemma 6.8 of Costantini and Kurtz (2019), together with Proposition 2.9, ensure that Condition 2.7 is satisfied by (A, D, B, Ξ) and any open set U. Let x 1 , x 2 , ..., x M be the corners of D, r 0 > 0 be such that By Lemmas 3.16 and 3.17, uniqueness holds for natural solutions of the stopped constrained martingale problems for (A, D, B, Ξ; U k ), for k = 1, ..., M. As for the stopped constrained martingale problem for (A, D, B, Ξ; U M +1 ), one can consider a domain , and argue as in Lemmas 3.16 and 3.17, but using Corollary 5.2 (Case 2) of Dupuis and Ishii (1993) and Theorem 6.12 of Costantini and Kurtz (2019), to obtain that uniqueness holds for natural solutions of the stopped constrained martingale problems for (A, D, B, Ξ; U M +1 ).
Then the assertion follows by Theorems 2.13 and 3.13.

A Proofs of Section 2
Proof of Theorem 2.3 The proof is a suitable modification of the proof of Lemma 4.5.16 of Ethier and Kurtz (1986): Let P U denote the distribution of (Y U , λ U 0 , Λ U 1 ), ν denote the distribution of Y U (θ U ) and P denote the distribution of a solution of the controlled martingale problem for (A, E 0 , B, Ξ) with initial distribution ν.let Q be the probability measure on D Then the distribution of (Y, λ 0 , Λ 1 )(• ∧ θ) under Q is P U .In particular θ as defined above agrees Q-a.s. with θ as defined in (2.6).

Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
s))dλ 0 (s) − (tn∨θ,t n+1 ∨θ]×U Bf (Y (s), u)Λ 1 (ds × du), R ∧ := f (Y (t n+1 ∧ θ)) − f (Y (t n ∧ θ)) − t n+1 ∧θ tn∧θ Af (Y (s))dλ 0 (s) − (tn∧θ,t n+1 ∧θ]×U ). However there are significant examples of controlled martingale problems for which Condition 2.7 is verified for a large class of open sets U although the assumptions of Proposition 2.9 are not satisfied.For instance, this is the case for diffusions with jump boundary conditions.Let E 0 be a bounded domain in R d with smooth boundary, E be a compact set in R d such that E 0 ⊆ • E , where E 0 and • E denote the closure of E 0 and the interior of E in the topology of R d respectively.Consider the operator Af