An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries
The Annals of Applied Probability
2007, Vol. 17, No. 2, 741–779
DOI: 10.1214/105051606000000899
c© Institute of Mathematical Statistics, 2007
AN INVARIANCE PRINCIPLE FOR SEMIMARTINGALE
REFLECTING BROWNIAN MOTIONS IN DOMAINS WITH
PIECEWISE SMOOTH BOUNDARIES1
By W. Kang and R. J. Williams
Carnegie Mellon University and University of California, San Diego
Semimartingale reflecting Brownian motions (SRBMs) living in
the closures of domains with piecewise smooth boundaries are of in-
terest in applied probability because of their role as heavy traffic ap-
proximations for some stochastic networks. In this paper, assuming
certain conditions on the domains and directions of reflection, a per-
turbation result, or invariance principle, for SRBMs is proved. This
provides sufficient conditions for a process that satisfies the definition
of an SRBM, except for small random perturbations in the defining
conditions, to be close in distribution to an SRBM. A crucial ingredi-
ent in the proof of this result is an oscillation inequality for solutions
of a perturbed Skorokhod problem. We use the invariance principle
to show weak existence of SRBMs under mild conditions. We also
use the invariance principle, in conjunction with known uniqueness
results for SRBMs, to give some sufficient conditions for validating
approximations involving (i) SRBMs in convex polyhedrons with a
constant reflection vector field on each face of the polyhedron, and
(ii) SRBMs in bounded domains with piecewise smooth boundaries
and possibly nonconstant reflection vector fields on the boundary
surfaces.
1. Introduction. Semimartingale reflecting Brownian motions (SRBMs)
living in the closures of domains with piecewise smooth boundaries are of
interest in applied probability because of their role as heavy traffic diffusion
approximations for some stochastic networks. The nonsmoothness of the
boundary for such a domain, combined with discontinuities in the oblique
directions of reflection at intersections of smooth boundary surfaces, present
Received May 2006; revised November 2006.
1Supported in part by NSF Grants DMS-03-05272 and DMS-06-04537.
AMS 2000 subject classifications. 60F17, 60J60, 60K25, 90B15, 93E03.
Key words and phrases. Semimartingale reflecting Brownian motion, piecewise smooth
domain, invariance principle, oscillation inequality, Skorokhod problem, stochastic net-
works.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2007, Vol. 17, No. 2, 741–779. This reprint differs from the original in pagination
and typographic detail.
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2 W. KANG AND R. J. WILLIAMS
challenges in the development of a rigorous theory of existence, uniqueness
and approximation for such SRBMs.
When the state space is an orthant and the direction of reflection is con-
stant on each boundary face, a necessary and sufficient condition for weak
existence and uniqueness of an SRBM is known [14]. This condition involves
a so-called completely-S condition on the matrix formed by the reflection
directions. An invariance principle for such SRBMs was established in [15]
and used in [16] to justify heavy traffic diffusion approximations for cer-
tain open multiclass queueing networks. Loosely speaking, the invariance
principle shows that, assuming uniqueness in law for the SRBM, a process
satisfying the definition of the SRBM, except for small perturbations in the
defining conditions, is close in distribution to the SRBM.
For more general domains with piecewise smooth boundaries, some con-
ditions for existence and uniqueness of SRBMs are known. In particular, for
convex polyhedrons with a constant direction of reflection on each boundary
face, necessary and sufficient conditions for weak existence and uniqueness of
SRBMs are known for simple convex polyhedrons (where precisely d faces
meet at each vertex in d-dimensions) and sufficient conditions are known
for nonsimple convex polyhedrons, see [4]. For a bounded domain that can
be represented as a finite intersection of domains, each of which has a C1-
boundary and an associated uniformly Lipschitz continuous reflection vector
field, sufficient conditions for strong existence and uniqueness were provided
by Dupuis and Ishii [6]; in fact, these authors study stochastic differential
equations with reflection which include SRBMs. Despite these existence and
uniqueness results, a general invariance principle for SRBMs living in the
closures of domains with piecewise smooth boundaries has not been proved
to date. (We note that for the special case when the directions of reflection
are normal, that is, perpendicular to the boundary, there are a number of
perturbation results for reflecting Brownian motions. Our emphasis here is
on treating a wide range of oblique reflection directions.)
Motivated by its potential for use in approximating heavily loaded stochas-
tic networks that are more general than open multiclass queueing networks,
in this paper, we formulate and prove an invariance principle for SRBMs
living in the closures of domains with piecewise smooth boundaries with
possibly nonconstant directions of reflection on each of the smooth bound-
ary surfaces. An application of the results of this paper to the analysis of
an internet congestion control model can be found in [13]. An outline of the
current paper is as follows.
The definition of an SRBM and assumptions on the domains and direc-
tions of reflection are given in Sections 2 and 3, respectively. Some sufficient
conditions for these assumptions to hold are provided in Section 3. Section
4 is devoted to proving the main result of this paper, namely, the invari-
ance principle. A key element for our proof of this result is an oscillation
INVARIANCE PRINCIPLE FOR SRBMS 3
inequality for solutions of a perturbed Skorokhod problem; this inequality,
which may be of independent interest, is proved in Section 4.1. In Section 5
we give some applications of the invariance principle. We prove weak ex-
istence of SRBMs under the conditions specified in Section 3. We also use
the invariance principle, in conjunction with known uniqueness results for
SRBMs, to give sufficient conditions for validating approximations involving
(i) SRBMs in convex polyhedrons with a constant reflection vector field on
each face of the polyhedron, and (ii) SRBMs in bounded domains with piece-
wise smooth boundaries and possibly nonconstant reflection vector fields on
the boundary surfaces.
Beyond its possible use in justifying SRBM approximations for stochastic
networks, the invariance principle might be used to justify numerical ap-
proximations to SRBMs. A further possible extension of the results stated
here would involve an invariance principle for stochastic differential equa-
tions with reflection. The oscillation inequality for the perturbed Skorokhod
problem and associated criteria for C-tightness described in Sections 4.1
and 4.2 are likely to be useful for this. We have not developed such an ex-
tension here as that would involve introduction of extra assumptions that
would make the result less relevant for potential applications to stochastic
networks. In particular, the approximating processes would involve stochas-
tic integrals driven by a Brownian motion, whereas in stochastic network
applications, the Brownian motion typically only appears in the limit.
1.1. Notation, terminology and preliminaries. Let N denote the set of all
positive integers, that is, N = {1,2, . . .}, R denote the set of real numbers,
which is also denoted by (−∞,∞), R+ denote the nonnegative half-line,
which is also denoted by [0,∞). For x ∈ R, we write |x| for the absolute
value of x, [x] for the largest integer less than or equal to x, x+ for the
positive part of x. For any positive integer d, we let Rd denote d-dimensional
Euclidean space, where any element in Rd is denoted by a column vector.
Let ‖ · ‖ denote the Euclidean norm on Rd, that is, ‖x‖ = (
i=1 x
1/2 for
x ∈Rd, and 〈·, ·〉 denote the inner product on Rd, that is, 〈x, y〉=
i=1 xiyi,
for x, y ∈ Rd. We note that for any x ∈ Rd, ‖x‖ ≤
i=1 |xi|. Let R
+ denote
the positive orthant in Rd, that is, Rd+ = {x ∈ R
d :xi ≥ 0,1 ≤ i ≤ d}. Let
B(S) denote the Borel σ-algebra on S ⊂ Rd, that is, the collection formed
by intersecting all Borel sets in Rd with S. Let dist(x,S) denote the distance
between x ∈Rd and S ⊂Rd, that is, dist(x,S) = inf{‖x−y‖ :y ∈ S}, with the
convention that dist(x,∅) =∞ for x ∈Rd. Let Ur(S) denote the closed set
{x ∈Rd : dist(x,S)≤ r} for any r > 0 and S ⊂Rd, where if S =∅, Ur(S) =∅
for all r > 0. Let Br(x) denote the closed ball {y ∈R
d :‖y− x‖ ≤ r} for any
x ∈Rd and r > 0. For any set S ⊂Rd, we write S for the closure of S, So for
the interior of S and ∂S = S \So. For a finite set S, |S| denotes the number
4 W. KANG AND R. J. WILLIAMS
of elements in S. For any v ∈Rd, v′ denotes the transpose of v. Inequalities
between vectors in Rd should be interpreted componentwise, that is, if u, v ∈
d, then u ≤ (<)v means that ui ≤ (<)vi for each i ∈ {1, . . . , d}. For any
matrix A, let A′ denote the transpose of A. For any function x :R+ → R
x(t−) denotes the left limit of x at t > 0 whenever x has a left limit at t;
unless explicitly stated otherwise, x(0−)≡ 0, where 0 is the zero vector in Rd.
For any function x :R+ → R
d, we let ∆x(t) = x(t)− x(t−) for t ∈ R+ when
x(t−) exists. We let 0 be the constant deterministic function x :R+ → R
such that x(t) = 0 for all t ∈R+.
A domain in Rd is an open connected subset of Rd. For each continuously
differentiable function f defined on some nonempty domain S ⊂Rd, ∇f(x)
is the gradient of f at x ∈ S. For each x ∈ Rd, a neighborhood Vx of x is a
bounded domain in Rd that contains x. For any nonempty domain S ⊂Rd,
we say that the boundary ∂S of S is C1, if for each x ∈ ∂S there exists a
Euclidean coordinate system Cx for R
d centered at x, an rx > 0, and a once
continuously differentiable function ϕx :R
d−1 →R such that ϕx(0) = 0 and
S ∩Brx(x) = {z = (z1, . . . , zd)
′ in Cx : zd >ϕx(z1, . . . , zd−1)} ∩Brx(x).
Then, for x ∈ ∂S, the inward unit normal to ∂S at z ∈ ∂S ∩Brx(x) is given
in the coordinate system Cx by
n(z) =
(1 + ‖∇ϕx(z1, . . . , zd−1)‖
2)1/2
(−∇ϕx(z1, . . . , zd−1)
where ∇ϕx(z1, . . . , zd−1) = (
, . . . ,
∂zd−1
)′(z1, . . . , zd−1). For any nonempty
convex set S ⊂Rd, we call a vector n ∈Rd \{0} an inward unit normal vector
to S at x ∈ ∂S if ‖n‖= 1 and 〈n, y − x〉 ≥ 0 for all y ∈ S. Note that such a
vector need not be unique.
All stochastic processes used in this paper will be assumed to have paths
that are right continuous with finite left limits (abbreviated henceforth as
r.c.l.l.). A process is called continuous if almost surely its sample paths
are continuous. We denote by D([0,∞),Rd) the space of r.c.l.l. functions
from [0,∞) into Rd and we endow this space with the usual Skorokhod
J1-topology (cf. Chapter 3 of [7]). We denote by C([0,∞),R
d) the space
of continuous functions from [0,∞) into Rd. The Borel σ-algebra on either
D([0,∞),Rd) or C([0,∞),Rd) will be denoted by Md. The abbreviation
u.o.c. will stand for uniformly on compacts and will be used to indicate
that a sequence of functions in D([0,∞),Rd) (or C([0,∞),Rd)) is converg-
ing uniformly on compact time intervals to a limit in D([0,∞),Rd) (or
C([0,∞),Rd)). Consider W 1,W 2, . . . ,W , each of which is a d-dimensional
process (possibly defined on different probability spaces). The sequence
{W n}∞n=1 is said to be tight if the probability measures induced by the
W n on the measurable space (D([0,∞),Rd),Md) form a tight sequence,
INVARIANCE PRINCIPLE FOR SRBMS 5
that is, they form a weakly relatively compact sequence in the space of
probability measures on (D([0,∞),Rd),Md). The notation “W n ⇒W” will
mean that, as n → ∞, the sequence of probability measures induced on
(D([0,∞),Rd),Md) by {W n} converges weakly to the probability measure
induced on the same space by W . We shall describe this in words by saying
that W n converges weakly (or in distribution) to W as n → ∞. The se-
quence of processes {W n}∞n=1 is called C-tight if it is tight, and if each weak
limit point, obtained as a weak limit along a subsequence, almost surely has
sample paths in C([0,∞),Rd). The following proposition provides a useful
criterion for checking C-tightness.
Proposition 1.1. Suppose that, for each n ∈N, W n is a d-dimensional
process defined on the probability space (Ωn,Fn, Pn). The sequence {W n}∞n=1
is C-tight if and only if the following two conditions hold:
(i) For each η > 0 and T ≥ 0, there exists a finite constant Mη,T > 0
such that
lim inf
0≤t≤T
‖W n(t)‖ ≤Mη,T
≥ 1− η.(1)
(ii) For each ε > 0, η > 0 and T > 0, there exists λ ∈ (0, T ) such that
lim sup
Pn{wT (W
n, λ)≥ ε} ≤ η,(2)
where for x ∈D([0,∞),Rd),
wT (x,λ) = sup
u,v∈[t,t+λ]
‖x(u)− x(v)‖ : 0≤ t < t+ λ≤ T
Proof. See Proposition VI.3.26 in [12]. �
A d-dimensional process W is said to be locally of bounded variation if
all sample paths of W are of bounded variation on each finite time interval.
For such a process W , we define V(W ) = {V(W )(t), t≥ 0} such that for each
t≥ 0,
V(W )(t) = ‖W (0)‖
+ sup
‖W (ti)−W (ti−1)‖ : 0 = t0 < t1 < · · ·< tl = t, l≥ 1
A triple (Ω,F ,{Ft, t ≥ 0}) will be called a filtered space if Ω is a set,
F is a σ-algebra of subsets of Ω, and {Ft, t≥ 0} is an increasing family of
sub-σ-algebras of F , that is, a filtration. Henceforth, the filtration {Ft, t≥ 0}
will be simply written as {Ft}. If P is a probability measure on (Ω,F), then
6 W. KANG AND R. J. WILLIAMS
(Ω,F ,{Ft}, P ) is called a filtered probability space. A d-dimensional process
X = {X(t), t ≥ 0} defined on (Ω,F , P ) is called {Ft}-adapted if for each
t ≥ 0, X(t) :Ω→ Rd is measurable when Ω is endowed with the σ-algebra
Given a filtered probability space (Ω,F ,{Ft}, P ), a vector µ ∈R
d, a d× d
symmetric, strictly positive definite matrix Γ, and a probability distribution
ν on (Rd,B(Rd)), an {Ft}-Brownian motion with drift vector µ, covariance
matrix Γ, and initial distribution ν, is a d-dimensional {Ft}-adapted process
defined on (Ω,F ,{Ft}, P ) such that the following hold under P :
(a) X is a d-dimensional Brownian motion whose sample paths are almost
surely continuous and that has initial distribution ν,
(b) {Xi(t)−Xi(0)− µit,Ft, t≥ 0} is a martingale for i= 1, . . . , d, and
(c) {(Xi(t)−Xi(0)−µit)(Xj(t)−Xj(0)−µjt)−Γijt,Ft, t≥ 0} is a mar-
tingale for i, j = 1, . . . , d.
In this definition, the filtration {Ft} may be larger than the one generated
by X ; however, for each t≥ 0, under P , the σ-algebra Ft is independent of
the increments of X from t onward. The latter follows from the martingale
properties of X . If ν = δx, the unit mass at x ∈ R
d, we say that X starts
from x.
2. Definition of an SRBM. Let G=
i∈I Gi be a nonempty domain in
d, where I is a nonempty finite index set and for each i ∈ I , Gi is a
nonempty domain in Rd. For simplicity, we assume that I = {1,2, . . . , I}
and then |I|= I. For each i ∈ I , let γi(·) be a vector valued function defined
from Rd into Rd. Fix µ ∈Rd, Γ a d × d symmetric and strictly positive
definite covariance matrix and ν a probability measure on (G,B(G)), where
B(G) denotes the σ-algebra of Borel subsets of the closure G of G.
Definition 2.1 (Semimartingale reflecting Brownian motion). A semi-
martingale reflecting Brownian motion (abbreviated as SRBM) associated
with the data (G,µ,Γ,{γi, i ∈ I}, ν) is an {Ft}-adapted, d-dimensional pro-
cess W defined on some filtered probability space (Ω,F ,{Ft}, P ) such that:
(i) P -a.s., W (t) =X(t) +
(0,t] γ
i(W (s))dYi(s) for all t≥ 0,
(ii) P -a.s., W has continuous paths and W (t) ∈G for all t≥ 0,
(iii) under P , X is a d-dimensional {Ft}-Brownian motion with drift
vector µ, covariance matrix Γ and initial distribution ν,
(iv) for each i ∈ I , Yi is an {Ft}-adapted, one-dimensional process such
that P -a.s.,
(a) Yi(0) = 0,
(b) Yi is continuous and nondecreasing,
INVARIANCE PRINCIPLE FOR SRBMS 7
(c) Yi(t) =
(0,t] 1{W (s)∈∂Gi∩∂G} dYi(s) for all t≥ 0.
We shall often refer to Y = {Yi, i ∈ I} as the “pushing process” associated
with the SRBM W . When ν = δx, we may alternatively say that W is an
SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}) that starts from x. We
will call (W,X,Y ) satisfying Definition 2.1 an extended SRBM associated
with the data (G,µ,Γ,{γi, i ∈ I}, ν).
Loosely speaking, an SRBM behaves like a Brownian motion in the inte-
rior of the domain G and it is confined to G by instantaneous “reflection” (or
“pushing”) at the boundary, where the allowed directions of “reflection” at
x ∈ ∂G are convex combinations of the vectors γi(x) for i such that x ∈ ∂Gi.
Under the assumptions imposed on G and {γi, i ∈ I} in Sections 3.1 and 3.2
below, at each point on the boundary of G there is an allowed direction of
reflection that can be used there which “points into the interior of G.” We
end this section by introducing a related set-valued function I(·) and show
a key property of it.
Definition 2.2. For each x∈Rd, let I(x) = {i ∈ I :x ∈ ∂Gi}.
The set-valued function I(·) has the following property called upper semi-
continuity on ∂G.
Lemma 2.1. For each x ∈ ∂G, there is an open neighborhood Vx of x in
d such that
I(y)⊂ I(x) for all y ∈ Vx.(4)
Proof. We prove this lemma by contradiction. Suppose that the func-
tion I(·) does not satisfy (4). Then there is a point x ∈ ∂G such that there
is no open neighborhood Vx of x such that I(y)⊂ I(x) for all y ∈ Vx. Since
the index set I is finite, there is an index k ∈ I \ I(x) and a sequence of
points {yn} ⊂R
d such that ‖yn−x‖<
and k ∈ I(yn) for each n≥ 1. Hence
yn ∈ ∂Gk for all n≥ 1. Since ∂Gk is closed and yn → x as n→∞, we con-
clude that x ∈ ∂Gk. This implies that k ∈ I(x), which is a contradiction, as
desired. �
3. Assumptions on the domain G and the reflection vector fields {γi}.
3.1. Assumptions on the domain G. We henceforth assume that the
domain G satisfies assumptions (A1)–(A3) below. In the case when G is
bounded, assumptions (A2)–(A3) follow from assumption (A1) (see Lem-
mas A.1 and A.2 in the Appendix for details). If the domain G is a convex
polyhedron satisfying assumption (A1), then assumptions (A2)–(A3) hold
by Lemma A.3 in the Appendix.
8 W. KANG AND R. J. WILLIAMS
(A1) G is a nonempty domain in Rd with representation
Gi,(5)
where for each i ∈ I , Gi is a nonempty domain, Gi 6=R
d, and the boundary
∂Gi of Gi is C
1. For each i ∈ I , we let ni(·) be the unit normal vector field
on ∂Gi that points into Gi.
(A2) For each ε ∈ (0,1) there exists R(ε) > 0 such that for each i ∈ I ,
x ∈ ∂Gi ∩ ∂G and y ∈G satisfying ‖x− y‖<R(ε), we have
〈ni(x), y − x〉 ≥−ε‖x− y‖.(6)
(A3) The function D : [0,∞)→ [0,∞] defined such that D(0) = 0 and
D(r) = sup
J 6=∅
(∂Gj ∩ ∂G)
Ur(∂Gj ∩ ∂G)
for r > 0, satisfies
D(r)→ 0 as r→ 0.(8)
Remark. Assumption (A2) is reminiscent of the uniform exterior cone
condition (cf. [9], page 195). We say that a region G ⊂ Rd satisfies a uni-
form exterior cone condition if for each x0 ∈ ∂G, there is a truncated closed
right circular cone Vx0 , with nonempty interior and vertex x0, satisfying
Vx0 ∩G= {x0}, and the truncated cones Vx0 are all congruent to some fixed
truncated closed right circular cone V . By comparing assumption (A2) with
the uniform exterior cone condition, we see that assumption (A2) implies
the uniform exterior cone condition. On the other hand, under assumption
(A1), assumption (A2) is implied by a family of uniform exterior cone condi-
tions where for each ε ∈ (0,1), the axis of the truncated closed right circular
cone at x ∈ ∂G is along the vector −ni(x) and all of the truncated closed
right circular cones are congruent to a truncated closed right circular cone
whose height and base radius are R(ε) and R(ε)( 1
− 1)1/2 respectively.
Assumption (A2) holds automatically if G is convex. We also note that as-
sumption (A2) is strictly weaker than the uniform exterior sphere condition.
The definition of the uniform exterior sphere condition is similar to that for
the uniform exterior cone condition where a closed ball with x0 on its bound-
ary takes the place of the truncated closed right circular cone Vx0 . It can
be checked that for the domain G = {(x, y) ∈ R2 :y < |x|α} with α ∈ (1,2),
the uniform exterior sphere condition fails to hold, but assumption (A2)
holds. In fact, at the point (0,0) ∈R2, there is no r > 0 and y ∈R2 such that
Br(y)∩ ∂G= {(0,0)}.
INVARIANCE PRINCIPLE FOR SRBMS 9
Remark. For the definition of D(·) in (A3), we adopt the convention
that the supremum over an empty set is zero and dist(x,∅) =∞. Since ∂Gi∩
∂G 6=∅ for at least one i ∈ I , the function D(·) satisfies limr→∞D(r) =∞.
Furthermore, D(r1)≤D(r2) whenever r1, r2 ∈ [0,∞) and r1 ≤ r2. Assump-
tion (A3) requires that for any nonempty subset J ⊂ I , the intersection of
tubular neighborhoods of the boundaries {∂Gj ∩ ∂G : j ∈ J } given by the
j∈J Ur(∂Gj ∩ ∂G) “converges” to the intersection of the boundaries
given by the set
j∈J (∂Gj ∩ ∂G) as r approaches 0. Property (8) need
not always hold. For example, let G1 = {(x, y) ∈ R
2 :y < e−x
2/2, x ∈ R} and
G2 = {(x, y) ∈ R
2 :y > 0, x ∈ R}. Then ∂G1 ∩ ∂G2 =∅. But for each r > 0,
Ur(∂G1)∩Ur(∂G2) 6=∅. Hence D(r) =∞ for each r > 0.
3.2. Assumptions on the reflection vector fields {γi}. We henceforth as-
sume that there are vector fields {γi(·), i ∈ I} satisfying assumptions (A4)–
(A5) below.
(A4) There is a constant L > 0 such that for each i ∈ I , γi(·) is a uni-
formly Lipschitz continuous function from Rd into Rd with Lipschitz con-
stant L and ‖γi(x)‖= 1 for each x ∈Rd.
(A5) There is a constant a ∈ (0,1), and vector valued functions b(·) =
(b1(·), . . . , bI(·)) and c(·) = (c1(·), . . . , cI(·)) from ∂G into R
+ such that for
each x ∈ ∂G,
i∈I(x) bi(x) = 1,
j∈I(x)
i∈I(x)
bi(x)n
i(x), γj(x)
≥ a,(9)
i∈I(x) ci(x) = 1,
j∈I(x)
i∈I(x)
ci(x)γ
i(x), nj(x)
≥ a.(10)
We note here for future use that by (A4), if we set ρ0 =
, then for
any x, y ∈ Rd satisfying ‖x − y‖ < ρ0, we have ‖γ
i(x) − γi(y)‖ < a/4 for
each i ∈ I . So for each 0< ρ < ρ0/4, by (9)–(10) and the normalization of
b(·), c(·), γi(·), nj(·) for i, j ∈ I , we obtain
j∈I(x)
y∈B4ρ(x)
i∈I(x)
bi(x)n
i(x), γj(y)
≥ a/2(11)
j∈I(x)
y∈B4ρ(x)
i∈I(x)
ci(x)γ
i(y), nj(x)
≥ a/2.(12)
10 W. KANG AND R. J. WILLIAMS
The use of B4ρ(x) here is related to the form in which this is used in Section
Remark. Assumption (A4) is equivalent to (3.4) in [6] whenG is bounded.
Property (10) means that, at each point x ∈ ∂G, there is a convex combi-
nation γ(x) =
i∈I(x) ci(x)γ
i(x) of the vectors {γi(x), i ∈ I(x)} that can
be used there such that γ(x) “points into” G. Property (9) is in a sense a
dual condition to property (10), where the roles of γi and ni are reversed
for i ∈ I(x). This property (9) is used in showing the oscillation inequality
in Theorem 4.1 below. Assumption (A5) is an analogue of Assumption 1.1
in [4]. When G is bounded, (10) is similar to condition (3.6) in [6] (we as-
sume some additional uniformity through the lack of dependence of a on
It is straightforward to see using the triangle inequality that the following
condition (A5)′ implies (A5).
(A5)′ There is a ∈ (0,1) and vector valued functions b, c from ∂G into RI+
such that for each x ∈ ∂G,
i∈I(x) bi(x) = 1, and for each i ∈ I(x),
bi(x)〈n
i(x), γi(x)〉 ≥ a+
j∈I(x)\{i}
bj(x)|〈n
j(x), γi(x)〉|,(13)
i∈I(x) ci(x) = 1, and for each i ∈ I(x),
ci(x)〈γ
i(x), ni(x)〉 ≥ a+
j∈I(x)\{i}
cj(x)|〈γ
j(x), ni(x)〉|.(14)
Condition (A5)′(ii) is similar to condition (3.8) in [6], although here we
assume additional uniformity through the lack of dependence of a on x. As
noted in [6], their condition (3.8) can be phrased in terms of a nonsingu-
lar M-matrix requirement [2]. (This is sometimes also called a generalized
Harrison–Reiman type of condition [10].) Since that nonsingular M-matrix
property is invariant under transpose, and this property for the transpose
corresponds to a local form of (A5)′(i), one might conjecture that there is an
equivalence between the existence of a nonnegative vector valued function b
such that (A5)′(i) holds for each x ∈ ∂G and the existence of a nonnegative
vector valued function c such that (A5)′(ii) holds for each x ∈ ∂G. Indeed
we have the following lemma. We have stated the two (equivalent) condi-
tions (i) and (ii) in specifying (A5)′ to preserve a parallel with (A5) and
since both properties can be useful in proofs. Furthermore, in light of the
following lemma, verifying either condition suffices for both to hold.
INVARIANCE PRINCIPLE FOR SRBMS 11
Lemma 3.1. There is a constant a ∈ (0,1) and a vector valued function
b :∂G → RI+ such that (A5)
′(i) holds for each x ∈ ∂G if and only if there
is a constant a ∈ (0,1) and a vector valued function c :∂G→ RI+ such that
(A5)′(ii) holds for each x ∈ ∂G.
Proof. We just prove the “if” part; the “only if” part can be proved
in a similar manner.
We suppose that there is a constant a ∈ (0,1) and a vector valued function
c :∂G → RI+ such that (A5)
′(ii) holds for each x ∈ ∂G. For fixed x ∈ ∂G,
consider the square matrix A(x) whose diagonal entries are given by the
(positive) elements 〈ni(x), γi(x)〉 for i ∈ I(x) and whose off-diagonal entries
are given by−|〈ni(x), γj(x)〉| for i ∈ I(x), j ∈ I(x), j 6= i. Let E be the square
matrix having the same dimensions as A(x) and whose entries are all equal
to one. By the theory of M-matrices (see [2], Chapter 6, especially condition
(M35)), condition (ii) of (A5)
′ implies that A(x)− a
E is a nonsingular M-
matrix, that is, A(x)− a
E has nonnegative diagonal entries and nonpositive
off-diagonal entries and it can be written in the form s(x)I −B(x) where
B(x) is a matrix with nonnegative entries and s(x)> 0 is a constant that is
strictly larger than the spectral radius of B(x).
Since the nonsingular M-matrix property is invariant under transpose (cf.
(G21) in Chapter 6 of [2]), then A
′(x)− a
E is also a nonsingular M-matrix.
Hence, there is a vector b̃(x) = (b̃i(x) : i ∈ I(x)) with nonnegative entries such
that (A′(x)− a
E)b̃(x)> 0 (cf. (I27) in Chapter 6 of [2]). We can extend b̃(x)
to an I-dimensional vector b(x) and normalize it so that
i∈I(x) bi(x) = 1.
Then (A5)′(i) holds with a
in place of a. �
4. Invariance principle. In this section we state and prove an invariance
principle for an SRBM living in the closure of a domain G with piecewise
smooth boundary and having associated reflection fields {γi, i ∈ I}, where
G, {γi, i ∈ I} satisfy assumptions (A1)–(A5) of Section 3. (These assump-
tions hold throughout this section.) We shall first state a preliminary result
called an oscillation inequality (see Theorem 4.1), then we use it to prove
a tightness result (see Theorem 4.2). Finally, we establish the invariance
principle (see Theorem 4.3).
4.1. Oscillation inequality. The following oscillation inequality is the key
to the proof of the tightness result claimed in Theorem 4.2. In this subsec-
tion, for any 0≤ t1 < t2 <∞ and any integer k ≥ 1, D([t1, t2],R
k) denotes
the set of functions w : [t1, t2]→R
k that are right continuous on [t1, t2) and
have finite left limits on (t1, t2]. For w ∈D([t1, t2],R
Osc(w, [t1, t2]) = sup{‖w(t)−w(s)‖ : t1 ≤ s < t≤ t2},(15)
Osc(w, [t1, t2)) = sup{‖w(t)−w(s)‖ : t1 ≤ s < t < t2}.(16)
12 W. KANG AND R. J. WILLIAMS
Note that we do not explicitly indicate the dependence on k in the notation.
Recall the constants a,L from assumptions (A4)–(A5), the functions R(·)
from assumption (A2) and D(·) from (7). Let ρ0 =
Theorem 4.1 (Oscillation inequality). There exists a nondecreasing func-
tion Π: (0,∞)→ (0,∞] satisfying Π(u)→ 0 as u→ 0, such that Π depends
only on the constants I, a and the function D(·), and such that whenever
0 < ρ <min{
R(a/4)
}, 0 < δ <
, 0 ≤ s < t < ∞, w,x ∈ D([s, t],Rd) and
y ∈D([s, t],RI) satisfy:
(i) w(u) ∈Bρ(x0)∩Uδ(G) for all u ∈ [s, t], for some x0 ∈G,
(ii) w(u) = w(s) + x(u) − x(s) +
(s,u] γ
i(w(v))dyi(v) for all u ∈
[s, t],
(iii) for each i ∈ I ,
(a) yi(s)≥ 0,
(b) yi is nondecreasing and ∆yi(u)≤ δ for all u ∈ (s, t],
(c) yi(u) = yi(s) +
(s,u] 1{w(v)∈Uδ(∂Gi∩∂G)} dyi(v) for all u ∈ [s, t],
(iv) D(Π(Osc(x, [s, t]) + δ))<
then we have that the following hold:
Osc(w, [s, t])≤Π(Osc(x, [s, t]) + δ),(17)
Osc(y, [s, t])≤Π(Osc(x, [s, t]) + δ).(18)
Proof. Let
Π0(u) = u for all u > 0.
Define Πm : (0,∞)→ (0,∞], m= 1, . . . , I, inductively such that
Πm(u) = Πm−1(u) + (I+2)u+
(D(Πm−1(u) + (I+2)u) + 2u).
Here the sum of any element of [0,∞) with ∞ is ∞ and D(∞) is defined
to equal ∞. For each m= 0,1, . . . , I, the function Πm is nondecreasing and
depends only on I, a and D(·). For each m= 1, . . . , I and u > 0, Πm−1(u)≤
Πm(u). By assumption (A3), we conclude (using an induction proof) that
Πm(u)→ 0 as u→ 0, for m= 0,1, . . . , I.
Let Π(·) = ΠI(·).
Fix 0 < ρ < min{
R(a/4)
}, 0 < δ <
, 0 ≤ s < t < ∞. Suppose that
w,x ∈D([s, t],Rd) and y ∈D([s, t],RI) satisfy (i)–(iv) in the statement of
Theorem 4.1. For each nonempty interval [t1, t2]⊂ [s, t], let
I[t1,t2] = {i ∈ I :w(u) ∈ Uδ(∂Gi ∩ ∂G) for some u ∈ [t1, t2]},
INVARIANCE PRINCIPLE FOR SRBMS 13
the indices of the boundary surfaces that w(·) comes close to in the time
interval [t1, t2]. For each 0 ≤ m ≤ I, define Tm = {[t1, t2] ⊂ [s, t] : |I[t1,t2]| ≤
m}. Note that under the partial ordering of set inclusion, Tm increases with
m. To prove the theorem, we will prove by induction that for each 0≤m≤ I
and each interval [t1, t2] ∈ Tm, (17)–(18) hold with [t1, t2] in place of [s, t] and
Πm(·) in place of Π(·). The result for m= I yields the theorem.
Suppose that m= 0. Then T0 = {[t1, t2]⊂ [s, t] : |I[t1,t2]|= 0}. Fix an inter-
val [t1, t2] ∈ T0. Since I[t1,t2] =∅ and (iii)(c) holds, the function y does not
increase on the time interval (t1, t2], that is, yi(t2)− yi(t1) = 0 for all i ∈ I .
Then, for t1 ≤ u < v ≤ t2,
w(v)−w(u) = x(v)− x(u).(19)
So in this case,
Osc(w, [t1, t2]) = Osc(x, [t1, t2])≤Osc(x, [t1, t2]) + δ,(20)
Osc(y, [t1, t2]) = 0≤Osc(x, [t1, t2]) + δ.(21)
Thus, (17)–(18) hold with Π0(·) in place of Π(·) and [t1, t2] in place of [s, t]
for each interval [t1, t2] ∈ T0.
For the induction step, let 1 ≤ m ≤ I and suppose that (17)–(18) hold
with Πm−1(·) in place of Π(·) and [t1, t2] in place of [s, t] for each interval
[t1, t2] ∈ Tm−1.
Now fix [t1, t2] ∈ Tm. If |I[t1,t2]| ≤ m − 1, then [t1, t2] ∈ Tm−1 and so by
the induction assumption we have that (17)–(18) hold with [t1, t2] in place
of [s, t] and Πm−1(·) [and hence Πm(·)] in place of Π(·). Thus, it suffices
to consider [t1, t2] ⊂ [s, t] such that |I[t1,t2]| =m. For i /∈ I[t1,t2], by (iii)(c),
yi(t2)− yi(t1) = 0, and so by (ii), for t1 ≤ u < v ≤ t2, we have
w(v)−w(u) = x(v)− x(u) +
i∈I[t1,t2]
(u,v]
γi(w(r))dyi(r).(22)
Let Πm(u) = Πm−1(u) + (I + 2)u for all u > 0, and η = Osc(x, [t1, t2]) + δ.
For any M ∈ (0,∞] and any nonempty set J ⊂ I , let
FMJ = {z ∈R
d : dist(z, ∂Gi ∩ ∂G)<M for all i ∈ J }.
Note that FMJ = ∅ when there is an i ∈ J such that ∂Gi ∩ ∂G = ∅. Since
Πm(·)≤Πm(·)≤Π(·), D(·) and Π(·) are nondecreasing, and Osc(x, [t1, t2])≤
Osc(x, [s, t]), we have by (iv) that
D(Πm(η))≤D(Πm(η))≤D(Π(η))<
.(23)
Note that this implies Πm(η)<∞ since D(∞) =∞.
We now consider two cases.
14 W. KANG AND R. J. WILLIAMS
Case 1. Suppose that w(r) ∈ F
Πm(η)
I[t1,t2]
for all r ∈ [t1, t2].
Fix u, v such that t1 ≤ u < v ≤ t2. Since we have that
w(v) ∈
j∈I[t1,t2]
Πm(η)
(∂Gj ∩ ∂G),
by the definition of D(·) and (23), there is z ∈
j∈I[t1,t2]
(∂Gj ∩ ∂G) such
‖w(v)− z‖ ≤D(Πm(η))<
.(24)
For each r ∈ [t1, t2], by (i) we have that w(r) ∈ Uδ(G), and so there is z
such that
‖w(r)− zr‖ ≤ 2δ.
Hence by (i) and (24) we have
‖zr − z‖ ≤ ‖zr −w(r)‖+ ‖w(r)− x0‖+ ‖x0 −w(v)‖+ ‖w(v)− z‖
≤ 2δ + ρ+ ρ+ ρ/2< 4ρ < R(a/4)
‖w(r)− z‖ ≤ ‖w(r)− x0‖+ ‖x0 −w(v)‖+ ‖w(v)− z‖
≤ ρ+ ρ+ ρ/2< 4ρ.
By (6) and (25) we have
〈nj(z), z − zr〉 ≤
‖z − zr‖ for each j ∈ I(z) and r ∈ [t1, t2].(27)
Note that I(z) ⊃ I[t1,t2]. Recalling the definition of b(·) from assumption
(A5), on dotting the vector
j∈I(z) bj(z)n
j(z) with both sides of (22) and
rearranging, we obtain
i∈I[t1,t2]
(u,v]
j∈I(z)
bj(z)n
j(z), γi(w(r))
dyi(r)
j∈I(z)
bj(z)〈n
j(z),w(v)−w(u)〉(28)
j∈I(z)
bj(z)〈n
j(z), x(v)− x(u)〉.
So by (11), (22), (24)–(28), and the fact that
j∈I(z) bj(z) = 1, bj(z)≥ 0 for
j ∈ I , we have
i∈I[t1,t2]
(yi(v)− yi(u))
INVARIANCE PRINCIPLE FOR SRBMS 15
i∈I[t1,t2]
(u,v]
j∈I(z)
bj(z)n
j(z), γi(w(r))
dyi(r)
j∈I(z)
bj(z)〈n
j(z),w(v)− z〉+
j∈I(z)
bj(z)〈n
j(z), z − zu〉
j∈I(z)
bj(z)〈n
j(z), zu −w(u)〉 −
j∈I(z)
bj(z)〈n
j(z), x(v)− x(u)〉
≤D(Πm(η)) +
‖z − zu‖+ 2δ + ‖x(v)− x(u)‖
≤D(Πm(η)) + 2δ + ‖x(v)− x(u)‖
(‖z −w(v)‖+ ‖w(v)−w(u)‖+ ‖w(u)− zu‖)
≤D(Πm(η)) + 2δ + ‖x(v)− x(u)‖
D(Πm(η)) + ‖x(v)− x(u)‖+
i∈I[t1,t2]
(yi(v)− yi(u)) + 2δ
{D(Πm(η)) + 2δ + ‖x(v)− x(u)‖}+
i∈I[t1,t2]
(yi(v)− yi(u)).
Hence
i∈I[t1,t2]
(yi(v)− yi(u))≤
{D(Πm(η)) + 2δ + ‖x(v)− x(u)‖}
{D(Πm(η)) + 2η}.
On multiplying through by 4
, we obtain
i∈I[t1,t2]
(yi(v)− yi(u))≤
{D(Πm(η)) + 2η} ≤Πm(η).(29)
Hence, by (29) and the fact that for any x ∈Rd, ‖x‖ ≤
i=1 |xi|, we have
Osc(y, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ),(30)
and by (22), (29) and the definitions of Πm(·) and Πm(·), we have
Osc(w, [t1, t2])≤Osc(x, [t1, t2]) +
{D(Πm(η)) + 2η}
≤Πm(Osc(x, [t1, t2]) + δ),
as desired.
16 W. KANG AND R. J. WILLIAMS
Case 2. Suppose that there is t3 ∈ [t1, t2] such that w(t3) /∈ F
Πm(η)
I[t1,t2]
Define σ = inf{u ∈ [t1, t2] :w(u) /∈ F
Πm(η)
I[t1,t2]
}. Then σ ≤ t2. For each u ∈
[t1, σ), w(u) ∈ F
Πm(η)
I[t1,t2]
and so by a similar analysis to that for Case 1, we
obtain for each v ∈ [t1, σ),
Osc(w, [t1, v])≤ η+
(D(Πm(η)) + 2η)
Osc(y, [t1, v])≤
(D(Πm(η)) + 2η).
By the right continuity of paths we have w(σ) /∈ F
Πm(η)
I[t1,t2]
. Then there is an
i ∈ I[t1,t2] such that dist(w(σ), ∂Gi ∩ ∂G) ≥ Πm(η), and it follows that w
does not reach Uδ(∂Gi ∩ ∂G) during the interval [σ, t2]. To see this, let
τ = inf{u ∈ [σ, t2] : dist(w(u), ∂Gi ∩ ∂G) ≤ δ} with the convention that the
infimum of an empty set is ∞. If τ ≤ t2, then by the right continuity of
w(·) and since Πm(η)> δ, we have τ > σ and dist(w(τ), ∂Gi∩∂G)≤ δ. Also,
since |I[t1,t2]|=m, we have [σ,u] ∈ Tm−1 for each u ∈ [σ, τ). By the induction
assumption and letting u→ τ , we have ‖w(τ−)−w(σ)‖ ≤Πm−1(η). By (ii),
(iii)(b) and since ‖γi(·)‖= 1, we have
‖∆w(τ)‖ ≤ ‖∆x(τ)‖+
∆yi(τ)≤Osc(x, [t1, t2]) + Iδ ≤ Iη.
Then simple manipulations yield
dist(w(σ), ∂Gi ∩ ∂G)≤ ‖w(σ)−w(τ−)‖+ ‖∆w(τ)‖+ dist(w(τ), ∂Gi ∩ ∂G)
≤Πm−1(η) + Iη+ δ
<Πm(η).
This contradicts the fact that dist(w(σ), ∂Gi∩∂G)≥Πm(η), and so confirms
that w does not reach Uδ(∂Gi ∩ ∂G) in [σ, t2]. Thus we must have [σ, t2] ∈
Tm−1. Hence we have by the induction assumption that
Osc(w, [t1, t2])≤ sup
v∈[t1,σ)
Osc(w, [t1, v]) + ‖∆w(σ)‖+Osc(w, [σ, t2])
≤ η +
(D(Πm(η)) + 2η) + Iη+Πm−1(η)
≤Πm(Osc(x, [t1, t2]) + δ)
INVARIANCE PRINCIPLE FOR SRBMS 17
Osc(y, [t1, t2])≤ sup
v∈[t1,σ)
Osc(y, [t1, v]) + ‖∆y(σ)‖+Osc(y, [σ, t2])
(D(Πm(η)) + 2η) + Iη+Πm−1(η)
≤Πm(Osc(x, [t1, t2]) + δ).
On combining all of the cases above, we have
Osc(w, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ),(31)
Osc(y, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ).(32)
This completes the induction step. �
Remark. The proof of the above theorem was inspired by the proof of
Lemma 4.3 of [4]. Because of the condition (i) in Theorem 4.1, the oscilla-
tion inequality given here is localized. Similar, but nonlocalized, oscillation
inequalities were proved in [15] when G = Rd+ and in [3] for a sequence of
convex polyhedrons; in these cases, the direction of reflection was constant
on each boundary face.
4.2. C-tightness result. Throughout this subsection and the next, we
suppose that the following assumption holds in addition to (A1)–(A5).
Assumption 4.1. There is a sequence of strictly positive constants {δn}∞n=1
such that for each positive integer n, there are processes W n, W̃ n,Xn, αn
having paths in D([0,∞),Rd) and processes Y n, Ỹ n, βn having paths in
D([0,∞),RI) defined on some probability space (Ωn,Fn, Pn) such that:
(i) Pn-a.s., W n = W̃ n + αn and W̃ n(t) ∈Uδn(G) for all t≥ 0,
(ii) Pn-a.s., W n(t) =Xn(t)+
(0,t] γ
i,n(W n(s−),W n(s))dY ni (s) for
all t≥ 0, where for each i ∈ I , γi,n :Rd ×Rd → Rd is Borel measurable and
‖γi,n(y,x)‖= 1 for all x, y ∈Rd,
(iii) Y n = Ỹ n+βn, where βn is locally of bounded variation and Pn-a.s.,
for each i ∈ I ,
(a) Ỹ ni (0) = 0,
(b) Ỹ ni is nondecreasing and ∆Ỹ
i (t)≤ δ
n for all t > 0,
(c) Ỹ ni (t) =
(0,t] 1{W̃n(s)∈Uδn (∂Gi∩∂G)}
dỸ ni (s),
(iv) δn → 0 as n → ∞, and for each ε > 0, there is ηε > 0 and nε > 0
such that for each i ∈ I , ‖γi,n(y,x)− γi(x)‖< ε whenever ‖x− y‖< ηε and
n≥ nε,
18 W. KANG AND R. J. WILLIAMS
(v) αn → 0 and V(βn)→ 0 in probability, as n→∞,
(vi) {Xn} is C-tight.
Remark. A simple case in which (iv) above holds is where γi,n(y,x)≡
γi(y). In (v), V(βn) is the total variation process for βn (cf. Section 1.1).
The following theorem will play an important role in the proof of the in-
variance principle. It will be used to show that a sequence of processes sat-
isfying suitably perturbed versions of the defining conditions for an SRBM
[cf. (i)–(vi) above] is C-tight.
Theorem 4.2 (C-tightness). Suppose that Assumption 4.1 holds. Define
Zn = (W n,Xn, Y n) for each n. Then the sequence of processes {Zn}∞n=1 is
C-tight.
Remark. Note that C-tightness of {W n}, {Xn} and {Y n} implies C-
tightness of {Zn} (see Chapter VI, Corollary 3.33 of [12] for details).
Proof of Theorem 4.2. References here to (i)–(vi) are to the condi-
tions in Assumption 4.1.
Simple algebraic manipulations yield Pn-a.s.,
W̃ n(t) = X̃n(t) +
(0,t]
γi,n(W n(s−),W n(s))dỸ ni (s)(33)
= X̃n(t) + Ṽ n(t) +
(0,t]
γi(W̃ n(s))dỸ ni (s),(34)
where
X̃n(t) =Xn(t) +
−αn(t) +
(0,t]
γi,n(W n(s−),W n(s))dβni (s)
Ṽ n(t) =
(0,t]
(γi,n(W n(s−),W n(s))− γi(W n(s)))dỸ ni (s)
(0,t]
(γi(W n(s))− γi(W̃ n(s)))dỸ ni (s).
The hypotheses on αn, the total variation process V(βn) of βn, and the
fact that ‖γi,n(y,x)‖ = 1 for all x, y ∈ Rd and each i ∈ I , imply that the
process
−αn(·) +
(0,·]
γi,n(W n(s−),W n(s))dβni (s)
INVARIANCE PRINCIPLE FOR SRBMS 19
converges to 0 in probability as n→∞. Combining this with the fact that
{Xn}∞n=1 is C-tight, we obtain that {X̃
n}∞n=1 is C-tight.
Recall the positive nondecreasing function Π(·) from Theorem 4.1, and
the constants a, L and functions R(·) and D(·) from assumptions (A1)–(A5)
in Section 3. Recall also that ρ0 =
Fix ρ, ε, η, T such that 0 < ρ <min{
R(a/4)
}, ε > 0, η > 0 and T > 0.
By assumption (A3), there is a constant r1 > 0 such that
D(r)<min
for all r ∈ (0, r1].(37)
Since Π(u) → 0 as u → 0, there are constants 0 < r3 < r2 < min{r1,
such that
Π(r)<
for all r ∈ (0, r3].(38)
By (iv), there are 0< ε̃ <min{
} and n0 > 0 such that for all n≥ n0,
‖y−x‖<2ε̃
‖γi,n(y,x)− γi(x)‖<
.(39)
By (iv)–(vi), and Proposition 1.1, there exist an integer n1 > n0, a con-
stant M̃η,T > 0 and λ̃ ∈ (0, T ), such that for all n≥ n1,
0≤s≤T
‖X̃n(s)‖ ≤ M̃η,T
≥ 1− η/2,(40)
Pn{wT (X̃
n, λ̃)≥ ε̃} ≤ η/4,(41)
0≤s≤T
‖αn(s)‖<
6ILr2
≥ 1− η/4,(42)
δn <min
8(1 + I)
.(43)
To prove C-tightness of {W̃ n} and {Ỹ n} (and hence of {W n}, {Y n}),
by Proposition 1.1, it suffices to show that there exists a constant Nη,T > 0
such that for all n≥ n1,
Pn{wT (W̃
n, λ̃)≥ ε} ≤ η,(44)
Pn{wT (Ỹ
n, λ̃)≥ ε} ≤ η,(45)
0≤s≤T
‖W̃ n(s)‖ ≤Nη,T
≥ 1− η,(46)
0≤s≤T
‖Ỹ n(s)‖ ≤Nη,T
≥ 1− η.(47)
20 W. KANG AND R. J. WILLIAMS
For each n≥ 1, let Fn be a set in Fn such that Pn(Fn) = 1 and on Fn,
properties (iii)(a)–(c) hold, (33)–(36) hold, and W̃ n(t) ∈ Uδn(G) for all t≥ 0.
Fix a t such that 0≤ t < t+ λ̃≤ T . Let
τn = inf{s≥ t :W̃ n(s) ∈ Uδn(∂Gi ∩ ∂G) for some i ∈ I}.(48)
For each n≥ n1, let
wT (X̃
n, λ̃)< ε̃, sup
0≤s≤T
‖αn(s)‖<
6ILr2
0≤s≤T
‖X̃n(s)‖ ≤ M̃η,T
Then by (40)–(42) and the definition of Fn,
P{Hn} ≥ 1− η.(50)
Fix ωn ∈Hn. By the definition of wT (x,λ) in (3), we have that,
r,s∈[t,t+λ̃]
‖X̃n(s,ωn)− X̃n(r,ωn)‖< ε̃.(51)
Now there are two cases to consider for n ≥ n1 and u, v fixed such that
t≤ u < v ≤ t+ λ̃.
Case 1. ωn ∈ {τn > v}. In this case, by (iii)(c), Ỹ n(·, ωn) does not increase
on the interval (u, v], that is, Ỹ ni (v,ω
n)− Ỹ ni (u,ω
n) = 0 for all i ∈ I . Then
by (34) and (36),
W̃ n(v,ωn)− W̃ n(u,ωn) = X̃n(v,ωn)− X̃n(u,ωn).(52)
Hence, by (51),
‖W̃ n(v,ωn)− W̃ n(u,ωn)‖ ≤ sup
r,s∈[t,t+λ̃]
‖X̃n(s,ωn)− X̃n(r,ωn)‖< ε̃ < ε/8,
and we also have
‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖= 0< ε/2.
Case 2. ωn ∈ {τn ≤ v}. Then there is an i ∈ I such that W̃ n(τn, ωn) ∈
Uδn(∂Gi ∩ ∂G), since the set Uδn(∂Gi ∩ ∂G) is closed and W̃
n(·, ωn) is right
continuous. It follows that there is some x0 ∈ ∂G (which depends on ω
such that W̃ n(τn, ωn) is in the closed ball Bδn(x0)⊂Bρ(x0). To apply the
INVARIANCE PRINCIPLE FOR SRBMS 21
oscillation inequality in Theorem 4.1, we first prove the following:
W̃ n(r,ωn) ∈Bρ(x0) for all r satisfying τ
n ≤ r ≤ v.(53)
For the proof of (53), let
ξn = inf{r ≥ τn :W̃ n(r,ωn) /∈Bρ(x0)} ∧ v.(54)
By the definition of ξn, W̃ n(r,ωn) ∈Bρ(x0) for each r ∈ [τ
n, ξn). In order to
apply the oscillation inequality in Theorem 4.1 on the time interval [τn, ξn),
we show that
D(Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn)) + δn))<
.(55)
For each r ∈ (0, T ], by (i)–(iii) and (33), (49), (43), we have that
‖W n(r−, ωn)−W n(r,ωn)‖
≤ ‖W̃ n(r−, ωn)− W̃ n(r,ωn)‖+ ‖αn(r−, ωn)−αn(r,ωn)‖
≤ ‖∆X̃n(r,ωn)‖+2 sup
0≤s≤T
‖αn(s)‖+ Iδn
≤ ε̃+
< 2ε̃.
Hence by (39), for each r ∈ (0, T ],
‖γi,n(W n(r−, ωn),W n(r,ωn))− γi(W n(r,ωn))‖ ≤
.(56)
By (36), (56), Assumption (A4), (i) and (49), we have that for any s1, s2
such that u≤ s1 < s2 ≤ v,
‖Ṽ n(s2, ω
n)− Ṽ n(s1, ω
(s1,s2]
‖γi,n(W n(r−, ωn),W n(r,ωn))
− γi(W n(r,ωn))‖dỸ ni (r,ω
(s1,s2]
‖γi(W n(r,ωn))− γi(W̃ n(r,ωn))‖dỸ ni (r,ω
(Ỹ ni (s2, ω
n)− Ỹ ni (s1, ω
(s1,s2]
L‖W n(r,ωn)− W̃ n(r,ωn)‖dỸ ni (r,ω
(Ỹ ni (s2, ω
n)− Ỹ ni (s1, ω
6ILr2
(Ỹ ni (s2, ω
n)− Ỹ ni (s1, ω
22 W. KANG AND R. J. WILLIAMS
‖Ỹ n(s2, ω
n)− Ỹ n(s1, ω
σn = inf{s≥ τn :Osc(Ỹ n(·, ωn), [τn, s))> r2}.(58)
Note that Osc(Ỹ n(·, ωn), [τn, s)) as a function of s defined on (τn,∞) is
left continuous with finite right limits and is nondecreasing. By the right
continuity of Ỹ n, we know that
Osc(Ỹ n(·, ωn), [τn, s))→ 0 as s ↓ τn.
Thus, σn > τn, Osc(Ỹ n(·, ωn), [τn, σn))≤ r2 and on {σ
n <∞}, Osc(Ỹ n(·, ωn),
[τn, σn]) ≥ r2. By (57), (51), (43), the choice of ε, and since t≤ τ
n ≤ ξn ≤
v ≤ t+ λ̃, we have
Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn
≤Osc(X̃n(·, ωn), [τn, ξn ∧ σn))
+Osc(Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn
≤Osc(X̃n(·, ωn), [τn, ξn ∧ σn))
Osc(Ỹ n(·, ωn), [τn, ξn ∧ σn)) + δn
≤ ε̃+
r2 + δ
n < r3.
Then by (38) and the monotonicity of D(·), we have
D(Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn))
≤D(r2)≤D(r1)<
We claim that
σn ≥ ξn.(61)
To prove (61), we proceed by contradiction and suppose that σn < ξn. Then
by (60), with x= X̃n(·, ωn) + Ṽ n(·, ωn) and δ = δn, condition (iv) of The-
orem 4.1 holds with [s, t] = [τn, σn − 1/m] for all m sufficiently large. By
applying Theorem 4.1 and letting m→∞, we obtain using (34), (38) and
(59) that,
Osc(Ỹ n(·, ωn), [τn, σn))
≤Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn)(62)
≤Π(r3)<
INVARIANCE PRINCIPLE FOR SRBMS 23
By (62), (iii)(b) and (43), we obtain that
Osc(Ỹ n(·, ωn), [τn, σn])≤
+ Iδn < r2.
This contradicts the fact that Osc(Ỹ n(·, ωn), [τn, σn]) ≥ r2 on {σ
n < ∞},
and so (61) holds and (55) follows by (60).
By applying Theorem 4.1 on [τn, ξn − 1/m] and then letting m→∞, we
obtain using (61), (59) and (38), that
Osc(W̃ n(·, ωn), [τn, ξn))
≤Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn)
and similarly,
Osc(Ỹ n(·, ωn), [τn, ξn))<
.(63)
Then we have
‖W̃ n(ξn−, ωn)− x0‖
≤ ‖W̃ n(ξn−, ωn)− W̃ n(τn, ωn)‖+ ‖W̃ n(τn, ωn)− x0‖
+ δn.
Using hypotheses (ii), (iii)(b), and (33), (51), we obtain
‖W̃ n(ξn, ωn)− W̃ n(ξn−, ωn)‖
≤ ‖X̃n(ξn, ωn)− X̃n(ξn−, ωn)‖
‖γi,n(W n(ξn−, ωn),W n(ξn, ωn))‖
× (Ỹ ni (ξ
n, ωn)− Ỹ ni (ξ
n−, ωn))
≤ ε̃+ Iδn.
Hence
‖W̃ n(ξn, ωn)− x0‖ ≤ ‖W̃
n(ξn−, ωn)− x0‖
+ ‖W̃ n(ξn, ωn)− W̃ n(ξn−, ωn)‖
+ δn + ε̃+ Iδn
≤ ε̃+ (I+1)δn +
< ρ/8 + ρ/8 + ρ/8< ρ/2.
24 W. KANG AND R. J. WILLIAMS
It follows from this that ξn = v and (53) holds, as desired.
Then, by (33), (51), (iii)(b), (iii)(c), (63) and (43), we have
‖W̃ n(v,ωn)− W̃ n(u,ωn)‖
≤ sup
r,s∈[u,v]
‖X̃n(s,ωn)− X̃n(r,ωn)‖+
(Ỹ ni (v,ω
n)− Ỹ ni (u,ω
≤ ε̃+
(Ỹ ni (v,ω
n)− Ỹ ni (u∨ τ
n, ωn))
(Ỹ ni (u ∨ τ
n, ωn)− Ỹ ni (u,ω
n))(64)
≤ ε̃+ IOsc(Ỹ n(·, ωn), [u∨ τn, v)) +
∆Ỹ ni (v,ω
n) + Iδn
≤ ε̃+ I
+ Iδn + Iδn <
‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖ ≤
(Ỹ ni (v,ω
n)− Ỹ ni (u,ω
(Ỹ ni (v,ω
n)− Ỹ ni (u∨ τ
n, ωn))
(Ỹ ni (u ∨ τ
n, ωn)− Ỹ ni (u,ω
Here we have used the fact that Ỹi does not increase on (u, τ
n ∨ u) and can
jump at most by δn at τn, by the definition of τn and (iii)(c).
On combining the results from Case 1 and Case 2, we obtain that for each
n≥ n1,
u,v∈[t,t+λ̃]
‖W̃ n(v,ωn)− W̃ n(u,ωn)‖ : 0≤ t≤ t+ λ̃≤ T
< ε(66)
u,v∈[t,t+λ̃]
‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖ : 0≤ t≤ t+ λ̃≤ T
< ε.(67)
Hence since ωn ∈Hn was arbitrary, by (50), we have that (44) and (45) hold
for all n≥ n1.
Next we show that there is a constant Nη,T > 0 such that (46) and (47)
hold for all n≥ n1. By (66)–(67) above, we have that for each n≥ n1, ω
INVARIANCE PRINCIPLE FOR SRBMS 25
Hn, t such that 0≤ t < t+ λ̃≤ T and t≤ u < v ≤ t+ λ̃,
‖W̃ n(v,ωn)− W̃ n(u,ωn)‖< ε(68)
‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖< ε.(69)
Then, for each 0≤ s≤ T , by (68), (69), (49) and (33), we have
‖W̃ n(s,ωn)‖ ≤ ‖W̃ n(s,ωn)− W̃ n(0, ωn)‖+ ‖W̃ n(0, ωn)‖
[T/λ̃]+1∑
‖W̃ n(iλ̃∧ s,ωn)− W̃ n((i− 1)λ̃ ∧ s,ωn)‖+ ‖X̃n(0, ωn)‖
≤ ([T/λ̃] + 1)ε+ M̃η,T
‖Ỹ n(s,ωn)‖ ≤ ‖Ỹ n(s,ωn)− Ỹ n(0, ωn)‖
[T/λ̃]+1∑
‖Ỹ n(iλ̃ ∧ s,ωn)− Ỹ n((i− 1)λ̃∧ s,ωn)‖
≤ ([T/λ̃] + 1)ε.
Here [T/λ̃] is the greatest integer less than or equal to T/λ̃. Let Nη,T =
([T/λ̃] + 1)ε+ M̃η,T . Then we obtain that for n≥ n1 and ω
n ∈Hn,
0≤s≤T
‖W̃ n(s,ωn)‖ ≤Nη,T(70)
0≤s≤T
‖Ỹ n(s,ωn)‖ ≤Nη,T .(71)
Then by (50), we have that (46) and (47) hold for all n≥ n1.
Finally by applying Proposition 1.1, we have the C-tightness of {W̃ n}
and {Ỹ n}. It then follows that {(W̃ n,Xn, Ỹ n)}∞n=1 is C-tight. Since Z
(W̃ n,Xn, Ỹ n) + (αn,0, βn) where αn,V(βn) → 0 in probability as n→∞,
then {Zn}∞n=1 is also C-tight. �
4.3. Invariance principle for SRBMs. The main theorem of the paper is
the following.
Theorem 4.3 (Invariance principle for SRBMs). Suppose that Assump-
tion 4.1 holds. Define Zn = (W n,Xn, Y n) for each n. Then the sequence of
26 W. KANG AND R. J. WILLIAMS
processes {Zn}∞n=1 is C-tight and any (weak) limit point of this sequence is
of the form Z = (W,X,Y ) where continuous d-dimensional processes W,X
and a continuous I-dimensional process Y are defined on some probability
space (Ω,F , P ) such that conditions (i), (ii) and (iv) of Definition 2.1 hold
with Ft = σ{Z(s) : 0≤ s≤ t}, t≥ 0.
If, in addition, the following conditions (vi)′ and (vii) hold, then any
weak limit point of the sequence {Zn}∞n=1 is an extended SRBM associated
with the data (G,µ,Γ,{γi, i ∈ I}, ν). If furthermore the following condition
(viii) holds, then W n ⇒W as n→∞ where W is an SRBM associated with
(G,µ,Γ,{γi, i ∈ I}, ν).
(vi)′ {Xn} converges in distribution to a d-dimensional Brownian mo-
tion with drift µ, covariance matrix Γ and initial distribution ν.
(vii) For each (weak) limit point Z = (W,X,Y ) of {Zn}∞n=1, {X(t) −
X(0)− µt, Ft, t≥ 0} is a martingale.
(viii) If a process W satisfies the properties in Definition 2.1, the law
of W is unique, that is, the law of an SRBM associated with the data
(G,µ,Γ,{γi, i ∈ I}, ν) is unique.
Remark. We note that (vi)′ implies that (vi) of Assumption 4.1 holds.
Proof of Theorem 4.3. By Theorem 4.2, we have that the sequence
{Zn}∞n=1 is C-tight. Let Z = (W,X,Y ) be a (weak) limit point of {Z
n}∞n=1,
that is, there is a subsequence {nk} of {n} such that Z
nk ⇒Z as k→∞. It
also follows that Z̃nk ≡ (W̃ nk ,Xnk , Ỹ nk)⇒Z as k→∞. By the C-tightness
of {Zn}, we obtain that Z has continuous paths a.s. For the purpose of
verifying that Z satisfies the listed properties in Definition 2.1, one may
invoke the Skorokhod representation theorem to assume, without loss of
generality, that Znk and Z̃nk converge u.o.c. to Z a.s. as k→∞ and V(βnk)
converges u.o.c. to 0 a.s. as k → ∞. With this simplification, it is easily
verified that the properties of {Znk} and {Z̃nk} imply that Z has properties
(ii) and (iv)(a)–(b) of Definition 2.1. For the verification of property (i) of
Definition 2.1, note that for each k, a.s. for each t≥ 0,
W nk(t) =Xnk(t) +
(0,t]
γi,nk(W nk(s−),W nk(s))dβ
i (s)
(0,t]
(γi,nk(W nk(s−),W nk(s))− γi(W nk(s)))dỸ
i (s)
(0,t]
γi(W nk(s))dỸ
i (s).
The sum of the first two terms on the right-hand side of the above equality
converges a.s. to X(t) as k → ∞. The third term on the right-hand side
INVARIANCE PRINCIPLE FOR SRBMS 27
converges a.s. to 0 as k→∞, by property (iv) and the fact that a.s.,
s∈(0,t]
‖W nk(s)−W nk(s−)‖
≤ sup
s∈(0,t]
‖∆Xnk(s)‖+ I sup
s∈(0,t]
‖∆Y nk(s)‖→ 0 as k→∞.
It remains to show that for each i ∈ I and t≥ 0, a.s.,
(0,t]
γi(W nk(s))dỸ
i (s)→
(0,t]
γi(W (s))dYi(s) as k→∞.
This follows directly from Lemma A.4.
For the verification of property (iv)(c) of Definition 2.1, it suffices to show
that for each i ∈ I , m= 1,2, . . . , a.s. for each t≥ 0,
Yi(t) =
(0,t]
fm(W (s))dYi(s),(72)
where {fm}
m=1 is a sequence of real valued continuous functions defined
on Rd such that for each m, the range of fm is [0,1], fm(x) = 1 for x ∈
U1/m(∂Gi ∩ ∂G) and fm(x) = 0 for x /∈ U2/m(∂Gi ∩ ∂G). The existence
of such a sequence of continuous functions {fm}
m=1 can be shown using
Urysohn’s lemma (cf. [8], page 122). Then (72) is a consequence of Lemma
A.4, property (iii) of Ỹ
i and the fact that δ
nk → 0 as k→∞. Indeed, a.s.,
for each t≥ 0,
Yi(t) = lim
i (t) = lim
(0,t]
{W̃nk (s)∈U
nk (∂Gi∩∂G)}
i (s)
= lim
(0,t]
fm(W̃
nk(s))dỸ
i (s)
(0,t]
fm(W (s))dYi(s).
Thus, Z satisfies properties (i), (ii) and (iv) of Definition 2.1 with Ft =
σ{Z(s) : 0≤ s≤ t}, t≥ 0.
Assuming properties (vi)′ and (vii) hold, Z satisfies (iii) of Definition
2.1. Then Z is an extended SRBM associated with the data (G,µ,Γ,{γi, i ∈
I}, ν). If in addition, property (viii) holds, then the law of W is unique. Since
each weak limit W is an SRBM associated with the data (G,µ,Γ,{γi, i ∈
I}, ν) and the law of such an SRBM is unique, then by a standard argument,
W n ⇒W as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈
I}, ν). �
Some sufficient conditions for (vii) to hold are given in Proposition 4.2 of
[15] for a simpler setting where G=Rd+. Two of those conditions generalize
to our setting here and can be proved in the same manner as in [15]. For
completeness, we state the ensuing result here.
28 W. KANG AND R. J. WILLIAMS
Proposition 4.1. Suppose that Assumption 4.1 and (vi)′ of Theorem
4.3 hold. If, in addition, one of the following conditions (I)–(II) holds, then
condition (vii) of Theorem 4.3 is satisfied, and any weak limit point of
{Zn}∞n=1 is an extended SRBM associated with (G,µ,Γ,{γ
i, i ∈ I}, ν).
(I) For any triple of d-dimensional {Ft}-adapted processes (W,X,Y )
defined on some filtered probability space (Ω,F ,{Ft}, P ) and satisfying con-
ditions (i), (ii) and (iv) of Definition 2.1 together with the condition that
X, under P , is a d-dimensional Brownian motion with drift vector µ, co-
variance matrix Γ and initial distribution ν, the pair (W,Y ) is adapted to
the filtration generated by X and the P -null sets.
(II) Xn = X̌n + εn1 , Y
n = Y̌ n + εn2 , W
n = W̌ n + εn3 , where ε
1 , ε
2 , ε
3 are
processes converging to 0 in probability as n→∞, and:
(a) {X̌n(t)− X̌n(0)}∞n=1 is uniformly integrable for each t≥ 0,
(b) there is a sequence of constants {µn}∞n=1 in R
d such that
limn→∞µ
n = µ,
(c) for each n, {X̌n(t)− X̌n(0)−µnt, t≥ 0} is a Pn-martingale with
respect to the filtration generated by (W̌ n, X̌n, Y̌ n).
In the rest of this work, we focus on applications of the invariance prin-
ciple and in particular on giving sufficient conditions for property (viii) of
Theorem 4.3 to hold.
5. Applications of the invariance principle. In Section 5.1, we prove weak
existence of SRBMs associated with data (G,µ,Γ,{γi, i ∈ I}, ν) satisfying
(A1)–(A5) of Section 3. This is accomplished by constructing a sequence of
approximations whose weak limit points are SRBMs. The invariance prin-
ciple is used to prove the C-tightness of the approximations and that any
weak limit point is an SRBM. In Sections 5.2 and 5.3, using known results
on uniqueness in law for SRBMs, we illustrate the invariance principle for
certain domains and directions of reflection.
5.1. Weak existence of SRBMs.
Theorem 5.1. Suppose that assumptions (A1)–(A5) of Section 3 hold.
Then there exists an SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}, ν).
Proof. We construct a sequence of approximations to an SRBM and
use the invariance principle to establish weak convergence along a subse-
quence to an SRBM.
In the following we will use R(·) from assumption (A2), L > 0 from as-
sumption (A4), a > 0 from assumption (A5), and ρ0 =
. Fix ε > 0 and
INVARIANCE PRINCIPLE FOR SRBMS 29
0 < ρ < min{
R(a/4)
}. By assumption (A3), there is a constant r1 > 0
such that
D(r)<min
for all r ∈ (0, r1].
Recall the properties of Π(·) from Theorem 4.1. Since Π(u)→ 0 as u→ 0,
there are constants 0< r3 < r2 <min{r1,
} such that
Π(r)<
for all r ∈ (0, r3].
Fix ε̃ and δ such that 0< ε̃ <min{
24ILr2
} and 0< 2δ <min{r3
8(1+I)
We will construct a d-dimensional stochastic processW δ and an I-dimensional
“pushing” process Y δ , such that W δ approximately satisfies the conditions
defining an SRBM for the data (G,µ,Γ,{γi, i ∈ I}, ν) (cf. Assumption 4.1).
The idea for this construction is to use a Brownian motion X with drift
vector µ, covariance matrix Γ and initial distribution ν. Away from ∂G, the
increments of W δ are determined by those of X . For any time t ≥ 0 such
that W δ(t−) ∈ ∂G, we add an instantaneous jump to W δ(t−) to obtain
W δ(t) ∈G. Here W δ(0−) =X(0). The size of the jump is such that W δ(t)
is a strictly positive distance (depending on δ) from the boundary of G.
The jump vector is obtained as a measurable function of W δ(t−). To ensure
the measurability, each point x on ∂G is associated with a nearby point x̄,
chosen in a measurable way from a fixed countable set of points in ∂G. The
jump vector for x is one associated with x̄. We now specify the mapping
x→ x̄ and the associated jump vector more precisely.
By assumption (A5)(ii), for each x ∈ ∂G, there is c(x) ∈RI+ such that
i∈I(x)
ci(x) = 1 and min
j∈I(x)
i∈I(x)
ci(x)γ
i(x), nj(x)
≥ a.(73)
By (73), Lemma 2.1 and the fact that ni(·) is continuous on ∂Gi for each
i ∈ I , we have that for each x ∈ ∂G there is rx ∈ (0, δ) such that for each
y ∈Brx(x)∩ ∂G,
I(y)⊂ I(x)(74)
j∈I(x)
i∈I(x)
ci(x)γ
i(x), nj(y)
.(75)
It follows, using the C1 nature of ∂Gi and the fact that n
i(y) is the inward
unit normal to ∂Gi at y ∈ ∂G for each i ∈ I(y), that (by choosing rx even
30 W. KANG AND R. J. WILLIAMS
smaller if necessary) for each x ∈ ∂G there is m(x)> 0 and rx ∈ (0, δ) such
that for each y ∈Brx(x)∩ ∂G, (74)–(75) hold and
y + λ
i∈I(x)
ci(x)γ
i(x) ∈G for all λ ∈ (0,m(x)).(76)
Let Borx(x) denote the interior of the closed ball Brx(x) for each x ∈ ∂G. The
collection {Borx(x) :x ∈ ∂G} is an open cover of ∂G and it follows that there
is a countable set {xk} such that ∂G⊂
kBrxk (xk) and {xk} ∩BN (0) is a
finite set for each integer N ≥ 1. We can further choose the set {xk} to be
minimal in the sense that for each strict subset C of {xk}, {Brx(x) :x ∈C}
does not cover ∂G. Let Dk = (Brxk (xk) \ (
i=1 Brxi (xi)) ∩ ∂G for each k.
Then Dk 6= ∅ for each k, {Dk} is a partition of ∂G, and for each x ∈ ∂G
there is a unique index i(x) such that x ∈Di(x). For each x ∈R
d, let
x, if x /∈ ∂G,
xi(x), if x ∈ ∂G.
Note that for all x ∈Rd,
‖x− x̄‖< δ.(77)
For each i ∈ I and x∈Rd, let
γi,δ(x) = γi(x̄).(78)
The mapping x → x̄ is Borel measurable on Rd and hence γi,δ is a Borel
measurable function from Rd into Rd.
We construct (W δ, Y δ) as follows. Let X defined on some filtered proba-
bility space (Ω,F ,{Ft}, P ) be a d-dimensional {Ft}-Brownian motion with
drift µ and covariance matrix Γ such that X is continuous surely and X(0)
has distribution ν. Let
τ1 = inf{t≥ 0 :X(t) ∈ ∂G}
W δ(t) =X(t), Y δ(t) = 0 for 0≤ t < τ1.
Note that W δ(τ1−) exists on {τ1 <∞} since X has continuous paths and
in the case that τ1 = 0, W
δ(0−)≡X(0). On {τ1 <∞}, define
Y δi (τ1) =
0, i /∈ I(W δ(τ1−)),
ci(W δ(τ1−))
m(W δ(τ1−))
, i ∈ I(W δ(τ1−)),
W δ(τ1) =X(τ1)
m(W δ(τ1−))
i∈I(W δ(τ1−))
ci(W δ(τ1−))γ
i,δ(W δ(τ1−))
INVARIANCE PRINCIPLE FOR SRBMS 31
So W δ, Y δ have been defined on [0, τ1) and at τ1 on {τ1 <∞}, such that:
(i) W δ(t) = X(t) +
i∈I γ
i,δ(W δ(0−))Y δi (0) +
(0,t]
γi,δ(W δ(s−))dY δi (s) for all t ∈ [0, τ1]∩ [0,∞), where W
δ(0−) =X(0),
(ii) W δ(t) ∈G for t ∈ [0, τ1]∩ [0,∞),
(iii) for i ∈ I ,
(a) Y δi (0)≥ 0,
(b) Y δi is nondecreasing on [0, τ1]∩ [0,∞),
(c) Y δi (t) = Y
i (0) +
(0,t] 1{W δ(s)∈U2δ(∂Gi∩∂G)} dY
i (s) for t ∈ [0, τ1] ∩
[0,∞),
(iv) ‖∆Y δ(t)‖ ≡ ‖Y δ(t)−Y δ(t−)‖ ≤ δ for t ∈ [0, τ1]∩ [0,∞), where Y
δ(0−)≡
Note that (iii)(c) above contains the expression W δ(s) ∈U2δ(∂Gi∩∂G). The
reader may wonder why 2δ appears instead of δ. The reason is that at a jump
time s of Y δi , W
δ(s−) ∈ ∂Gi ∩ ∂G and so
dist(W δ(s), ∂Gi ∩ ∂G)≤ ‖W
δ(s)−W δ(s−)‖+ ‖W δ(s−)−W δ(s−)‖
≤ δ + δ = 2δ.
Proceeding by induction, we assume that for some n ≥ 2, τ1 ≤ · · · ≤ τn−1
have been defined, and W δ, Y δ have been defined on [0, τn−1) and at τn−1
on {τn−1 <∞}, such that (i)–(iv) above hold with τn−1 in place of τ1. Then
we define τn =∞ on {τn−1 =∞}, and on {τn−1 <∞} we define
τn = inf{t≥ τn−1 :W
δ(τn−1) +X(t)−X(τn−1) ∈ ∂G}.
For τn−1 ≤ t < τn, let
Y δ(t) = Y δ(τn−1),
W δ(t) =W δ(τn−1) +X(t)−X(τn−1),
and on {τn <∞}, let
Y δi (τn) =
Y δi (τn−1), i /∈ I(W
δ(τn−)),
Y δi (τn−1) + ci(W
δ(τn−))
m(W δ(τn−))
, i ∈ I(W δ(τn−)),
W δ(τn) =W
δ(τn−)
m(W δ(τn−))
i∈I(W δ(τn−))
ci(W δ(τn−))γ
i,δ(W δ(τn−))
32 W. KANG AND R. J. WILLIAMS
In this way, W δ, Y δ have been defined on [0, τn) and at τn on {τn <∞} such
that (i)–(iv) hold with τn in place of τ1.
By construction {τn}
n=1 is a nondecreasing sequence of stopping times.
Let τ = limn→∞ τn. On {τ =∞}, the construction of (W
δ, Y δ) is complete.
We now show that {τ < ∞} = ∅. In fact, if {τ < ∞} 6= ∅, let ω ∈ {τ <
∞}. The above construction gives (W δ(·, ω), Y δ(·, ω)) on the time interval
[0, τ(ω)). For each t ∈ [0, τ(ω)), we have
W δ(t,ω) =X(t,ω) +
γi,δ(W δ(0−, ω))Y δi (0, ω)
(0,t]
γi,δ(W δ(s−, ω))dY δi (s,ω).
SinceX is continuous on [0,∞), ‖γi,δ(x)‖= 1 for each x ∈Rd and
i∈I Y
i (0,
ω)≤ δ, there are constants λ̃ ∈ (0, τ(ω)) and M̃ > 0 (depending on ω) such
wτ(ω)(X(·, ω) + γ
i,δ(W δ(0−, ω))Y δi (0, ω), λ̃)< ε̃(80)
0≤t≤τ(ω)
∥∥∥∥∥X(·, ω) +
γi,δ(W δ(0−, ω))Y δi (0, ω)
∥∥∥∥∥≤ M̃,(81)
where w
(·, ·) is defined in (3). By the choice of ε̃, δ made at the beginning of
this proof, (77)–(78) and the uniform Lipschitz property of the γi(·), i ∈ I , it
follows that (39) and (43) hold with γi,δ(y) and 2δ in place of γi,n(y,x) and
δn, respectively. Then by similar pathwise analysis to that used in Case 1 and
2 of the proof of Theorem 4.2, with W̃ n =W n =W δ, αn = 0, γi,n(y,x) =
γi,δ(y) for each i ∈ I and x, y ∈ Rd, Xn = X +
i∈I γ
i,δ(W δ(0−))Y δi (0),
Y n = Y δ , Ỹ n = Y δ − Y δ(0), βn = Y δ(0) and δn = 2δ, we obtain that (71)
holds for any T < τ(ω) with ωn = ω, Nη,T = ([τ(ω)/λ̃] + 1)ε+ M̃ . It follows
that supi∈I sups∈[0,τ(ω)) Y
i (s,ω) is finite. By the nondecreasing property of
Y δi (·, ω) on [0, τ(ω)) for each i ∈ I , Y
i (τ(ω)−, ω) exists and is finite for each
i ∈ I . Then by (79) and the continuity of X , we see that W δ(τ(ω)−, ω) exists
and is finite. By the construction of Y δ and the fact that
i∈I(x) ci(x) = 1
for all x∈ ∂G, we have that
Y δi (τ(ω)−, ω) =
m(W δ(τn(ω)−, ω))
∧ δ.(82)
Since τn(ω) ↑ τ(ω) as n → ∞ and W
δ(τ(ω)−, ω) exists, it follows that
{W δ(τn(ω)−, ω)}
n=1 converges to W
δ(τ(ω)−, ω) ∈ ∂G as n → ∞. Conse-
quently, {W δ(τn(ω)−, ω)}
n=1 is a bounded sequence in ∂G and so by the
INVARIANCE PRINCIPLE FOR SRBMS 33
definition of the sets {Dk} which form a partition of ∂G, there is a finite set
C such that
{W δ(τn(ω)−, ω)}
n=1 ⊂
Hence,
m(W δ(τn(ω)−, ω))≤ inf
m(xk)> 0,(83)
and so the right-hand side of (82) is infinite. On the other hand, since
supi∈I sups∈[0,τ(ω)) Y
i (s,ω) is finite, the left-hand side of (82) is finite. This
yields the desired contradiction and so {τ < ∞} = ∅ and we have con-
structed (W δ, Y δ) on [0,∞).
From the construction above, we can see that W δ and Y δ are well-defined
stochastic processes with sample paths in D([0,∞),Rd) and D([0,∞),RI).
They are adapted to the filtration generated by X and satisfy (i)–(iv) above
with [0,∞) in place of [0, τ1].
Consider a sequence of sufficiently small δ’s, denoted by {δn}, such that
δn ↓ 0 as n → ∞. For each δn, let (W δ
, Y δ
) be the pair constructed as
above for the same process X . By the above properties and the fact that for
each i ∈ I and x, y ∈Rd,
‖γi,δ
(y)− γi(x)‖ ≤ ‖γi(ȳ)− γi(x)‖ ≤ L‖ȳ− x‖ ≤ L(δn + ‖y − x‖),
we obtain that Assumption 4.1 holds with W̃ n =W n =W δ
, αn = 0, γi,n(y,x) =
(y) for each i ∈ I and x, y ∈Rd, Xn =X+
i∈I γ
i,δn(W δ
(0−))Y δ
i (0),
Y n = Y δ
, Ỹ n = Y δ
(0), βn = Y δ
(0) and 2δn in place of δn. By invok-
ing the first part of Theorem 4.3, we obtain that {Zδ
}∞n=1 = {(W
δn ,Xδ
)}∞n=1 is C-tight and any weak limit point Z of this sequence satisfies
conditions (i), (ii) and (iv) of Definition 2.1 with Ft = σ{Z(s) : 0 ≤ s ≤ t},
t≥ 0. Note that condition (vi)′ of Theorem 4.3 holds trivially. Furthermore,
= {Xδ
(t)−Xδ
(0)− µt, t≥ 0}= {X(t)−X(0)− µt, t≥ 0} is a mar-
tingale with respect to the filtration generated by X . Since W δ
, Y δ
adapted to this filtration, it follows that M δ
is a martingale with respect
to the filtration generated by W δ
, Y δ
(which in fact is the same as
that generated by X). For each t≥ 0, Xδ
(t)−Xδ
(0) =X(t)−X(0) and
so trivially this forms a uniformly integrable sequence as n varies. It fol-
lows from Proposition 4.1 that condition (vii) of Theorem 4.3 holds. Hence,
any weak limit point of {Zδ
}∞n=1 is an extended SRBM with the data
(G,µ,Γ,{γi, i ∈ I}, ν). �
5.2. SRBMs in convex polyhedrons with constant reflection fields. Exis-
tence and uniqueness in law for SRBMs living in convex polyhedrons with a
constant reflection field on each boundary face has been studied by Dai and
34 W. KANG AND R. J. WILLIAMS
Williams [4]. In this subsection, we state a consequence of our invariance
principle using the results in [4] to establish uniqueness in law. In this case,
G is defined in terms of I (I≥ 1) d-dimensional unit vectors {ni, i ∈ I} and
an I-dimensional vector β = (β1, . . . , βI)
′ such that
G≡ {x ∈Rd : 〈ni, x〉 ≥ βi for all i ∈ I}.(84)
It is assumed that G is nonempty and that the set {(n1, β1), . . . , (n
I, βI)} is
minimal in the sense that no proper subset defines G. For each i ∈ I , let Fi
denote the boundary face: {x ∈G : 〈ni, x〉= βi}. Then, n
i is the inward unit
normal to Fi. A constant vector field γ
i of unit length specifies the direction
of reflection associated with Fi.
Definition 5.2. For each ∅ 6=K⊂ I , define FK =
i∈KFi. Let F∅ =G.
A set K⊂ I is maximal if K 6=∅, FK 6=∅ and FK 6= FK̄ for any K̄ ⊃ K such
that K̄ 6=K.
In [4], Dai and Williams introduced the following assumption.
Assumption 5.1. For each maximal K⊂ I ,
(S.a) there is a positive linear combination n=
i∈K bin
i (bi > 0 ∀i ∈K)
of the {ni, i ∈K} such that 〈n,γi〉> 0 for all i ∈K,
(S.b) there is a positive linear combination γ =
i∈K ciγ
i (ci > 0 ∀i ∈K)
of the {γi, i ∈K} such that 〈ni, γ〉> 0 for all i ∈K.
Remark. For the given G and constant vector fields {γi, i ∈ I}, As-
sumption 5.1 is equivalent to assumption (A5).
Definition 5.3. The convex polyhedron G is simple if for each K⊂ I
such that K 6=∅ and FK 6=∅, exactly |K| distinct faces contain FK.
Remark. The polyhedron G is simple if and only if K is maximal for
every K such that ∅ 6= K ⊂ I and FK 6= ∅. It is shown in [4] that when G
is simple, (S.a) holds for all maximal K⊂ I if and only if (S.b) holds for all
maximal K⊂ I.
Dai and Williams [4] showed that Assumption 5.1 is sufficient for exis-
tence and uniqueness in law of SRBMs living in G with the reflection fields
{γi, i ∈ I} and fixed starting point. [They also showed that condition (S.b)
holding for all maximal K⊂ I is necessary for existence of an SRBM starting
from each point in G. Consequently, when G is simple, Assumption 5.1 is
necessary and sufficient for existence of an SRBM starting from each point
in G.] This yields the following consequence of our invariance principle.
INVARIANCE PRINCIPLE FOR SRBMS 35
Theorem 5.4. Let G be a nonempty domain such that G is a convex
polyhedron of the form (84) (with minimal description), and let {γi, i ∈ I} be
a family of constant vector fields of unit length satisfying Assumption 5.1.
Suppose that Assumption 4.1 and (vi)′, (vii) of Theorem 4.3 hold. Then
W n ⇒W as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈
I}, ν).
Proof. Clearly (A1) holds. Assumptions (A2)–(A3) hold by Lemma
A.3. Since for each i ∈ I , γi(·) is a constant vector field of unit length, as-
sumption (A4) holds trivially. Assumption (A5) is implied by Assumption
5.1. Hence by Theorem 4.3, the only thing that we have to check is condition
(viii) of Theorem 4.3, that is, uniqueness in law for SRBMs in convex poly-
hedrons with constant reflection fields of unit length. But this is proved in
Theorem 1.3 of [4] for a fixed starting point in G and follows by a standard
conditioning argument for the initial distribution ν. �
5.3. SRBMs in bounded domains with piecewise smooth boundaries. Dupuis
and Ishii [6] have established sufficient conditions for the existence and path-
wise uniqueness of reflecting diffusions living in the closures of bounded
domains with piecewise smooth boundaries. In this subsection, we state a
consequence of our invariance principle using the results in [6] to establish
uniqueness in law.
Theorem 5.5. Let G be a bounded domain and {γi, i ∈ I} be a family
of reflection fields that satisfy assumptions (A1)–(A4) and (A5)′ in Section
3. We further assume that for each i ∈ I , γi(·) is once continuously differ-
entiable with locally Lipschitz continuous first partial derivatives. Suppose
that Assumption 4.1 and (vi)′, (vii) of Theorem 4.3 hold. Then W n ⇒W
as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈ I}, ν).
Remark. We remind the reader that in view of Lemma 3.1, to verify
condition (A5)′, one only needs to show that (i) or (ii) holds for all x ∈ ∂G.
However, as can be seen from the proof below, both forms of the condition
can be useful.
Proof of Theorem 5.5. This theorem follows from Theorem 4.3 and
uniqueness in law for the associated SRBMs. The latter follows by a standard
argument from the pathwise uniqueness established in Corollary 5.2 of [6] for
their Case 2. The conditions required for that case are satisfied in particular
because (A5)′(ii) implies condition (3.8) of [6]. That condition (3.8) readily
implies condition (3.6) of [6]; and, by [5], under the additional smoothness
assumptions imposed on the γi in the statement of our theorem, condition
(3.8) also implies condition (3.7) in [6]. In addition, (A5)′(i) implies that
36 W. KANG AND R. J. WILLIAMS
for each x ∈ ∂G, 〈γi(x), ni(x)〉> 0 for each i ∈ I(x), and furthermore, since
(A5)′ implies (A5), we have by (A5)(i) that the origin does not belong to
the convex hull of the {γi(x) : i ∈ I(x)}. �
APPENDIX: AUXILIARY LEMMAS
Lemma A.1. Suppose that G is bounded. If assumption (A1) holds, then
assumption (A2) holds.
Proof. To see this, suppose G is bounded and assumption (A1) holds.
Fix ε ∈ (0,1). For each i ∈ I and z ∈ ∂Gi ∩ ∂G, by the C
1 property of ∂Gi,
there is a neighborhood Vz of z and a constant R(ε, i, z) > 0 such that for
all x ∈ Vz ∩ ∂Gi ∩ ∂G and y ∈Gi such that ‖x− y‖<R(ε, i, z),
〈ni(x), y − x〉 ≥−ε‖y − x‖.(85)
Assumption (A2) then follows by a standard compactness argument. �
Lemma A.2. Suppose that G is a nonempty bounded domain satisfying
(5), where for each i ∈ I , Gi is a nonempty domain. Then assumption (A3)
holds.
Proof. We prove the lemma by contradiction. Suppose that assumption
(A3) does not hold. Then, since there are only finite many J ⊂ I , J 6=∅,
there is an ε > 0, a nonempty set J ⊂ I , a sequence {rn} ⊂ (0,∞) with rn →
0 as n→∞, a sequence {xn} ⊂R
d such that for each n, xn ∈
j∈J Urn(∂Gj∩
∂G) and dist(xn,
j∈J (∂Gj ∩ ∂G)) > ε. But since G is bounded, {xn} is
bounded and without loss of generality we may assume that xn → x as
n→∞ for some x ∈Rd. It follows that x ∈
j∈J (∂Gj ∩ ∂G), since for each
j ∈ J ,
dist(x,∂Gj ∩ ∂G)≤ ‖xn − x‖+dist(xn, ∂Gj ∩ ∂G)≤ ‖xn − x‖+ rn → 0
as n→∞. This is inconsistent with xn → x and dist(xn,
j∈J (∂Gj ∩∂G))>
Lemma A.3. Suppose (A1) holds where
Gi = {x ∈R
d : 〈ni, x〉> βi} for i ∈ I,(86)
{ni, i ∈ I} is a finite collection of d-dimensional vectors of unit length, and
for I= |I|, β = (β1, . . . , βI)
′ is an I-dimensional vector. (Thus, G is a convex
polyhedron.) Assume that for each i ∈ I , ∂Gi ∩ ∂G 6=∅. Then assumptions
(A2) and (A3) hold.
INVARIANCE PRINCIPLE FOR SRBMS 37
Proof. Assumption (A2) holds automatically since G is convex. In or-
der to show that assumption (A3) holds, we just need to show that for each
J ⊂ I with J 6=∅,
(∂Gj ∩ ∂G)
Ur(∂Gj ∩ ∂G)
→ 0(87)
as r→ 0. Fix J ⊂ I such that J 6=∅. Then
j∈J (∂Gj ∩∂G) is the collection
of all solutions x ∈Rd to the following system of linear inequalities:
〈ni, x〉 ≥ βi for all i ∈ I,
〈−ni, x〉 ≥ −βi for all i ∈ J .
Suppose that
j∈J (∂Gj ∩∂G) 6=∅, that is, (LS) has at least one solution.
By a theorem of Hoffman [11], with supporting lemmas proved by Agmon
[1], there is a constant C > 0 (depending only on {ni, i ∈ I} and not on β)
such that for any x ∈Rd there exists a solution x0 ∈R
d of (LS) with
‖x− x0‖ ≤C
(βi − 〈n
i, x〉)+ +
(−βi − 〈−n
i, x〉)+
.(88)
For r > 0, any x ∈
j∈J Ur(∂Gj ∩ ∂G) satisfies the following:
〈ni, x〉 ≥ βi − r for all i ∈ I,
(r-LS)
〈−ni, x〉 ≥ −βi − r for all i ∈ J .
Then by (88), there is x0 ∈
j∈J (∂Gj ∩ ∂G) such that
(∂Gj ∩ ∂G)
≤ ‖x− x0‖ ≤ 2C|I|r.
It follows that (87) holds when
j∈J (∂Gj ∩ ∂G) 6=∅.
Now suppose that
j∈J (∂Gj ∩∂G) =∅, that is, (LS) has no solution. We
shall use an argument by contradiction to show that
j∈J Ur(∂Gj ∩∂G) =∅
for all r sufficiently small. Suppose that this is not true. Then we have that⋂
j∈J Ur(∂Gj ∩ ∂G) 6= ∅ for all r ∈ (0,∞). As we have seen before, any
j∈J Ur(∂Gj ∩ ∂G) is a solution to (r-LS). We now construct a Cauchy
sequence. Let x1 ∈
j∈J U1/2(∂Gj ∩ ∂G). Then x1 is a solution to (
-LS).
Since ( 1
-LS) has at least one solution, by the theorem of Hoffman [11] (using
the fact that the constant C depends only on {ni, i ∈ I}), we conclude that
there is a solution x2 to (
-LS) such that ‖x1−x2‖ ≤
, where C ′ = 2C|I|.
Continuing in this manner, we can obtain a sequence {xn}
n=1 such that for
each n ≥ 1, ‖xn − xn+1‖ ≤
and xn+1 is a solution of (
-LS). The
38 W. KANG AND R. J. WILLIAMS
sequence {xn}
n=1 is Cauchy. Hence, there is an x
∗ ∈Rd such that xn → x
as n→ ∞, and x∗ is a solution to (LS). This contradicts the supposition
j∈J (∂Gj ∩ ∂G) =∅. Thus we have that
j∈J Ur(∂Gj ∩ ∂G) =∅ for
all r sufficiently small, and for such r,
(∂Gj ∩ ∂G)
Ur(∂Gj ∩ ∂G)
by convention.
Combining the above we see that for each J ⊂ I with J 6=∅, (87) holds
and hence assumption (A3) holds. �
Remark. In fact, under the assumptions of Lemma A.3, there is a con-
stant C > 0 such that D(u) ≤ Cu for each u ≥ 0 and D(·) defined as in
assumption (A3).
Lemma A.4. Given T > 0, functions φ,{φn}∞n=1 in D([0,∞),R
d), and
χ,{χn}∞n=1 in D([0,∞),R), suppose that sup0≤s≤T ‖φ
n(s)− φ(s)‖ → 0 and
sup0≤s≤T |χ
n(s)−χ(s)| → 0 as n→∞. Assume that χn is nondecreasing for
each n. Then for any sequence of real valued continuous functions {fn}∞n=1
defined on Rd such that fn converges uniformly on each compact set to a
continuous function f :Rd →R, we have
(0,t]
fn(φn(s))dχn(s)→
(0,t]
f(φ(s))dχ(s) as n→∞,(89)
uniformly for t ∈ [0, T ].
Proof. By replacing χn(·) and χ(·) by χn(·)− χn(0) and χ(·)− χ(0),
respectively, we may assume that χn(0) = χ(0) = 0. It is straightforward to
see by the uniform convergence of {χn} to χ on [0, T ] that χ inherits the
nondecreasing property of the {χn}.
By the triangle inequality,
0≤t≤T
(0,t]
fn(φn(s))dχn(s)−
(0,t]
f(φ(s))dχ(s)
≤ sup
0≤t≤T
(0,t]
(fn(φn(s))− f(φ(s)))dχn(s)
∣∣∣∣(90)
+ sup
0≤t≤T
(0,t]
f(φ(s))d(χn(s)− χ(s))
∣∣∣∣.
For the first term on the right-hand side of the above inequality, we have
0≤t≤T
(0,t]
(fn(φn(s))− f(φ(s)))dχn(s)
≤ sup
0≤s≤T
|fn(φn(s))− f(φ(s))|χn(T ),
INVARIANCE PRINCIPLE FOR SRBMS 39
where the right-hand side member above tends to zero as n → ∞ by the
uniform convergence of φn to φ on [0, T ] (which implies uniform boundedness
of {φn} on [0, T ]), the uniform convergence of fn to f on compact sets, the
continuity of f , and the convergence of χn(T ) to χ(T ). For the second term,
note that since f(φ(·)) ∈D([0,∞),R), by Theorem 3.5.6, Proposition 3.5.3
and Remark 3.5.4 of [7], there is a sequence of step functions {zk}∞k=1 of the
zk(·) =
zk(tki )1[tk
)(·),(91)
where 1 ≤ lk < ∞, 0 = t
1 < t
2 < · · · < t
< ∞ and sup0≤s≤T |f(φ(s)) −
zk(s)| → 0 as k→∞. Then
0≤t≤T
(0,t]
f(φ(s))d(χn(s)− χ(s))
≤ sup
0≤t≤T
(0,t]
(f(φ(s))− zk(s))d(χn(s)− χ(s))
+ sup
0≤t≤T
(0,t]
zk(s)d(χn(s)− χ(s))
≤ sup
0≤s≤T
|f(φ(s))− zk(s)|(χn(T ) + χ(T ))
+ sup
0≤t≤T
|zk(tki )||(χ
n − χ)((tki+1 ∧ t)−)− (χ
n − χ)((tki ∧ t)−)|.
For fixed k, the last term above can be made as small as we like for all n
sufficiently large since χn → χ uniformly on [0, T ]. The desired result follows.
Remark. The proof of Lemma A.4 is a modification of the proof of the
related Lemma 2.4 in [4]. The difference in assumptions is that in [4] it is
assumed that φn → φ in the J1-topology rather than uniformly on [0, T ],
χn, χ ∈C([0,∞),R+) rather than χ
n, χ ∈D([0,∞),R), and there is a single
function f rather than a sequence {fn}.
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[4] Dai, J. G. and Williams, R. J. (1995). Existence and uniqueness of semimartingale
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Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, Pennsylvania 15213
E-mail: weikang@andrew.cmu.edu
Department of Mathematics
University of California at San Diego
9500 Gilman Drive
La Jolla, California 92093
E-mail: williams@math.ucsd.edu
http://www.ams.org/mathscinet-getitem?mr=1346729
http://www.ams.org/mathscinet-getitem?mr=2222685
http://www.ams.org/mathscinet-getitem?mr=1096165
http://www.ams.org/mathscinet-getitem?mr=1207237
http://www.ams.org/mathscinet-getitem?mr=0838085
http://www.ams.org/mathscinet-getitem?mr=1681462
http://www.ams.org/mathscinet-getitem?mr=0473443
http://www.ams.org/mathscinet-getitem?mr=0606992
http://www.ams.org/mathscinet-getitem?mr=0051275
http://www.ams.org/mathscinet-getitem?mr=0959133
http://www.ams.org/mathscinet-getitem?mr=1231926
http://www.ams.org/mathscinet-getitem?mr=1663755
http://www.ams.org/mathscinet-getitem?mr=1663759
mailto:weikang@andrew.cmu.edu
mailto:williams@math.ucsd.edu
	Introduction
	Notation, terminology and preliminaries
	Definition of an SRBM
	Assumptions on the domain G and the reflection vector fields {i}
	Assumptions on the domain G
	Assumptions on the reflection vector fields {i}
	Invariance principle
	Oscillation inequality
	C-tightness result
	Invariance principle for SRBMs
	Applications of the invariance principle
	Weak existence of SRBMs
	SRBMs in convex polyhedrons with constant reflection fields
	SRBMs in bounded domains with piecewise smooth boundaries
	Appendix: Auxiliary lemmas
	References
	Author's addresses

  Semimartingale reflecting Brownian motions (SRBMs) living in the closures of
domains with piecewise smooth boundaries are of interest in applied probability
because of their role as heavy traffic approximations for some stochastic
networks. In this paper, assuming certain conditions on the domains and
directions of reflection, a perturbation result, or invariance principle, for
SRBMs is proved. This provides sufficient conditions for a process that
satisfies the definition of an SRBM, except for small random perturbations in
the defining conditions, to be close in distribution to an SRBM. A crucial
ingredient in the proof of this result is an oscillation inequality for
solutions of a perturbed Skorokhod problem. We use the invariance principle to
show weak existence of SRBMs under mild conditions. We also use the invariance
principle, in conjunction with known uniqueness results for SRBMs, to give some
sufficient conditions for validating approximations involving (i) SRBMs in
convex polyhedrons with a constant reflection vector field on each face of the
polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries
and possibly nonconstant reflection vector fields on the boundary surfaces.

Introduction. Semimartingale reflecting Brownian motions (SRBMs)
living in the closures of domains with piecewise smooth boundaries are of
interest in applied probability because of their role as heavy traffic diffusion
approximations for some stochastic networks. The nonsmoothness of the
boundary for such a domain, combined with discontinuities in the oblique
directions of reflection at intersections of smooth boundary surfaces, present
Received May 2006; revised November 2006.
1Supported in part by NSF Grants DMS-03-05272 and DMS-06-04537.
AMS 2000 subject classifications. 60F17, 60J60, 60K25, 90B15, 93E03.
Key words and phrases. Semimartingale reflecting Brownian motion, piecewise smooth
domain, invariance principle, oscillation inequality, Skorokhod problem, stochastic net-
works.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2007, Vol. 17, No. 2, 741–779. This reprint differs from the original in pagination
and typographic detail.
http://arxiv.org/abs/0704.0405v1
http://www.imstat.org/aap/
http://dx.doi.org/10.1214/105051606000000899
http://www.imstat.org
http://www.ams.org/msc/
http://www.imstat.org
http://www.imstat.org/aap/
http://dx.doi.org/10.1214/105051606000000899
2 W. KANG AND R. J. WILLIAMS
challenges in the development of a rigorous theory of existence, uniqueness
and approximation for such SRBMs.
When the state space is an orthant and the direction of reflection is con-
stant on each boundary face, a necessary and sufficient condition for weak
existence and uniqueness of an SRBM is known [14]. This condition involves
a so-called completely-S condition on the matrix formed by the reflection
directions. An invariance principle for such SRBMs was established in [15]
and used in [16] to justify heavy traffic diffusion approximations for cer-
tain open multiclass queueing networks. Loosely speaking, the invariance
principle shows that, assuming uniqueness in law for the SRBM, a process
satisfying the definition of the SRBM, except for small perturbations in the
defining conditions, is close in distribution to the SRBM.
For more general domains with piecewise smooth boundaries, some con-
ditions for existence and uniqueness of SRBMs are known. In particular, for
convex polyhedrons with a constant direction of reflection on each boundary
face, necessary and sufficient conditions for weak existence and uniqueness of
SRBMs are known for simple convex polyhedrons (where precisely d faces
meet at each vertex in d-dimensions) and sufficient conditions are known
for nonsimple convex polyhedrons, see [4]. For a bounded domain that can
be represented as a finite intersection of domains, each of which has a C1-
boundary and an associated uniformly Lipschitz continuous reflection vector
field, sufficient conditions for strong existence and uniqueness were provided
by Dupuis and Ishii [6]; in fact, these authors study stochastic differential
equations with reflection which include SRBMs. Despite these existence and
uniqueness results, a general invariance principle for SRBMs living in the
closures of domains with piecewise smooth boundaries has not been proved
to date. (We note that for the special case when the directions of reflection
are normal, that is, perpendicular to the boundary, there are a number of
perturbation results for reflecting Brownian motions. Our emphasis here is
on treating a wide range of oblique reflection directions.)
Motivated by its potential for use in approximating heavily loaded stochas-
tic networks that are more general than open multiclass queueing networks,
in this paper, we formulate and prove an invariance principle for SRBMs
living in the closures of domains with piecewise smooth boundaries with
possibly nonconstant directions of reflection on each of the smooth bound-
ary surfaces. An application of the results of this paper to the analysis of
an internet congestion control model can be found in [13]. An outline of the
current paper is as follows.
The definition of an SRBM and assumptions on the domains and direc-
tions of reflection are given in Sections 2 and 3, respectively. Some sufficient
conditions for these assumptions to hold are provided in Section 3. Section
4 is devoted to proving the main result of this paper, namely, the invari-
ance principle. A key element for our proof of this result is an oscillation
INVARIANCE PRINCIPLE FOR SRBMS 3
inequality for solutions of a perturbed Skorokhod problem; this inequality,
which may be of independent interest, is proved in Section 4.1. In Section 5
we give some applications of the invariance principle. We prove weak ex-
istence of SRBMs under the conditions specified in Section 3. We also use
the invariance principle, in conjunction with known uniqueness results for
SRBMs, to give sufficient conditions for validating approximations involving
(i) SRBMs in convex polyhedrons with a constant reflection vector field on
each face of the polyhedron, and (ii) SRBMs in bounded domains with piece-
wise smooth boundaries and possibly nonconstant reflection vector fields on
the boundary surfaces.
Beyond its possible use in justifying SRBM approximations for stochastic
networks, the invariance principle might be used to justify numerical ap-
proximations to SRBMs. A further possible extension of the results stated
here would involve an invariance principle for stochastic differential equa-
tions with reflection. The oscillation inequality for the perturbed Skorokhod
problem and associated criteria for C-tightness described in Sections 4.1
and 4.2 are likely to be useful for this. We have not developed such an ex-
tension here as that would involve introduction of extra assumptions that
would make the result less relevant for potential applications to stochastic
networks. In particular, the approximating processes would involve stochas-
tic integrals driven by a Brownian motion, whereas in stochastic network
applications, the Brownian motion typically only appears in the limit.
1.1. Notation, terminology and preliminaries. Let N denote the set of all
positive integers, that is, N = {1,2, . . .}, R denote the set of real numbers,
which is also denoted by (−∞,∞), R+ denote the nonnegative half-line,
which is also denoted by [0,∞). For x ∈ R, we write |x| for the absolute
value of x, [x] for the largest integer less than or equal to x, x+ for the
positive part of x. For any positive integer d, we let Rd denote d-dimensional
Euclidean space, where any element in Rd is denoted by a column vector.
Let ‖ · ‖ denote the Euclidean norm on Rd, that is, ‖x‖ = (
i=1 x
1/2 for
x ∈Rd, and 〈·, ·〉 denote the inner product on Rd, that is, 〈x, y〉=
i=1 xiyi,
for x, y ∈ Rd. We note that for any x ∈ Rd, ‖x‖ ≤
i=1 |xi|. Let R
+ denote
the positive orthant in Rd, that is, Rd+ = {x ∈ R
d :xi ≥ 0,1 ≤ i ≤ d}. Let
B(S) denote the Borel σ-algebra on S ⊂ Rd, that is, the collection formed
by intersecting all Borel sets in Rd with S. Let dist(x,S) denote the distance
between x ∈Rd and S ⊂Rd, that is, dist(x,S) = inf{‖x−y‖ :y ∈ S}, with the
convention that dist(x,∅) =∞ for x ∈Rd. Let Ur(S) denote the closed set
{x ∈Rd : dist(x,S)≤ r} for any r > 0 and S ⊂Rd, where if S =∅, Ur(S) =∅
for all r > 0. Let Br(x) denote the closed ball {y ∈R
d :‖y− x‖ ≤ r} for any
x ∈Rd and r > 0. For any set S ⊂Rd, we write S for the closure of S, So for
the interior of S and ∂S = S \So. For a finite set S, |S| denotes the number
4 W. KANG AND R. J. WILLIAMS
of elements in S. For any v ∈Rd, v′ denotes the transpose of v. Inequalities
between vectors in Rd should be interpreted componentwise, that is, if u, v ∈
d, then u ≤ (<)v means that ui ≤ (<)vi for each i ∈ {1, . . . , d}. For any
matrix A, let A′ denote the transpose of A. For any function x :R+ → R
x(t−) denotes the left limit of x at t > 0 whenever x has a left limit at t;
unless explicitly stated otherwise, x(0−)≡ 0, where 0 is the zero vector in Rd.
For any function x :R+ → R
d, we let ∆x(t) = x(t)− x(t−) for t ∈ R+ when
x(t−) exists. We let 0 be the constant deterministic function x :R+ → R
such that x(t) = 0 for all t ∈R+.
A domain in Rd is an open connected subset of Rd. For each continuously
differentiable function f defined on some nonempty domain S ⊂Rd, ∇f(x)
is the gradient of f at x ∈ S. For each x ∈ Rd, a neighborhood Vx of x is a
bounded domain in Rd that contains x. For any nonempty domain S ⊂Rd,
we say that the boundary ∂S of S is C1, if for each x ∈ ∂S there exists a
Euclidean coordinate system Cx for R
d centered at x, an rx > 0, and a once
continuously differentiable function ϕx :R
d−1 →R such that ϕx(0) = 0 and
S ∩Brx(x) = {z = (z1, . . . , zd)
′ in Cx : zd >ϕx(z1, . . . , zd−1)} ∩Brx(x).
Then, for x ∈ ∂S, the inward unit normal to ∂S at z ∈ ∂S ∩Brx(x) is given
in the coordinate system Cx by
n(z) =
(1 + ‖∇ϕx(z1, . . . , zd−1)‖
2)1/2
(−∇ϕx(z1, . . . , zd−1)
where ∇ϕx(z1, . . . , zd−1) = (
, . . . ,
∂zd−1
)′(z1, . . . , zd−1). For any nonempty
convex set S ⊂Rd, we call a vector n ∈Rd \{0} an inward unit normal vector
to S at x ∈ ∂S if ‖n‖= 1 and 〈n, y − x〉 ≥ 0 for all y ∈ S. Note that such a
vector need not be unique.
All stochastic processes used in this paper will be assumed to have paths
that are right continuous with finite left limits (abbreviated henceforth as
r.c.l.l.). A process is called continuous if almost surely its sample paths
are continuous. We denote by D([0,∞),Rd) the space of r.c.l.l. functions
from [0,∞) into Rd and we endow this space with the usual Skorokhod
J1-topology (cf. Chapter 3 of [7]). We denote by C([0,∞),R
d) the space
of continuous functions from [0,∞) into Rd. The Borel σ-algebra on either
D([0,∞),Rd) or C([0,∞),Rd) will be denoted by Md. The abbreviation
u.o.c. will stand for uniformly on compacts and will be used to indicate
that a sequence of functions in D([0,∞),Rd) (or C([0,∞),Rd)) is converg-
ing uniformly on compact time intervals to a limit in D([0,∞),Rd) (or
C([0,∞),Rd)). Consider W 1,W 2, . . . ,W , each of which is a d-dimensional
process (possibly defined on different probability spaces). The sequence
{W n}∞n=1 is said to be tight if the probability measures induced by the
W n on the measurable space (D([0,∞),Rd),Md) form a tight sequence,
INVARIANCE PRINCIPLE FOR SRBMS 5
that is, they form a weakly relatively compact sequence in the space of
probability measures on (D([0,∞),Rd),Md). The notation “W n ⇒W” will
mean that, as n → ∞, the sequence of probability measures induced on
(D([0,∞),Rd),Md) by {W n} converges weakly to the probability measure
induced on the same space by W . We shall describe this in words by saying
that W n converges weakly (or in distribution) to W as n → ∞. The se-
quence of processes {W n}∞n=1 is called C-tight if it is tight, and if each weak
limit point, obtained as a weak limit along a subsequence, almost surely has
sample paths in C([0,∞),Rd). The following proposition provides a useful
criterion for checking C-tightness.
Proposition 1.1. Suppose that, for each n ∈N, W n is a d-dimensional
process defined on the probability space (Ωn,Fn, Pn). The sequence {W n}∞n=1
is C-tight if and only if the following two conditions hold:
(i) For each η > 0 and T ≥ 0, there exists a finite constant Mη,T > 0
such that
lim inf
0≤t≤T
‖W n(t)‖ ≤Mη,T
≥ 1− η.(1)
(ii) For each ε > 0, η > 0 and T > 0, there exists λ ∈ (0, T ) such that
lim sup
Pn{wT (W
n, λ)≥ ε} ≤ η,(2)
where for x ∈D([0,∞),Rd),
wT (x,λ) = sup
u,v∈[t,t+λ]
‖x(u)− x(v)‖ : 0≤ t < t+ λ≤ T
Proof. See Proposition VI.3.26 in [12]. �
A d-dimensional process W is said to be locally of bounded variation if
all sample paths of W are of bounded variation on each finite time interval.
For such a process W , we define V(W ) = {V(W )(t), t≥ 0} such that for each
t≥ 0,
V(W )(t) = ‖W (0)‖
+ sup
‖W (ti)−W (ti−1)‖ : 0 = t0 < t1 < · · ·< tl = t, l≥ 1
A triple (Ω,F ,{Ft, t ≥ 0}) will be called a filtered space if Ω is a set,
F is a σ-algebra of subsets of Ω, and {Ft, t≥ 0} is an increasing family of
sub-σ-algebras of F , that is, a filtration. Henceforth, the filtration {Ft, t≥ 0}
will be simply written as {Ft}. If P is a probability measure on (Ω,F), then
6 W. KANG AND R. J. WILLIAMS
(Ω,F ,{Ft}, P ) is called a filtered probability space. A d-dimensional process
X = {X(t), t ≥ 0} defined on (Ω,F , P ) is called {Ft}-adapted if for each
t ≥ 0, X(t) :Ω→ Rd is measurable when Ω is endowed with the σ-algebra
Given a filtered probability space (Ω,F ,{Ft}, P ), a vector µ ∈R
d, a d× d
symmetric, strictly positive definite matrix Γ, and a probability distribution
ν on (Rd,B(Rd)), an {Ft}-Brownian motion with drift vector µ, covariance
matrix Γ, and initial distribution ν, is a d-dimensional {Ft}-adapted process
defined on (Ω,F ,{Ft}, P ) such that the following hold under P :
(a) X is a d-dimensional Brownian motion whose sample paths are almost
surely continuous and that has initial distribution ν,
(b) {Xi(t)−Xi(0)− µit,Ft, t≥ 0} is a martingale for i= 1, . . . , d, and
(c) {(Xi(t)−Xi(0)−µit)(Xj(t)−Xj(0)−µjt)−Γijt,Ft, t≥ 0} is a mar-
tingale for i, j = 1, . . . , d.
In this definition, the filtration {Ft} may be larger than the one generated
by X ; however, for each t≥ 0, under P , the σ-algebra Ft is independent of
the increments of X from t onward. The latter follows from the martingale
properties of X . If ν = δx, the unit mass at x ∈ R
d, we say that X starts
from x.
2. Definition of an SRBM. Let G=
i∈I Gi be a nonempty domain in
d, where I is a nonempty finite index set and for each i ∈ I , Gi is a
nonempty domain in Rd. For simplicity, we assume that I = {1,2, . . . , I}
and then |I|= I. For each i ∈ I , let γi(·) be a vector valued function defined
from Rd into Rd. Fix µ ∈Rd, Γ a d × d symmetric and strictly positive
definite covariance matrix and ν a probability measure on (G,B(G)), where
B(G) denotes the σ-algebra of Borel subsets of the closure G of G.
Definition 2.1 (Semimartingale reflecting Brownian motion). A semi-
martingale reflecting Brownian motion (abbreviated as SRBM) associated
with the data (G,µ,Γ,{γi, i ∈ I}, ν) is an {Ft}-adapted, d-dimensional pro-
cess W defined on some filtered probability space (Ω,F ,{Ft}, P ) such that:
(i) P -a.s., W (t) =X(t) +
(0,t] γ
i(W (s))dYi(s) for all t≥ 0,
(ii) P -a.s., W has continuous paths and W (t) ∈G for all t≥ 0,
(iii) under P , X is a d-dimensional {Ft}-Brownian motion with drift
vector µ, covariance matrix Γ and initial distribution ν,
(iv) for each i ∈ I , Yi is an {Ft}-adapted, one-dimensional process such
that P -a.s.,
(a) Yi(0) = 0,
(b) Yi is continuous and nondecreasing,
INVARIANCE PRINCIPLE FOR SRBMS 7
(c) Yi(t) =
(0,t] 1{W (s)∈∂Gi∩∂G} dYi(s) for all t≥ 0.
We shall often refer to Y = {Yi, i ∈ I} as the “pushing process” associated
with the SRBM W . When ν = δx, we may alternatively say that W is an
SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}) that starts from x. We
will call (W,X,Y ) satisfying Definition 2.1 an extended SRBM associated
with the data (G,µ,Γ,{γi, i ∈ I}, ν).
Loosely speaking, an SRBM behaves like a Brownian motion in the inte-
rior of the domain G and it is confined to G by instantaneous “reflection” (or
“pushing”) at the boundary, where the allowed directions of “reflection” at
x ∈ ∂G are convex combinations of the vectors γi(x) for i such that x ∈ ∂Gi.
Under the assumptions imposed on G and {γi, i ∈ I} in Sections 3.1 and 3.2
below, at each point on the boundary of G there is an allowed direction of
reflection that can be used there which “points into the interior of G.” We
end this section by introducing a related set-valued function I(·) and show
a key property of it.
Definition 2.2. For each x∈Rd, let I(x) = {i ∈ I :x ∈ ∂Gi}.
The set-valued function I(·) has the following property called upper semi-
continuity on ∂G.
Lemma 2.1. For each x ∈ ∂G, there is an open neighborhood Vx of x in
d such that
I(y)⊂ I(x) for all y ∈ Vx.(4)
Proof. We prove this lemma by contradiction. Suppose that the func-
tion I(·) does not satisfy (4). Then there is a point x ∈ ∂G such that there
is no open neighborhood Vx of x such that I(y)⊂ I(x) for all y ∈ Vx. Since
the index set I is finite, there is an index k ∈ I \ I(x) and a sequence of
points {yn} ⊂R
d such that ‖yn−x‖<
and k ∈ I(yn) for each n≥ 1. Hence
yn ∈ ∂Gk for all n≥ 1. Since ∂Gk is closed and yn → x as n→∞, we con-
clude that x ∈ ∂Gk. This implies that k ∈ I(x), which is a contradiction, as
desired. �
3. Assumptions on the domain G and the reflection vector fields {γi}.
3.1. Assumptions on the domain G. We henceforth assume that the
domain G satisfies assumptions (A1)–(A3) below. In the case when G is
bounded, assumptions (A2)–(A3) follow from assumption (A1) (see Lem-
mas A.1 and A.2 in the Appendix for details). If the domain G is a convex
polyhedron satisfying assumption (A1), then assumptions (A2)–(A3) hold
by Lemma A.3 in the Appendix.
8 W. KANG AND R. J. WILLIAMS
(A1) G is a nonempty domain in Rd with representation
Gi,(5)
where for each i ∈ I , Gi is a nonempty domain, Gi 6=R
d, and the boundary
∂Gi of Gi is C
1. For each i ∈ I , we let ni(·) be the unit normal vector field
on ∂Gi that points into Gi.
(A2) For each ε ∈ (0,1) there exists R(ε) > 0 such that for each i ∈ I ,
x ∈ ∂Gi ∩ ∂G and y ∈G satisfying ‖x− y‖<R(ε), we have
〈ni(x), y − x〉 ≥−ε‖x− y‖.(6)
(A3) The function D : [0,∞)→ [0,∞] defined such that D(0) = 0 and
D(r) = sup
J 6=∅
(∂Gj ∩ ∂G)
Ur(∂Gj ∩ ∂G)
for r > 0, satisfies
D(r)→ 0 as r→ 0.(8)
Remark. Assumption (A2) is reminiscent of the uniform exterior cone
condition (cf. [9], page 195). We say that a region G ⊂ Rd satisfies a uni-
form exterior cone condition if for each x0 ∈ ∂G, there is a truncated closed
right circular cone Vx0 , with nonempty interior and vertex x0, satisfying
Vx0 ∩G= {x0}, and the truncated cones Vx0 are all congruent to some fixed
truncated closed right circular cone V . By comparing assumption (A2) with
the uniform exterior cone condition, we see that assumption (A2) implies
the uniform exterior cone condition. On the other hand, under assumption
(A1), assumption (A2) is implied by a family of uniform exterior cone condi-
tions where for each ε ∈ (0,1), the axis of the truncated closed right circular
cone at x ∈ ∂G is along the vector −ni(x) and all of the truncated closed
right circular cones are congruent to a truncated closed right circular cone
whose height and base radius are R(ε) and R(ε)( 1
− 1)1/2 respectively.
Assumption (A2) holds automatically if G is convex. We also note that as-
sumption (A2) is strictly weaker than the uniform exterior sphere condition.
The definition of the uniform exterior sphere condition is similar to that for
the uniform exterior cone condition where a closed ball with x0 on its bound-
ary takes the place of the truncated closed right circular cone Vx0 . It can
be checked that for the domain G = {(x, y) ∈ R2 :y < |x|α} with α ∈ (1,2),
the uniform exterior sphere condition fails to hold, but assumption (A2)
holds. In fact, at the point (0,0) ∈R2, there is no r > 0 and y ∈R2 such that
Br(y)∩ ∂G= {(0,0)}.
INVARIANCE PRINCIPLE FOR SRBMS 9
Remark. For the definition of D(·) in (A3), we adopt the convention
that the supremum over an empty set is zero and dist(x,∅) =∞. Since ∂Gi∩
∂G 6=∅ for at least one i ∈ I , the function D(·) satisfies limr→∞D(r) =∞.
Furthermore, D(r1)≤D(r2) whenever r1, r2 ∈ [0,∞) and r1 ≤ r2. Assump-
tion (A3) requires that for any nonempty subset J ⊂ I , the intersection of
tubular neighborhoods of the boundaries {∂Gj ∩ ∂G : j ∈ J } given by the
j∈J Ur(∂Gj ∩ ∂G) “converges” to the intersection of the boundaries
given by the set
j∈J (∂Gj ∩ ∂G) as r approaches 0. Property (8) need
not always hold. For example, let G1 = {(x, y) ∈ R
2 :y < e−x
2/2, x ∈ R} and
G2 = {(x, y) ∈ R
2 :y > 0, x ∈ R}. Then ∂G1 ∩ ∂G2 =∅. But for each r > 0,
Ur(∂G1)∩Ur(∂G2) 6=∅. Hence D(r) =∞ for each r > 0.
3.2. Assumptions on the reflection vector fields {γi}. We henceforth as-
sume that there are vector fields {γi(·), i ∈ I} satisfying assumptions (A4)–
(A5) below.
(A4) There is a constant L > 0 such that for each i ∈ I , γi(·) is a uni-
formly Lipschitz continuous function from Rd into Rd with Lipschitz con-
stant L and ‖γi(x)‖= 1 for each x ∈Rd.
(A5) There is a constant a ∈ (0,1), and vector valued functions b(·) =
(b1(·), . . . , bI(·)) and c(·) = (c1(·), . . . , cI(·)) from ∂G into R
+ such that for
each x ∈ ∂G,
i∈I(x) bi(x) = 1,
j∈I(x)
i∈I(x)
bi(x)n
i(x), γj(x)
≥ a,(9)
i∈I(x) ci(x) = 1,
j∈I(x)
i∈I(x)
ci(x)γ
i(x), nj(x)
≥ a.(10)
We note here for future use that by (A4), if we set ρ0 =
, then for
any x, y ∈ Rd satisfying ‖x − y‖ < ρ0, we have ‖γ
i(x) − γi(y)‖ < a/4 for
each i ∈ I . So for each 0< ρ < ρ0/4, by (9)–(10) and the normalization of
b(·), c(·), γi(·), nj(·) for i, j ∈ I , we obtain
j∈I(x)
y∈B4ρ(x)
i∈I(x)
bi(x)n
i(x), γj(y)
≥ a/2(11)
j∈I(x)
y∈B4ρ(x)
i∈I(x)
ci(x)γ
i(y), nj(x)
≥ a/2.(12)
10 W. KANG AND R. J. WILLIAMS
The use of B4ρ(x) here is related to the form in which this is used in Section
Remark. Assumption (A4) is equivalent to (3.4) in [6] whenG is bounded.
Property (10) means that, at each point x ∈ ∂G, there is a convex combi-
nation γ(x) =
i∈I(x) ci(x)γ
i(x) of the vectors {γi(x), i ∈ I(x)} that can
be used there such that γ(x) “points into” G. Property (9) is in a sense a
dual condition to property (10), where the roles of γi and ni are reversed
for i ∈ I(x). This property (9) is used in showing the oscillation inequality
in Theorem 4.1 below. Assumption (A5) is an analogue of Assumption 1.1
in [4]. When G is bounded, (10) is similar to condition (3.6) in [6] (we as-
sume some additional uniformity through the lack of dependence of a on
It is straightforward to see using the triangle inequality that the following
condition (A5)′ implies (A5).
(A5)′ There is a ∈ (0,1) and vector valued functions b, c from ∂G into RI+
such that for each x ∈ ∂G,
i∈I(x) bi(x) = 1, and for each i ∈ I(x),
bi(x)〈n
i(x), γi(x)〉 ≥ a+
j∈I(x)\{i}
bj(x)|〈n
j(x), γi(x)〉|,(13)
i∈I(x) ci(x) = 1, and for each i ∈ I(x),
ci(x)〈γ
i(x), ni(x)〉 ≥ a+
j∈I(x)\{i}
cj(x)|〈γ
j(x), ni(x)〉|.(14)
Condition (A5)′(ii) is similar to condition (3.8) in [6], although here we
assume additional uniformity through the lack of dependence of a on x. As
noted in [6], their condition (3.8) can be phrased in terms of a nonsingu-
lar M-matrix requirement [2]. (This is sometimes also called a generalized
Harrison–Reiman type of condition [10].) Since that nonsingular M-matrix
property is invariant under transpose, and this property for the transpose
corresponds to a local form of (A5)′(i), one might conjecture that there is an
equivalence between the existence of a nonnegative vector valued function b
such that (A5)′(i) holds for each x ∈ ∂G and the existence of a nonnegative
vector valued function c such that (A5)′(ii) holds for each x ∈ ∂G. Indeed
we have the following lemma. We have stated the two (equivalent) condi-
tions (i) and (ii) in specifying (A5)′ to preserve a parallel with (A5) and
since both properties can be useful in proofs. Furthermore, in light of the
following lemma, verifying either condition suffices for both to hold.
INVARIANCE PRINCIPLE FOR SRBMS 11
Lemma 3.1. There is a constant a ∈ (0,1) and a vector valued function
b :∂G → RI+ such that (A5)
′(i) holds for each x ∈ ∂G if and only if there
is a constant a ∈ (0,1) and a vector valued function c :∂G→ RI+ such that
(A5)′(ii) holds for each x ∈ ∂G.
Proof. We just prove the “if” part; the “only if” part can be proved
in a similar manner.
We suppose that there is a constant a ∈ (0,1) and a vector valued function
c :∂G → RI+ such that (A5)
′(ii) holds for each x ∈ ∂G. For fixed x ∈ ∂G,
consider the square matrix A(x) whose diagonal entries are given by the
(positive) elements 〈ni(x), γi(x)〉 for i ∈ I(x) and whose off-diagonal entries
are given by−|〈ni(x), γj(x)〉| for i ∈ I(x), j ∈ I(x), j 6= i. Let E be the square
matrix having the same dimensions as A(x) and whose entries are all equal
to one. By the theory of M-matrices (see [2], Chapter 6, especially condition
(M35)), condition (ii) of (A5)
′ implies that A(x)− a
E is a nonsingular M-
matrix, that is, A(x)− a
E has nonnegative diagonal entries and nonpositive
off-diagonal entries and it can be written in the form s(x)I −B(x) where
B(x) is a matrix with nonnegative entries and s(x)> 0 is a constant that is
strictly larger than the spectral radius of B(x).
Since the nonsingular M-matrix property is invariant under transpose (cf.
(G21) in Chapter 6 of [2]), then A
′(x)− a
E is also a nonsingular M-matrix.
Hence, there is a vector b̃(x) = (b̃i(x) : i ∈ I(x)) with nonnegative entries such
that (A′(x)− a
E)b̃(x)> 0 (cf. (I27) in Chapter 6 of [2]). We can extend b̃(x)
to an I-dimensional vector b(x) and normalize it so that
i∈I(x) bi(x) = 1.
Then (A5)′(i) holds with a
in place of a. �
4. Invariance principle. In this section we state and prove an invariance
principle for an SRBM living in the closure of a domain G with piecewise
smooth boundary and having associated reflection fields {γi, i ∈ I}, where
G, {γi, i ∈ I} satisfy assumptions (A1)–(A5) of Section 3. (These assump-
tions hold throughout this section.) We shall first state a preliminary result
called an oscillation inequality (see Theorem 4.1), then we use it to prove
a tightness result (see Theorem 4.2). Finally, we establish the invariance
principle (see Theorem 4.3).
4.1. Oscillation inequality. The following oscillation inequality is the key
to the proof of the tightness result claimed in Theorem 4.2. In this subsec-
tion, for any 0≤ t1 < t2 <∞ and any integer k ≥ 1, D([t1, t2],R
k) denotes
the set of functions w : [t1, t2]→R
k that are right continuous on [t1, t2) and
have finite left limits on (t1, t2]. For w ∈D([t1, t2],R
Osc(w, [t1, t2]) = sup{‖w(t)−w(s)‖ : t1 ≤ s < t≤ t2},(15)
Osc(w, [t1, t2)) = sup{‖w(t)−w(s)‖ : t1 ≤ s < t < t2}.(16)
12 W. KANG AND R. J. WILLIAMS
Note that we do not explicitly indicate the dependence on k in the notation.
Recall the constants a,L from assumptions (A4)–(A5), the functions R(·)
from assumption (A2) and D(·) from (7). Let ρ0 =
Theorem 4.1 (Oscillation inequality). There exists a nondecreasing func-
tion Π: (0,∞)→ (0,∞] satisfying Π(u)→ 0 as u→ 0, such that Π depends
only on the constants I, a and the function D(·), and such that whenever
0 < ρ <min{
R(a/4)
}, 0 < δ <
, 0 ≤ s < t < ∞, w,x ∈ D([s, t],Rd) and
y ∈D([s, t],RI) satisfy:
(i) w(u) ∈Bρ(x0)∩Uδ(G) for all u ∈ [s, t], for some x0 ∈G,
(ii) w(u) = w(s) + x(u) − x(s) +
(s,u] γ
i(w(v))dyi(v) for all u ∈
[s, t],
(iii) for each i ∈ I ,
(a) yi(s)≥ 0,
(b) yi is nondecreasing and ∆yi(u)≤ δ for all u ∈ (s, t],
(c) yi(u) = yi(s) +
(s,u] 1{w(v)∈Uδ(∂Gi∩∂G)} dyi(v) for all u ∈ [s, t],
(iv) D(Π(Osc(x, [s, t]) + δ))<
then we have that the following hold:
Osc(w, [s, t])≤Π(Osc(x, [s, t]) + δ),(17)
Osc(y, [s, t])≤Π(Osc(x, [s, t]) + δ).(18)
Proof. Let
Π0(u) = u for all u > 0.
Define Πm : (0,∞)→ (0,∞], m= 1, . . . , I, inductively such that
Πm(u) = Πm−1(u) + (I+2)u+
(D(Πm−1(u) + (I+2)u) + 2u).
Here the sum of any element of [0,∞) with ∞ is ∞ and D(∞) is defined
to equal ∞. For each m= 0,1, . . . , I, the function Πm is nondecreasing and
depends only on I, a and D(·). For each m= 1, . . . , I and u > 0, Πm−1(u)≤
Πm(u). By assumption (A3), we conclude (using an induction proof) that
Πm(u)→ 0 as u→ 0, for m= 0,1, . . . , I.
Let Π(·) = ΠI(·).
Fix 0 < ρ < min{
R(a/4)
}, 0 < δ <
, 0 ≤ s < t < ∞. Suppose that
w,x ∈D([s, t],Rd) and y ∈D([s, t],RI) satisfy (i)–(iv) in the statement of
Theorem 4.1. For each nonempty interval [t1, t2]⊂ [s, t], let
I[t1,t2] = {i ∈ I :w(u) ∈ Uδ(∂Gi ∩ ∂G) for some u ∈ [t1, t2]},
INVARIANCE PRINCIPLE FOR SRBMS 13
the indices of the boundary surfaces that w(·) comes close to in the time
interval [t1, t2]. For each 0 ≤ m ≤ I, define Tm = {[t1, t2] ⊂ [s, t] : |I[t1,t2]| ≤
m}. Note that under the partial ordering of set inclusion, Tm increases with
m. To prove the theorem, we will prove by induction that for each 0≤m≤ I
and each interval [t1, t2] ∈ Tm, (17)–(18) hold with [t1, t2] in place of [s, t] and
Πm(·) in place of Π(·). The result for m= I yields the theorem.
Suppose that m= 0. Then T0 = {[t1, t2]⊂ [s, t] : |I[t1,t2]|= 0}. Fix an inter-
val [t1, t2] ∈ T0. Since I[t1,t2] =∅ and (iii)(c) holds, the function y does not
increase on the time interval (t1, t2], that is, yi(t2)− yi(t1) = 0 for all i ∈ I .
Then, for t1 ≤ u < v ≤ t2,
w(v)−w(u) = x(v)− x(u).(19)
So in this case,
Osc(w, [t1, t2]) = Osc(x, [t1, t2])≤Osc(x, [t1, t2]) + δ,(20)
Osc(y, [t1, t2]) = 0≤Osc(x, [t1, t2]) + δ.(21)
Thus, (17)–(18) hold with Π0(·) in place of Π(·) and [t1, t2] in place of [s, t]
for each interval [t1, t2] ∈ T0.
For the induction step, let 1 ≤ m ≤ I and suppose that (17)–(18) hold
with Πm−1(·) in place of Π(·) and [t1, t2] in place of [s, t] for each interval
[t1, t2] ∈ Tm−1.
Now fix [t1, t2] ∈ Tm. If |I[t1,t2]| ≤ m − 1, then [t1, t2] ∈ Tm−1 and so by
the induction assumption we have that (17)–(18) hold with [t1, t2] in place
of [s, t] and Πm−1(·) [and hence Πm(·)] in place of Π(·). Thus, it suffices
to consider [t1, t2] ⊂ [s, t] such that |I[t1,t2]| =m. For i /∈ I[t1,t2], by (iii)(c),
yi(t2)− yi(t1) = 0, and so by (ii), for t1 ≤ u < v ≤ t2, we have
w(v)−w(u) = x(v)− x(u) +
i∈I[t1,t2]
(u,v]
γi(w(r))dyi(r).(22)
Let Πm(u) = Πm−1(u) + (I + 2)u for all u > 0, and η = Osc(x, [t1, t2]) + δ.
For any M ∈ (0,∞] and any nonempty set J ⊂ I , let
FMJ = {z ∈R
d : dist(z, ∂Gi ∩ ∂G)<M for all i ∈ J }.
Note that FMJ = ∅ when there is an i ∈ J such that ∂Gi ∩ ∂G = ∅. Since
Πm(·)≤Πm(·)≤Π(·), D(·) and Π(·) are nondecreasing, and Osc(x, [t1, t2])≤
Osc(x, [s, t]), we have by (iv) that
D(Πm(η))≤D(Πm(η))≤D(Π(η))<
.(23)
Note that this implies Πm(η)<∞ since D(∞) =∞.
We now consider two cases.
14 W. KANG AND R. J. WILLIAMS
Case 1. Suppose that w(r) ∈ F
Πm(η)
I[t1,t2]
for all r ∈ [t1, t2].
Fix u, v such that t1 ≤ u < v ≤ t2. Since we have that
w(v) ∈
j∈I[t1,t2]
Πm(η)
(∂Gj ∩ ∂G),
by the definition of D(·) and (23), there is z ∈
j∈I[t1,t2]
(∂Gj ∩ ∂G) such
‖w(v)− z‖ ≤D(Πm(η))<
.(24)
For each r ∈ [t1, t2], by (i) we have that w(r) ∈ Uδ(G), and so there is z
such that
‖w(r)− zr‖ ≤ 2δ.
Hence by (i) and (24) we have
‖zr − z‖ ≤ ‖zr −w(r)‖+ ‖w(r)− x0‖+ ‖x0 −w(v)‖+ ‖w(v)− z‖
≤ 2δ + ρ+ ρ+ ρ/2< 4ρ < R(a/4)
‖w(r)− z‖ ≤ ‖w(r)− x0‖+ ‖x0 −w(v)‖+ ‖w(v)− z‖
≤ ρ+ ρ+ ρ/2< 4ρ.
By (6) and (25) we have
〈nj(z), z − zr〉 ≤
‖z − zr‖ for each j ∈ I(z) and r ∈ [t1, t2].(27)
Note that I(z) ⊃ I[t1,t2]. Recalling the definition of b(·) from assumption
(A5), on dotting the vector
j∈I(z) bj(z)n
j(z) with both sides of (22) and
rearranging, we obtain
i∈I[t1,t2]
(u,v]
j∈I(z)
bj(z)n
j(z), γi(w(r))
dyi(r)
j∈I(z)
bj(z)〈n
j(z),w(v)−w(u)〉(28)
j∈I(z)
bj(z)〈n
j(z), x(v)− x(u)〉.
So by (11), (22), (24)–(28), and the fact that
j∈I(z) bj(z) = 1, bj(z)≥ 0 for
j ∈ I , we have
i∈I[t1,t2]
(yi(v)− yi(u))
INVARIANCE PRINCIPLE FOR SRBMS 15
i∈I[t1,t2]
(u,v]
j∈I(z)
bj(z)n
j(z), γi(w(r))
dyi(r)
j∈I(z)
bj(z)〈n
j(z),w(v)− z〉+
j∈I(z)
bj(z)〈n
j(z), z − zu〉
j∈I(z)
bj(z)〈n
j(z), zu −w(u)〉 −
j∈I(z)
bj(z)〈n
j(z), x(v)− x(u)〉
≤D(Πm(η)) +
‖z − zu‖+ 2δ + ‖x(v)− x(u)‖
≤D(Πm(η)) + 2δ + ‖x(v)− x(u)‖
(‖z −w(v)‖+ ‖w(v)−w(u)‖+ ‖w(u)− zu‖)
≤D(Πm(η)) + 2δ + ‖x(v)− x(u)‖
D(Πm(η)) + ‖x(v)− x(u)‖+
i∈I[t1,t2]
(yi(v)− yi(u)) + 2δ
{D(Πm(η)) + 2δ + ‖x(v)− x(u)‖}+
i∈I[t1,t2]
(yi(v)− yi(u)).
Hence
i∈I[t1,t2]
(yi(v)− yi(u))≤
{D(Πm(η)) + 2δ + ‖x(v)− x(u)‖}
{D(Πm(η)) + 2η}.
On multiplying through by 4
, we obtain
i∈I[t1,t2]
(yi(v)− yi(u))≤
{D(Πm(η)) + 2η} ≤Πm(η).(29)
Hence, by (29) and the fact that for any x ∈Rd, ‖x‖ ≤
i=1 |xi|, we have
Osc(y, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ),(30)
and by (22), (29) and the definitions of Πm(·) and Πm(·), we have
Osc(w, [t1, t2])≤Osc(x, [t1, t2]) +
{D(Πm(η)) + 2η}
≤Πm(Osc(x, [t1, t2]) + δ),
as desired.
16 W. KANG AND R. J. WILLIAMS
Case 2. Suppose that there is t3 ∈ [t1, t2] such that w(t3) /∈ F
Πm(η)
I[t1,t2]
Define σ = inf{u ∈ [t1, t2] :w(u) /∈ F
Πm(η)
I[t1,t2]
}. Then σ ≤ t2. For each u ∈
[t1, σ), w(u) ∈ F
Πm(η)
I[t1,t2]
and so by a similar analysis to that for Case 1, we
obtain for each v ∈ [t1, σ),
Osc(w, [t1, v])≤ η+
(D(Πm(η)) + 2η)
Osc(y, [t1, v])≤
(D(Πm(η)) + 2η).
By the right continuity of paths we have w(σ) /∈ F
Πm(η)
I[t1,t2]
. Then there is an
i ∈ I[t1,t2] such that dist(w(σ), ∂Gi ∩ ∂G) ≥ Πm(η), and it follows that w
does not reach Uδ(∂Gi ∩ ∂G) during the interval [σ, t2]. To see this, let
τ = inf{u ∈ [σ, t2] : dist(w(u), ∂Gi ∩ ∂G) ≤ δ} with the convention that the
infimum of an empty set is ∞. If τ ≤ t2, then by the right continuity of
w(·) and since Πm(η)> δ, we have τ > σ and dist(w(τ), ∂Gi∩∂G)≤ δ. Also,
since |I[t1,t2]|=m, we have [σ,u] ∈ Tm−1 for each u ∈ [σ, τ). By the induction
assumption and letting u→ τ , we have ‖w(τ−)−w(σ)‖ ≤Πm−1(η). By (ii),
(iii)(b) and since ‖γi(·)‖= 1, we have
‖∆w(τ)‖ ≤ ‖∆x(τ)‖+
∆yi(τ)≤Osc(x, [t1, t2]) + Iδ ≤ Iη.
Then simple manipulations yield
dist(w(σ), ∂Gi ∩ ∂G)≤ ‖w(σ)−w(τ−)‖+ ‖∆w(τ)‖+ dist(w(τ), ∂Gi ∩ ∂G)
≤Πm−1(η) + Iη+ δ
<Πm(η).
This contradicts the fact that dist(w(σ), ∂Gi∩∂G)≥Πm(η), and so confirms
that w does not reach Uδ(∂Gi ∩ ∂G) in [σ, t2]. Thus we must have [σ, t2] ∈
Tm−1. Hence we have by the induction assumption that
Osc(w, [t1, t2])≤ sup
v∈[t1,σ)
Osc(w, [t1, v]) + ‖∆w(σ)‖+Osc(w, [σ, t2])
≤ η +
(D(Πm(η)) + 2η) + Iη+Πm−1(η)
≤Πm(Osc(x, [t1, t2]) + δ)
INVARIANCE PRINCIPLE FOR SRBMS 17
Osc(y, [t1, t2])≤ sup
v∈[t1,σ)
Osc(y, [t1, v]) + ‖∆y(σ)‖+Osc(y, [σ, t2])
(D(Πm(η)) + 2η) + Iη+Πm−1(η)
≤Πm(Osc(x, [t1, t2]) + δ).
On combining all of the cases above, we have
Osc(w, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ),(31)
Osc(y, [t1, t2])≤Πm(Osc(x, [t1, t2]) + δ).(32)
This completes the induction step. �
Remark. The proof of the above theorem was inspired by the proof of
Lemma 4.3 of [4]. Because of the condition (i) in Theorem 4.1, the oscilla-
tion inequality given here is localized. Similar, but nonlocalized, oscillation
inequalities were proved in [15] when G = Rd+ and in [3] for a sequence of
convex polyhedrons; in these cases, the direction of reflection was constant
on each boundary face.
4.2. C-tightness result. Throughout this subsection and the next, we
suppose that the following assumption holds in addition to (A1)–(A5).
Assumption 4.1. There is a sequence of strictly positive constants {δn}∞n=1
such that for each positive integer n, there are processes W n, W̃ n,Xn, αn
having paths in D([0,∞),Rd) and processes Y n, Ỹ n, βn having paths in
D([0,∞),RI) defined on some probability space (Ωn,Fn, Pn) such that:
(i) Pn-a.s., W n = W̃ n + αn and W̃ n(t) ∈Uδn(G) for all t≥ 0,
(ii) Pn-a.s., W n(t) =Xn(t)+
(0,t] γ
i,n(W n(s−),W n(s))dY ni (s) for
all t≥ 0, where for each i ∈ I , γi,n :Rd ×Rd → Rd is Borel measurable and
‖γi,n(y,x)‖= 1 for all x, y ∈Rd,
(iii) Y n = Ỹ n+βn, where βn is locally of bounded variation and Pn-a.s.,
for each i ∈ I ,
(a) Ỹ ni (0) = 0,
(b) Ỹ ni is nondecreasing and ∆Ỹ
i (t)≤ δ
n for all t > 0,
(c) Ỹ ni (t) =
(0,t] 1{W̃n(s)∈Uδn (∂Gi∩∂G)}
dỸ ni (s),
(iv) δn → 0 as n → ∞, and for each ε > 0, there is ηε > 0 and nε > 0
such that for each i ∈ I , ‖γi,n(y,x)− γi(x)‖< ε whenever ‖x− y‖< ηε and
n≥ nε,
18 W. KANG AND R. J. WILLIAMS
(v) αn → 0 and V(βn)→ 0 in probability, as n→∞,
(vi) {Xn} is C-tight.
Remark. A simple case in which (iv) above holds is where γi,n(y,x)≡
γi(y). In (v), V(βn) is the total variation process for βn (cf. Section 1.1).
The following theorem will play an important role in the proof of the in-
variance principle. It will be used to show that a sequence of processes sat-
isfying suitably perturbed versions of the defining conditions for an SRBM
[cf. (i)–(vi) above] is C-tight.
Theorem 4.2 (C-tightness). Suppose that Assumption 4.1 holds. Define
Zn = (W n,Xn, Y n) for each n. Then the sequence of processes {Zn}∞n=1 is
C-tight.
Remark. Note that C-tightness of {W n}, {Xn} and {Y n} implies C-
tightness of {Zn} (see Chapter VI, Corollary 3.33 of [12] for details).
Proof of Theorem 4.2. References here to (i)–(vi) are to the condi-
tions in Assumption 4.1.
Simple algebraic manipulations yield Pn-a.s.,
W̃ n(t) = X̃n(t) +
(0,t]
γi,n(W n(s−),W n(s))dỸ ni (s)(33)
= X̃n(t) + Ṽ n(t) +
(0,t]
γi(W̃ n(s))dỸ ni (s),(34)
where
X̃n(t) =Xn(t) +
−αn(t) +
(0,t]
γi,n(W n(s−),W n(s))dβni (s)
Ṽ n(t) =
(0,t]
(γi,n(W n(s−),W n(s))− γi(W n(s)))dỸ ni (s)
(0,t]
(γi(W n(s))− γi(W̃ n(s)))dỸ ni (s).
The hypotheses on αn, the total variation process V(βn) of βn, and the
fact that ‖γi,n(y,x)‖ = 1 for all x, y ∈ Rd and each i ∈ I , imply that the
process
−αn(·) +
(0,·]
γi,n(W n(s−),W n(s))dβni (s)
INVARIANCE PRINCIPLE FOR SRBMS 19
converges to 0 in probability as n→∞. Combining this with the fact that
{Xn}∞n=1 is C-tight, we obtain that {X̃
n}∞n=1 is C-tight.
Recall the positive nondecreasing function Π(·) from Theorem 4.1, and
the constants a, L and functions R(·) and D(·) from assumptions (A1)–(A5)
in Section 3. Recall also that ρ0 =
Fix ρ, ε, η, T such that 0 < ρ <min{
R(a/4)
}, ε > 0, η > 0 and T > 0.
By assumption (A3), there is a constant r1 > 0 such that
D(r)<min
for all r ∈ (0, r1].(37)
Since Π(u) → 0 as u → 0, there are constants 0 < r3 < r2 < min{r1,
such that
Π(r)<
for all r ∈ (0, r3].(38)
By (iv), there are 0< ε̃ <min{
} and n0 > 0 such that for all n≥ n0,
‖y−x‖<2ε̃
‖γi,n(y,x)− γi(x)‖<
.(39)
By (iv)–(vi), and Proposition 1.1, there exist an integer n1 > n0, a con-
stant M̃η,T > 0 and λ̃ ∈ (0, T ), such that for all n≥ n1,
0≤s≤T
‖X̃n(s)‖ ≤ M̃η,T
≥ 1− η/2,(40)
Pn{wT (X̃
n, λ̃)≥ ε̃} ≤ η/4,(41)
0≤s≤T
‖αn(s)‖<
6ILr2
≥ 1− η/4,(42)
δn <min
8(1 + I)
.(43)
To prove C-tightness of {W̃ n} and {Ỹ n} (and hence of {W n}, {Y n}),
by Proposition 1.1, it suffices to show that there exists a constant Nη,T > 0
such that for all n≥ n1,
Pn{wT (W̃
n, λ̃)≥ ε} ≤ η,(44)
Pn{wT (Ỹ
n, λ̃)≥ ε} ≤ η,(45)
0≤s≤T
‖W̃ n(s)‖ ≤Nη,T
≥ 1− η,(46)
0≤s≤T
‖Ỹ n(s)‖ ≤Nη,T
≥ 1− η.(47)
20 W. KANG AND R. J. WILLIAMS
For each n≥ 1, let Fn be a set in Fn such that Pn(Fn) = 1 and on Fn,
properties (iii)(a)–(c) hold, (33)–(36) hold, and W̃ n(t) ∈ Uδn(G) for all t≥ 0.
Fix a t such that 0≤ t < t+ λ̃≤ T . Let
τn = inf{s≥ t :W̃ n(s) ∈ Uδn(∂Gi ∩ ∂G) for some i ∈ I}.(48)
For each n≥ n1, let
wT (X̃
n, λ̃)< ε̃, sup
0≤s≤T
‖αn(s)‖<
6ILr2
0≤s≤T
‖X̃n(s)‖ ≤ M̃η,T
Then by (40)–(42) and the definition of Fn,
P{Hn} ≥ 1− η.(50)
Fix ωn ∈Hn. By the definition of wT (x,λ) in (3), we have that,
r,s∈[t,t+λ̃]
‖X̃n(s,ωn)− X̃n(r,ωn)‖< ε̃.(51)
Now there are two cases to consider for n ≥ n1 and u, v fixed such that
t≤ u < v ≤ t+ λ̃.
Case 1. ωn ∈ {τn > v}. In this case, by (iii)(c), Ỹ n(·, ωn) does not increase
on the interval (u, v], that is, Ỹ ni (v,ω
n)− Ỹ ni (u,ω
n) = 0 for all i ∈ I . Then
by (34) and (36),
W̃ n(v,ωn)− W̃ n(u,ωn) = X̃n(v,ωn)− X̃n(u,ωn).(52)
Hence, by (51),
‖W̃ n(v,ωn)− W̃ n(u,ωn)‖ ≤ sup
r,s∈[t,t+λ̃]
‖X̃n(s,ωn)− X̃n(r,ωn)‖< ε̃ < ε/8,
and we also have
‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖= 0< ε/2.
Case 2. ωn ∈ {τn ≤ v}. Then there is an i ∈ I such that W̃ n(τn, ωn) ∈
Uδn(∂Gi ∩ ∂G), since the set Uδn(∂Gi ∩ ∂G) is closed and W̃
n(·, ωn) is right
continuous. It follows that there is some x0 ∈ ∂G (which depends on ω
such that W̃ n(τn, ωn) is in the closed ball Bδn(x0)⊂Bρ(x0). To apply the
INVARIANCE PRINCIPLE FOR SRBMS 21
oscillation inequality in Theorem 4.1, we first prove the following:
W̃ n(r,ωn) ∈Bρ(x0) for all r satisfying τ
n ≤ r ≤ v.(53)
For the proof of (53), let
ξn = inf{r ≥ τn :W̃ n(r,ωn) /∈Bρ(x0)} ∧ v.(54)
By the definition of ξn, W̃ n(r,ωn) ∈Bρ(x0) for each r ∈ [τ
n, ξn). In order to
apply the oscillation inequality in Theorem 4.1 on the time interval [τn, ξn),
we show that
D(Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn)) + δn))<
.(55)
For each r ∈ (0, T ], by (i)–(iii) and (33), (49), (43), we have that
‖W n(r−, ωn)−W n(r,ωn)‖
≤ ‖W̃ n(r−, ωn)− W̃ n(r,ωn)‖+ ‖αn(r−, ωn)−αn(r,ωn)‖
≤ ‖∆X̃n(r,ωn)‖+2 sup
0≤s≤T
‖αn(s)‖+ Iδn
≤ ε̃+
< 2ε̃.
Hence by (39), for each r ∈ (0, T ],
‖γi,n(W n(r−, ωn),W n(r,ωn))− γi(W n(r,ωn))‖ ≤
.(56)
By (36), (56), Assumption (A4), (i) and (49), we have that for any s1, s2
such that u≤ s1 < s2 ≤ v,
‖Ṽ n(s2, ω
n)− Ṽ n(s1, ω
(s1,s2]
‖γi,n(W n(r−, ωn),W n(r,ωn))
− γi(W n(r,ωn))‖dỸ ni (r,ω
(s1,s2]
‖γi(W n(r,ωn))− γi(W̃ n(r,ωn))‖dỸ ni (r,ω
(Ỹ ni (s2, ω
n)− Ỹ ni (s1, ω
(s1,s2]
L‖W n(r,ωn)− W̃ n(r,ωn)‖dỸ ni (r,ω
(Ỹ ni (s2, ω
n)− Ỹ ni (s1, ω
6ILr2
(Ỹ ni (s2, ω
n)− Ỹ ni (s1, ω
22 W. KANG AND R. J. WILLIAMS
‖Ỹ n(s2, ω
n)− Ỹ n(s1, ω
σn = inf{s≥ τn :Osc(Ỹ n(·, ωn), [τn, s))> r2}.(58)
Note that Osc(Ỹ n(·, ωn), [τn, s)) as a function of s defined on (τn,∞) is
left continuous with finite right limits and is nondecreasing. By the right
continuity of Ỹ n, we know that
Osc(Ỹ n(·, ωn), [τn, s))→ 0 as s ↓ τn.
Thus, σn > τn, Osc(Ỹ n(·, ωn), [τn, σn))≤ r2 and on {σ
n <∞}, Osc(Ỹ n(·, ωn),
[τn, σn]) ≥ r2. By (57), (51), (43), the choice of ε, and since t≤ τ
n ≤ ξn ≤
v ≤ t+ λ̃, we have
Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn
≤Osc(X̃n(·, ωn), [τn, ξn ∧ σn))
+Osc(Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn
≤Osc(X̃n(·, ωn), [τn, ξn ∧ σn))
Osc(Ỹ n(·, ωn), [τn, ξn ∧ σn)) + δn
≤ ε̃+
r2 + δ
n < r3.
Then by (38) and the monotonicity of D(·), we have
D(Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn))
≤D(r2)≤D(r1)<
We claim that
σn ≥ ξn.(61)
To prove (61), we proceed by contradiction and suppose that σn < ξn. Then
by (60), with x= X̃n(·, ωn) + Ṽ n(·, ωn) and δ = δn, condition (iv) of The-
orem 4.1 holds with [s, t] = [τn, σn − 1/m] for all m sufficiently large. By
applying Theorem 4.1 and letting m→∞, we obtain using (34), (38) and
(59) that,
Osc(Ỹ n(·, ωn), [τn, σn))
≤Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn)(62)
≤Π(r3)<
INVARIANCE PRINCIPLE FOR SRBMS 23
By (62), (iii)(b) and (43), we obtain that
Osc(Ỹ n(·, ωn), [τn, σn])≤
+ Iδn < r2.
This contradicts the fact that Osc(Ỹ n(·, ωn), [τn, σn]) ≥ r2 on {σ
n < ∞},
and so (61) holds and (55) follows by (60).
By applying Theorem 4.1 on [τn, ξn − 1/m] and then letting m→∞, we
obtain using (61), (59) and (38), that
Osc(W̃ n(·, ωn), [τn, ξn))
≤Π(Osc(X̃n(·, ωn) + Ṽ n(·, ωn), [τn, ξn ∧ σn)) + δn)
and similarly,
Osc(Ỹ n(·, ωn), [τn, ξn))<
.(63)
Then we have
‖W̃ n(ξn−, ωn)− x0‖
≤ ‖W̃ n(ξn−, ωn)− W̃ n(τn, ωn)‖+ ‖W̃ n(τn, ωn)− x0‖
+ δn.
Using hypotheses (ii), (iii)(b), and (33), (51), we obtain
‖W̃ n(ξn, ωn)− W̃ n(ξn−, ωn)‖
≤ ‖X̃n(ξn, ωn)− X̃n(ξn−, ωn)‖
‖γi,n(W n(ξn−, ωn),W n(ξn, ωn))‖
× (Ỹ ni (ξ
n, ωn)− Ỹ ni (ξ
n−, ωn))
≤ ε̃+ Iδn.
Hence
‖W̃ n(ξn, ωn)− x0‖ ≤ ‖W̃
n(ξn−, ωn)− x0‖
+ ‖W̃ n(ξn, ωn)− W̃ n(ξn−, ωn)‖
+ δn + ε̃+ Iδn
≤ ε̃+ (I+1)δn +
< ρ/8 + ρ/8 + ρ/8< ρ/2.
24 W. KANG AND R. J. WILLIAMS
It follows from this that ξn = v and (53) holds, as desired.
Then, by (33), (51), (iii)(b), (iii)(c), (63) and (43), we have
‖W̃ n(v,ωn)− W̃ n(u,ωn)‖
≤ sup
r,s∈[u,v]
‖X̃n(s,ωn)− X̃n(r,ωn)‖+
(Ỹ ni (v,ω
n)− Ỹ ni (u,ω
≤ ε̃+
(Ỹ ni (v,ω
n)− Ỹ ni (u∨ τ
n, ωn))
(Ỹ ni (u ∨ τ
n, ωn)− Ỹ ni (u,ω
n))(64)
≤ ε̃+ IOsc(Ỹ n(·, ωn), [u∨ τn, v)) +
∆Ỹ ni (v,ω
n) + Iδn
≤ ε̃+ I
+ Iδn + Iδn <
‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖ ≤
(Ỹ ni (v,ω
n)− Ỹ ni (u,ω
(Ỹ ni (v,ω
n)− Ỹ ni (u∨ τ
n, ωn))
(Ỹ ni (u ∨ τ
n, ωn)− Ỹ ni (u,ω
Here we have used the fact that Ỹi does not increase on (u, τ
n ∨ u) and can
jump at most by δn at τn, by the definition of τn and (iii)(c).
On combining the results from Case 1 and Case 2, we obtain that for each
n≥ n1,
u,v∈[t,t+λ̃]
‖W̃ n(v,ωn)− W̃ n(u,ωn)‖ : 0≤ t≤ t+ λ̃≤ T
< ε(66)
u,v∈[t,t+λ̃]
‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖ : 0≤ t≤ t+ λ̃≤ T
< ε.(67)
Hence since ωn ∈Hn was arbitrary, by (50), we have that (44) and (45) hold
for all n≥ n1.
Next we show that there is a constant Nη,T > 0 such that (46) and (47)
hold for all n≥ n1. By (66)–(67) above, we have that for each n≥ n1, ω
INVARIANCE PRINCIPLE FOR SRBMS 25
Hn, t such that 0≤ t < t+ λ̃≤ T and t≤ u < v ≤ t+ λ̃,
‖W̃ n(v,ωn)− W̃ n(u,ωn)‖< ε(68)
‖Ỹ n(v,ωn)− Ỹ n(u,ωn)‖< ε.(69)
Then, for each 0≤ s≤ T , by (68), (69), (49) and (33), we have
‖W̃ n(s,ωn)‖ ≤ ‖W̃ n(s,ωn)− W̃ n(0, ωn)‖+ ‖W̃ n(0, ωn)‖
[T/λ̃]+1∑
‖W̃ n(iλ̃∧ s,ωn)− W̃ n((i− 1)λ̃ ∧ s,ωn)‖+ ‖X̃n(0, ωn)‖
≤ ([T/λ̃] + 1)ε+ M̃η,T
‖Ỹ n(s,ωn)‖ ≤ ‖Ỹ n(s,ωn)− Ỹ n(0, ωn)‖
[T/λ̃]+1∑
‖Ỹ n(iλ̃ ∧ s,ωn)− Ỹ n((i− 1)λ̃∧ s,ωn)‖
≤ ([T/λ̃] + 1)ε.
Here [T/λ̃] is the greatest integer less than or equal to T/λ̃. Let Nη,T =
([T/λ̃] + 1)ε+ M̃η,T . Then we obtain that for n≥ n1 and ω
n ∈Hn,
0≤s≤T
‖W̃ n(s,ωn)‖ ≤Nη,T(70)
0≤s≤T
‖Ỹ n(s,ωn)‖ ≤Nη,T .(71)
Then by (50), we have that (46) and (47) hold for all n≥ n1.
Finally by applying Proposition 1.1, we have the C-tightness of {W̃ n}
and {Ỹ n}. It then follows that {(W̃ n,Xn, Ỹ n)}∞n=1 is C-tight. Since Z
(W̃ n,Xn, Ỹ n) + (αn,0, βn) where αn,V(βn) → 0 in probability as n→∞,
then {Zn}∞n=1 is also C-tight. �
4.3. Invariance principle for SRBMs. The main theorem of the paper is
the following.
Theorem 4.3 (Invariance principle for SRBMs). Suppose that Assump-
tion 4.1 holds. Define Zn = (W n,Xn, Y n) for each n. Then the sequence of
26 W. KANG AND R. J. WILLIAMS
processes {Zn}∞n=1 is C-tight and any (weak) limit point of this sequence is
of the form Z = (W,X,Y ) where continuous d-dimensional processes W,X
and a continuous I-dimensional process Y are defined on some probability
space (Ω,F , P ) such that conditions (i), (ii) and (iv) of Definition 2.1 hold
with Ft = σ{Z(s) : 0≤ s≤ t}, t≥ 0.
If, in addition, the following conditions (vi)′ and (vii) hold, then any
weak limit point of the sequence {Zn}∞n=1 is an extended SRBM associated
with the data (G,µ,Γ,{γi, i ∈ I}, ν). If furthermore the following condition
(viii) holds, then W n ⇒W as n→∞ where W is an SRBM associated with
(G,µ,Γ,{γi, i ∈ I}, ν).
(vi)′ {Xn} converges in distribution to a d-dimensional Brownian mo-
tion with drift µ, covariance matrix Γ and initial distribution ν.
(vii) For each (weak) limit point Z = (W,X,Y ) of {Zn}∞n=1, {X(t) −
X(0)− µt, Ft, t≥ 0} is a martingale.
(viii) If a process W satisfies the properties in Definition 2.1, the law
of W is unique, that is, the law of an SRBM associated with the data
(G,µ,Γ,{γi, i ∈ I}, ν) is unique.
Remark. We note that (vi)′ implies that (vi) of Assumption 4.1 holds.
Proof of Theorem 4.3. By Theorem 4.2, we have that the sequence
{Zn}∞n=1 is C-tight. Let Z = (W,X,Y ) be a (weak) limit point of {Z
n}∞n=1,
that is, there is a subsequence {nk} of {n} such that Z
nk ⇒Z as k→∞. It
also follows that Z̃nk ≡ (W̃ nk ,Xnk , Ỹ nk)⇒Z as k→∞. By the C-tightness
of {Zn}, we obtain that Z has continuous paths a.s. For the purpose of
verifying that Z satisfies the listed properties in Definition 2.1, one may
invoke the Skorokhod representation theorem to assume, without loss of
generality, that Znk and Z̃nk converge u.o.c. to Z a.s. as k→∞ and V(βnk)
converges u.o.c. to 0 a.s. as k → ∞. With this simplification, it is easily
verified that the properties of {Znk} and {Z̃nk} imply that Z has properties
(ii) and (iv)(a)–(b) of Definition 2.1. For the verification of property (i) of
Definition 2.1, note that for each k, a.s. for each t≥ 0,
W nk(t) =Xnk(t) +
(0,t]
γi,nk(W nk(s−),W nk(s))dβ
i (s)
(0,t]
(γi,nk(W nk(s−),W nk(s))− γi(W nk(s)))dỸ
i (s)
(0,t]
γi(W nk(s))dỸ
i (s).
The sum of the first two terms on the right-hand side of the above equality
converges a.s. to X(t) as k → ∞. The third term on the right-hand side
INVARIANCE PRINCIPLE FOR SRBMS 27
converges a.s. to 0 as k→∞, by property (iv) and the fact that a.s.,
s∈(0,t]
‖W nk(s)−W nk(s−)‖
≤ sup
s∈(0,t]
‖∆Xnk(s)‖+ I sup
s∈(0,t]
‖∆Y nk(s)‖→ 0 as k→∞.
It remains to show that for each i ∈ I and t≥ 0, a.s.,
(0,t]
γi(W nk(s))dỸ
i (s)→
(0,t]
γi(W (s))dYi(s) as k→∞.
This follows directly from Lemma A.4.
For the verification of property (iv)(c) of Definition 2.1, it suffices to show
that for each i ∈ I , m= 1,2, . . . , a.s. for each t≥ 0,
Yi(t) =
(0,t]
fm(W (s))dYi(s),(72)
where {fm}
m=1 is a sequence of real valued continuous functions defined
on Rd such that for each m, the range of fm is [0,1], fm(x) = 1 for x ∈
U1/m(∂Gi ∩ ∂G) and fm(x) = 0 for x /∈ U2/m(∂Gi ∩ ∂G). The existence
of such a sequence of continuous functions {fm}
m=1 can be shown using
Urysohn’s lemma (cf. [8], page 122). Then (72) is a consequence of Lemma
A.4, property (iii) of Ỹ
i and the fact that δ
nk → 0 as k→∞. Indeed, a.s.,
for each t≥ 0,
Yi(t) = lim
i (t) = lim
(0,t]
{W̃nk (s)∈U
nk (∂Gi∩∂G)}
i (s)
= lim
(0,t]
fm(W̃
nk(s))dỸ
i (s)
(0,t]
fm(W (s))dYi(s).
Thus, Z satisfies properties (i), (ii) and (iv) of Definition 2.1 with Ft =
σ{Z(s) : 0≤ s≤ t}, t≥ 0.
Assuming properties (vi)′ and (vii) hold, Z satisfies (iii) of Definition
2.1. Then Z is an extended SRBM associated with the data (G,µ,Γ,{γi, i ∈
I}, ν). If in addition, property (viii) holds, then the law of W is unique. Since
each weak limit W is an SRBM associated with the data (G,µ,Γ,{γi, i ∈
I}, ν) and the law of such an SRBM is unique, then by a standard argument,
W n ⇒W as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈
I}, ν). �
Some sufficient conditions for (vii) to hold are given in Proposition 4.2 of
[15] for a simpler setting where G=Rd+. Two of those conditions generalize
to our setting here and can be proved in the same manner as in [15]. For
completeness, we state the ensuing result here.
28 W. KANG AND R. J. WILLIAMS
Proposition 4.1. Suppose that Assumption 4.1 and (vi)′ of Theorem
4.3 hold. If, in addition, one of the following conditions (I)–(II) holds, then
condition (vii) of Theorem 4.3 is satisfied, and any weak limit point of
{Zn}∞n=1 is an extended SRBM associated with (G,µ,Γ,{γ
i, i ∈ I}, ν).
(I) For any triple of d-dimensional {Ft}-adapted processes (W,X,Y )
defined on some filtered probability space (Ω,F ,{Ft}, P ) and satisfying con-
ditions (i), (ii) and (iv) of Definition 2.1 together with the condition that
X, under P , is a d-dimensional Brownian motion with drift vector µ, co-
variance matrix Γ and initial distribution ν, the pair (W,Y ) is adapted to
the filtration generated by X and the P -null sets.
(II) Xn = X̌n + εn1 , Y
n = Y̌ n + εn2 , W
n = W̌ n + εn3 , where ε
1 , ε
2 , ε
3 are
processes converging to 0 in probability as n→∞, and:
(a) {X̌n(t)− X̌n(0)}∞n=1 is uniformly integrable for each t≥ 0,
(b) there is a sequence of constants {µn}∞n=1 in R
d such that
limn→∞µ
n = µ,
(c) for each n, {X̌n(t)− X̌n(0)−µnt, t≥ 0} is a Pn-martingale with
respect to the filtration generated by (W̌ n, X̌n, Y̌ n).
In the rest of this work, we focus on applications of the invariance prin-
ciple and in particular on giving sufficient conditions for property (viii) of
Theorem 4.3 to hold.
5. Applications of the invariance principle. In Section 5.1, we prove weak
existence of SRBMs associated with data (G,µ,Γ,{γi, i ∈ I}, ν) satisfying
(A1)–(A5) of Section 3. This is accomplished by constructing a sequence of
approximations whose weak limit points are SRBMs. The invariance prin-
ciple is used to prove the C-tightness of the approximations and that any
weak limit point is an SRBM. In Sections 5.2 and 5.3, using known results
on uniqueness in law for SRBMs, we illustrate the invariance principle for
certain domains and directions of reflection.
5.1. Weak existence of SRBMs.
Theorem 5.1. Suppose that assumptions (A1)–(A5) of Section 3 hold.
Then there exists an SRBM associated with the data (G,µ,Γ,{γi, i ∈ I}, ν).
Proof. We construct a sequence of approximations to an SRBM and
use the invariance principle to establish weak convergence along a subse-
quence to an SRBM.
In the following we will use R(·) from assumption (A2), L > 0 from as-
sumption (A4), a > 0 from assumption (A5), and ρ0 =
. Fix ε > 0 and
INVARIANCE PRINCIPLE FOR SRBMS 29
0 < ρ < min{
R(a/4)
}. By assumption (A3), there is a constant r1 > 0
such that
D(r)<min
for all r ∈ (0, r1].
Recall the properties of Π(·) from Theorem 4.1. Since Π(u)→ 0 as u→ 0,
there are constants 0< r3 < r2 <min{r1,
} such that
Π(r)<
for all r ∈ (0, r3].
Fix ε̃ and δ such that 0< ε̃ <min{
24ILr2
} and 0< 2δ <min{r3
8(1+I)
We will construct a d-dimensional stochastic processW δ and an I-dimensional
“pushing” process Y δ , such that W δ approximately satisfies the conditions
defining an SRBM for the data (G,µ,Γ,{γi, i ∈ I}, ν) (cf. Assumption 4.1).
The idea for this construction is to use a Brownian motion X with drift
vector µ, covariance matrix Γ and initial distribution ν. Away from ∂G, the
increments of W δ are determined by those of X . For any time t ≥ 0 such
that W δ(t−) ∈ ∂G, we add an instantaneous jump to W δ(t−) to obtain
W δ(t) ∈G. Here W δ(0−) =X(0). The size of the jump is such that W δ(t)
is a strictly positive distance (depending on δ) from the boundary of G.
The jump vector is obtained as a measurable function of W δ(t−). To ensure
the measurability, each point x on ∂G is associated with a nearby point x̄,
chosen in a measurable way from a fixed countable set of points in ∂G. The
jump vector for x is one associated with x̄. We now specify the mapping
x→ x̄ and the associated jump vector more precisely.
By assumption (A5)(ii), for each x ∈ ∂G, there is c(x) ∈RI+ such that
i∈I(x)
ci(x) = 1 and min
j∈I(x)
i∈I(x)
ci(x)γ
i(x), nj(x)
≥ a.(73)
By (73), Lemma 2.1 and the fact that ni(·) is continuous on ∂Gi for each
i ∈ I , we have that for each x ∈ ∂G there is rx ∈ (0, δ) such that for each
y ∈Brx(x)∩ ∂G,
I(y)⊂ I(x)(74)
j∈I(x)
i∈I(x)
ci(x)γ
i(x), nj(y)
.(75)
It follows, using the C1 nature of ∂Gi and the fact that n
i(y) is the inward
unit normal to ∂Gi at y ∈ ∂G for each i ∈ I(y), that (by choosing rx even
30 W. KANG AND R. J. WILLIAMS
smaller if necessary) for each x ∈ ∂G there is m(x)> 0 and rx ∈ (0, δ) such
that for each y ∈Brx(x)∩ ∂G, (74)–(75) hold and
y + λ
i∈I(x)
ci(x)γ
i(x) ∈G for all λ ∈ (0,m(x)).(76)
Let Borx(x) denote the interior of the closed ball Brx(x) for each x ∈ ∂G. The
collection {Borx(x) :x ∈ ∂G} is an open cover of ∂G and it follows that there
is a countable set {xk} such that ∂G⊂
kBrxk (xk) and {xk} ∩BN (0) is a
finite set for each integer N ≥ 1. We can further choose the set {xk} to be
minimal in the sense that for each strict subset C of {xk}, {Brx(x) :x ∈C}
does not cover ∂G. Let Dk = (Brxk (xk) \ (
i=1 Brxi (xi)) ∩ ∂G for each k.
Then Dk 6= ∅ for each k, {Dk} is a partition of ∂G, and for each x ∈ ∂G
there is a unique index i(x) such that x ∈Di(x). For each x ∈R
d, let
x, if x /∈ ∂G,
xi(x), if x ∈ ∂G.
Note that for all x ∈Rd,
‖x− x̄‖< δ.(77)
For each i ∈ I and x∈Rd, let
γi,δ(x) = γi(x̄).(78)
The mapping x → x̄ is Borel measurable on Rd and hence γi,δ is a Borel
measurable function from Rd into Rd.
We construct (W δ, Y δ) as follows. Let X defined on some filtered proba-
bility space (Ω,F ,{Ft}, P ) be a d-dimensional {Ft}-Brownian motion with
drift µ and covariance matrix Γ such that X is continuous surely and X(0)
has distribution ν. Let
τ1 = inf{t≥ 0 :X(t) ∈ ∂G}
W δ(t) =X(t), Y δ(t) = 0 for 0≤ t < τ1.
Note that W δ(τ1−) exists on {τ1 <∞} since X has continuous paths and
in the case that τ1 = 0, W
δ(0−)≡X(0). On {τ1 <∞}, define
Y δi (τ1) =
0, i /∈ I(W δ(τ1−)),
ci(W δ(τ1−))
m(W δ(τ1−))
, i ∈ I(W δ(τ1−)),
W δ(τ1) =X(τ1)
m(W δ(τ1−))
i∈I(W δ(τ1−))
ci(W δ(τ1−))γ
i,δ(W δ(τ1−))
INVARIANCE PRINCIPLE FOR SRBMS 31
So W δ, Y δ have been defined on [0, τ1) and at τ1 on {τ1 <∞}, such that:
(i) W δ(t) = X(t) +
i∈I γ
i,δ(W δ(0−))Y δi (0) +
(0,t]
γi,δ(W δ(s−))dY δi (s) for all t ∈ [0, τ1]∩ [0,∞), where W
δ(0−) =X(0),
(ii) W δ(t) ∈G for t ∈ [0, τ1]∩ [0,∞),
(iii) for i ∈ I ,
(a) Y δi (0)≥ 0,
(b) Y δi is nondecreasing on [0, τ1]∩ [0,∞),
(c) Y δi (t) = Y
i (0) +
(0,t] 1{W δ(s)∈U2δ(∂Gi∩∂G)} dY
i (s) for t ∈ [0, τ1] ∩
[0,∞),
(iv) ‖∆Y δ(t)‖ ≡ ‖Y δ(t)−Y δ(t−)‖ ≤ δ for t ∈ [0, τ1]∩ [0,∞), where Y
δ(0−)≡
Note that (iii)(c) above contains the expression W δ(s) ∈U2δ(∂Gi∩∂G). The
reader may wonder why 2δ appears instead of δ. The reason is that at a jump
time s of Y δi , W
δ(s−) ∈ ∂Gi ∩ ∂G and so
dist(W δ(s), ∂Gi ∩ ∂G)≤ ‖W
δ(s)−W δ(s−)‖+ ‖W δ(s−)−W δ(s−)‖
≤ δ + δ = 2δ.
Proceeding by induction, we assume that for some n ≥ 2, τ1 ≤ · · · ≤ τn−1
have been defined, and W δ, Y δ have been defined on [0, τn−1) and at τn−1
on {τn−1 <∞}, such that (i)–(iv) above hold with τn−1 in place of τ1. Then
we define τn =∞ on {τn−1 =∞}, and on {τn−1 <∞} we define
τn = inf{t≥ τn−1 :W
δ(τn−1) +X(t)−X(τn−1) ∈ ∂G}.
For τn−1 ≤ t < τn, let
Y δ(t) = Y δ(τn−1),
W δ(t) =W δ(τn−1) +X(t)−X(τn−1),
and on {τn <∞}, let
Y δi (τn) =
Y δi (τn−1), i /∈ I(W
δ(τn−)),
Y δi (τn−1) + ci(W
δ(τn−))
m(W δ(τn−))
, i ∈ I(W δ(τn−)),
W δ(τn) =W
δ(τn−)
m(W δ(τn−))
i∈I(W δ(τn−))
ci(W δ(τn−))γ
i,δ(W δ(τn−))
32 W. KANG AND R. J. WILLIAMS
In this way, W δ, Y δ have been defined on [0, τn) and at τn on {τn <∞} such
that (i)–(iv) hold with τn in place of τ1.
By construction {τn}
n=1 is a nondecreasing sequence of stopping times.
Let τ = limn→∞ τn. On {τ =∞}, the construction of (W
δ, Y δ) is complete.
We now show that {τ < ∞} = ∅. In fact, if {τ < ∞} 6= ∅, let ω ∈ {τ <
∞}. The above construction gives (W δ(·, ω), Y δ(·, ω)) on the time interval
[0, τ(ω)). For each t ∈ [0, τ(ω)), we have
W δ(t,ω) =X(t,ω) +
γi,δ(W δ(0−, ω))Y δi (0, ω)
(0,t]
γi,δ(W δ(s−, ω))dY δi (s,ω).
SinceX is continuous on [0,∞), ‖γi,δ(x)‖= 1 for each x ∈Rd and
i∈I Y
i (0,
ω)≤ δ, there are constants λ̃ ∈ (0, τ(ω)) and M̃ > 0 (depending on ω) such
wτ(ω)(X(·, ω) + γ
i,δ(W δ(0−, ω))Y δi (0, ω), λ̃)< ε̃(80)
0≤t≤τ(ω)
∥∥∥∥∥X(·, ω) +
γi,δ(W δ(0−, ω))Y δi (0, ω)
∥∥∥∥∥≤ M̃,(81)
where w
(·, ·) is defined in (3). By the choice of ε̃, δ made at the beginning of
this proof, (77)–(78) and the uniform Lipschitz property of the γi(·), i ∈ I , it
follows that (39) and (43) hold with γi,δ(y) and 2δ in place of γi,n(y,x) and
δn, respectively. Then by similar pathwise analysis to that used in Case 1 and
2 of the proof of Theorem 4.2, with W̃ n =W n =W δ, αn = 0, γi,n(y,x) =
γi,δ(y) for each i ∈ I and x, y ∈ Rd, Xn = X +
i∈I γ
i,δ(W δ(0−))Y δi (0),
Y n = Y δ , Ỹ n = Y δ − Y δ(0), βn = Y δ(0) and δn = 2δ, we obtain that (71)
holds for any T < τ(ω) with ωn = ω, Nη,T = ([τ(ω)/λ̃] + 1)ε+ M̃ . It follows
that supi∈I sups∈[0,τ(ω)) Y
i (s,ω) is finite. By the nondecreasing property of
Y δi (·, ω) on [0, τ(ω)) for each i ∈ I , Y
i (τ(ω)−, ω) exists and is finite for each
i ∈ I . Then by (79) and the continuity of X , we see that W δ(τ(ω)−, ω) exists
and is finite. By the construction of Y δ and the fact that
i∈I(x) ci(x) = 1
for all x∈ ∂G, we have that
Y δi (τ(ω)−, ω) =
m(W δ(τn(ω)−, ω))
∧ δ.(82)
Since τn(ω) ↑ τ(ω) as n → ∞ and W
δ(τ(ω)−, ω) exists, it follows that
{W δ(τn(ω)−, ω)}
n=1 converges to W
δ(τ(ω)−, ω) ∈ ∂G as n → ∞. Conse-
quently, {W δ(τn(ω)−, ω)}
n=1 is a bounded sequence in ∂G and so by the
INVARIANCE PRINCIPLE FOR SRBMS 33
definition of the sets {Dk} which form a partition of ∂G, there is a finite set
C such that
{W δ(τn(ω)−, ω)}
n=1 ⊂
Hence,
m(W δ(τn(ω)−, ω))≤ inf
m(xk)> 0,(83)
and so the right-hand side of (82) is infinite. On the other hand, since
supi∈I sups∈[0,τ(ω)) Y
i (s,ω) is finite, the left-hand side of (82) is finite. This
yields the desired contradiction and so {τ < ∞} = ∅ and we have con-
structed (W δ, Y δ) on [0,∞).
From the construction above, we can see that W δ and Y δ are well-defined
stochastic processes with sample paths in D([0,∞),Rd) and D([0,∞),RI).
They are adapted to the filtration generated by X and satisfy (i)–(iv) above
with [0,∞) in place of [0, τ1].
Consider a sequence of sufficiently small δ’s, denoted by {δn}, such that
δn ↓ 0 as n → ∞. For each δn, let (W δ
, Y δ
) be the pair constructed as
above for the same process X . By the above properties and the fact that for
each i ∈ I and x, y ∈Rd,
‖γi,δ
(y)− γi(x)‖ ≤ ‖γi(ȳ)− γi(x)‖ ≤ L‖ȳ− x‖ ≤ L(δn + ‖y − x‖),
we obtain that Assumption 4.1 holds with W̃ n =W n =W δ
, αn = 0, γi,n(y,x) =
(y) for each i ∈ I and x, y ∈Rd, Xn =X+
i∈I γ
i,δn(W δ
(0−))Y δ
i (0),
Y n = Y δ
, Ỹ n = Y δ
(0), βn = Y δ
(0) and 2δn in place of δn. By invok-
ing the first part of Theorem 4.3, we obtain that {Zδ
}∞n=1 = {(W
δn ,Xδ
)}∞n=1 is C-tight and any weak limit point Z of this sequence satisfies
conditions (i), (ii) and (iv) of Definition 2.1 with Ft = σ{Z(s) : 0 ≤ s ≤ t},
t≥ 0. Note that condition (vi)′ of Theorem 4.3 holds trivially. Furthermore,
= {Xδ
(t)−Xδ
(0)− µt, t≥ 0}= {X(t)−X(0)− µt, t≥ 0} is a mar-
tingale with respect to the filtration generated by X . Since W δ
, Y δ
adapted to this filtration, it follows that M δ
is a martingale with respect
to the filtration generated by W δ
, Y δ
(which in fact is the same as
that generated by X). For each t≥ 0, Xδ
(t)−Xδ
(0) =X(t)−X(0) and
so trivially this forms a uniformly integrable sequence as n varies. It fol-
lows from Proposition 4.1 that condition (vii) of Theorem 4.3 holds. Hence,
any weak limit point of {Zδ
}∞n=1 is an extended SRBM with the data
(G,µ,Γ,{γi, i ∈ I}, ν). �
5.2. SRBMs in convex polyhedrons with constant reflection fields. Exis-
tence and uniqueness in law for SRBMs living in convex polyhedrons with a
constant reflection field on each boundary face has been studied by Dai and
34 W. KANG AND R. J. WILLIAMS
Williams [4]. In this subsection, we state a consequence of our invariance
principle using the results in [4] to establish uniqueness in law. In this case,
G is defined in terms of I (I≥ 1) d-dimensional unit vectors {ni, i ∈ I} and
an I-dimensional vector β = (β1, . . . , βI)
′ such that
G≡ {x ∈Rd : 〈ni, x〉 ≥ βi for all i ∈ I}.(84)
It is assumed that G is nonempty and that the set {(n1, β1), . . . , (n
I, βI)} is
minimal in the sense that no proper subset defines G. For each i ∈ I , let Fi
denote the boundary face: {x ∈G : 〈ni, x〉= βi}. Then, n
i is the inward unit
normal to Fi. A constant vector field γ
i of unit length specifies the direction
of reflection associated with Fi.
Definition 5.2. For each ∅ 6=K⊂ I , define FK =
i∈KFi. Let F∅ =G.
A set K⊂ I is maximal if K 6=∅, FK 6=∅ and FK 6= FK̄ for any K̄ ⊃ K such
that K̄ 6=K.
In [4], Dai and Williams introduced the following assumption.
Assumption 5.1. For each maximal K⊂ I ,
(S.a) there is a positive linear combination n=
i∈K bin
i (bi > 0 ∀i ∈K)
of the {ni, i ∈K} such that 〈n,γi〉> 0 for all i ∈K,
(S.b) there is a positive linear combination γ =
i∈K ciγ
i (ci > 0 ∀i ∈K)
of the {γi, i ∈K} such that 〈ni, γ〉> 0 for all i ∈K.
Remark. For the given G and constant vector fields {γi, i ∈ I}, As-
sumption 5.1 is equivalent to assumption (A5).
Definition 5.3. The convex polyhedron G is simple if for each K⊂ I
such that K 6=∅ and FK 6=∅, exactly |K| distinct faces contain FK.
Remark. The polyhedron G is simple if and only if K is maximal for
every K such that ∅ 6= K ⊂ I and FK 6= ∅. It is shown in [4] that when G
is simple, (S.a) holds for all maximal K⊂ I if and only if (S.b) holds for all
maximal K⊂ I.
Dai and Williams [4] showed that Assumption 5.1 is sufficient for exis-
tence and uniqueness in law of SRBMs living in G with the reflection fields
{γi, i ∈ I} and fixed starting point. [They also showed that condition (S.b)
holding for all maximal K⊂ I is necessary for existence of an SRBM starting
from each point in G. Consequently, when G is simple, Assumption 5.1 is
necessary and sufficient for existence of an SRBM starting from each point
in G.] This yields the following consequence of our invariance principle.
INVARIANCE PRINCIPLE FOR SRBMS 35
Theorem 5.4. Let G be a nonempty domain such that G is a convex
polyhedron of the form (84) (with minimal description), and let {γi, i ∈ I} be
a family of constant vector fields of unit length satisfying Assumption 5.1.
Suppose that Assumption 4.1 and (vi)′, (vii) of Theorem 4.3 hold. Then
W n ⇒W as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈
I}, ν).
Proof. Clearly (A1) holds. Assumptions (A2)–(A3) hold by Lemma
A.3. Since for each i ∈ I , γi(·) is a constant vector field of unit length, as-
sumption (A4) holds trivially. Assumption (A5) is implied by Assumption
5.1. Hence by Theorem 4.3, the only thing that we have to check is condition
(viii) of Theorem 4.3, that is, uniqueness in law for SRBMs in convex poly-
hedrons with constant reflection fields of unit length. But this is proved in
Theorem 1.3 of [4] for a fixed starting point in G and follows by a standard
conditioning argument for the initial distribution ν. �
5.3. SRBMs in bounded domains with piecewise smooth boundaries. Dupuis
and Ishii [6] have established sufficient conditions for the existence and path-
wise uniqueness of reflecting diffusions living in the closures of bounded
domains with piecewise smooth boundaries. In this subsection, we state a
consequence of our invariance principle using the results in [6] to establish
uniqueness in law.
Theorem 5.5. Let G be a bounded domain and {γi, i ∈ I} be a family
of reflection fields that satisfy assumptions (A1)–(A4) and (A5)′ in Section
3. We further assume that for each i ∈ I , γi(·) is once continuously differ-
entiable with locally Lipschitz continuous first partial derivatives. Suppose
that Assumption 4.1 and (vi)′, (vii) of Theorem 4.3 hold. Then W n ⇒W
as n→∞ where W is an SRBM associated with (G,µ,Γ,{γi, i ∈ I}, ν).
Remark. We remind the reader that in view of Lemma 3.1, to verify
condition (A5)′, one only needs to show that (i) or (ii) holds for all x ∈ ∂G.
However, as can be seen from the proof below, both forms of the condition
can be useful.
Proof of Theorem 5.5. This theorem follows from Theorem 4.3 and
uniqueness in law for the associated SRBMs. The latter follows by a standard
argument from the pathwise uniqueness established in Corollary 5.2 of [6] for
their Case 2. The conditions required for that case are satisfied in particular
because (A5)′(ii) implies condition (3.8) of [6]. That condition (3.8) readily
implies condition (3.6) of [6]; and, by [5], under the additional smoothness
assumptions imposed on the γi in the statement of our theorem, condition
(3.8) also implies condition (3.7) in [6]. In addition, (A5)′(i) implies that
36 W. KANG AND R. J. WILLIAMS
for each x ∈ ∂G, 〈γi(x), ni(x)〉> 0 for each i ∈ I(x), and furthermore, since
(A5)′ implies (A5), we have by (A5)(i) that the origin does not belong to
the convex hull of the {γi(x) : i ∈ I(x)}. �
APPENDIX: AUXILIARY LEMMAS
Lemma A.1. Suppose that G is bounded. If assumption (A1) holds, then
assumption (A2) holds.
Proof. To see this, suppose G is bounded and assumption (A1) holds.
Fix ε ∈ (0,1). For each i ∈ I and z ∈ ∂Gi ∩ ∂G, by the C
1 property of ∂Gi,
there is a neighborhood Vz of z and a constant R(ε, i, z) > 0 such that for
all x ∈ Vz ∩ ∂Gi ∩ ∂G and y ∈Gi such that ‖x− y‖<R(ε, i, z),
〈ni(x), y − x〉 ≥−ε‖y − x‖.(85)
Assumption (A2) then follows by a standard compactness argument. �
Lemma A.2. Suppose that G is a nonempty bounded domain satisfying
(5), where for each i ∈ I , Gi is a nonempty domain. Then assumption (A3)
holds.
Proof. We prove the lemma by contradiction. Suppose that assumption
(A3) does not hold. Then, since there are only finite many J ⊂ I , J 6=∅,
there is an ε > 0, a nonempty set J ⊂ I , a sequence {rn} ⊂ (0,∞) with rn →
0 as n→∞, a sequence {xn} ⊂R
d such that for each n, xn ∈
j∈J Urn(∂Gj∩
∂G) and dist(xn,
j∈J (∂Gj ∩ ∂G)) > ε. But since G is bounded, {xn} is
bounded and without loss of generality we may assume that xn → x as
n→∞ for some x ∈Rd. It follows that x ∈
j∈J (∂Gj ∩ ∂G), since for each
j ∈ J ,
dist(x,∂Gj ∩ ∂G)≤ ‖xn − x‖+dist(xn, ∂Gj ∩ ∂G)≤ ‖xn − x‖+ rn → 0
as n→∞. This is inconsistent with xn → x and dist(xn,
j∈J (∂Gj ∩∂G))>
Lemma A.3. Suppose (A1) holds where
Gi = {x ∈R
d : 〈ni, x〉> βi} for i ∈ I,(86)
{ni, i ∈ I} is a finite collection of d-dimensional vectors of unit length, and
for I= |I|, β = (β1, . . . , βI)
′ is an I-dimensional vector. (Thus, G is a convex
polyhedron.) Assume that for each i ∈ I , ∂Gi ∩ ∂G 6=∅. Then assumptions
(A2) and (A3) hold.
INVARIANCE PRINCIPLE FOR SRBMS 37
Proof. Assumption (A2) holds automatically since G is convex. In or-
der to show that assumption (A3) holds, we just need to show that for each
J ⊂ I with J 6=∅,
(∂Gj ∩ ∂G)
Ur(∂Gj ∩ ∂G)
→ 0(87)
as r→ 0. Fix J ⊂ I such that J 6=∅. Then
j∈J (∂Gj ∩∂G) is the collection
of all solutions x ∈Rd to the following system of linear inequalities:
〈ni, x〉 ≥ βi for all i ∈ I,
〈−ni, x〉 ≥ −βi for all i ∈ J .
Suppose that
j∈J (∂Gj ∩∂G) 6=∅, that is, (LS) has at least one solution.
By a theorem of Hoffman [11], with supporting lemmas proved by Agmon
[1], there is a constant C > 0 (depending only on {ni, i ∈ I} and not on β)
such that for any x ∈Rd there exists a solution x0 ∈R
d of (LS) with
‖x− x0‖ ≤C
(βi − 〈n
i, x〉)+ +
(−βi − 〈−n
i, x〉)+
.(88)
For r > 0, any x ∈
j∈J Ur(∂Gj ∩ ∂G) satisfies the following:
〈ni, x〉 ≥ βi − r for all i ∈ I,
(r-LS)
〈−ni, x〉 ≥ −βi − r for all i ∈ J .
Then by (88), there is x0 ∈
j∈J (∂Gj ∩ ∂G) such that
(∂Gj ∩ ∂G)
≤ ‖x− x0‖ ≤ 2C|I|r.
It follows that (87) holds when
j∈J (∂Gj ∩ ∂G) 6=∅.
Now suppose that
j∈J (∂Gj ∩∂G) =∅, that is, (LS) has no solution. We
shall use an argument by contradiction to show that
j∈J Ur(∂Gj ∩∂G) =∅
for all r sufficiently small. Suppose that this is not true. Then we have that⋂
j∈J Ur(∂Gj ∩ ∂G) 6= ∅ for all r ∈ (0,∞). As we have seen before, any
j∈J Ur(∂Gj ∩ ∂G) is a solution to (r-LS). We now construct a Cauchy
sequence. Let x1 ∈
j∈J U1/2(∂Gj ∩ ∂G). Then x1 is a solution to (
-LS).
Since ( 1
-LS) has at least one solution, by the theorem of Hoffman [11] (using
the fact that the constant C depends only on {ni, i ∈ I}), we conclude that
there is a solution x2 to (
-LS) such that ‖x1−x2‖ ≤
, where C ′ = 2C|I|.
Continuing in this manner, we can obtain a sequence {xn}
n=1 such that for
each n ≥ 1, ‖xn − xn+1‖ ≤
and xn+1 is a solution of (
-LS). The
38 W. KANG AND R. J. WILLIAMS
sequence {xn}
n=1 is Cauchy. Hence, there is an x
∗ ∈Rd such that xn → x
as n→ ∞, and x∗ is a solution to (LS). This contradicts the supposition
j∈J (∂Gj ∩ ∂G) =∅. Thus we have that
j∈J Ur(∂Gj ∩ ∂G) =∅ for
all r sufficiently small, and for such r,
(∂Gj ∩ ∂G)
Ur(∂Gj ∩ ∂G)
by convention.
Combining the above we see that for each J ⊂ I with J 6=∅, (87) holds
and hence assumption (A3) holds. �
Remark. In fact, under the assumptions of Lemma A.3, there is a con-
stant C > 0 such that D(u) ≤ Cu for each u ≥ 0 and D(·) defined as in
assumption (A3).
Lemma A.4. Given T > 0, functions φ,{φn}∞n=1 in D([0,∞),R
d), and
χ,{χn}∞n=1 in D([0,∞),R), suppose that sup0≤s≤T ‖φ
n(s)− φ(s)‖ → 0 and
sup0≤s≤T |χ
n(s)−χ(s)| → 0 as n→∞. Assume that χn is nondecreasing for
each n. Then for any sequence of real valued continuous functions {fn}∞n=1
defined on Rd such that fn converges uniformly on each compact set to a
continuous function f :Rd →R, we have
(0,t]
fn(φn(s))dχn(s)→
(0,t]
f(φ(s))dχ(s) as n→∞,(89)
uniformly for t ∈ [0, T ].
Proof. By replacing χn(·) and χ(·) by χn(·)− χn(0) and χ(·)− χ(0),
respectively, we may assume that χn(0) = χ(0) = 0. It is straightforward to
see by the uniform convergence of {χn} to χ on [0, T ] that χ inherits the
nondecreasing property of the {χn}.
By the triangle inequality,
0≤t≤T
(0,t]
fn(φn(s))dχn(s)−
(0,t]
f(φ(s))dχ(s)
≤ sup
0≤t≤T
(0,t]
(fn(φn(s))− f(φ(s)))dχn(s)
∣∣∣∣(90)
+ sup
0≤t≤T
(0,t]
f(φ(s))d(χn(s)− χ(s))
∣∣∣∣.
For the first term on the right-hand side of the above inequality, we have
0≤t≤T
(0,t]
(fn(φn(s))− f(φ(s)))dχn(s)
≤ sup
0≤s≤T
|fn(φn(s))− f(φ(s))|χn(T ),
INVARIANCE PRINCIPLE FOR SRBMS 39
where the right-hand side member above tends to zero as n → ∞ by the
uniform convergence of φn to φ on [0, T ] (which implies uniform boundedness
of {φn} on [0, T ]), the uniform convergence of fn to f on compact sets, the
continuity of f , and the convergence of χn(T ) to χ(T ). For the second term,
note that since f(φ(·)) ∈D([0,∞),R), by Theorem 3.5.6, Proposition 3.5.3
and Remark 3.5.4 of [7], there is a sequence of step functions {zk}∞k=1 of the
zk(·) =
zk(tki )1[tk
)(·),(91)
where 1 ≤ lk < ∞, 0 = t
1 < t
2 < · · · < t
< ∞ and sup0≤s≤T |f(φ(s)) −
zk(s)| → 0 as k→∞. Then
0≤t≤T
(0,t]
f(φ(s))d(χn(s)− χ(s))
≤ sup
0≤t≤T
(0,t]
(f(φ(s))− zk(s))d(χn(s)− χ(s))
+ sup
0≤t≤T
(0,t]
zk(s)d(χn(s)− χ(s))
≤ sup
0≤s≤T
|f(φ(s))− zk(s)|(χn(T ) + χ(T ))
+ sup
0≤t≤T
|zk(tki )||(χ
n − χ)((tki+1 ∧ t)−)− (χ
n − χ)((tki ∧ t)−)|.
For fixed k, the last term above can be made as small as we like for all n
sufficiently large since χn → χ uniformly on [0, T ]. The desired result follows.
Remark. The proof of Lemma A.4 is a modification of the proof of the
related Lemma 2.4 in [4]. The difference in assumptions is that in [4] it is
assumed that φn → φ in the J1-topology rather than uniformly on [0, T ],
χn, χ ∈C([0,∞),R+) rather than χ
n, χ ∈D([0,∞),R), and there is a single
function f rather than a sequence {fn}.
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Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, Pennsylvania 15213
E-mail: weikang@andrew.cmu.edu
Department of Mathematics
University of California at San Diego
9500 Gilman Drive
La Jolla, California 92093
E-mail: williams@math.ucsd.edu
http://www.ams.org/mathscinet-getitem?mr=1346729
http://www.ams.org/mathscinet-getitem?mr=2222685
http://www.ams.org/mathscinet-getitem?mr=1096165
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http://www.ams.org/mathscinet-getitem?mr=1681462
http://www.ams.org/mathscinet-getitem?mr=0473443
http://www.ams.org/mathscinet-getitem?mr=0606992
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http://www.ams.org/mathscinet-getitem?mr=0959133
http://www.ams.org/mathscinet-getitem?mr=1231926
http://www.ams.org/mathscinet-getitem?mr=1663755
http://www.ams.org/mathscinet-getitem?mr=1663759
mailto:weikang@andrew.cmu.edu
mailto:williams@math.ucsd.edu
	Introduction
	Notation, terminology and preliminaries
	Definition of an SRBM
	Assumptions on the domain G and the reflection vector fields {i}
	Assumptions on the domain G
	Assumptions on the reflection vector fields {i}
	Invariance principle
	Oscillation inequality
	C-tightness result
	Invariance principle for SRBMs
	Applications of the invariance principle
	Weak existence of SRBMs
	SRBMs in convex polyhedrons with constant reflection fields
	SRBMs in bounded domains with piecewise smooth boundaries
	Appendix: Auxiliary lemmas
	References
	Author's addresses
