A uniqueness theorem for the martingale problem describing a diffusion in media with membranes
We formulate a martingale problem that describes a diffusion process in a multidimensional Euclidean space with a membrane located on a given mooth surface and having the properties of skewing and delaying. The theorem on the existence of no more than one solution to the problem is proved.
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| Цитувати: | A uniqueness theorem for the martingale problem describing a diffusion in media with membranes / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 1–9. — Бібліогр.: 9 назв.— англ. |
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irk-123456789-45472009-12-07T12:00:35Z A uniqueness theorem for the martingale problem describing a diffusion in media with membranes Aryasova, O.V. Portenko, M.I. We formulate a martingale problem that describes a diffusion process in a multidimensional Euclidean space with a membrane located on a given mooth surface and having the properties of skewing and delaying. The theorem on the existence of no more than one solution to the problem is proved. 2008 Article A uniqueness theorem for the martingale problem describing a diffusion in media with membranes / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 1–9. — Бібліогр.: 9 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4547 519.21 en Інститут математики НАН України |
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We formulate a martingale problem that describes a diffusion process in a multidimensional Euclidean space with a membrane located on a given mooth surface and having the properties of skewing and delaying. The theorem on the existence of no more than one solution to the problem is proved. |
| format |
Article |
| author |
Aryasova, O.V. Portenko, M.I. |
| spellingShingle |
Aryasova, O.V. Portenko, M.I. A uniqueness theorem for the martingale problem describing a diffusion in media with membranes |
| author_facet |
Aryasova, O.V. Portenko, M.I. |
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Aryasova, O.V. |
| title |
A uniqueness theorem for the martingale problem describing a diffusion in media with membranes |
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A uniqueness theorem for the martingale problem describing a diffusion in media with membranes |
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A uniqueness theorem for the martingale problem describing a diffusion in media with membranes |
| title_fullStr |
A uniqueness theorem for the martingale problem describing a diffusion in media with membranes |
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A uniqueness theorem for the martingale problem describing a diffusion in media with membranes |
| title_sort |
uniqueness theorem for the martingale problem describing a diffusion in media with membranes |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/4547 |
| citation_txt |
A uniqueness theorem for the martingale problem describing a diffusion in media with membranes / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 1–9. — Бібліогр.: 9 назв.— англ. |
| work_keys_str_mv |
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| first_indexed |
2025-07-02T07:45:52Z |
| last_indexed |
2025-07-02T07:45:52Z |
| _version_ |
1836520416934363136 |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 1–9
UDC 519.21
OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
A UNIQUENESS THEOREM FOR THE MARTINGALE PROBLEM
DESCRIBING A DIFFUSION IN MEDIA WITH MEMBRANES
We formulate a martingale problem that describes a diffusion process in a multidi-
mensional Euclidean space with a membrane located on a given smooth surface and
having the properties of skewing and delaying. The theorem on the existence of no
more than one solution to the problem is proved.
Introduction
Let S be a given closed bounded surface in Rd that divides the space Rd into two open
parts: the interior domain Di and the exterior one De, D is the union of them. The
surface S is assumed to be smooth enough (see Section 1 for the precise assumptions)
so that there is a well-defined normal at any point of S. By ν(x) for x ∈ S, we denote
the unit vector of the outward normal to S at the point x. Let A(x), x ∈ S, be a given
real-valued continuous function and, for each y ∈ Rd, let b(y) be a symmetric positive
definite linear operator in Rd. The function (b(y))y∈Rd is supposed to be bounded and
Hölder continuous. For x ∈ S, the vector N(x) = b(x)ν(x) is called the co-normal vector
to S at the point x. Consider the stochastic differential equation in Rd
(1) dx(t) = A(x(t))N(x(t))1IS(x(t))dt + b(x(t))1/21ID(x(t))dw(t),
where (w(t))t≥0 is a standard Wiener process in Rd, 1IΓ is the indicator function of a set
Γ ⊂ Rd. As was shown in [1], this equation has infinitely many solutions. Consequently,
if a solution to (1) is treated as that to the corresponding martingale problem, the latter
turns out not to be well-posed.
Each solution constructed in [1] is determined by a representation of the function
A(x), x ∈ S, in the form A(x) = q(x)
r(x) , where q(·) and r(·) are continuous functions on
S taking their values in [−1, 1] and (0,+∞), respectively. Thus, the formulation of the
well-posed martingale problem must involve these functions.
A solution to (1) was constructed in [1] as a continuous Markov process (x(t))t≥0
in Rd obtained from a d-dimensional diffusion process with its diffusion operator b(·)
and a zero drift vector by two transformations. The first transformation is skewing the
diffusion process on S. The skew is determined by the function q(·). As a result, one
get a continuous Markov process (x0(t))t≥0 in Rd such that its trajectories satisfy the
stochastic differential equation (see [2], Ch. 3)
(2) dx0(t) = q(x0(t))δS(x0(t))N(x0(t))dt + b(x0(t))1/2dw(t),
where (δS(x))x∈Rd is a generalized function on Rd that acts on a test function (ϕ(x))x∈Rd
according to the following rule:
〈δS , ϕ〉 =
∫
S
ϕ(x)dσ
2000 AMS Mathematics Subject Classification. Primary 60J60, 60J35.
Key words and phrases. Diffusion process, martingale problem, uniqueness of solution.
1
2 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
(the integral in this equality is a surface integral).
To do the second transformation determined by a given function r(·) : S → (0,∞),
one should put, for t ≥ 0,
ζt = inf
{
s : s+
∫ s
0
r(x0(τ))δS(x0(τ))dτ ≥ t
}
and define
x(t) = x0(ζt), t ≥ 0.
Here, the functional
ηt =
∫ t
0
r(x0(τ))δS(x0(τ))dτ, t ≥ 0,
of the process (x0(t))t≥0 is well defined as an additive homogeneous continuous func-
tional (see [2], Ch. 3). As is known (see [3], Theorem 10.11), the process (x(t))t≥0 is a
continuous Markov process in Rd as a result of the random change of time for the process
(x0(t))t≥0.
The following observation gives us a suggestion how to formulate correctly the mar-
tingale problem for the process (x(t))t≥0 corresponding to a given pair of functions q(·)
and r(·). Namely, fix an orthonormal basis in Rd and denote, by xj for j = 1, 2, . . . , d
the coordinates of a vector x ∈ Rd and by bjk(x) for j, k = 1, 2, . . . , d, the elements of
the matrix of the operator b(x) in that basis. For a given continuous bounded function
ϕ on Rd with real values, we put u(t, x, ϕ) = Exϕ(x(t)), t ≥ 0 and x ∈ Rd. Then this
function is continuous in the arguments t ≥ 0 and x ∈ Rd and turns out to satisfy the
following conditions:
1) it satisfies the equation
(3)
∂u
∂t
=
1
2
d∑
i,j=1
bij(x)
∂2u
∂xi∂xj
in the domain t > 0, x ∈ D;
2) it satisfies the equation
r(x)
∂u
∂t
=
1 + q(x)
2
∂u(t, x+)
∂N(x)
− 1 − q(x)
2
∂u(t, x−)
∂N(x)
for t > 0, x ∈ S;
3) the initial condition
u(0+, x) = ϕ(x)
holds for all x ∈ Rd.
In Section 1, we give a correct form of the martingale problem desired. Our aim is
to show that the solution to that problem is unique. We obtain the statement from the
uniqueness theorem for the boundary process (see Section 2 for the precise definition)
by the Strook–Varadhan method from [4]. The particular case of an identity diffusion
matrix and S being a hyperplane was investigated in [5]. One can also find there some
further discussion of the topic.
1. The martingale problem
From now on, we assume that, for each x ∈ Rd, b(x) = (bij(x))d
i,j=1 is a symmetric
d× d-matrix satisfying the following conditions we call the conditions J :
1) there are two positive constants C1 and C2, 0 < C1 ≤ C2, such that
C1|θ|2 ≤ (b(x)θ, θ) ≤ C2|θ|2
is valid for all θ, x ∈ Rd.
A UNIQUENESS THEOREM FOR THE MARTINGALE PROBLEM 3
2) for all x, x′ ∈ Rd, i, j = 1, 2, . . . , d,
(4) |bij(x) − bij(x′)| ≤ L|x− x′|α,
where L and α are positive constants, α ≤ 1.
Suppose S belongs to the class H2+κ for some κ ∈ (0, 1) (see [6], Ch. 4, § 4). By δ,
we denote the minimal one of the numbers α from (4) and κ.
Suppose a continuous function q(·) : S → [−1, 1] and a continuous bounded function
r(·) : S → [0,+∞) are fixed.
Ω stands for the space of all continuous Rd-valued functions on [0,+∞), and Mt
denotes the σ-algebra generated by x(u) for 0 ≤ u ≤ t. If t = ∞, Mt will be denoted by
M.
We say that a function f belongs to the class F if
1) f is continuous and bounded in (t, x) on [0,+∞) × Rd;
2) f has a continuous and bounded derivative with respect to t on [0,+∞) × Rd;
3) f has continuous and bounded derivatives with respect to x on [0,+∞)×D up to
the second order;
4) for all t ∈ [0,+∞) and x ∈ S, there exist the non-tangent limits
∂f(t, x+)
∂N(x)
and
∂f(t, x−)
∂N(x)
from the side De and Di, respectively, and the function
Kf(t, x) =
1 + q(x)
2
∂f(t, x+)
∂N(x)
− 1 − q(x)
2
∂f(t, x−)
∂N(x)
is continuous and bounded on [0,+∞) × S.
Definition 1. Given x ∈ Rd, a probability measure Px on M is a solution to the
submartingale problem starting from x if
1) Px{x(0) = x} = 1;
2) the process
Xf(t) = f(t, x(t)) −
∫ t
0
1ID(x(u))
⎛⎝∂f
∂u
+
1
2
d∑
i,j=1
bij(x(u))
∂2f
∂xi∂xj
⎞⎠ (u, x(u))du, t ≥ 0,
is a Px-submartingale whenever f belongs to F and satisfies the inequality
r(x)
∂f(t, x)
∂t
+Kf(t, x) ≥ 0 for t ≥ 0 and x ∈ S.
Remark 1. One can verify that the transition probability of the process (x(t))t≥0 de-
scribed in the Introduction is a solution to the submartingale problem ([1]).
Define the function φ on Rd by the equality φ(x) = d(x, S) := inf{d(x, y) : y ∈ S},
where d(·, ·) is the Euclidean metric on Rd . Then
1) S = {x ∈ Rd : φ(x) = 0}, D = {x ∈ Rd : φ(x) > 0},
2) Kφ(x) ≡ 1 on S.
Remark 2. The function φ does not belong to the class F because of its unboundedness.
To overcome this, we choose, for each m ≥ 1, a non-increasing infinitely differentiable
function ηm defined on [0,∞) and having a compact support such that 0 ≤ ηm ≤ 1, ηm ≡
1 on [0,m], ηm ≡ 0 off [0,m + 1] and the derivatives of ηm up to the second order are
uniformly bounded. Set φm(x) = ηm(d(x, S)) ·φ(x), x ∈ Rd. Then φm belongs to F .
Hence, Xφm is a Px-submartingale. Clearly, φm(x) → φ(x) monotonically as m→ ∞ and∑d
i,j=1 bij(x)
∂2φm(x)
∂xi∂xj
tends to 0 boundedly. So Xφ(t) = φ(x(t)) is a Px-submartingale.
4 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
The following proposition gives a reformulation of the submartingale problem into a
martingale one.
Proposition 1. Given x ∈ Rd, the probability measure Px on M solves the submartin-
gale problem starting from x iff Px{x(0) = x} = 1 and there exists a continuous non-
decreasing (Mt)-adapted process γ(t), t ≥ 0, such that
1) γ(0) = 0, Eγ(t) < +∞ for all t ≥ 0;
2) γ(t) =
∫ t
0
1IS(x(u))dγ(u), t ≥ 0;
3) the process
f(t, x(t)) −
∫ t
0
1ID(x(u))
⎛⎝∂f
∂u
+
1
2
d∑
i,j=1
bij(x(u))
∂2f
∂xi∂xj
⎞⎠ (u, x(u))du−
−
∫ t
0
(r
∂f
∂u
+Kf)(u, x(u))dγ(u), t ≥ 0,
is a Px-martingale for any f belonging to F.
If Px is such a solution, then γ(t) is uniquely determined, up to Px-equivalence, by
the condition that
φ(x(t)) − γ(t), t ≥ 0,
is a Px-martingale.
Proof. The existence of a solution to this problem was established in [7]. The proof of
the last statement is similar to that of Theorem 2.5 in [4].
Corollary 1. For each x ∈ Rd, t ≥ 0, the equality∫ t
0
1IS(x(u))du =
∫ t
0
r(x(u))dγ(u)
holds Px-almost surely.
Corollary 2. If x ∈ S, then Px{γ(t) > 0, t > 0} = 1.
These assertions can be verified like Corollaries 1,2 in [5].
2. A uniqueness theorem for a boundary process
Let Px be a solution to the submartingale problem starting from x ∈ S. Then there
exists a process γ(t), t ≥ 0, that has the properties stated in Proposition 1. For θ ≥ 0,
we put τ(θ) = sup{t ≥ 0 : γ(t) ≤ θ}. Define T (ω) = limt→+∞ γ(t). Assume that
T (ω) = +∞ a.s. Then the process y(θ) = x(τ(θ)) is defined for all 0 ≤ θ < ∞. It
is not hard to see that the process τ(θ) and, consequently, the process y(θ) are right-
continuous processes having no discontinuities of the second kind, and the latter takes
on its values on S. Since the starting point is on S, we have γ(t) > 0 for t > 0 almost
surely, i.e., τ(0) = 0 and y(0) = x. Following Strook and Varadhan [4], we define a
(d+ 1)-dimensional process (τ(θ), y(θ)), θ ≥ 0, and call it the boundary process starting
from x. If T (ω) <∞ with positive probability, we put (τ(θ), y(θ)) = ∞ for θ ≥ T (ω).
Further on we denote, by C1,2
0 ([0,+∞)×S), the class of functions on [0,+∞)×S that
have compact supports with respect to t and together with their first t-derivative and
two x-derivatives are continuous and bounded, C∞
0 ([0,+∞) × S) stands for the class of
infinitely differentiable functions on [0,+∞) × S having compact supports with respect
to t.
A UNIQUENESS THEOREM FOR THE MARTINGALE PROBLEM 5
Proposition 2. For each h ∈ C1,2
0 ([0,+∞) × S), there exists a function Hh such that
(i) it belongs to the class F ;
(ii) it is a solution to the equation
(5)
∂U
∂t
+
1
2
d∑
i,j=1
bij(x)
∂2U
∂xi∂xj
= 0
on both [0,+∞) ×Di and [0,+∞) ×De;
(iii) the relations
(6) Hh(t, x+) = h(t, x),
(7) Hh(t, x−) = h(t, x),
hold true for all t ≥ 0 and x ∈ S.
Proof. Assume that Θ is a domain in Rd. Set ΘT = (0, T ) × Θ and denote, by ΘT , its
closure. Let Hδ/2+1, δ+2(ΘT ) be a corresponding Hölder space (see [6]), Hδ/2+1, δ+2
T (ΘT )
stands for the set of all functions from Hδ/2+1, δ+2(ΘT ) which together with their first
derivatives with respect to t are equal to zero at the point t = T . Notice that for all
T > 0, h ∈ Hδ/2+1, δ+2(ST ). In addition, there exists T0 > 0 such that h = 0 if
t ≥ T0. Therefore, h ∈ H
δ/2+1, δ+2
T0
(ST0). By analogy to Theorem 5.2 in [6], there is a
uniquely defined function hi
T0
∈ H
δ/2+1, δ+2
T0
(D
i
T0
) which satisfy Eq. (5) in the domains
(0, T0)×Di
T0
and boundary conditions (6). Remark that if h(t, x) ≡ 0 on [A,B]×S, where
A and B are some constants, 0 < A < B, then the function equal to zero on [A,B]×D i
is
the unique solution to problem (5), (6) belonging to Hδ/2+1, δ+2
B ([A,B]×D i
B). Similarly,
there is a uniquely defined function he
T0
∈ H
δ/2+1, δ+2
T0
(D
e
T0
) that is a solution to (5), (7)
in (0, T0) ×De
T0
. Now we define the function
Hh =
⎧⎪⎨⎪⎩
hi
T0
on D
i
T0
,
he
T0
on D
e
T0
,
0, otherwise.
The function Hh has all the required properties, and this completes the proof.
Proposition 2 implies that, for each h ∈ C1,2
0 ([0,+∞) × S),
(K̃h)(t, x) = r(x)
∂(Hh)(t, x)
∂t
+
+
[
1 + q(x)
2
∂(Hh)(t, x+)
∂N(x)
− 1 − q(x)
2
∂(Hf)(t, x−)
∂N(x)
]
is well defined as a continuous and bounded function on [0,+∞) × S.
Proposition 3. Suppose a probability measure Px solves the submartingale problem
starting from x ∈ S. Then the relation Px{(τ(0), y(0)) = (0, x)} = 1 holds, and, for
any function h ∈ C1,2
0 ([0,+∞) × S), the process
h(τ(θ), y(θ)) −
∫ θ
0
(K̃h)(τ(u), y(u))du, θ ≥ 0,
is a Px-martingale with respect to the filtration
(
Mτ(θ)
)
θ≥0
.
Proof. The proof follows that of Theorem 4.1 in [4].
6 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
Denote, by D([0,+∞), [0,+∞)× S), the class of [0,+∞)× S-valued right-continuous
functions on [0,+∞) with no discontinuities of the second kind.
Definition 2. The uniqueness theorem is valid for the boundary process if, for any given
x ∈ S, there is only one probability measure Qx on the space D([0,+∞), [0,+∞) × S)
such that
1) Qx{τ(0) = 0, y(0) = x} = 1;
2) (τ(θ), y(θ)) = ∞ if θ > T (ω);
3) for any function h ∈ C1,2
0 ([0,+∞) × S), the process
h(τ(θ), y(θ)) −
∫ θ
0
(K̃h)(τ(u), y(u))du, θ ≥ 0,
is a Qx-martingale relative to the natural σ-algebras M̃θ, θ ≥ 0, in D([0,+∞), [0,+∞)×
S).
To prove the uniqueness theorem for the boundary process, we will make use of the
following lemma.
Lemma. For each λ > 0, ψ ∈ C∞
0 ([0,+∞) × S), the equation
(8) λf − K̃f = ψ
has a unique solution in the class of all continuous functions on [0,∞)×S having compact
supports with respect to t and such that there exists a function Hf on [0,∞)×Rd satisfying
conditions (i)-(iii) of Proposition 2.
Proof. We first prove the Lemma in the case of r being identically equal to 0 on S.
Let g0(t, x, y), t > 0, x ∈ Rd, y ∈ Rd, be the fundamental solution to Eq. (3) (see [8],
Ch. I). The process (x0(t))t≥0 solving Eq. (2) possesses a transition probability density.
We denote it by G0(t, x, y), t > 0, x ∈ Rd, y ∈ Rd. As is proved in [1], for t > 0, x ∈ Rd,
and y ∈ Rd, the representation
G0(t, x, y) = g0(t, x, y) +
∫ t
0
dτ
∫
S
Ṽ (τ, x, z)
∂g0(t− τ, z, y)
∂N(z)
q(z)dσz
takes place, where, for t > 0, x ∈ Rd, and y ∈ S, Ṽ (t, x, y) is the solution to the integral
equation
Ṽ (t, x, y) = g0(t, x, y) +
∫ t
0
dτ
∫
S
Ṽ (τ, x, z)
∂g0(t− τ, z, y)
∂N(z)
q(z)dσz.
In addition, the equality
(9) Ṽ (t, x, y) =
1
2
[G0(t, x, y+) +G0(t, x, y−)]
is valid for t > 0, x ∈ Rd, and y ∈ S, and the equality
(10)
1 + q(x)
2
∂G0(t, x+, y)
∂N(x)
− 1 − q(x)
2
∂G0(t, x−, y)
∂N(x)
= 0
h0lds for t > 0, x ∈ S, and y ∈ Rd.
For λ ≥ 0, we define a function Gλ of the arguments t > 0, x ∈ Rd, and y ∈ Rd by the
relation
Ex(ϕ(x0(t)) exp{−ληt}) =
∫
Rd
ϕ(y)Gλ(t, x, y)dy
A UNIQUENESS THEOREM FOR THE MARTINGALE PROBLEM 7
that must be fulfilled for all t > 0, x ∈ Rd, and ϕ being a bounded measurable function
on Rd. Then (see [1]) such a function exists, and it can be found as a solution to the pair
of equations
(11) Gλ(t, x, y) = G0(t, x, y) − λ
∫ t
0
dτ
∫
S
Ṽ (τ, x, z)Gλ(t− τ, z, y)r(z)dσz ,
(12) Gλ(t, x, y) = G0(t, x, y) − λ
∫ t
0
dτ
∫
S
Gλ(τ, x, z)G0(t− τ, z, y)r(z)dσz
in the domain t > 0, x ∈ Rd, and y ∈ Rd. Moreover, there is no more than one solution
to these equations satisfying the inequality
(13) Gλ(t, x, y) ≤ G0(t, x, y).
Then, for all t ≥ 0, x ∈ Rd, we can define the function
Vλ(t, x) =
∫ ∞
t
dτ
∫
S
G1
λ(τ − t, x, y)ψ(τ, y)dσy ,
where G1
λ(t, x, y) is the solution to the pair of equations (11), (12) satisfying inequality
(13) for r(x) ≡ 1.
Notice that Gλ as a function of the third argument has a jump at the points of S.
Namely, for t > 0, x ∈ Rd, and y ∈ S, Eqs. (11) can be rewritten as
Gλ(t, x, y) = Ṽ (t, x, y) − λ
∫ t
0
dτ
∫
S
Ṽ (τ, x, z)Gλ(t− τ, z, y)r(z)dσz .
From this, we can write the following relation for the function Vλ(t, x) :
(14) Vλ(t, x) =
∫ ∞
t
dτ
∫
S
Ṽ (τ−t, x, y)ψ(τ, y)dσy−λ
∫ ∞
t
dτ
∫
S
Ṽ (τ−t, x, y)Vλ(τ, y)dσy.
The function Vλ(t, x) has the following properties:
1) Vλ(t, x) satisfies conditions (i), (ii) of Proposition 2;
2) the equality
(15) λVλ(t, x) −
[
1 + q(x)
2
∂Vλ(t, x+)
∂N(x)
− 1 − q(x)
2
∂Vλ(t, x−)
∂N(x)
]
= ψ(t, x)
holds for t ≥ 0 and x ∈ S.
Property 1) is easily justified. Applying the theorem on the jump of the co-normal
derivative of a single-layer potential to (14) [6] or, more precisely, its version for the
integrals over [t,∞) instead of the ones over [0, t], we get the relations
∂Vλ(t, x±)
∂N(x)
= ∓ψ(t, x) ± λVλ(t, x) +
∫ ∞
t
dτ
∫
S
∂Ṽ (τ − t, x, y)
∂N(x)
(ψ(τ, y) − λVλ(τ, y))dσy
valid for t > 0 and x ∈ S.
Taking into account (9),(10), we arrive at formula (15).
Obviously, the restriction of the function Vλ(t, x) on [0,∞)×S is a solution to Eq. (8)
in the required class. We now show that there is no more than one such solution. Assume
that f1(t, x) and f2(t, x) are two solutions from this class. Put f̂(t, x) = f1(t, x)−f2(t, x).
Then there exists a function Hf̂ , and the relation
(16) λf̂(t, x) =
[
1 + q(x)
2
∂Hf̂(t, x+)
∂N(x)
− 1 − q(x)
2
∂Hf̂(t, x−)
∂N(x)
]
8 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
is fulfilled for t > 0, x ∈ S. Choose T0 > 0 such that, for all t ≥ T0, x ∈ S, f1(t, x) =
f2(t, x) = 0. If inf
t∈[0,T0], x∈S
Hf̂(t, x) = β < 0, then, according to the maximum principle,
there exists a point (t0, x0) ∈ [0, T0) × S such that Hf̂(t0, x0) = f̂(t0, x0) = β. This
yeilds the inequalities
(17)
∂Hf̂(t0, x0−)
∂N(x0)
≤ 0,
∂Hf̂(t0, x0+)
∂N(x0)
≥ 0.
From (17), we have that the right-hand side of (16) is non-negative at the point
(t0, x0). But this contradicts the assertion that f̂(t0, x0) < 0. Thus, inf
t∈[0,T0], x∈S
f̂(t, x) =
inf
t∈[0,T0], x∈Rd
Hf̂(t, x) ≥ 0. We can get that sup
t∈[0,T0], x∈S
f̂(t, x) ≤ 0 in the same manner.
So, f̂(t, x) ≡ 0 on (t, x) ∈ [0,∞)×S. This completes the proof for r being equal to 0. In
the case of non-negative r, we get the assertion from the previous one arguing as in [4,
pp. 194-196].
Proposition 4. Let r and q be given continuous real-valued functions on S such that
r is bounded and non-negative, |q| ≤ 1. Then the uniqueness theorem is valid for the
boundary process.
Proof. We follow the proof of Theorem 5.2 in [4]. The martingale property can be easily
extended to continuous functions on [0,∞)×S having compact supports with respect to t
in the manner of Remark 2. Given x ∈ S, for any measure Rx on D([0,+∞), [0,+∞)×S)
being a solution to the submartingale problem starting from x, we have the relation
ERx [f(τ(θ), y(θ))] = f(0, x) + ERx
[∫ θ
0
(K̃f)(τ(u), y(u)))du
]
.
Performing the Laplace transformation, we get, for λ > 0, the equality∫ ∞
0
e−λuERx [λf(τ(u), x(τ(u))) − (K̃f)(τ(u), x(τ(u)))]du = f(0, x).
Then, for ψ ∈ C∞
0 ([0,∞) × S), the integral
∫∞
0
e−λuERxψ(τ(u), x(τ(u)))du is uniquely
determined, provided the equation λf − K̃f = ψ has the unique solution for each
ψ ∈ C∞
0 ([0,∞) × S). But this condition is true because of the Lemma. The assertion of
the Proposition follows in the way of Corollary 6.2.4 in [9].
3. The main result
Theorem. Let S be a closed bounded surface in Rd which belongs to the class H2+δ for
some δ ∈ (0, 1), q and r be given continuous functions on S taking values in [−1, 1] and
[0,+∞), respectively, and r is bounded. For y ∈ Rd, let b(y) be a symmetric d×d-matrix
satisfying the conditions J . Then, for each x ∈ Rd, there exists a unique solution to the
submartingale problem.
Proof. This assertion follows from Proposition 5 by the arguments of Theorem 4.2 in [4].
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