One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution
A class of stochastic differential equations in a multidimensional Euclidean space such that the property of a solution to be unique (in a weak sense) fails for it is considered. We present the correct formulation of the corresponding martingale problem and prove the uniqueness of its solution.
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2005
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| Cite this: | One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 14–28. — Бібліогр.: 12 назв.— англ. |
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| author | Aryasova, O.V. Portenko, M.I. |
| author_facet | Aryasova, O.V. Portenko, M.I. |
| citation_txt | One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 14–28. — Бібліогр.: 12 назв.— англ. |
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| description | A class of stochastic differential equations in a multidimensional Euclidean space such that the property of a solution to be unique (in a weak sense) fails for it is considered. We present the correct formulation of the corresponding martingale problem and prove the uniqueness of its solution.
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Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 14–28
UDC 519.21
OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
ONE CLASS OF MULTIDIMENSIONAL STOCHASTIC
DIFFERENTIAL EQUATIONS HAVING NO PROPERTY
OF WEAK UNIQUENESS OF A SOLUTION
A class of stochastic differential equations in a multidimensional Euclidean space
such that the property of a solution to be unique (in a weak sense) fails for it is
considered. We present the correct formulation of the corresponding martingale
problem and prove the uniqueness of its solution.
Introduction
Let ν be a fixed unit vector in a d-dimensional Euclidean space Rd. By S, denote the
hyperplane in Rd that is orthogonal to the vector ν. By D+ and D−, denote the half-
spaces into which the hyperplane S divides the space Rd : D+ = {x ∈ Rd : (x, ν) > 0}
and D− = {x ∈ Rd : (x, ν) < 0}, and put D = D− ∪ D+. The indicator function of a set
Γ ⊂ Rd is denoted by 1IΓ(x), x ∈ Rd, and the identity operator in Rd is denoted by I.
For a given real-valued continuous function A(x), x ∈ S, consider the stochastic dif-
ferential equation in Rd
(1) dx(t) = νA(x(t))1IS(x(t))dt + dξ(t),
where (ξ(t))t≥0 is an Rd-valued continuous square integrable martingale, whose charac-
teristic is given by
(2) 〈ξ〉t = I
∫ t
0
1ID(x(s))ds, t ≥ 0.
We will construct infinitely many solutions to this equation. Each solution corresponds
to a representation of the function A(·) in the form
(3) A(x) =
q(x)
r(x)
, x ∈ S,
where q(·) is a continuous function on S with its values in the interval [−1, 1], and
r(·) is a continuous bounded function on S with positive values (it is clear that such
a representation is not unique). For such a pair of the functions q(·) and r(·), the
desired solution will be constructed as a continuous Markov process in Rd obtained from
a d-dimensional Wiener process by two transformations: its skewing on S and random
changing time (these transformations are characterized by the functions q(·) and r(·)
respectively).
In particular, if A(x) ≡ 0 (there is no skewing) we should put q(x) ≡ 0 and choose a
non-negative function r(·) arbitrarily. The corresponding solution to Eq. (1) (and even
a more general one, see Section 1) is a Wiener process in Rd for which the points of the
hyperplane S are sticky.
2000 AMS Mathematics Subject Classification. Primary 60J60, 60J35.
Key words and phrases. Stochastic differential equation, martingale problem, uniqueness of solution.
14
ONE CLASS OF MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 15
Another particular case assumes A(x) to be strictly positive for all x ∈ S. One can
choose q(x) ≡ +1 and r(x) ≡ A(x)−1 in this case; the corresponding solution to Eq.
(1) has the following property: its part in the region D+ ∪ S is a Wiener process slowly
reflected on S in the direction ν (maximal skewing is equal to reflecting); this solution
can be singled out by the requirement (x(t), ν) ≥ 0 for all t > 0 if (x(0), ν) ≥ 0; such a
solution in the one-dimensional situation was investigated by R. Chitashvili, see [2].
A.V. Skorokhod was the first one who used the random change of time in order to
construct a slowly reflected process (in the one-dimensional case) from such a one, for
which the reflection was instantaneous (see [5], §24). The main distinction of our con-
struction is admitting the boundary to be permeable. The one-dimensional situation
was expounded in [7] in a manner quite similar to that of Section 1 of this article. Some
analytical methods were used in [8] for constructing much more general processes than
those considered here.
The problem of solving Eq. (1) can be formulated as the martingale problem: for
any x ∈ Rd, we search for a probabilistic measure Px on the space of all continuous
functions x(·) : [0, +∞) → Rd with usual filtration (see Section 2 for details). Moreover,
Px({x(0) = x}) = 1, and the process
(4) ξ(t) = x(t) − x(0) − ν
∫ t
0
A(x(τ))1IS(x(τ))dτ, t ≥ 0,
is a square integrable martingale with its characteristic given by (2). The above discussion
shows that such a problem is not a well-posed one: there are infinitely many solutions
to it.
In Section 2, we show the correct form of the martingale problem [it involves the
functions q(·) and r(·) from representation (3)]. The existence of a solution to this
problem (and even to more general one) was given in [1]. We prove the uniqueness
theorem for our problem making use of the method by Stroock–Varadhan from [11] that
has been become a classical one.
We emphasize that Eq. (1) is a degenerate one: its diffusion operator vanishes at the
points of the hyperplane S. As mentioned in [4], p. 153, ”the equation does not determine
the amount of time which the solution spends at the zeros of the diffusion coefficient.
Thus the sojourn time at the zeros of this coefficient appears as additional degree of
freedom for the solution”. We now can add to these words the following ones: the extent
of skewing is another degree of freedom for the solution; the equation determines only
the ratio of these two coefficients.
1. Constructing the solutions to Eq. (1).
1.1. The case of A(x) ≡ 0.
Let (w(t))t≥0 be a standard Wiener process in Rd. By g0(t, x, y) for t > 0, x ∈ Rd,
and y ∈ Rd, we denote its transition probability density (with respect to the Lebesgue
measure in Rd)
(5) g0(t, x, y) = (2πt)−d/2 exp
{
− 1
2t
|y − x|2
}
.
For a given continuous bounded function r(·) : S → [0, +∞), define a W -functional (see
[3], Chapter 6) (η̃t)t≥0 of the process w(·) such that for t ≥ 0 and x ∈ Rd
Exη̃t =
∫ t
0
dτ
∫
S
g0(τ, x, y)r(y)dσy
16 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
(here, the inner integral on the right-hand side is a surface integral). For a given λ ≥ 0,
we determine the function gλ(t, x, y) from the relation
Exϕ(w(t)) exp{−λη̃t} =
∫
Rd
ϕ(y)gλ(t, x, y)dy
valid for t > 0, x ∈ Rd, and any real-valued bounded measurable function ϕ on Rd (the set
of all such functions with the norm ‖ϕ‖ = supx∈Rd |ϕ(x)| forms a Banach space denoted
by B; the subspace of B consisting of all continuous functions also forms a Banach space
denoted by C). The function gλ can be found as a solution to each of two integral
equations (see [10])
(6) gλ(t, x, y) = g0(t, x, y) − λ
∫ t
0
dτ
∫
S
g0(τ, x, z)gλ(t − τ, z, y)r(z)dσz,
(7) 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,
where t > 0, λ ≥ 0, x ∈ Rd and y ∈ Rd. Moreover, there is exactly one solution to (6)
and (7) satisfying the inequality
0 ≤ gλ(t, x, y) ≤ g0(t, x, y)
for all t > 0, x ∈ Rd, and y ∈ Rd.
For t ≥ 0, we now put
ζ̃t = inf{s : s + η̃s ≥ t}, w̃(t) = w(ζ̃t).
It is well known (see [3], Theorem 10.11) that (w̃(t))t≥0 is a standard Markov process.
Denote its transition probability by P̃ (t, x, Γ) :
Exϕ(w̃(t)) =
∫
Rd
ϕ(y)P̃ (t, x, dy), t > 0, x ∈ Rd, ϕ ∈ B.
The following calculation is similar to that given in [6], Chapter II, §6 (we assume
that ϕ ∈ C): ∫ ∞
0
e−λt
(∫
Rd
ϕ(y)P̃ (t, x, dy)
)
dt = Ex
∫ ∞
0
e−λtϕ(w(ζ̃t))dt =
=
∫ ∞
0
e−λtEx(ϕ(w(t)) exp{−λη̃t})dt + Ex
∫ ∞
0
e−λ(t+ηt)ϕ(w(t))dη̃t =
=
∫ ∞
0
e−λt
(∫
Rd
ϕ(y)gλ(t, x, y)dy
)
dt+
+
∫ ∞
0
e−λt
(∫
S
ϕ(y)r(y)gλ(t, x, y)dσy
)
dt.
Putting
Qλ(x, y) =
∫ ∞
0
e−λtgλ(t, x, y)dt
for λ > 0, x ∈ Rd, and y ∈ Rd, we arrive at the formula∫ ∞
0
e−λt
(∫
Rd
ϕ(y)P̃ (t, x, dy)
)
dt =
(8) =
∫
Rd
Qλ(x, y)ϕ(y)dy +
∫
S
Qλ(x, y)r(y)ϕ(y)dσy
ONE CLASS OF MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 17
valid for all λ > 0, x ∈ Rd, and ϕ ∈ B. In addition, Eqs. (6) and (7) imply the relations
for the function Qλ
Qλ(x, y) = g̃0(λ, x, y) − λ
∫
S
g̃0(λ, x, z)Qλ(z, y)r(z)dσz ,
(9) Qλ(x, y) = g̃0(λ, x, y) − λ
∫
S
Qλ(x, z)g̃0(λ, z, y)r(z)dσz
which are true for λ > 0, x ∈ Rd, and y ∈ Rd. Here, we put
g̃0(λ, x, y) =
∫ ∞
0
e−λtg0(t, x, y)dt.
These formulae allow us to calculate Exϕ(w̃(t)) for some functions ϕ. For example, if
we put ϕ0(y) ≡ 1, then (9) implies∫
Rd
Qλ(x, y)dy =
1
λ
−
∫
S
Qλ(x, z)r(z)dσz .
Taking into account (8), this yields the identity P̃ (t, x, Rd) ≡ 1.
For a fixed θ ∈ Rd, we put ϕ1(x) = (x, θ). It follows from (8) and (9) that∫
Rd
ϕ1(y)P̃ (t, x, dy) = ϕ1(x), t > 0, x ∈ Rd.
This means that Ex(w̃(t) − w̃(0)) ≡ 0. Therefore, the process ξ(t) = w̃(t) − w̃(0), t ≥ 0,
is a martingale. Let us find out its square characteristic. With that end in view, we
calculate Ex(w̃(t), θ)2. Putting ϕ2(x) = (x, θ)2 for a fixed θ ∈ Rd and taking into account
that ∫
Rd
ϕ2(y)g̃0(λ, x, y)dy = λ−1ϕ2(x) + λ−2|θ|2, λ > 0, x ∈ Rd,
relations (8) and (9) yield∫ ∞
0
e−λtExϕ2(w̃(t))dt = λ−1ϕ2(x)+
(10) +λ−1|θ|2
[
λ−1 −
∫
S
Qλ(x, z)r(z)dσz
]
.
Denote, by S+(r), the set of those x ∈ S for which r(x) > 0 and, by S+(r)c, its comple-
ment: S+(r)c = Rd \ S+(r). Using formula (8), we can write∫ ∞
0
e−λtEx1IS+(r)c(w̃(t))dt =
∫
S+(r)c
Qλ(x, y)dy =
=
[∫
Rd
Qλ(x, y)dy +
∫
S
Qλ(x, y)r(y)dσy
]
−
∫
S
Qλ(x, y)r(y)dσy =
= λ−1 −
∫
S
Qλ(x, y)r(y)dσy .
This equality and (10) imply the relation∫ ∞
0
e−λtExϕ2(w̃(t))dt = λ−1ϕ2(x)+
+|θ|2
∫ ∞
0
e−λtEx
(∫ t
0
1IS+(r)c(w̃(s))ds
)
dt
18 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
that can be rewritten in the form
Ex(w̃(t) − w̃(0), θ)2 = |θ|2Ex
∫ t
0
1IS+(r)c(w̃(s))ds.
Since this equality holds true for all t > 0, x ∈ Rd, and θ ∈ Rd, we can assert that the
square characteristic of the martingale ξ(t) = w̃(t) − w̃(0), t ≥ 0, is given by
(11) 〈ξ〉t = I
∫ t
0
1IS+(r)c(w̃(s))ds, t ≥ 0.
So, we have just proved that in the case of S+(r)
= ∅ the stochastic integral equation
(12) w̃(t) = w̃(0) + ξ(t), t ≥ 0,
driven by a square integrable martingale ξ(·) with characteristic (11) has infinitely many
solutions.
Indeed, let us consider another continuous bounded function r̂(·) on S with non-
negative values such that S+(r̂) = S+(r)
= ∅ (for example, r̂(x) = βr(x) for x ∈ S with
an arbitrary positive constant β
= 1) and construct the process ŵ(·) for it in the same
way as the process w̃(·) has been constructed for the function r(·). The process ŵ(·) is
also a solution to Eq. (12).
1.2. The general case.
We now try to do something like that in Section 1.1, but this time we start from the
process that can be called a Wiener process skewed on the hyperplane S. Fix a continuous
function q(·) on S with its values in the interval [−1, 1] and put
G0(t, x, y) = g0(t, x, y) +
∫ t
0
dτ
∫
S
g0(τ, x, z)
∂g0(t − τ, z, y)
∂νz
q(z)dσz
for t > 0, x ∈ Rd, and y ∈ Rd. It is well known that there exists a continuous Markov
process (x0(t))t≥0 such that
Exϕ(x0(t)) =
∫
Rd
ϕ(y)G0(t, x, y)dy
for t > 0, x ∈ Rd, and ϕ ∈ B (see, for example, [9]). Moreover, for a given continuous
bounded function r(·) on S with non-negative values, there exists a W -functional (ηt)t≥0
of the process x0(·) such that for all t > 0 and x ∈ Rd
Exηt =
∫ t
0
dτ
∫
S
G0(τ, x, y)r(y)dσy =
∫ t
0
dτ
∫
S
g0(τ, x, y)r(y)dσy
(it is not difficult to see that the equality G0(t, x, y) = g0(t, x, y) is held for t > 0, x ∈ Rd,
and y ∈ S). We now define a function Gλ(t, x, y) for λ ≥ 0,
t > 0, x ∈ Rd and y ∈ Rd in such a way that the relation
Exϕ(x0(t)) exp{−ληt} =
∫
Rd
ϕ(y)Gλ(t, x, y)dy
holds true for t > 0, x ∈ Rd, and ϕ ∈ B. It is known (see, for example, [10]) that such
a function exists and it can be found as a solution to each one of the following pair of
equations
(13) Gλ(t, x, y) = G0(t, x, y) − λ
∫ t
0
dτ
∫
S
g0(τ, x, z)Gλ(t − τ, z, y)r(z)dσz ,
ONE CLASS OF MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 19
(14) 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 .
Moreover, there is no more than one solution to these equations satisfying the inequality
(15) Gλ(t, x, y) ≤ G0(t, x, y)
for all t > 0, λ ≥ 0, x ∈ Rd and y ∈ Rd.
Proposition 1. For t > 0, λ > 0, x ∈ Rd, and y ∈ Rd, the formula
(16) Gλ(t, x, y) = gλ(t, x, y) +
∫ t
0
dτ
∫
S
gλ(τ, x, z)
∂g0(t − τ, z, y)
∂νz
q(z)dσz
holds true.
The proof of this statement is elementary, and we omit it.
Remark 1. Proposition 1 means that the transformations of skewing and killing consid-
ered above are commutative.
Remark 2. One can see that Gλ(t, x, y) coincides with gλ(t, x, y) for t > 0, x ∈ Rd,
and y ∈ S, because of
∂g0(t, z, y)
∂νz
= 0 for z ∈ S and y ∈ S. Nevertheless, the well-known
theorem on the jump of the normal derivative of a single-layer potential shows that
Gλ(t, x, y±) = (1 ± q(y))gλ(t, x, y)
for y ∈ S, where Gλ(t, x, y+) and Gλ(t, x, y−) are the non-tangent limits of Gλ(t, x, z),
as z → y from the sides D+ and D−, respectively.
Putting
Rλ(x, y) =
∫ ∞
0
e−λtGλ(t, x, y)dt
for λ > 0, x ∈ Rd, and y ∈ Rd and taking into account Remark 2, relation (16) yields the
formula
(17) Rλ(x, y) = Qλ(x, y) +
∫
S
Qλ(x, z)
∂g̃0(λ, z, y)
∂νz
q(z)dσz
valid for λ > 0, x ∈ Rd, y ∈ Rd, and if y ∈ S we have Rλ(x, y) = Qλ(x, y).
For t ≥ 0, we now put
ζt = inf{s : s + ηs ≥ t}, x(t) = x0(ζt).
Theorem 10.11 from [3] allow us, as above, to assert that the process (x(t))t≥0 is a
standard Markov process. Denote, by P (t, x, dy), its transition probability
Exϕ(x(t)) =
∫
Rd
ϕ(y)P (t, x, dy), t ≥ 0, x ∈ Rd, ϕ ∈ B.
Using the calculations similar to those in Section 1.1, we arrive at the formula∫ ∞
0
e−λtExϕ(x(t))dt =
∫
Rd
Rλ(x, y)ϕ(y)dy+
(18) +
∫
S
Qλ(x, y)ϕ(y)r(y)dσy
that is fulfilled for λ > 0, x ∈ Rd and ϕ ∈ C.
Formulae (17) and (18) allow us to give a martingale characterization of the process
x(·). In what follows up to the end of this section, we assume that the function r(·) takes
on only strictly positive values.
20 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
For a fixed θ ∈ Rd, we put ϕ1(x) = (x, θ), as above. Since∫
Rd
(y, θ)
∂g̃0(λ, z, y)
∂νz
dy = λ−1(θ, ν),
we have ∫ ∞
0
e−λtExϕ1(x(t))dt =
∫
Rd
Rλ(x, y)ϕ1(y)dy+
+
∫
S
Qλ(x, y)ϕ1(y)r(y)dσy =
∫
Rd
Qλ(x, y)ϕ1(y)dy+
+
∫
S
Qλ(x, y)ϕ1(y)r(y)dσy + λ−1(θ, ν)
∫
S
Qλ(x, z)q(z)dσz .
As shown in Section 1.1, the sum of the first two terms here is equal to λ−1(x, θ). So we
obtain ∫ ∞
0
e−λtExϕ1(x(t))dt =
= λ−1ϕ1(x) + λ−1(θ, ν)
∫
S
Qλ(x, z)A(z)r(z)dσz .
This equality means that
(19) Ex(x(t) − x(0), θ) = (θ, ν)Ex
∫ t
0
A(x(τ))1IS(x(τ))dτ,
because of the equality∫ ∞
0
e−λt
(
Ex
∫ t
0
A(x(τ))1IS(x(τ))dτ
)
dt =
= λ−1
∫
S
Qλ(x, y)A(y)r(y)dσy
that is a simple consequence of (18).
If we put, for t ≥ 0,
ξ(t) = x(t) − x(0) − ν
∫ t
0
A(x(τ))1IS(x(τ))dτ,
then equality (19) will mean that the process (ξ(t)))t≥0 is a martingale. We now find
out the square characteristic of this martingale.
Since
Ex(ξ(t), θ)2 = Ex(x(t) − x(0), θ)2 + 2(ν, θ)(x, θ)Ex
∫ t
0
A(x(τ))1IS(x(τ))dτ−
−2(ν, θ)Ex
∫ t
0
(x(τ), θ)A(x(τ))1IS(x(τ))dτ
and
Ex(x(t) − x(0), θ)2 = Ex(x(t), θ)2 − (x, θ)2−
−2(x, θ)(ν, θ)Ex
∫ t
0
A(x(τ))1IS(x(τ))dτ,
we obtain the formula
Ex(ξ(t), θ)2 = Ex(x(t), θ)2 − (x, θ)2−
(20) −2(ν, θ)Ex
∫ t
0
(x(τ), θ)A(x(τ))1IS(x(τ))dτ.
ONE CLASS OF MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 21
Let us calculate Ex(x(t), θ)2. We have∫ ∞
0
e−λtEx(x(t), θ)2dt =
∫
Rd
Rλ(x, y)(y, θ)2dy+
+
∫
S
Qλ(x, y)(y, θ)2r(y)dσy =
∫
Rd
Qλ(x, y)(y, θ)2dy+
(21) +
2(ν, θ)
λ
∫
S
Qλ(x, z)(z, θ)q(z)dσz +
∫
S
Qλ(x, y)(y, θ)2r(y)dσy .
It is not difficult to establish the relation∫ ∞
0
e−λt
(
Ex
∫ t
0
(x(τ), θ)A(x(τ))1IS(x(τ))dτ
)
dt =
= λ−1
∫
S
Qλ(x, y)(y, θ)q(y)dσy .
Taking into account this equality and (21), relation (20) yields∫ ∞
0
e−λtEx(ξ(t), θ)2dt =
∫
Rd
Qλ(x, y)(y, θ)2dy − λ−1(x, θ)2+
(22) +
∫
S
Qλ(x, y)(y, θ)2r(y)dy.
We can now write ∫
Rd
Qλ(x, y)(y, θ)2dy +
∫
S
Qλ(x, y)(y, θ)2r(y)dσy =
=
∫ ∞
0
e−λtEx(w̃(t), θ)2dt =
(x, θ)2
λ
+
|θ|2
λ
[
1
λ
−
∫
S
Qλ(x, z)r(z)dσz
]
=
=
|θ|2
λ
∫
Rd
Qλ(x, z)dz +
(x, θ)2
λ
.
Substituting this into (22), we get∫ ∞
0
e−λtEx(ξ(t), θ)2dt =
|θ|2
λ
∫
Rd
Qλ(x, z)dz =
=
|θ|2
λ
∫
Rd
1ID(z)Qλ(x, z)dz =
= |θ|2
∫ ∞
0
e−λt
(
Ex
∫ t
0
1ID(x(τ))dτ
)
dt,
because of ∫ ∞
0
e−λt
(
Ex
∫ t
0
1ID(x(τ))dτ
)
dt =
1
λ
∫ ∞
0
e−λtEx1ID(x(t))dt =
=
1
λ
[∫
Rd
Rλ(x, y)1ID(y)dy +
∫
S
Qλ(x, y)1ID(y)r(y)dσy
]
=
=
1
λ
∫
Rd
Qλ(x, y)1ID(y)dy.
We have thus proved that, for t > 0, θ ∈ Rd, and x ∈ Rd, the equality
Ex(ξ(t), θ)2 = |θ|2Ex
∫ t
0
1ID(x(τ))dτ
22 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
is held true. This means that the square characteristic of the martingale ξ(·) is given by
formula (2). Let us formulate the obtained result as a proposition.
Proposition 2. For a continuous function q(·) : S → [−1, 1] and a continuous bounded
function r(·) : S → (0, +∞), the Markov process (x(t))t≥0 constructed above is such that
its trajectories satisfy Eq. (1) with A(x) = q(x)/r(x) for x ∈ S.
Remark 3. As shown in [9], the process (x0(t))t≥0 is a solution to the stochastic differ-
ential equation
dx0(t) = νq(x0(t))δS(x0(t))dt + dw(t),
where w(·) is a standard Wiener process in Rd, and δS(x), x ∈ Rd, is a generalized
function determined by the relation
〈δS , ϕ〉 =
∫
S
ϕ(x)dσ
valid for any compactly supported function ϕ ∈ C. The process (x(t))t≥0 is obtained
from (x0(t))t≥0 by a random change of time generated by the functional∫ t
0
v(x0(s))ds, t ≥ 0,
where v(x) = 1+r(x)δS(x), x ∈ Rd. It is well known (see [3], Chapter 10) that in order to
obtain the local characteristics of the process x(·), the corresponding ones of the process
x0(·) should be multiplied by v(x)−1. So, we have that the diffusion coefficients of the
process x(·) should be given, for x ∈ Rd, by
ν
q(x)δS(x)
1 + r(x)δS(x)
= ν
q(x)
r(x)
1IS(x),
I
1
1 + r(x)δS(x)
= I · 1ID(x).
Exactly these characteristics were obtained as a result of the calculations in proving
Proposition 2. As we can see, the diffusion coefficients of the process x(·) are not gener-
alized functions. The arguments of Remark 3 are heuristical, but they lead to the right
formulae.
Remark 4. For a given ϕ ∈ C, the function
Uλ(x, ϕ) =
∫ ∞
0
e−λtExϕ(x(t))dt, λ > 0, x ∈ Rd,
is continuous in the argument x ∈ Rd, satisfies the equation
λUλ(x, ϕ) − ϕ(x) =
1
2
ΔUλ(x, ϕ)
in the region D (Δ stands for the Laplace operator) and the equation
1 + q(x)
2r(x)
∂Uλ(x+, ϕ)
∂ν
− 1 − q(x)
2r(x)
∂Uλ(x−, ϕ)
∂ν
= λUλ(x, ϕ) − ϕ(x)
for x ∈ S, where ∂Uλ(x+,ϕ)
∂ν and ∂Uλ(x−,ϕ)
∂ν mean the non-tangent limits of ∂Uλ(z,ϕ)
∂ν , as
z → x, x ∈ S, from the sides D+ and D−, respectively.
ONE CLASS OF MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 23
2. The uniqueness theorem.
2.1. The martingale problem.
Throughout this section a continuous function q(·) : S → [−1, 1] and a continuous
bounded function r(·) : S → [0, +∞) will be fixed.
We say that a function f : [0, +∞) × Rd → R1 satisfies condition F if
1) f is a continuous and bounded function in the arguments (t, x) ∈ [0, +∞)× Rd;
2) the first t-derivative of f on [0, +∞)×Rd and the first two x-derivatives on [0, +∞)×
D are continuous and bounded;
3) for all t ∈ [0, +∞) and x ∈ S, there exist ∂f(t,x±)
∂ν such that the function
Kf(t, x) =
1 + q(x)
2
∂f(t, x+)
∂ν
− 1 − q(x)
2
∂f(t, x−)
∂ν
is continuous and bounded on [0, +∞) × S.
Denote, by Ω, the space of all continuous functions x(·) : [0, +∞) → Rd and, by
Mt = σ{x(u) : 0 ≤ u ≤ t}, the smallest σ-algebra of subsets of Ω which makes each of
the maps x → x(u) from Ω to Rd Mt-measurable for u ∈ [0, t] and put M = σ{x(u) :
0 ≤ u < +∞}.
Definition 1. Given x ∈ Rd, a probability measure Px on M is a solution of 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))(fu +
1
2
Δf)(u, x(u))du, t ≥ 0,
is a Px-submartingale for any f satisfying condition F and the inequality
r(x)ft(t, x) + Kf(t, x) ≥ 0 for t ≥ 0 and x ∈ S.
Here and below, we put ft(t, x) = ∂f(t,x)
∂t .
It is not difficult to verify that the measure Px on Ω that corresponds to the Markov
process (x(t))t≥0 constructed in Section 1 is a solution to this submartingale problem.
Define the function φ on Rd by the equality φ(x) = |(x, ν)|, x ∈ Rd. Then
1) φ satisfies condition F ;
2) S = {x ∈ Rd : φ(x) = 0}, D = {x ∈ Rd : φ(x) > 0},
3) Kφ(x) ≡ 1 on S.
Proposition 3. 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 α(t), t ≥ 0, such that
1) α(0) = 0, Eα(t) < +∞ for all t ≥ 0;
2) α(t) =
∫ t
0
1IS(x(u))dα(u), t ≥ 0;
3) the process
f(t, x(t)) −
∫ t
0
1ID(x(u))(fu +
1
2
Δf)(u, x(u))du−
−
∫ t
0
(rfu + Kf)(u, x(u))dα(u), t ≥ 0,
is a Px-martingale for any f satisfying condition F.
If Px is such a solution, then α(t) is uniquely determined, up to Px-equivalence, by
the condition that
φ(x(t)) − α(t)
24 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
is a Px-martingale.
Proof. The proof is similar to that of Theorem 2.5 in [11].
Corollary 1. For each x ∈ Rd, t ≥ 0, the equality∫ t
0
1IS(x(u))du =
∫ t
0
r(x(u))dα(u)
is held Px-almost surely.
Proof. Let now f(t) = t. Then the process
Y (t) = t −
∫ t
0
1ID(x(u))du −
∫ t
0
r(x(u))dα(u) =
=
∫ t
0
1IS(x(u))du −
∫ t
0
r(x(u))dα(u), t ≥ 0,
is a Px-martingale. It has continuous paths of bounded variation, and Y (0) = 0. There-
fore, the process Y (t) is almost surely zero.
Corollary 2. If x ∈ S, then Px{α(t) > 0, t > 0} = 1.
Proof. The process X(t) = φ(x(t)) − α(t) is a martingale relative to Mt. It is easy to
see that X(t) is a Px-martingale with respect to Mt+ = ∩ε>0Mt+ε. Indeed, for all
A ∈ Ms+, ε > 0, s ≥ 0, t > 0, s + ε ≤ t, we have
(23)
∫
A
EPx(X(t)/Ms+)dPx =
∫
A
X(t)dPx =
∫
A
X(s + ε)dPx.
The process X(t) is continuous and locally bounded. Passing to the limit in (23) as
ε ↓ 0, we obtain
EPx(X(t)/Ms+) = X(s).
Define τ0 = sup{t > 0 : α(t) = 0}. Since {α(t) > 0} ∈ Mt, we get {τ0 < t} ∈ Mt, and
hence τ0 is a stopping time relative to Mt+.
The process
X(t ∧ τ0) = φ(x(t ∧ τ0)) − α(t ∧ τ0) = φ(x(t ∧ τ0)), t ≥ 0,
is a Px-martingale with respect to M(t∧τ0)+ such that EPxX(t ∧ τ0) = 0 for all t ≥ 0.
This implies x(t) ∈ S for t ∈ [0, τ0]. But∫ τ0
0
1IS(x(u))du =
∫ τ0
0
r(x(u))dα(u) = 0 Px-a.s.
by Corollary 1. Therefore, τ0 = 0 almost surely.
2.2. A boundary process.
Let Px be a solution to the submartingale problem starting from x ∈ S. Then there
exists the process α(t), t ≥ 0, with properties stated in Proposition 3. For θ ≥ 0, we put
τ(θ) = sup{t ≥ 0 : α(t) ≤ θ}. It is well known that limt→+∞ α(t) = +∞, so τ(θ) < +∞
for all θ ≥ 0. In addition, τ(θ) is a stopping time relative to Mt+. Since the starting
point is on S we have α(t) > 0 for t > 0 almost surely, i.e. τ(0) = 0 and x(τ(0)) = x.
Moreover, x(τ(θ)) must be on S for all θ ≥ 0.
Define the (d + 1)-dimensional process (τ(θ), x(τ(θ))), θ ≥ 0, we call the boundary
process starting from x. This process is a right-continuous and has no discontinuities of
the second kind.
We denote, by C1,2
0 ([0, +∞)×S), the class of functions on [0, +∞)×S having compact
support which together with their first t-derivative and two x-derivatives are continuous
ONE CLASS OF MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 25
and, by C∞
0 ([0, +∞) × S), the class of infinitely differentiable functions on [0, +∞) × S
with compact support.
For f ∈ C1,2
0 ([0, +∞) × S), we put
Hf(t, x) =
∫ ∞
t
dτ
∫
S
∂g0(τ − t, x, y)
∂νx
f(τ, y)dσy,
where t ≥ 0, x ∈ Rd, g0(t, x, y) defined by formula (5).
Proposition 4. The function Hf satisfies condition F.
Proof. It is easy to observe that the function Hf together with its first t-derivative are
continuous and bounded on [0, +∞) × Rd, and its first two x-derivatives are continuous
and bounded on [0, +∞)×D. We will show that ∂(Hf)(t,x±)
∂ν , t ≥ 0, x ∈ S, are continuous
and bounded.
We can write Hf = J1 + J2, where
J1 =
∫ ∞
t
dτ
∫
S
∂g0(τ − t, x, y)
∂νx
[f(τ, y) − f(t, xS) − (∇xS f(t, xS), y − xS)]dσy ,
J2 = f(t, xS)
∫ ∞
t
|(x, ν)|e
−
(x, ν)2
2(τ − t)√
2π(τ − t)3
dτ.
Here, t ≥ 0, x ∈ D, xS = x − ν(x, ν) is a projection of x on S.
It is easily to see that J2 = f(t, xS). Formally for t ≥ 0, x ∈ D,
∂J1
∂ν
=
∫ ∞
t
(
1 − (x, ν)2
τ − t
)
sign(x, ν)√
2π(τ − t)3
e
−
(x, ν)2
2(τ − t)Φ(t, τ, xS)dτ,
where
Φ(t, τ, xS) =
∫
S
e
−
|y − xS |2
2(τ − t)√
(2π(τ − t))d−1
[f(τ, y) − f(t, xS) − (∇xS f(t, xS), y − xS)]dσy.
Since f has a compact support in t, there exists T > 0 such that Φ(t, τ, xS) = 0 for
t ≥ T , and the estimate
(24) |Φ(t, τ, xS)| ≤ C
∫
S
e
−
|y − xS |2
2(τ − t)√
(2π(τ − t))d−1
(|y − xS |2 + (τ − t))dσy ≤ C(τ − t)
is valid for t ∈ [0, T ], τ > t, and some positive constant C.
Using (24), we arrive at the conclusion that, for t ≥ 0, x ∈ D, there exists ∂I
∂ν , and we
can pass to the limit as (x, ν) → 0 ± .
The statement of Proposition 4 follows immediately.
Remark 5. The function Hf is a solution of the equation
∂U
∂t
+
1
2
ΔU = 0 in the region [0, +∞) × D,
and the relation (Hf)(t, x±) = f(t, x) is true for all t ≥ 0, x ∈ S.
According to Proposition 4, we can define
(K̃f)(t, x) = r(x)(Hf)t(t, x)+
26 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
+
[
1 + q(x)
2
∂(Hf)(t, x+)
∂ν
− 1 − q(x)
2
∂(Hf)(t, x−)
∂ν
]
as a continuous and bounded function on [0, +∞)× S.
Proposition 5. Let a probability measure Px solve the submartingale problem for a given
x ∈ S. Then the relation Px{(τ(0), x(τ(0))) = (0, x)} = 1 is held, and the process
f(τ(θ), x(τ(θ))) −
∫ θ
0
(K̃f)(τ(u), x(τ(u)))du
is a Px-martingale with respect to the filtration
(Mτ(θ)
)
θ≥0
for any function
f ∈ C1,2
0 ([0, +∞) × S).
Proof. Proposition 5 can be proved analogously to Theorem 4.1 of [11].
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, x(τ(0)) = x} = 1;
2) the process
f(τ(θ), x(τ(θ))) −
∫ θ
0
(K̃f)(u, x(τ(u)))du, θ ≥ 0,
is a Qx-martingale relative to
(Mτ(θ)
)
θ≥0
for any function f ∈ C1,2
0 ([0, +∞) × S).
Proposition 6. 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 [11].
Given x ∈ S, let Rx be a solution of the submartingale problem starting from x. Then
f(τ(θ), x(τ(θ))) −
∫ θ
0
(K̃f)(τ(u), x(τ(u)))du
is a Rx-martingale with respect to
(Mτ(θ)
)
θ≥0
, and
ERx [f(τ(θ), x(τ(θ)))] = f(0, x) + ERx
[∫ θ
0
(K̃f)(τ(u), x(τ(u)))du
]
.
Performing the Laplace transformation, we get the equality∫ ∞
0
e−λuERx [λf(τ(u), x(τ(u))) − (K̃f)(τ(u), x(τ(u)))]du = f(0, x)
for λ > 0.
If we know that the equation λf − K̃f = g has the unique solution for each g ∈
C∞
0 ([0,∞) × S) (this fact is proved in Lemma 1 below), then the integral∫ ∞
0
e−λuERxg(τ(u), x(τ(u)))du
is uniquely determined. This yields the uniqueness of Rx in the same way as in Corollary
6.2.4 of [12].
ONE CLASS OF MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS 27
Lemma 1. For each λ > 0, g ∈ C∞
0 ([0, +∞) × S), there is only one solution of the
equation
(25) λf − K̃f = g.
Proof. We prove Lemma 1 in two steps. At the first one, we deal with r being equal to
zero on S identically. Denote, by G1
λ(t, x, y), the solution to the pair of equations (13),
(14), in which we put r(x) ≡ 1, satisfying inequality (15). Let
Vλ(t, x) =
∫ ∞
t
dτ
∫
S
G1
λ(τ − t, x, y)ψ(τ, y)dσy ,
where t ≥ 0, x ∈ Rd, ψ is a continuous function on [0, +∞) × S with compact support.
As a consequence of (13), we can write the following relation for the function Vλ(t, x) :
Vλ(t, x) =
∫ ∞
t
dτ
∫
S
g0(τ − t, x, y)ψ(τ, y)dσy−
(26) −λ
∫ ∞
t
dτ
∫
S
g0(τ − t, x, y)Vλ(τ, y)dσy .
It is not hard to verify that the function Vλ(t, x) has the properties:
1) Vλ(t, x) satisfies the heat equation
∂u
∂t
+
1
2
Δu = 0 in the region [0, +∞)× D;
2) Vλ(t, x) is a continuous function of (t, x) in the region t ≥ 0, x ∈ Rd;
3) for each λ ≥ 0, supx∈Rd |Vλ(t, x)| → 0 as t → ∞;
4) for each t ≥ 0, Vλ(t, x) → 0 as |x| → ∞.
Using the theorem on the jump of the normal derivative of a single-layer potential
(mentioned above), relation (26) yields
5) the equality
(27)
∂Vλ(t, x±)
∂ν
= ∓ψ(t, x) ± λVλ(t, x)
valid for t ≥ 0 and x ∈ S.
The maximum principle implies the uniqueness of a function satisfying conditions 1)
– 5).
Equality (27) is equivalent to the equation
(28) λVλ(t, x) −
[
1 + q(x)
2
∂Vλ(t, x+)
∂ν
− 1 − q(x)
2
∂Vλ(t, x−)
∂ν
]
= ψ(t, x),
where t ≥ 0, x ∈ S.
Moreover, the function Vλ(t, x) on [0, +∞)×Rd is defined by its values on [0, +∞)×S
according to the formula
Vλ(t, x) =
∫ ∞
t
dτ
∫
S
∂g0(τ − t, x, y)
∂νx
Vλ(τ, y)dσy .
Thus, Eq. (28) coincides with (25) in the case where r is equal to zero on S, and we
get the statement of Lemma 1 in this case.
If r is non-negative, we get the result from the previous one in the same way as in
[11], pp.194–196.
28 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
Theorem. Let q and r be given continuous functions on S with their values on [−1, 1]
and [0, +∞), respectively, and r is bounded. Then, for each x ∈ Rd, there is only one
solution to the submartingale problem.
Proof. This assertion follows from Proposition 5 by the arguments of Theorem 4.2 in
[11].
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E-mail : oaryasova@mail.ru
E-mail : portenko@imath.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4422 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-11-25T23:31:38Z |
| publishDate | 2005 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Aryasova, O.V. Portenko, M.I. 2009-11-09T13:05:31Z 2009-11-09T13:05:31Z 2005 One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 14–28. — Бібліогр.: 12 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4422 519.21 A class of stochastic differential equations in a multidimensional Euclidean space such that the property of a solution to be unique (in a weak sense) fails for it is considered. We present the correct formulation of the corresponding martingale problem and prove the uniqueness of its solution. en Інститут математики НАН України One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution Article published earlier |
| spellingShingle | One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution Aryasova, O.V. Portenko, M.I. |
| title | One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution |
| title_full | One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution |
| title_fullStr | One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution |
| title_full_unstemmed | One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution |
| title_short | One class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution |
| title_sort | one class of multidimensional stochastic differential equations having no property of weak uniqueness of a solution |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4422 |
| work_keys_str_mv | AT aryasovaov oneclassofmultidimensionalstochasticdifferentialequationshavingnopropertyofweakuniquenessofasolution AT portenkomi oneclassofmultidimensionalstochasticdifferentialequationshavingnopropertyofweakuniquenessofasolution |