Uniqueness in law of solutions of stochastic differential inclusions
The paper deals with one-dimensional homogeneous stochastic differential inclusions without drift with Borel measurable mapping at the right side. We give the conditions for uniqueness in law and existence of unique in law weak solutions of the inclusions with locally unbounded right sides.
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irk-123456789-44822009-11-20T12:00:35Z Uniqueness in law of solutions of stochastic differential inclusions Lepeyev, A.N. The paper deals with one-dimensional homogeneous stochastic differential inclusions without drift with Borel measurable mapping at the right side. We give the conditions for uniqueness in law and existence of unique in law weak solutions of the inclusions with locally unbounded right sides. 2007 Article Uniqueness in law of solutions of stochastic differential inclusions / A.N. Lepeyev // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 110-121. — Бібліогр.: 7 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4482 en Інститут математики НАН України |
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The paper deals with one-dimensional homogeneous stochastic differential inclusions without drift with Borel measurable mapping at the right side. We give the conditions for uniqueness in law and existence of unique in law weak solutions of the inclusions with locally unbounded right sides. |
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Article |
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Lepeyev, A.N. |
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Lepeyev, A.N. Uniqueness in law of solutions of stochastic differential inclusions |
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Lepeyev, A.N. |
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Lepeyev, A.N. |
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Uniqueness in law of solutions of stochastic differential inclusions |
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Uniqueness in law of solutions of stochastic differential inclusions |
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Uniqueness in law of solutions of stochastic differential inclusions |
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Uniqueness in law of solutions of stochastic differential inclusions |
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Uniqueness in law of solutions of stochastic differential inclusions |
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uniqueness in law of solutions of stochastic differential inclusions |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4482 |
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Uniqueness in law of solutions of stochastic differential inclusions / A.N. Lepeyev // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 110-121. — Бібліогр.: 7 назв.— англ. |
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2025-07-02T07:42:59Z |
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2025-07-02T07:42:59Z |
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1836520235613552640 |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.110-121
ANDREI N. LEPEYEV
UNIQUENESS IN LAW OF SOLUTIONS OF
STOCHASTIC DIFFERENTIAL INCLUSIONS
The paper deals with one-dimensional homogeneous stochastic dif-
ferential inclusions without drift with Borel measurable mapping at
the right side. We give the conditions for uniqueness in law and ex-
istence of unique in law weak solutions of the inclusions with locally
unbounded right sides.
1. Introduction
The following stochastic differential inclusion (SDI) is considered
dXt ∈ B(Xt) dWt, t ≥ 0, (1)
where B : IR → comp(IR) - multi-valued Borel measurable mapping,
comp(IR) - the set of all non-empty compact subsets of IR with Haus-
dorff metric ρ(A, B) = max(β(A, B), β(B, A)), ∀A, B ∈ comp(IR), where
β(A, B) = supx∈A(infy∈B |x− y|) - excess of A over B, W - one-dimensional
Wiener process.
One can suspect the uniqueness concept for the SDI solutions to be am-
biguous, because the definition of stochastic integral of multi-valued map-
pings implies the set of possibly different processes (see [2]). But despite
that, the uniqueness of SDI solutions is very important due to the fact that
availability of uniqueness helps in description of the SDI solution set.
Inclusion (1) was investigated in paper [7]. The necessary and sufficient
conditions were given for the existence of weak and explicit weak solutions
of the SDI, including the case of non-trivial solutions. This paper is the
natural continuation of the investigation.
The aim of the paper is to give the conditions for uniqueness in law
and existence of unique in law weak solutions of the inclusions with locally
unbounded right sides. For this purpose, we use a selectionwise approach
2000 Mathematics Subject Classifications: 34A60, 60G44, 60H10, 60J65, 60H99
Key words and phrases. Stochastic differential equations, stochastic differential in-
clusions, measurable coefficients
110
UNIQUENESS IN LAW OF SDI SOLUTIONS 111
that requires introduction of several intermediate definitions of uniqueness
in law in terms of their selections. The main results are necessary, sufficient,
and necessary and sufficient conditions for existence of unique in law explicit
weak solutions. On the one hand, the results express the way of classifying
the solutions and, on the other hand, given some additional conditions,
they guarantee the existence of unique in law weak solutions in the general
context.
In addition to notions from paper [7], the article uses the definitions and
results obtained for stochastic differential equations by H.-J. Engelbert and
W. Schmidt (see [4] and [5]). We also use the methods they have developed.
2. Preliminaries
Let (Ω,F ,P) be a complete probability space with filtration IF = (Ft)t≥0.
As usually, assume that filtration IF satisfies the natural conditions, i.e. it
is right-continuous and F0 contains all P - zero subsets of F . If stochastic
process (Xt)t≥0 is IF-adapted, we write (X, IF).
The first part of the section is devoted to notions from the theory, de-
veloped for investigation of stochastic differential equations (SDEs). The
notions are also important for inclusions due to the fact that an inclusion
is a generalization of an equation. For instance, the considered SDI (1) is
the natural generalization of the following SDE:
Xt = X0 +
∫ t
0
b(Xs) dWs, t ≥ 0, (2)
where diffusion coefficient b : IR → IR is a Borel measurable function and
W is a Wiener process.
Measurable process (X, IF), defined on probability space (Ω,F ,P), pos-
sesses the representation property, if any martingale (Y, IF) can be repre-
sented in the form of stochastic Ito integral driven by (X, IF)
Yt = Y0 +
∫ t
0
f(s)dXs, ∀t ≥ 0,
where (f, IF) – some predictable process.
Let X be a weak solution of SDE (2) with Borel measurable diffusion
coefficient v, i.e. there exist probability space (Ω,F ,P) with filtration IF
and Wiener process (W v, IF) such that (X, IF) is measurable and (2) is
valid P-a.s. for all t ≥ 0. The quadratic variation can be defined by
〈X〉t = Av
t =
∫ t
0 v2(Xs)ds, ∀t ≥ 0 and the process reverse to the quadratic
variation can be defined by T v
t = inf{s ≥ 0 : Av
s > t}, ∀t ≥ 0.
Proposition 1. [3, Theorem 2, Proposition 7] If Av
∞ = limt→+∞ Av
t is
IFW v
-predictable stopping time (where IFW v
is the filtration generated by W v
and completed in the natural way) and for all fixed t ≥ 0 variable value T v
t
112 ANDREI N. LEPEYEV
is FW v
∞ -measurable (where sigma-algebra FW v
∞ = ∪t≥0FW v
t ), then (X, IFX)
possesses the representation property.
The next result guarantees the availability of representation property
for certain processes. We will use sets Nv = {x ∈ IR|v(x) = 0} and
Mv = {x ∈ IR| ∫U(x,C) v
−2(y) dy = ∞, ∀C > 0}, where U(x, C) is an open
ball with the center in x and radius C. The sets can be defined for any
Borel measurable function v : IR → IR.
Proposition 2. [4, Lemma 2] Let Nv ⊆ Mv. Then:
1) T v
t =
∫ t
0 v−2(W v
s )ds, ∀t < Av
∞,P-a.s.;
2) Av
∞ is IFW v
-predictable stopping time.
The representation property ensures some important properties of the
martingale distributions. An extremal point of some convex set of measures
is measure P such that if there exist measures Q and S (Q �= S) from this
set of measures and P = λQ + (1 − λ)S, λ ∈ [0, 1], then λ can just have
values of either 0 or 1.
Proposition 3. [3, p.326] The next properties are equivalent:
i) process (X, IF) on probability space (Ω,F ,P) possesses the represen-
tation property for continuous local martingales;
ii) measure P is an extremal point of the convex set of such probability
measures on F∞ that (X, IF) is a continuous local martingale with
respect to them.
Let (X, IF) be a continuous local martingale, the local time of the process
is function LX from [0, +∞)× IR×Ω to IR such that, for any non-negative
Borel function g and all t ≥ 0, P-a.s.
∫ t
0
g(Xs)d〈X〉s =
∫
IR
g(y)LX(t, y)dy. (3)
The local time of Wiener process possesses some important properties,
which describe the process itself and the integrals driven by the process.
The next property is used in the paper.
Proposition 4. [5, Lemma 2.24] For local time of Wiener process,
LW (t, x) > 0 holds for all t ≥ 0, l ×P-a.e..
The existence conditions of the paper will guarantee the solutions ex-
istence for every initial distribution, that means: for every probabilistic
measure P̄ on (IR,B(IR)) there exists solution X with initial value X0, such
that the following holds:
P({X0 ∈ B}) = P̄ (B), ∀B ∈ B(IR).
Stochastic process (X, IF), defined on probability space (Ω,F ,P) with
filtration IF = (Ft)t≥0, is called a weak solution of SDI (1), if there exist
UNIQUENESS IN LAW OF SDI SOLUTIONS 113
Wiener process (W, IF) with W 0 = 0 and measurable process (u, IF), such
that u(t, ω) ∈ B(X(t, ω)) l+×P-a.e. and the following holds P- a.s. ∀t ≥ 0
Xt = X0 +
∫ t
0
u(s) dW s.
Stochastic process (X, IF), defined on probability space (Ω,F ,P) with fil-
tration IF = (Ft)t≥0, is called an explicit weak solution of SDI (1), if there
exist Wiener process (W, IF) with W 0 = 0 and Borel measurable function
v : IR → IR, such that v(x) ∈ B(x), ∀x ∈ IR and the following holds
P- a.s. ∀t ≥ 0
Xt = X0 +
∫ t
0
v(Xs) dW s.
The definitions were considered more in detail in paper [7]. Using excess β of
elements from comp(IR), the paper also introduced some sets and selections,
which will be used in this article as well. We call function
bint(x) =
{
β(0, B(x)), β(0, B(x)) ∈ B(x);
−β(0, B(x)), β(0, B(x)) /∈ B(x);
(4)
the internal characteristic selection of mapping B and function
bext(x) =
{
β(B(x), 0), β(B(x), 0) ∈ B(x);
−β(B(x), 0), β(B(x), 0) /∈ B(x);
(5)
the external characteristic selection of mapping B. Additionally, paper [7]
introduced sets
MB =
{
x ∈ IR
∣∣∣∣∣
∫
U(x,C)
β(0, B(y))−2 dy = ∞, ∀C > 0
}
,
MB =
{
x ∈ IR
∣∣∣∣∣
∫
U(x,C)
β(B(y), 0)−2 dy = ∞, ∀C > 0
}
,
NB = {x ∈ IR|{0} ∈ B(x)},
NB = {x ∈ IR|B(x) = {0}}.
Using the sets we can define the optimal characteristic selection of mapping
B as function
bopt(x) =
{
0, 0 ∈ B(x) and x ∈MB;
bext(x), otherwise .
(6)
This selection is an improvement of that one introduced in paper [7]. The
following relations are valid for the selection: Nbopt =NB ∪ (MB ∩NB) ⊆NB;
Mbopt = Cl(NB ∩MB) ∪MB =MB, since MB is closed and Cl(NB ∩MB) ⊆
Cl(MB); Nbopt \ Mbopt =NB \MB; Mbopt \ Nbopt =MB \ (NB ∪ (MB ∩ NB)) =
114 ANDREI N. LEPEYEV
MB \NB, where Cl stands for the closure of a set. All the results from paper
[7] are valid for this new selection, due to the relations above.
Proposition 5. [7, Theorem 1]Stochastic differential inclusion (1) has weak
and explicit weak solutions for every initial distribution if and only if the
following holds
MB ⊆NB. (7)
The proof of the result contains the main way of constructing a solution
of inclusion (1), which basically can be described in five steps:
1) Take selection v : IR → IR equal to bopt and choose a Wiener process
W̃ with the needed initial distribution.
2) Using the Wiener process the following processes can be introduces
for a given selection v:
T v
t =
∫ t+
0
v−2(W̃s)ds, Av
t = inf{s ≥ 0 : T v
s > t}, ∀t ≥ 0. (8)
3) Process X can be introduced as a composition of the Wiener process
and process A.
4) Having introduced the set of the first entry of the Wiener process to
Mv
U(Mv) = inf{s ≥ 0 : W̃s ∈ Mv} (9)
and using condition (7), one can show the existence of quadratic variation
of process X.
5) From Doob Theorem (see [6, Theorem II.7.1′]), the existence of the
quadratic variation guaranties the existence of a filtrated probability space
with a Wiener process where process X satisfies the conditions of the weak
solution definition.
Proposition 6. [7, Corollary 2] If stochastic differential inclusion (1) has
weak solutions for every initial distribution, then the inclusion has explicit
weak solutions with respect to selection bopt for every initial distribution.
Proposition 7. [7, Theorem 3] Stochastic differential inclusion (1) has
explicit weak solutions with respect to every Borel measurable selection for
every initial distribution if and only if the following holds
MB ⊆NB. (10)
Multi-valued mapping B : IR → P(IR ∪ {±∞}) is locally integrable in
the wide sense (locally integrable in the narrow sense), if its every Borel
measurable selection is locally integrable (there exists its Borel measurable
locally integrable selection). Using the notions, the existence conditions for
non-trivial weak and explicit weak solutions were also given in paper [7].
UNIQUENESS IN LAW OF SDI SOLUTIONS 115
Proposition 8. [7, Theorem 4] Stochastic differential inclusion (1) has
weak and explicit weak non-trivial solutions for every initial distribution if
and only if mapping B−2 is locally integrable over IR in the narrow sense.
Proposition 9. [7, Theorem 5] Stochastic differential inclusion (1) has
explicit weak non-trivial solutions with respect to every Borel measurable
selection for every initial distribution if and only if mapping B−2 is locally
integrable over IR in the wide sense.
Let us go on to uniqueness concepts. In correspondence with the SDE
theory, we say that the uniqueness in law holds for SDI (1) if any two
solutions (X1, IF1) and (X2, IF2) with coinciding initial distributions possess
the same image law on the space of continuous functions over IR.
Definition 1. Additionally, we say that the uniqueness in law in the class
of explicit solutions holds for SDI (1) if any two explicit solutions (X1, IF1)
and (X2, IF2) with coinciding initial distributions possess the same image
law on the space of continuous functions over IR.
Separating the solutions according to their selections, we can introduce
the following definitions.
Definition 2. We say that the uniqueness in law with respect to selection v
holds for SDI (1) if any two explicit solutions (X1, IF1) and (X2, IF2) with
respect to explicit selection v with coinciding initial distributions possess
the same image law on the space of continuous functions over IR.
Definition 3. We say that the selectionwise uniqueness in law holds for
SDI (1) if uniqueness in law holds for every explicit Borel measurable selec-
tion of the SDI right side.
Remarks 1. The notions of uniqueness with respect to a selection and the
selectionwise uniqueness does not make any sense for non-explicit solutions,
due to the fact that every non-explicit selection corresponds to one solution.
2. Uniqueness in law implies uniqueness in law in the class of explicit
solutions, which, in its turn, implies the selectionwise uniqueness in law.
3. If the right side of SDI (1) is single-valued then the definitions of the
uniqueness in law coincide and equal the definition of uniqueness in law of
SDE solutions.
The following result is crucial for the following calculations.
Lemma 1. If there exists set V ⊂ IR such that l{V } > 0, then there exists
point x0 ∈ IR such that for every open ball U(x0, C) inequality
l{V ∩ U(x0, C)} > 0 holds.
Proof. Sigma-additivity of Lebesgue measure implies the existence of z ∈ Z
such that Vz = V ∩ [z, z + 1] has non-zero measure. We will prove that
∃x0 ∈ [z, z + 1] such that ∀C ∈ (0, 1) : l{U(x0, C) ∩ V } > 0. Using
the rule of contraries, assume that, for every point x ∈ [z, z + 1], there
exists radius Cx such that l{U(x, Cx) ∩ V } = 0. The set of open balls
U(x, Cx) is an open cover of [z, z+1]. The compactness implies the existence
116 ANDREI N. LEPEYEV
of finite subcover {m ∈ 1, n|U(xm, Cxm)} of [z, z + 1]. We obtained that
l{∪m∈1,nU(xm, Cxm)∩V } = 0, but, on the other hand, l{∪m∈1,nU(xm, Cxm)∩
V } ≥ l{[z, z + 1] ∩ V } > 0. Lemma is proved.
3. Uniqueness Theorems
Let us start with the seletionwise uniqueness in law.
Theorem 1. Stochastic differential inclusion (1) has selectionwise unique
in law explicit weak solutions with respect to every Borel measurable selection
for every initial distribution if and only if (10) holds and
NB ⊆MB. (11)
Proof. Necessity. Let unique in law explicit weak solutions exist with re-
spect to every selection and for every initial distribution. Condition (10)
holds from Proposition 7. The next step is to prove relation (11). Using
the rule of contraries, assume the existence of x0 ∈ (NB \MB). From the
statement of the theorem, there exists weak solution X of SDI (1), defined
on probability space (Ω,F ,P) with filtration IF, with respect to selection
v = bint and initial value X0 = x0. Moreover, sinceMB ⊆NB ⊆NB, then, for
this selection, we can construct the solution in the way described by Propo-
sition 5, where original Wiener process W̃ can be taken with initial value x0.
In one’s turn, quadratic variation 〈X〉 P-a.s. coincides with process Av de-
fined in (8). Using the continuity of Wiener process trajectories, one can
conclude that the corresponding set U(Mv), defined in (9), is separated from
zero, but U(Mv) = Av
∞ P-a.s. (the last equality follows from Lemma 1 in
[4]) and, hence, Av
∞ > 0 P-a.s., that implies non-triviality of (X, IF). On
the other hand, since v(x0) = bint(x0) = 0 P-a.s., then X̄t = x0 ∀t ≥ 0 is a
trivial solution of SDI (1) with respect to selection v, hence the condition
of selectionwise uniqueness in law fails. The same proof can be conducted
for selection v = bext, hence, NB ⊆MB and NB ⊆MB ⊆NB ⊆MB .
Sufficiency. From Proposition 7, condition (10) guarantees the existence
of explicit weak solutions with respect to every Borel measurable selec-
tion for every initial distribution. To finish the proof we need to show,
that condition (11) implies selectionwise uniqueness in law. Let (X, IF)
be an arbitrary solution of SDI (1) with respect to some explicit selec-
tion v. Similarly to the necessity proof of Theorem 1 in [7] (Proposition 5)
we can introduce process Av, his inverse T v and Wiener process (W v, IFv)
stopped at Av
∞, where (W v
t )t≥0 = (XT v
t
)t≥0, and the corresponding filtration
IFv = (F v
t )t≥0 = (FT v
t
)t≥0.
Conditions of Proposition 2 are valid for our case, due to the relation
Nv ⊆ NB ⊆MB ⊆ Mv and, hence, process T v is IFW v
-adapted and Av
∞ is
IFW v
-predictable stopping time. From Proposition 1, we can conclude that
UNIQUENESS IN LAW OF SDI SOLUTIONS 117
(X, IFX) possesses the representation property for continuous local martin-
gale. That is why, any solution of inclusion (1) with respect to the same
explicit selection v possesses the representation property.
Now let (X, IF) and (X ′, IF′) be two weak solutions on probability spaces
(Ω,F ,P) and (Ω′,F ′,P′) with the same initial distribution and with respect
to the same selection v. From Proposition IV.2.1 in [6], the existence of
solutions means the existence of the corresponding distributions Q and Q′
on measurable space (C(IR+, IR),B(C(IR+, IR))). To prove the uniqueness
in law, we should prove the coincidence of the distributions. Using the
coordinate method, let us introduce process
Zt(w) = w(t), ∀t ≥ 0, w ∈ C(IR+, IR).
Measure P̄ can be defined on (C(IR+, IR),B(C(IR+, IR))) in the following
way
P̄(C) = λQ(C) + (1 − λ)Q′(C), ∀C ∈ B(C(IR+, IR))
for an arbitrary, but fixed λ ∈ [0, 1]. Proposition IV.2.1 in [6] implies, that
process Z is a local continuous martingale with respect to filtration IFZ and
measures Q, Q′ and P̄, and P̄-a.s.
〈Z〉t =
∫ t
0
v2(Zs)ds.
Using the Doob Theorem (see [6, Theorem II.7.1′]) one can conclude, that
there exists Wiener process W such that Z is the solution of equation (2)
with diffusion coefficient v. Hence, process Z with distribution P̄ possesses
the representation property and, from Proposition 3, it is an extremal point
of the convex set of probability measures on FZ
∞, for which process (Z, IFZ)
is a local martingale. Since (Z, IFZ) is also a local martingale for measures
Q and Q′, and λ is an arbitrary, then Q = P̄ = Q′. The proof is completed.
Corollary 1. The selectiowise uniqueness in law holds for SDI (1), if it
satisfies condition (11).
Using Proposition 9, we can deduce the following statement.
Corollary 2. Stochastic differential inclusion (1) has selectionwise unique
in law non-trivial explicit weak solutions with respect to every Borel mea-
surable selection for every initial distribution if and only if
NB =MB = Ø. (12)
Corollary 3. Stochastic differential inclusion (1) does not have trivial
solutions if it satisfies condition (12).
118 ANDREI N. LEPEYEV
Now we will consider the question of the uniqueness in law with respect
to a specific selection.
Theorem 2. If (7) holds and
NB ⊆MB, (13)
then there exists explicit selection v such that stochastic differential inclu-
sion (1) has unique in law with respect to the selection explicit weak solutions
for every initial distribution.
Proof. Let explicit weak solutions of inclusion (1) exist, then Proposition 5
implies the existence of explicit weak solutions with respect to selection
v = bopt for every initial distribution. The proof of the uniqueness repeats
the steps of the Theorem 1 proof, using Proposition 2 and the fact that
Nv =NB ∪ (MB ∩NB) ⊆MB = Mv. The proof is completed.
Corollary 4. If conditions (7) and (13) are satisfied, then stochastic differ-
ential inclusion (1) has unique in law with respect to selection bopt explicit
weak solutions for every initial distribution.
Corollary 5. If condition (13) is satisfied, then there exists explicit selec-
tion v such that the uniqueness in law with respect to selection v holds for
SDI (1).
Using Proposition 8, we can deduce the following statement.
Corollary 6. If condition (13) is satisfied and B−2 is locally integrable
over IR in the narrow sense, then there exists explicit selection v such that
stochastic differential inclusion (1) has unique in law with respect to the
selection non-trivial explicit weak solutions for every initial distribution.
Theorem 3. If there exists explicit selection v such that stochastic differ-
ential inclusion (1) has unique in law with respect to the selection explicit
weak solutions for every initial distribution, then conditions (7) and (11)
are satisfied.
Proof. Basically, we should follow the proof of necessity in Theorem 1.
Statement MB ⊆ NB holds from Proposition 5. In order to prove state-
ment (11), we use the rule of contraries. Assume the existence of point
x0 ∈ (NB \MB). Let us take an arbitrary Wiener process (W̃ , ĨF) with
initial value W̃0 = x0. Analogously to Proposition 5, we can construct
solution (X, IF) with respect to selection v = bopt and initial distribution
X0 = x0 and the quadratic variation 〈X〉 will coincide with Av P-a.s.. Set
U(Mv) for the constructed Wiener process is separated from zero, since
the trajectories of Wiener process are continuous, set Mv =MB is closed
UNIQUENESS IN LAW OF SDI SOLUTIONS 119
and does not contain x0. Additionally, U(Mv) = Av
∞ P-a.s. and hence
Av
∞ > 0 P-a.s., that implies non-triviality of (X, IF). On the other hand,
since v(x0) = bopt(x0) = 0 P-a.s., then X̄t = x0, ∀t ≥ 0 is a trivial solution of
inclusion (1) with respect to selection v, hence, the condition of Definition 2
fails. Thus, relation (11) holds. The proof is completed.
Corollary 7. If there exists selection v such that stochastic differential in-
clusion (1) has unique in law with respect to the selection non-trivial explicit
weak solutions for every initial distribution, then condition (11) is satisfied
and B−2 is locally integrable over IR in the narrow sense.
Theorem 4. For every initial distribution, stochastic differential inclu-
sion (1) has unique in law non-trivial explicit weak solution if and only if
(12) holds and mapping B is almost everywhere single-valued, i.e.
B(x) = bext(x) = bint(x), l − a.e. (14)
Proof. Sufficiency. From Corollary 2 condition (12) ensures the existence of
non-trivial explicit weak solutions with respect to every Borel measurable
selection for every initial distribution as well as the selectionwise uniqueness
in law of the solutions. Let us prove that condition (14) implies coinci-
dence of the finite dimensional distributions of the solutions. Selectionwise
uniqueness in law means, that finite dimensional distributions of every ex-
plicit solution with respect to selection v coincide with finite dimensional
distribution of the explicit solution constructed in the way of Proposition 1
for this selection v. Let us take any almost everywhere equal selections v
and v′. Statement Nv = Mv = Nv′ = Mv′ = Ø implies that the correspond-
ing processes T v
t and T v′
t are continuous for all t (see proof of Theorem 1 in
[7]) and P-a.s. identical. That is why, for quadratic variations of processes
Xt = WAv
t
and X ′
t = WAv′
t
with respect to selections v and v′ for all t ≥ 0,
almost sure holds
〈X〉t = Av
t = Av′
t = 〈X ′〉t .
The accomplished equality also holds almost sure with the measure of, pos-
sibly, extended probability space, where there exists Wiener process W̄ and
the conditions of the weak solution definitions are satisfied (see [6, Theo-
rem II.7.1′]). Hence, the finite dimensional distributions of the constructed
solutions coincide.
Necessity. Condition (12) holds from Corollary 2. At the same time, the
condition guarantees the correctness of the construction of solutions with re-
spect to the selections in the way of Proposition 5. To prove condition (14),
we will use the rule of contraries. Let condition (14) fail, hence, selec-
tions bint and bext differ on some set V such that l{V } > 0. Hence, from
Lemma 1, there exists point x0 such that, for every open ball U(x0, C),
120 ANDREI N. LEPEYEV
l{V ∩U(x0, C)} > 0. Using any Wiener process W with initial value x0, we
can construct processes T bint and T bext . For all t ≥ 0 P-a.s.
T bint
t =
∫ t
0
b−2
int(Ws)ds =
∫
IR
b−2
int(y)LW (t, y)dy,
where the last equality follows from the definition of the local time (3).
Similar, for all t ≥ 0 P-a.s.
T bext
t =
∫
IR
b−2
ext(y)LW (t, y)dy,
that implies, for all t ≥ 0 P-a.s.
T bint
t − T bext
t =
∫
IR
(b−2
int(y) − b−2
ext(y))LW (t, y)dy > 0,
where the last inequality follows from Proposition 4.
Using the continuity of processes T bint and T bext for quadratic variations
of solutions Xt = W
A
bint
t
and X ′
t = W
A
bext
t
with respect to selections bint and
bext for all t ≥ 0 almost sure holds
〈X〉t = Abint
t < Abext
t = 〈X ′〉t.
The inequality also holds almost sure with the measure of, possibly, ex-
tended probability space, where there exists Wiener process W̄ and the con-
ditions of the definitions of explicit weak solution are satisfied (see [6,The-
orem II.7.1′]). The proof is completed.
Remark 4. In order to state the uniqueness in law for all weak solutions
(not just in the class of explicit solutions), one have to additionally impose
a condition, which can guarantee the existence of an explicit selection cor-
responding to the selection of the weak solution (as an example, Theorem
9.5.3 of [1] can be used). The condition means that any weak solution can
be expressed in the form of an explicit weak solution with an appropriate ex-
plicit selection. Hence, given the condition, the uniqueness in law coincides
with the uniqueness in law in the class of explicit solutions.
4. Examples
Example 1. Let us consider stochastic differential inclusion (1) with right
side
B(x) =
{ |x|, x ∈ (−∞, 1) ∪ (1, +∞);
{0, 1}, x = 1.
For this equation,MB = MB =NB = {0} and NB = {0, 1}. Hence, from
Theorem 2, there exists a solution with respect to selection bopt(x) = |x|
UNIQUENESS IN LAW OF SDI SOLUTIONS 121
and the solution is unique in low for the selection. But condition (11) of
selectionwise uniqueness is not satisfied and it can be verified, that SDI (1),
for initial value X0 = 1, has two solutions (trivial and non-trivial) with
respect to selection bint(x) =
{ |x|, x ∈ (−∞, 1) ∪ (1, +∞);
0, x = 1.
Example 2. Stochastic differential inclusion (1) with right side
B(x) =
⎧⎪⎨
⎪⎩
Arcth(x), x ∈ (−∞,−1) ∪ (1, +∞);
Arth(x), x ∈ (−1, +1);
[1, 2], x ∈ {−1, 1};
has selectionwise unique in law explicit weak solutions with respect to every
Borel measurable selection for every initial distribution, due to MB =MB =
NB =NB = {0}.
Example 3. For stochastic differential inclusion (1) with right side
B(x) =
{
1
x
, x ∈ IR \ {0};
[1, 2], x = 0;
MB =MB =NB =NB = Ø holds and the right side is single-valued almost
everywhere on IR, except one point. Hence, for every initial distribution,
this inclusion has a non-trivial explicit weak solution unique in law in the
class of explicit solutions.
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Department of Functional Analysis, Belarusian State University,
pr. Nezavisimosti 4a, 220030, Minsk, Belarus
E-mail address: ALepeev@iba.by
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