On stochastic differential inclusions with unbounded right sides
The paper deals with one-dimensional homogeneous stochastic differential inclusions without drift with a Borel measurable right side. Using a new method of explicit solutions, the necessary and sufficient conditions for the existence of weak solutions of the inclusions with locally unbounded right s...
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Lepeyev, A.N. 2009-11-10T14:51:45Z 2009-11-10T14:51:45Z 2006 On stochastic differential inclusions with unbounded right sides / A.N. Lepeyev // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С.94–105. — Бібліогр.: 23 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4445 519.21 The paper deals with one-dimensional homogeneous stochastic differential inclusions without drift with a Borel measurable right side. Using a new method of explicit solutions, the necessary and sufficient conditions for the existence of weak solutions of the inclusions with locally unbounded right sides are given. en Інститут математики НАН України On stochastic differential inclusions with unbounded right sides Article published earlier |
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On stochastic differential inclusions with unbounded right sides |
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On stochastic differential inclusions with unbounded right sides Lepeyev, A.N. |
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On stochastic differential inclusions with unbounded right sides |
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On stochastic differential inclusions with unbounded right sides |
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On stochastic differential inclusions with unbounded right sides |
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On stochastic differential inclusions with unbounded right sides |
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on stochastic differential inclusions with unbounded right sides |
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Lepeyev, A.N. |
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Lepeyev, A.N. |
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The paper deals with one-dimensional homogeneous stochastic differential inclusions without drift with a Borel measurable right side. Using a new method of explicit solutions, the necessary and sufficient conditions for the existence of weak solutions of the inclusions with locally unbounded right sides are given.
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0321-3900 |
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On stochastic differential inclusions with unbounded right sides / A.N. Lepeyev // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С.94–105. — Бібліогр.: 23 назв.— англ. |
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AT lepeyevan onstochasticdifferentialinclusionswithunboundedrightsides |
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2025-11-24T05:32:34Z |
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1850441536975667200 |
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Theory of Stochastic Processes
Vol. 12 (28), no. 1–2, 2006, pp. 94–105
UDC 519.21
ANDREI N. LEPEYEV
ON STOCHASTIC DIFFERENTIAL INCLUSIONS
WITH UNBOUNDED RIGHT SIDES
The paper deals with one-dimensional homogeneous stochastic differential inclusions
without drift with a Borel measurable right side. Using a new method of explicit
solutions, the necessary and sufficient conditions for the existence of weak solutions
of the inclusions with locally unbounded right sides are given.
Introduction
The following driftless homogeneous stochastic differential inclusion (SDI) is investi-
gated:
dXt ∈ B(Xt) dWt, t ≥ 0, (1)
where B : R → comp(R) is a multi-valued Borel measurable mapping, comp(R) - the set
of all non-empty compact subsets of R with the Hausdorff metric which is defined ∀A, B ∈
comp(R) by ρ(A, B) = max(β(A, B), β(B, A)), where β(A, B) = supx∈A(infy∈B |x − y|)
is the excess of A over B, and W is a one-dimensional Wiener process.
The theory of differential inclusions originated, basically, from two problems. The
first one is the generalization of the equations without solutions to inclusions, which have
solutions, and treatment of the solutions of the equations in the sense of the solutions
of the corresponding inclusions. This theory was systematized in [13] by A.F. Filippov,
who introduced the main properties, which the solutions of the inclusions should satisfy
to. The main property is the preservation of the solution set for the equations having
solutions in the classic context. The second problem arose in the models described by
the differential equations with multi-valued right sides (cf. [3]). The solutions of the
equations appeared to be well-defined in terms of the corresponding inclusions.
The investigation of multi-valued random processes (cf. [14]) motivated the creation of
a technique for stochastic differential equations with multi-valued right sides, that led to
the use of the ideas developed in the theory of differential inclusions in stochastic models
(cf. [6]). Stochastic differential inclusions as a separate theory were introduced by P. Kree
in [19]. N.U. Ahmed, E. Cepa, G. Da Prato, H. Frankovska, M. Kisielewicz, A.A. Levakov,
M. Michta, J. Motyl, R. Petterson who considered the stochastic differential inclusions
with different right sides and gave sufficient conditions for the existence of weak solutions
of SDIs (cf. [1,5,7,17,18,20,23]). The known results contain the existence conditions for
weak solutions of stochastic differential inclusions, whose right sides are required to be
bounded at least locally.
This paper deals with the existence conditions for weak solutions of SDIs with lo-
cally unbounded right sides. The main result is the necessary and sufficient existence
conditions for weak solutions of SDIs for every initial distribution.
2000 AMS Mathematics Subject Classification. Primary 34A60, 60G44, 60H10, 60J65, 60H99.
Key words and phrases. Stochastic differential equations, stochastic differential inclusions, measur-
able coefficients.
94
ON STOCHASTIC DIFFERENTIAL INCLUSIONS 95
Preliminaries
Solutions of stochastic differential equations with unbounded coefficients.
The main idea of the paper is to express the solutions of stochastic differential inclu-
sions in terms of the theory of stochastic differential equations (SDEs) and to use the
developed technique of the theory. That is why, at first we will recall several important
facts from the SDEs theory.
Let (Ω,F ,P) be a complete probability space with filtration F = (Ft)t≥0. As usually,
assume, that the filtration F satisfies the natural conditions, e.c. it is right-continuous
and F0 contains all P - zero subsets of F . For a stochastic process (Xt)t≥0 defined on
(Ω,F ,P), we will write (X, F), if X is F-adapted.
SDI (1) is, actually, the generalization of SDE of the following type:
dXt = b(Xt) dWt, t ≥ 0, (2)
where b : R → R - Borel measurable diffusion coefficient, W - one-dimensional Wiener
process.
It is known that if the process X is a stochastic integral driven by a Wiener process
(for example, it is a solution of Eq. (2), then it has continuous modification (if X is a
solution of Eq. (2), then its continuous modification is also a solution of (2)). Taking
that into consideration, we will deal with the trajectory space C(R+, R) equipped with
the Borel σ-algebra of cylindric sets B(C(R+, R)).
A stochastic process (X, F), defined on the probability space (Ω,F ,P) with a filtration
F = (Ft)t≥0 and trajectories in C(R+, R) is called a weak solution of SDE (2) with initial
distribution X0, if there exists a Wiener process (W, F) with W 0 = 0 such that P- a.s.
for all t ≥ 0
Xt = X0 +
∫ t
0
b(Xs) dW s.
It should be emphasized that the initial distribution X0 in the solution definition can
be an arbitrary random value in the space, where the solution exists. The existence con-
ditions of the paper will guarantee the existence of solutions for every initial distribution,
that means: for every probabilistic measure P̄ on (R,B(R)), there exists a solution X
with initial distribution X0 such that
P({X0 ∈ C}) = P̄ (C), ∀C ∈ B(R).
The idea of the proof of our main result is based on the Engelbert—Schmidt theorem
(cf. [12], Theorem 4.17) which states that the weak solutions of Eq. (2) exist for every
initial distribution if and only if the following holds:
Mb ⊆ Nb, (3)
where the sets in (3) are defined for every measurable function v : R → R by
Mv =
{
x ∈ R
∣∣∣∣∣
∫
U(x)
v−2(y) dy = ∞, ∀U(x) - open neighborhood of the point x
}
(4)
Nv = {x ∈ R|v(x) = 0} (5)
The solutions of SDE (2) are always non-exploding (cf. [12], Proposition 4.11).
Random time change.
Now we will focus attention on the method of random time change. Assume that a
filtrated probability space is given. Any increasing right-continuous family A = (At) of
P-a.s. finite F-stopping times is called F-time change. Having defined the right-inverse
process Tt = inf{s ≥ 0 : As > t}, one can give the main property of a random time
96 ANDREI N. LEPEYEV
change ATt = t P-a.s. ∀t ≥ 0. Additionally, if the process A is strictly increasing, then
the process T is continuous and TAt = t P-a.s. ∀t ≥ 0.
Now we will give some generalization of the random time change that was introduced
by H.J. Engelbert and W. Schmidt (cf. [10]). Let A = (At) be a right-continuous in-
creasing process with values in [0, +∞] and A∞ = limt→∞ At. Having defined the inverse
increasing process Tt = inf{s ≥ 0 : As > t}, we can pathwisely construct the Lebesgue
integral
∫ t
0
ZsdAs =
∫ t∧T∞
0
ZsdAs, ∀t ≥ 0, where Z is an arbitrary measurable process.
Lemma 1 ([10], Lemma 1.6). For every non-negative measurable process Z, the fol-
lowing holds: ∫ t
0
ZsdTs =
∫ Tt
0
ZAsds, ∀t ≥ 0.
Different kinds of random time changes proved to be helpful in the solution of various
problems of stochastic analysis. Let us consider a time change which will be used in
this article. Let us be given with an arbitrary Borel measurable function v : R → R
and a Wiener process (W̃ , F̃) on the probability space (Ω,F ,P) with arbitrary initial
distribution. We can define the increasing process
T v
t =
∫ t+
0
v−2(W̃s)ds, ∀t ≥ 0 (6)
and its right-inverse process
Av
t = inf{s ≥ 0 : T v
s > t}. (7)
Additionally, let U(Mv) be the first entry time of W̃ in Mv,
U(Mv) = inf{s ≥ 0 : W̃s ∈ Mv}. (8)
The following properties will be used in the proof of the main results of the paper.
Lemma 2 ([11],Lemma 1,[9],Theorem 3). Under the definitions above, P-a.s.∫ t
0
v−2(W̃s)ds < +∞, ∀t < U(Mv);
∫ U(Mv)+
0
v−2(W̃s)ds = +∞;
A∞ = U(Mv); and T v
∞ = +∞.
Right sides of stochastic differential inclusions.
In this section, we will closely investigate the right sides of SDI (1). The following
notations of measure will be used in the whole paper: l is the Lebesgue measure on R,
and l+ is a Lebesgue measure on R+ = [0,∞).
Let X and Y be arbitrary sets. The function v : X → Y is called the selection of a
multi-valued mapping B of the set X into some set of subsets from Y , if v(x) ∈ B(x), ∀x ∈
X (cf. [4]). The definition can be extended. Let (X, ν) be a space with measure ν, and
let Y be an arbitrary set. Then the function u : X → Y is called ν-a.e. selection of the
mapping B of the set X into some set of subsets from Y , if u(x) ∈ B(x), for ν-almost all
x ∈ X (cf. [22]).
Selections of the multi-valued mapping B(x) can be interpreted in two ways
1) Explicit selections of the mapping B : R → comp(R) such functions v : R → R that
v(x) ∈ B(x).
2) Composition selections B(X) : R+ × Ω → comp(R) (l+ × P-a.e. composition
selections) - such functions u : R+ × Ω → R that u(t, ω) ∈ B(X(t, ω)).
The measurability of the considered multi-valued mappings is meant in the sense of
the measurability of functions with values in the space (comp(R), ρ) with the Hausdorff
ON STOCHASTIC DIFFERENTIAL INCLUSIONS 97
metric. The corresponding σ-algebra of Borel subsets B(comp(R)) is generated by open
sets of the form [A]ε = {B ∈ comp(R)|ρ(A, B) < ε}, ∀A ∈ comp(R), ε > 0. Let us
denote op(R) and cl(R) - all open and closed sets from R generated by the Euclidian
metric.
Proposition 1 ([4], Theorem 3.2.). The multi-valued mapping B : R → comp(R) is
Borel measurable if and only if
B−(U) = {x ∈ R|B(x) ∩ U = ∅} ∈ B(R)
holds either for all U ∈ op(R) or for all U ∈ cl(R).
Let us define the internal characteristic selection of the mapping B
bint(x) =
{
β(0, B(x)), β(0, B(x)) ∈ B(x);
−β(0, B(x)), β(0, B(x)) /∈ B(x);
and external characteristic selection of the mapping B
bext(x) =
{
β(B(x), 0), β(B(x), 0) ∈ B(x);
−β(B(x), 0), β(B(x), 0) /∈ B(x).
Proposition 2. The internal and external characteristic selections of a Borel measurable
multi-valued mapping are Borel measurable functions.
Proof. Let U(x, r) ⊂ R denote an open ball around x ∈ R with radius r. For every r > 0,
{x ∈ R|β(0, B(x)) < r} = {x ∈ R|B(x) ∩ U(0, r) = ∅} = B−(U(0, r))
holds, and
{x ∈ R|β(B(x), 0) < r} = R \ {x ∈ R|B(x) ∩ (R \ U(0, r)) = ∅} = R \ (B−(R \ U(0, r)).
The functions β(0, B(x)) and β(B(x), 0) are Borel measurable from Proposition 1 and
the fact that the mapping B is Borel measurable. Hence, the statement of Proposition
2 holds due to the fact that the internal and external characteristic selections are the
compositions of Borel measurable functions.
Remark 1. a) The functions bint and bext are really selections of the mapping B due to
∀x ∈ R : bint(x) ∈ B(x), bext(x) ∈ B(x).
b) For every explicit selection v of the mapping B, the following holds: |bint(x)| ≤
|v(x)| ≤ |bext(x)|, ∀x ∈ R.
Additionally, let us define the optimal characteristic selection of the mapping B:
bopt(x) =
{
0, 0 ∈ B(x);
bext(x), 0 /∈ B(x);
Remark 2. a) The function bopt is a Borel measurable selection of the mapping B, because
∀x ∈ R : bopt(x) ∈ B(x) and it is a composition of Borel measurable functions.
b) |bint(x)| ≤ |bopt(x)| ≤ |bext(x)|, ∀x ∈ R.
It is worthwhile to provide a few facts on the integrability of multi-valued mappings.
The Lebesgue integral of multi-valued mappings was described in detail by R.J. Auman
in [3]. Let B be a multi-valued mapping of R into comp(R) and IB be the set of all
Lebesgue integrable selections of B. Then the integral of B over interval [0, t] ⊂ R, t > 0
is the set of integrals of all integrable selections∫ t
0
B(s)ds = {
∫ t
0
u(s)ds : u ∈ IB}.
98 ANDREI N. LEPEYEV
It is evident that, for the complete description of the integral, one can deal only with
Borel measurable selections. A stochastic integral driven by the Wiener process of multi-
valued mappings can be defined in the same way. Namely, let a multi-valued process
B : R+ × Ω → comp(R) be defined on the probability space (Ω,F ,P) with filtration F,
and let the process be measurable and F-adapted. Then we can define the set IB of all
(B(R+)×F)-measurable, F-adapted processes u such that u(t, w) is l+×P-a.e. selection
of the mapping B(t, w), and E(
∫ +∞
0
|u(s)|2ds) < ∞. The stochastic integral driven by a
Wiener process over the interval [0, t] ⊂ R, t > 0 of the multi-valued mapping B can be
defined by ∫ t
0
B(s)dWs = {
∫ t
0
u(s)dWs : u ∈ IB}.
Taking into consideration the fact that the right side of the stochastic differential inclusion
(1) is the composition B(X) : R+ × Ω → R and that the selections can be interpreted
in two ways, we have a right, for fixed measurable F-adapted process X , to consider IB
- the set of all explicit Borel measurable selections v of the mapping B : R → comp(R)
such that E(
∫ +∞
0
|v(Xs)|2ds) < ∞ and introduce the set
{
∫ t
0
v(Xs)dWs : v ∈ IB}.
Using the excess β between sets from compR, we can define four sets which will be
used in the statements of the theorems given below:
MB =
{
x ∈ R
∣∣∣∣∣
∫
U(x)
β(0, B(y))−2 dy = ∞, ∀U(x) - open neighborhood of point x
}
,
MB =
{
x ∈ R
∣∣∣∣∣
∫
U(x)
β(B(y), 0)−2 dy = ∞, ∀U(x) - open neighborhood of point x
}
,
NB = {x ∈ R|{0} ∈ B(x)},
NB = {x ∈ R|B(x) = {0}}.
Remark 3. a) The sets NB andNB are Borel ones, that is easily seen from the equalities
{x ∈ R|{0} ∈ B(x)} = {x ∈ R|bint = 0}, {x ∈ R|B(x) = {0}} = {x ∈ R|bext = 0} and
Proposition 2.
b) The following relations are valid: MB ⊆MB, NB ⊆NB.
c) For the sets Mv and Nv defined in (4) and (5) corresponding to the selections bint,
bext, and bopt, we can deduce
Mbint =MB, Mbext =MB, Nbint =NB, Nbext =NB, Nbopt = Nbint ,
Mbext =MB ∪ (MB ∩NB) ⊆MB, MB \NB = Mbopt \ Nbopt ,
which, particularly, implies that the setsMB andMB are closed.
d) For every explicit selection v of the mapping B,
NB ⊆ Nv ⊆NB, MB ⊆ Mv ⊆MB, (Mv \ Nv) ⊇ (MB \NB).
ON STOCHASTIC DIFFERENTIAL INCLUSIONS 99
Weak solutions of stochastic differential inclusions
Definitions and properties of weak solutions.
Let us recall (cf. [1]) that the stochastic process (X, F) defined on the probability
space (Ω,F ,P) with filtration F = (Ft)t≥0 and trajectories in C(R+, R) is called a weak
solution of SDI (1) with initial distribution X0, if there exist a Wiener process (W, F)
with W 0 = 0 and a measurable F-adapted l+ × P-a.e. selection u : R+ × Ω → R of the
composition B(X) : R+ × Ω → comp(R) such that the following holds P- a.s. for all
t ≥ 0:
Xt = X0 +
∫ t
0
u(s) dW s. (9)
The following definition introduces a subset of the set of SDI weak solutions.
Definition 1. A stochastic process (X, F) defined on the probability space (Ω,F ,P)
with filtration F = (Ft)t≥0 and trajectories in C(R+, R) is called an explicit weak solution
of SDI (1) with initial distribution X0, if there exist a Wiener process (W, F) with W 0 = 0
and a Borel measurable explicit selection v : R → R of the mapping B : R → comp(R)
such that the following holds P- a.s. for all t ≥ 0:
Xt = X0 +
∫ t
0
v(Xs) dW s. (10)
Proposition 3. An explicit weak solution of SDI (1) is a weak solution of SDI (1).
Proof. Let the stochastic process (X, F) with trajectories in C(R+, R) on the probability
space (Ω,F ,P) with filtration F = (Ft)t≥0 and the Wiener process (W, F) be an explicit
weak solution of SDI (1) with respect to some explicit selection v. Then the same
process X is a weak solution on the same probability space with the same filtration and
Wiener process. Namely, we can take the selection u = v(X) which is a (B(R+) × F)-
measurable F-adapted process due to the facts that the function v is Borel measurable,
and X is (B(R+)×F)-measurable and F-adapted. For the selection, the following holds:
u(t, ω) = v(X(t, ω)) ∈ B(X(t, ω)), ∀t ≥ 0, ω ∈ Ω.
The next statement is evident, but it should be emphasized due to its importance.
Proposition 4. Every explicit weak solution of the stochastic differential inclusion (1)
with respect to some selection v is a weak solution of the stochastic differential equation
(2) with diffusion coefficient v. On the other hand, every weak solution of the stochastic
differential equation (2) with some diffusion coefficient v is an explicit weak solution
of the stochastic differential inclusion (1) with the right side that possesses the explicit
selection v.
Proof. The proof is easily accomplished from the definitions of an SDI explicit weak
solution and an SDE weak solution.
The stochastic process (X, F) is trivial if Xt = X0, ∀t ≥ 0,P-a.s. Otherwise, the
process is non-trivial.
Proposition 5. The stochastic differential inclusion (1) has trivial weak solutions with
initial distribution X0 if and only if 0 ∈ B(X(0, ω)) P-a.s., e.c. bint(X0) = 0 P-a.s..
Proof. The sufficiency is easily accomplished if we take the selection u(t) ≡ 0. To prove
the necessity, let us take an arbitrary trivial weak solution X with respect to some
selection u with initial distribution X0. Then u(t, ω) = 0 l+ × P-a.e.; hence, the
definition of a SDI weak solution implies 0 ∈ B(X(t, ω)) l+ × P-a.e.. But
B(X(t, ω)) = B(X(0, ω)), ∀t ≥ 0,P-a.s.,
therefore, 0 ∈ B(X(0, ω)) P-a.s. or, in other words, bint(X0) = 0 P-a.s..
100 ANDREI N. LEPEYEV
Necessary and sufficient conditions for the existence of weak solutions of
stochastic differential inclusions.
Theorem 1. The stochastic differential inclusion (1) has weak and explicit weak solu-
tions for every initial distribution if and only if
MB ⊆NB. (11)
Proof. We will start with the necessity. Our objective is to prove that condition (11)
implies the existence of weak solutions for every initial distribution with respect to the
optimal characteristic selection. For the simplicity of notations, let us denote the selection
v ≡ bopt.
Point a) of Remark 2 shows that the selection v satisfies the conditions of the definition
of an explicit weak solution of SDI (1). On the other hand, since the selection v is single-
valued, we can consider a stochastic differential equation of type (2) with the diffusion
coefficient b ≡ v. From Proposition 4, all the weak solutions of SDE (2) will be the
explicit weak solutions of SDI (1).
We will use the method of random time change.
Let us take an arbitrary Wiener process (W̃ , F̃) on some probability space (Ω,F ,P)
with filtration F̃ = (F̃t)t≥0 and an arbitrary probability measure P̄ on R as the initial
distribution of the Wiener process. Using W̃ , we can introduce the processes Av and T v
and the set U(Mv), as defined in (6), (7), and (8).
The process Av = (Av
t )t≥0 is a continuous increasing family of F̃-stopping times.
From Lemma 2, the process Av
t is finite P-a.s. for all t ≥ 0. We can define a process
Xt = W̃Av
t
, ∀t ≥ 0 and a filtration F = (Ft)t≥0 = (F̃Av
t
)t≥0 and can conclude that (X, F)
is a continuous local martingale with square characteristic 〈X〉 = Av (cf. [16]).
Let us give the explicit form of Av. The set (Nv \Mv) has the zero Lebesgue measure
(from the definition of the set Mv). Hence, W̃ ∈ Mv ∪ (R\Nv) l+×P-a.e. Using Lemma
2, we can conclude that v(W̃s) = 0 for all s < Av
∞ l+ × P-a.e.. Lemma 2 also implies
W̃Av∞ ∈ Mv if {Av
∞ < +∞}. Due to condition (11) and point c) of Remark 3, we have
Mv ⊆ Nv, and if {Av
∞ < +∞}, then P-a.s.
v2(W̃Av∞) = 0. (12)
Using the property and the definition of the process T v, we obtain that P-a.s.
Av
t =
∫ Av
t
0
v2(W̃s)dT v
s =
∫ T v
Av
t
0
v2(Xs)ds,
where the last equality follows from Lemma 1. Hence, P-a.s. Av
t =
∫ t
0 v2(Xs)ds if
Av
t < Av∞ and
∫ ∞
0 v2(Xs)ds if Av
t = Av∞. On the other hand, if Av
t = Av∞, then
Xs = WAv∞ for all s ≥ t, and we can conclude from (12) that v2(Xs) = 0 for all s ≥ t,
which implies P-a.s.
Av
t =
∫ t
0
v2(Xs)ds, ∀t ≥ 0.
From the Doob theorem (cf. [15], Theorem II.7.1′), there exists a Wiener process (W̄ , F)
(on a possibly extended probability space) such that (X, F) is a weak solution of SDE
(2) with initial distribution P̄ .
This process X is an explicit weak solution of SDI (1) from Proposition 4. From
Proposition 3, this process is a weak solution of SDI (1).
ON STOCHASTIC DIFFERENTIAL INCLUSIONS 101
Let (X, F) be an arbitrary weak solution of inclusion (1) with respect to some selection
u, e.c. (9) holds. Then a square variation 〈X〉 = Au has the form
Au
t =
∫ t
0
u2(s)ds, ∀t ≥ 0 P − a.s.
Let us define a process T u
t = inf{s ≥ 0 : Au
s > t}, that is right-inverse to the process
Au, and let us introduce a process (Wu
t )t≥0 = (XT u
t
)t≥0 with filtration F
u = (Fu
t )t≥0 =
(FT u
t
)t≥0, where X∞ = limt→∞ Xt if {Au∞ < +∞}. We can conclude that (Wu, Fu)
is a continuous local martingale such that 〈Wu〉t = t ∧ Au∞ and the initial distribution
W v
0 = X0. Hence, (Wu, Fu) is the Wiener process stopped in Au
∞ (cf. the proof of Lemma
2 in [8]). Using Lemma 1, we get∫ t∧Au
∞
0
b−2
ext(W
u
s )ds =
∫ Au
T u
t
0
b−2
ext(W
u
s )ds =
∫ T u
t
0
b−2
ext(Xs)dAu
s =
∫ T u
t
0
b−2
ext(Xs)u2(s)ds ≤ T u
t . (13)
The last inequality follows from the fact that since u(t, ω) ∈ B(X(t, ω)) for l+×P-almost
all (t, ω) ∈ R+ × Ω, bext(X(t, ω)) ≥ u(t, ω) for l+ × P-almost all (t, ω) ∈ R+ × Ω.
Let us take any point x0 /∈ NB and consider a weak solution X of SDI (1) with
initial distribution X0 = x0 that is non-trivial from Proposition 5. The non-triviality
of the solution implies P({Au∞ > 0}) > 0. Therefore, there exists t > 0 such that
P({Au
∞ > t}) > 0. Taking into consideration T u
t < +∞ if {Au
∞ > t} and inequality (13),
we can conclude
P({
∫ t
0
b−2
ext(W
u
s )ds < +∞, Au
∞ > t}) > 0. (14)
Hence, from Theorem 1 in [9], there exists an open neighborhood G of the point x0 such
that the function b−2
ext is integrable over G, and x0 /∈MB.
From the proof of Theorem 1, it is seen that condition (11) is valid if weak solutions
exist only for constant initial distributions x0 ∈ R.
Corollary 1. SDI (1) has weak and explicit weak solutions for every constant initial
distribution x0 ∈ R if and only if the right side of the inclusion satisfies condition (11).
Corollary 2. If stochastic differential inclusion (1) has weak solutions for every initial
distribution, then the inclusion has explicit weak solutions with respect to the selection
bopt for every initial distribution.
Conducting the proof of the sufficiency for the selections bint and bext, one can conclude
Theorem 2. If
MB ⊆NB (15)
or
MB ⊆NB, (16)
then the stochastic differential inclusion (1) has weak and explicit weak solutions for every
initial distribution.
Corollary 3. For every initial distribution, condition (15) guarantees the existence of
an explicit weak solution of the stochastic differential inclusion (1) with respect to the
internal characteristic selection bint. At the same time, for every initial distribution,
condition (16) guarantees the existence of an explicit weak solution of the stochastic
differential inclusion (1) with respect to the external characteristic selection bext.
102 ANDREI N. LEPEYEV
Lemma 3. If the right side of inclusion (1) is closed (its graph is a closed subset of R
2),
then it satisfies condition (15).
Proof. We will denote U(x, C) - open ball around x ∈ R with radius C > 0. Since the
family of such open balls is the fundamental system of neighborhoods of R, we can use
the open balls in the definitions of the sets MB and MB. Let x∗ ∈ MB. We use the
rule of contraries. Assume that x∗ /∈NB, e.c. (x∗, 0) does not belong to the graph. The
closure of the graph implies ∃U(0, C1), U(x∗, C2) such that U(0, C1) ∩ {y|y ∈ B(x), x ∈
U(x∗, C2)} = ∅ and, for all y ∈ B(x), x ∈ U(x∗, C2), the inequality |y| ≥ C1 holds.
Therefore, β(0, B(x)) ≥ C1 or β(0, B(x))−2 ≤ C−2
1 . Hence,
∫
U(x∗,C2)
β(0, B(x))−2 dx <
∞. Thus, we have a conflict with the definition of the setMB.
Corollary 4. The stochastic differential inclusion (1) has weak and explicit weak solu-
tions for every initial distribution, if its right side B is closed.
The next theorem clarifies the existence of explicit solutions for every explicit Borel
measurable selection of the SDI right side. It is known that the selections always exist
for every multi-valued Borel measurable mapping (cf. [4], Theorem III.6).
Theorem 3. The stochastic differential inclusion (1) has explicit weak solutions with
respect to every Borel measurable selection for every initial distribution if and only if
MB ⊆NB.
Proof. The proof of the sufficiency can be conducted similarly to that in Theorem 1. One
needs to take an arbitrary explicit Borel measurable selection v of the mapping B and
use the fact that Mv ⊆MB ⊆NB ⊆ Nv, which implies W̃ ∈ Mv ∪ (R \ Nv) l+ × P-a.e.
and if {Av∞ < +∞}, then P-a.s. v2(W̃Av∞) = 0.
The proof of the sufficiency can be conducted by the steps of the necessity proof of
Theorem 1. Having taken an arbitrary point x0 /∈NB, one can investigate the behavior
of explicit weak solutions with respect to the selection
v(x) =
{
bext(x), x = x0;
bint(x), x = x0.
Using the arguments of the proof of Theorem 1, one can conclude that a solution with
initial distribution x0 with respect to the selection v is non-trivial. Therefore,
P({
∫ t
0
b−2
ext(W
v
s )ds < +∞, Av
∞ > t}) > 0.
Hence, Lemma 1 in [9] implies the existence of an open neighborhood G of x0 such that
the function b−2
ext is integrable over G, and x0 /∈MB.
Remark 4. If the right side of SDI (1) is single-valued, then the existence conditions from
Theorems 1, 2, and 3 are coincide and equal to condition (3) (the Engelbert—Schmidt
theorem).
The next definition will use the set P(R ∪ {±∞}) of all non-empty subsets from
R ∪ {±∞}.
Definition 2. The multi-valued mapping B : R → P(R ∪ {±∞}) 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 selec-
tion).
ON STOCHASTIC DIFFERENTIAL INCLUSIONS 103
The upcoming statements will be dealing with the local integrability of the multi-
valued mapping B−2 which can be defined for every x ∈ R by
B−2(x) := {z =
1
y2
∈ R+ ∪ {+∞}, ∀y ∈ B(x)}.
Remark 5. The local integrability in the narrow sense of a multi-valued mapping is
equivalent to the local integrability of its internal characteristic selection. At the same
time, the local integrability in the wide sense of a multi-valued mapping is equivalent to
the local integrability of its external characteristic selection.
Theorem 4. The stochastic differential inclusion (1) has weak and explicit weak non-
trivial solutions for every initial distribution if and only if the mapping B−2 is locally
integrable over R in the narrow sense.
Proof. The proof mostly repeats the steps of the proof of Theorem 1, that is why we will
stop on the principal points about the non-triviality of solutions. The construction of a
solution can be conducted for the selection v(x) ≡ bext(x). By Remark 5, the function
v−2 is locally integrable, that implies Mv =MB = ∅ and, hence, Av∞ = U(Mv) = ∞.
Therefore, the constructed solution cannot be trivial, otherwise would be: v(Xt) =
0, ∀t ≥ 0,P-a.s. and Av
t = 0, ∀t ≥ 0,P-a.s., that is, we have a conflict with Av∞ = ∞.
The proof of the necessity is easily accomplished due to the existence of non-trivial
solutions for every initial distribution, which implies the fulfilment of (13) and, hence,
(14).
Combining the proofs of Theorems 3 and 4, we can formulate the following:
Theorem 5. The stochastic differential inclusion (1) has explicit weak non-trivial solu-
tions with respect to every Borel measurable selection for every initial distribution if and
only if the mapping B−2 is locally integrable over R in the wide sense.
Examples.
1. The theorems of this paper allow us to investigate SDIs of type (1) with locally
unbounded right sides. For instance, let the right side of SDI (1) be the countable union
of straight strophoids
B(x) = ±x
√
2a + x
2a − x
, a ∈ Z, x ∈ R.
It can be easily verified that the right side satisfies the conditions of Theorem 1, hence,
the inclusion has explicit weak solutions (from Proposition 3, they are weak solutions as
well) for every initial distributions. Furthermore, the right side satisfies the conditions
of Theorem 3. Hence, the inclusion has weak and explicit weak solutions with respect to
every Borel measurable selection of B for every initial distributions. On the other hand,
by Theorem 4, there exists a probabilistic measure P̄ such that there is no non-trivial
weak solution of the inclusion with P̄ as its initial distribution.
2. SDI (1) with the right side
B(x) =
{
[|x| 14 , |x| 15 ], |x| ≤ 1;
[|x| 14 , |x| 13 ], |x| > 1;
possesses non-trivial explicit weak solutions (by Proposition 3, they are weak solutions
as well) for every initial distribution from Theorem 4 due to the local integrability of the
mapping B−2 in the narrow sense. Its external characteristic selection
bext(x) =
{
|x| 15 , |x| ≤ 1;
|x| 13 , |x| > 1;
104 ANDREI N. LEPEYEV
is locally integrable in power (-2). Moreover, the inclusion possesses non-trivial explicit
weak solutions with respect to every Borel measurable selection of B for every initial
distributions from Theorem 5, because the internal characteristic selection bint(x) = |x| 14
is locally integrable in power (-2).
3. Consider SDE of type (2) with the diffusion coefficient
b(x) =
⎧⎪⎨
⎪⎩
Arcth(x), x ∈ (−∞,−1) ∪ (1, +∞);
Arth(x), x ∈ (−1, 0) ∪ (0, +1);
1, x ∈ {−1, 0, +1};
where Arth, Arcth - area hyperbolic tangent and cotangent, correspondingly. This equa-
tion does not have a weak solution for the initial distribution x0 = 0 from Theorem 1 in
[2].
Using one of the standard procedures (cf. [12,21]), we can generalize the equation. For
instance, let us construct the right part B of the corresponding inclusion as a minimal
close hull of the diffusion coefficient b excluding values {±∞}. As a result, we obtain
SDI with the right side
B(x) =
⎧⎪⎪⎨
⎪⎪⎩
Arcth(x), x ∈ (−∞,−1) ∪ (1, +∞);
Arth(x), x ∈ (−1, 0) ∪ (0, +1);
{0, 1}, x = 0;
1, x ∈ {−1, 1};
Weak solutions of the initial SDE will be treated in the sense of weak solutions of the
constructed SDI. The weak solutions exist for every initial distribution, since, for this
right side, MB =NB = {0}.
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E-mail : ALepeev@iba.by
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