An example of a stochastic differential equation with the property of weak non-uniqueness of a solution
A family of one-dimensional diffusion processes is constructed such that each one of this family is a weak solution to some stochastic differential equation. It turns out that the property of weak uniqueness of a solution to this equation is failed.
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| Zitieren: | An example of a stochastic differential equation with the property of weak non-uniqueness of a solution / B.I. Kopytko, M.I. Portenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 68–76. — Бібліогр.: 13 назв.— англ. |
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irk-123456789-44422009-11-11T12:00:31Z An example of a stochastic differential equation with the property of weak non-uniqueness of a solution Kopytko, B.I. Portenko, M.I. A family of one-dimensional diffusion processes is constructed such that each one of this family is a weak solution to some stochastic differential equation. It turns out that the property of weak uniqueness of a solution to this equation is failed. 2006 Article An example of a stochastic differential equation with the property of weak non-uniqueness of a solution / B.I. Kopytko, M.I. Portenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 68–76. — Бібліогр.: 13 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4442 519.21 en Інститут математики НАН України |
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A family of one-dimensional diffusion processes is constructed such that each one of
this family is a weak solution to some stochastic differential equation. It turns out
that the property of weak uniqueness of a solution to this equation is failed. |
| format |
Article |
| author |
Kopytko, B.I. Portenko, M.I. |
| spellingShingle |
Kopytko, B.I. Portenko, M.I. An example of a stochastic differential equation with the property of weak non-uniqueness of a solution |
| author_facet |
Kopytko, B.I. Portenko, M.I. |
| author_sort |
Kopytko, B.I. |
| title |
An example of a stochastic differential equation with the property of weak non-uniqueness of a solution |
| title_short |
An example of a stochastic differential equation with the property of weak non-uniqueness of a solution |
| title_full |
An example of a stochastic differential equation with the property of weak non-uniqueness of a solution |
| title_fullStr |
An example of a stochastic differential equation with the property of weak non-uniqueness of a solution |
| title_full_unstemmed |
An example of a stochastic differential equation with the property of weak non-uniqueness of a solution |
| title_sort |
example of a stochastic differential equation with the property of weak non-uniqueness of a solution |
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Інститут математики НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/4442 |
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An example of a stochastic differential equation with the property of weak non-uniqueness of a solution / B.I. Kopytko, M.I. Portenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 68–76. — Бібліогр.: 13 назв.— англ. |
| work_keys_str_mv |
AT kopytkobi anexampleofastochasticdifferentialequationwiththepropertyofweaknonuniquenessofasolution AT portenkomi anexampleofastochasticdifferentialequationwiththepropertyofweaknonuniquenessofasolution AT kopytkobi exampleofastochasticdifferentialequationwiththepropertyofweaknonuniquenessofasolution AT portenkomi exampleofastochasticdifferentialequationwiththepropertyofweaknonuniquenessofasolution |
| first_indexed |
2025-07-02T07:41:11Z |
| last_indexed |
2025-07-02T07:41:11Z |
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1836520122003488768 |
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Theory of Stochastic Processes
Vol. 12 (28), no. 1–2, 2006, pp. 68–76
UDC 519.21
BOGDAN I. KOPYTKO AND MYKOLA I. PORTENKO
AN EXAMPLE OF A STOCHASTIC DIFFERENTIAL
EQUATION WITH THE PROPERTY OF
WEAK NON-UNIQUENESS OF A SOLUTION
A family of one-dimensional diffusion processes is constructed such that each one of
this family is a weak solution to some stochastic differential equation. It turns out
that the property of weak uniqueness of a solution to this equation is failed.
Introduction
For some given constants b1 > 0 and b2 > 0, let b(x) for x ∈ R (R is a real line) denote
the function on R taking on the value b1 for x < 0, the value b2 for x > 0, and the value
zero for x = 0. Given a real constant A, consider the following stochastic differential
equation
(1) dx(t) = A1I{0}(x(t))dt + dξ(t),
where 1IΓ(x), Γ ⊂ R, x ∈ R, stands for an indicator function and ξ(t), t ≥ 0, is a square
integrable martingale, whose characteristic is given by the integral
(2) 〈ξ〉t =
∫ t
0
b(x(s))ds, t ≥ 0.
We will show that, for any given A (the constants b1 and b2 will be fixed throughout
this paper), there exist infinitely many diffusion processes satisfying Eq. (1); each one
of them is determined by fixing two parameters: q ∈ [−1, 1] and r > 0. The first one
characterizes the property of the point x = 0 to be “non-symmetric”, and the second one
characterizes the property of this point to be “sticky” for the corresponding process.
More precisely, if A = 0, then we should put q = q0 ≡ (b1 − b2)/(b1 + b2) and an
arbitrary positive number could be taken for the value of r. If A > 0, then the value of
q could be chosen arbitrarily from the interval (q0, 1] and, after that, we should put
(3) r =
(
√
b1 +
√
b2)[(1 + q)b2 − (1 − q)b1]
2A[(1 + q)
√
b2 + (1 − q)
√
b1]
.
If A < 0, then the value of r should be determined by the same formula, but this time
the value of q could be taken arbitrarily from the interval [−1, q0) (see Section 2 below).
In particular, if A > 0 and b1 = b2 = 1, the solution of (1) corresponding to q = 1
and r = A−1 has the following property: its part in the region [0, +∞) coincides with
the slowly reflecting Brownian motion (see [1]). This solution can be singled out by the
2000 AMS Mathematics Subject Classification. Primary 60J60, 60J35.
Key words and phrases. Stochastic differential equations, square integrable martingales, (generalized)
diffusion processes, random changes of time, (instantaneously or slowly) reflecting Brownian motions.
This article was supported (in part) by the Ministry of Education and Science of Ukraine, project
No. 01.07/103.
68
AN EXAMPLE OF A STOCHASTIC DIFFERENTIAL EQUATION 69
requirement of the property x(t) ≥ 0 for all t ≥ 0. As proven in [1], such a solution is
weakly unique, though there is no pathwise uniqueness in this situation.
Some examples of stochastic differential equations, for which the property of weak
uniqueness was failed, were constructed in [5], [8], [13]. Our example (described below
in Section 2) is a result of a random change of time in a (generalized) diffusion process
that itself can be described as a solution to the stochastic differential equation
(4) dx̃(t) = q̃δ(x̃(t))dt + dξ̃(t),
where δ(x), x ∈ R, is some kind of a (non-symmetric) Dirac function (see Section 1); the
constant q̃ is given by the right-hand side of (3) multiplied by A in the case of A
= 0
and q̃ = 0 if A = 0 (in this case, q = q0); ξ̃(t), t ≥ 0, is a square integrable martingale
with its characteristic given by
(5) 〈ξ̃〉t =
∫ t
0
b(x̃(s))ds, t ≥ 0
(see Remark 1 in Section 1).
A.V. Skorokhod was first to make use of a random change of time in order to construct
a slowly reflecting stochastic process from such one, for which the reflection was instanta-
neous (see [7], §24). The main distinction of our construction is admitting the boundary
to be permeable: notice that the point x = 0 is reflecting to the right (respectively, to the
left) for the solution of Eq. (4) in the case of q = +1 (respectively, q = −1); in the rest
cases (that is, for q ∈ (−1, 1)), the point x = 0 is “transparent” for the corresponding
process to move into either direction.
It is interesting to have a look at our example from the point of view of the general
theory of one-dimensional stochastic differential equations developed in [6]. It turns out
that Eq. (1) for A = q̃/r and Eq. (4) can be written in the form
(6) y(t) = y(0) + q̃Ly(·)(t, 0) +
∫ t
0
b(y(s))1/2dw(s), t ≥ 0,
where w(·) is a standard Wiener process in R and Ly(·)(t, y) is the (non-symmetric) local
time for the unknown process y(·) at the point y ∈ R on the interval [0, t] (see [6]).
As shown in [6], this equation has many weak solutions, and the so-called fundamental
solution can be pointed out such that any other solution with the strong Markov property
is obtained from the fundamental one by a random change of time. The solution x̃(·)
to Eq. (4) appears to be the fundamental solution to Eq. (6), and our construction can
thus serve as an illustrative example to the theory of [6].
Notice, finally, that the processes considered in this paper were investigated from
various points of view in [3], [4], [9], [10], [11].
1. The process that is a result of
pasting together two Brownian motions
We fix numbers b1 > 0, b2 > 0, and q ∈ [−1, 1] and put
d = (1 − q)
√
b1 + (1 + q)
√
b2, q1 = d−1(1 − q)
√
b1, q2 = d−1(1 + q)
√
b2,
c = 2(
√
b1 +
√
b2)−1, q̃ = c−1(q2
√
b2 − q1
√
b1).
Denote, by D1 and D2, respectively, the left and right halves of a real line: Di = {x ∈
R : (−1)ix > 0}, i = 1, 2. We define a function g̃(t, x, y) for t > 0, x ∈ R, and y ∈ R by
70 BOGDAN I. KOPYTKO AND MYKOLA I. PORTENKO
setting
g̃(t, x,y) = (2πbjt)−1/2
[
exp
{
− 1
2t
(
y√
bj
− x√
bi
)2}
+
+ (q2 − q1) sign y exp
{
− 1
2t
( |y|√
bj
+
|x|√
bi
)2}]
if x ∈ Di ∪ {0}, i = 1, 2, and y ∈ Dj, j = 1, 2. For y = 0, we set
g̃(t, x, 0) =
c
2
[
√
b1g̃(t, x, 0−) +
√
b2g̃(t, x, 0+)], t > 0, x ∈ R.
It is not difficult to verify that the function g̃ possesses the following properties:
1) g̃(t, x, y) ≥ 0 for all t > 0, x ∈ R, and y ∈ R;
2)
∫
R
g̃(t, x, y)dy = 1 for all t > 0 and x ∈ R;
3) g̃(s + t, x, y) =
∫
R
g̃(s, x, z)g̃(t, z, y)dz for all t > 0, s > 0, x ∈ R, and y ∈ R;
4) supx∈R
∫
R
(y − x)4g(t, x, y)dy = O(t2) as t ↓ 0.
These properties imply the existence of a continuous Markov process (x̃(t),M̃t, Px)
in R (we use notation from [2]) such that the relation
Exϕ(x̃(t)) =
∫
R
ϕ(y)g̃(t, x, y)dy
holds true for t > 0, x ∈ R, and an arbitrary bounded Borel function ϕ on R.
By some not very difficult computations, one can get the following formulae:
a) P0({x̃(t) ∈ Di}) = qi for t > 0 and i = 1, 2;
b) Ex(x̃(t) − x̃(0)) = q̃ft(x) for t > 0 and x ∈ R, where we put
ft(x) = c
∫ t
0
exp
{
− x2
2biτ
}
dτ√
2πτ
for t > 0 and x ∈ Di ∪ {0}, i = 1, 2;
c) for t > 0 and x ∈ Di, i = 1, 2,
Ex(x̃(t) − x̃(0))2
= tbi − 2q̃xft(x) + (−1)i+12q3−i(b2 − b1)
√
2
π
∫ t
0
exp
{
− x2
2biτ
}√
t − τ
dτ√
2πτ
;
in the case of x = 0, we have that, for t > 0,
E0x̃(t)2 = t(q1b1 + q2b2).
The relations in c) show that, for x ∈ R,
(7) lim
t↓0
t−1Ex(x̃(t) − x̃(0))2 = b̃(x),
where
b̃(x) =
2∑
i=1
bi1IDi(x) + (q1b1 + q2b2)1I{0}(x).
So, the function b̃(·) is the diffusion coefficient of the process x̃(t), t ≥ 0. In order to
calculate the drift coefficient of this process, we note that
(8) lim
t↓0
∫
R
t−1ft(x)ϕ(x)dx =
c
2
[
√
b1ϕ(0−) +
√
b2ϕ(0+)]
AN EXAMPLE OF A STOCHASTIC DIFFERENTIAL EQUATION 71
for an arbitrary bounded Borel function ϕ on R such that the limits
ϕ(0±) = lim
x→0±
ϕ(x)
exist. Relation (8) means that the function t−1ft converges, as t ↓ 0, to a non-symmetric
Dirac function δ, whose action on a test function ϕ is determined by the right-hand side
of (8). Taking into account relation b), we can say that the drift coefficient
ã(x) = lim
t↓0
t−1Ex(x̃(t) − x̃(0)), x ∈ R,
of the process x̃(t), t ≥ 0, coincides with the generalized function q̃δ(x). In other words,
if q̃
= 0, the process x̃(·) is a generalized diffusion process.
We now give a martingale characterization of the process (x̃(t),M̃t, Px). We observe
at first that the function ft(x) defined above satisfies the relation∫
R
ft(y)g̃(s, x, y)dy = ft+s(x) − fs(x)
for all t ≥ 0, s > 0, and x ∈ R. In addition, the relation
lim
t↓0
sup
x∈R
ft(x) = 0
is fulfilled. This implies (see Theorem 6.6 in [2]) the existence of an additive homogeneous
continuous non-negative functional (ηt)t≥0 of the process (x̃(t),M̃t, Px) such that Exηt =
ft(x) for all t ≥ 0 and x ∈ R. Moreover, as shown in [2], this functional is the limit for
the family of integral functionals:
ηt = l.i.m.
h↓0
∫ t
0
h−1fh(x̃(s))ds, t ≥ 0.
Since h−1fh → δ, as h ↓ 0, the functional ηt can be written in the form
ηt =
∫ t
0
δ(x̃(s))ds, t ≥ 0.
We now put, for t ≥ 0,
ξ̃(t) = x̃(t) − x̃(0) − q̃ηt.
Relation b) above shows that Exξ̃(t) = 0 for any t ≥ 0 and x ∈ R. This relation means
that the process ξ̃(·) is a martingale with respect to (M̃t, Px) for every x ∈ R. It is
evident that this martingale is square integrable, and relations c) above show that
Exξ̃(t)2 = Ex
∫ t
0
b̃(x̃(s))ds, t ≥ 0, x ∈ R.
Therefore, the characteristic of the martingale ξ̃(·) can be written in the form
〈ξ̃〉t =
∫ t
0
b̃(x̃(s))ds, t ≥ 0.
All these statements allow us to assert that there exists a one-dimensional standard
Wiener process (w(t))t≥0 (it is a functional of the process x̃(·)) such that
x̃(t) = x̃(0) + q̃ηt +
∫ t
0
b̃(x̃(s))1/2dw(s), t ≥ 0.
In other words, the process x̃(·) is the solution to the stochastic differential equation
dx̃(t) = q̃δ(x̃(t))dt + b̃(x̃(t))1/2dw(t).
72 BOGDAN I. KOPYTKO AND MYKOLA I. PORTENKO
Remark 1. This stochastic differential equation can be written in the form (4) (that is,
with the function b(·) instead of b̃(·)), since the time that the process x̃(·) spends at the
point x = 0 is a.s. equal to zero.
Remark 2. The process (x̃(t),M̃t, Px) described above is slightly different from those
considered in [3], [4], [9], [11].
2. The random change of time
We now fix a positive number r and put
ζt = inf{s ≥ 0 : s + rηs ≥ t}, x(t) = x̃(ζt),Mt = M̃ζt
for t ≥ 0.
Proposition 1. The process (x(t),Mt, Px) is a continuous strong Markov process in
R such that, for t > 0, x ∈ R, and any continuous bounded function ϕ on R, the relation
(9) Exϕ(x(t)) = ϕ(0)h(t, x) +
∫
R
g(t, x, y)ϕ(y)dy
holds true, where
h(t, x) =
2√
2πt
∫ ∞
0
exp
{
− 2θ
cr
− 1
2t
(
θ +
|x|√
bi
)2}
dθ, x ∈ Di ∪ {0}, i = 1, 2,
g(t, x, y) =
1√
2πtbj
[
exp
{
− 1
2t
(
y√
bj
− x√
bi
)2}
− exp
{
− 1
2t
( |y|√
bj
+
|x|√
bi
)2}]
+
+
4(1 + q sign y)
cdr
√
2πt
∫ ∞
0
exp
{
− 2θ
cr
− 1
2t
(
θ +
|x|√
bi
+
|y|√
bj
)2}
dθ
for x ∈ Di ∪ {0}, i = 1, 2 and y ∈ Dj , j = 1, 2; for y = 0, we have
g(t, x, 0) =
c
2
[
√
b1g(t, x, 0−) +
√
b2g(t, x, 0+)].
Proof. The fact that the process (x(t),Mt, Px) is a strong Markov one is a consequence
of Theorem 10.11 from [2]. To prove (9), we define the function gp(t, x, y) for p > 0, t >
0, x ∈ R, y ∈ R from the relation
Exϕ(x̃(t)) exp{−prηt} =
∫
R
ϕ(y)gp(t, x, y)dy
valid for all p > 0, t > 0, x ∈ R, and any continuous bounded function ϕ on R. The
function gp can be found from the equation (see [12])
(10) gp(t, x, y) = g̃(t, x, y) − pr
∫ t
0
g̃(τ, x, 0)gp(t − τ, 0, y)dτ,
where p > 0, t > 0, x ∈ R, y ∈ R (r is a fixed parameter).
We put
Rp(x, y) =
∫ ∞
0
e−ptgp(t, x, y)dt, Gp(x, y) =
∫ ∞
0
e−ptg̃(t, x, y)dt
for p > 0, x ∈ R, and y ∈ R. Then Eq. (10) implies the following equation for the function
Rp
Rp(x, y) = Gp(x, y) − prGp(x, 0)Rp(0, y).
AN EXAMPLE OF A STOCHASTIC DIFFERENTIAL EQUATION 73
It is not difficult to find out the solution of this equation. It can be written down in the
following form (for p > 0, x ∈ Di ∪ {0}, i = 1, 2, and y ∈ Dj, j = 1, 2):
(11)
Rp(x, y) =
1√
2pbj
[
exp
{
−
∣∣∣∣ y√
bj
− x√
bi
∣∣∣∣√2p
}
−
− exp
{
−
( |y|√
bj
+
|x|√
bi
)√
2p
}]
+
+
4(1 + q sign y)
cdr
√
2p
∫ ∞
0
exp
{
− 2θ
cr
−
(
θ +
|x|√
bi
+
|y|√
bj
)√
2p
}
dθ.
If y = 0, we have
(12)
Rp(x, 0) =
c
2
[Rp(x, 0+)
√
b2 + Rp(x, 0−)
√
b1] =
=
2
r
√
2p
∫ ∞
0
exp
{
− 2θ
cr
−
(
θ +
|x|√
bi
)√
2p
}
dθ.
Making use of the following well-known formula∫ ∞
0
(2πt)−1/2 exp
{ − pt − α2
2t
}
dt = (2p)−1/2 exp{−|α|
√
2p}
valid for all real α and p > 0, we can rewrite (11) and (12) in the form
(13) Rp(x, y) =
∫ ∞
0
e−ptg(t, x, y)dt, rRp(x, 0) =
∫ ∞
0
e−pth(t, x)dt.
On the other hand, for p > 0, x ∈ R, and a continuous bounded function ϕ on R, we
have ∫ ∞
0
e−ptExϕ(x(t))dt = Ex
∫ ∞
0
e−ptϕ(x̃(ζt))dt =
= Ex
∫ ∞
0
e−p(t+rηt)ϕ(x̃(t))dt + rEx
∫ ∞
0
e−p(t+rηt)ϕ(x̃(t))dηt =
=
∫ ∞
0
e−ptExϕ(x̃(t))e−prηtdt + r
∫ ∞
0
e−ptExϕ(x̃(t))δ(x̃(t))e−prηtdt.
Taking into account the definition of Rp, we get∫ ∞
0
e−ptExϕ(x(t))dt =
∫
R
ϕ(y)Rp(x, y)dy + rϕ(0)Rp(x, 0).
This equality and (13) imply (9). The proposition is proved.
Denote, by P (t, x, Γ) for t > 0, x ∈ R, and Γ ∈ B(R), the transition probability of the
process (x(t),Mt, Px). Then we have the equality
P (t, x, Γ) = h(t, x)1IΓ(0) +
∫
Γ
g(t, x, y)dy.
Remark 3. It is not difficult to see that the relations
lim
r↓0
h(t, x) = 0, lim
r↓0
g(t, x, y) = g̃(t, x, y)
are held for all t > 0, x ∈ R, and y ∈ R. On the other hand, we have the relations
lim
r↑+∞
h(t, x) =
2√
2πt
∫ ∞
|x|/√bi
exp
{
− θ2
2t
}
dθ, x ∈ Di ∪ {0}, i = 1, 2;
lim
r↑+∞
g(t, x, y) =
74 BOGDAN I. KOPYTKO AND MYKOLA I. PORTENKO
=
1√
2πtbj
[
exp
{
− 1
2t
(
y√
bj
− x√
bi
)2}
− exp
{
− 1
2t
( |y|√
bj
+
|x|√
bi
)2}]
for t > 0, x ∈ Di ∪ {0}, i = 1, 2, and y ∈ Dj , j = 1, 2. These relations mean that, for
r = +∞, the part of the process x(·) in Di ∪ {0}, i = 1, 2, coincides with the Brownian
motion there (that is, with the process
√
biw(·), where w(·) is a standard Wiener process)
with the absorbing point x = 0.
Let us now fix a finite r > 0 and take note of the following properties of the corre-
sponding process x(·) :
A) P0({x(t) ∈ Di}) = qi(1 − h(t, 0)), t > 0, i = 1, 2.
B) Ex
∫ t
0
1I{0}(x(s))ds =
∫ t
0
h(s, x)ds, t ≥ 0, x ∈ R.
C) The process (x(t),Mt, Px) is a classical diffusion process with the drift coefficient
a(x) = q
r1I{0}(x) and the diffusion coefficient b(x) =
∑2
i=1 bi1IDi(x). This assertion follows
from the relations:
(i) limt↓0 t−1P (t, x, (x − ε, x + ε)c) = 0
for all ε > 0 and x ∈ R, where (x − ε, x + ε)c = R \ (x − ε, x + ε);
(ii) limt↓0 t−1
∫
R
(y − x)P (t, x, dy) = a(x)
for all x ∈ R;
(iii) limt↓0 t−1
∫
R
(y − x)2P (t, x, dy) = b(x)
for all x ∈ R.
Remark 4. Property C) shows that the process (x(t),Mt, Px) is an example of the dif-
fusion process, whose local characteristics (that is, the coefficients a(·) and b(·)) do not
determine its transition probability uniquely (see [11]).
We now give a martingale characterization of the process (x(t),Mt, Px). For t ≥ 0,
we put
ξ(t) = x(t) − x(0) − q̃
r
∫ t
0
1I{0}(x(s))ds.
Proposition 2. The process (ξ(t))t≥0 is a square integrable martingale with respect to
(Mt, Px) for every x ∈ R, and its square characteristic is determined by formula (2).
Proof. It is sufficient to verify that the equalities
(14) Exξ(t) = 0, Exξ(t)2 = Ex
∫ t
0
b(x(s))ds
hold true for all x ∈ R and t ≥ 0.
The first relation in (14) is equivalent to the following one:∫
R
yg(t, x, y)dy = x +
q̃
r
∫ t
0
h(s, x)ds.
Its validity is established by taking the Laplace transformation. To prove the second
relation in (14), we notice that
(15) Exξ(t)2 = Ex(x(t) − x(0))2 +
2q̃x
r
∫ t
0
h(s, x)ds,
since
Ex(x(t) − x(0))
∫ t
0
1I{0}(x(s))ds = −x
∫ t
0
h(s, x)ds+
+
q̃
2r
Ex
( ∫ t
0
1I{0}(x(s))ds
)2
.
AN EXAMPLE OF A STOCHASTIC DIFFERENTIAL EQUATION 75
Taking into account the relation
Ex(x(t) − x(0))2 = Exx(t)2 − x2 − 2q̃x
r
∫ t
0
h(s, x)ds,
we get
Exξ(t)2 = Exx(t)2 − x2
from (15). Noticing that
Exx(t)2 =
∫
R
y2P (t, x, dy) =
∫
R
y2g(t, x, y)dy,
we arrive at the conclusion that the second relation in (14) is equivalent to the equality∫
R
y2g(t, x, y)dy − x2 =
∫ t
0
ds
∫
R
b(y)g(s, x, y)dy.
The validity of this equality can be verified by taking the Laplace transformation. The
proposition is proved.
As a consequence of Proposition 2, we have that the paths of the process (x(t),Mt, Px)
are the solution to the stochastic differential equation (1) with A = q̃/r. As mentioned
in Introduction, there are infinitely many possibilities to choose q ∈ [−1, 1] and r > 0 in
such a way that the ratio q̃/r is equal to a fixed constant A. It is obvious that A = 0 for
q = q0 (q̃ = 0 for q = q0), A > 0 for q ∈ (q0, 1], and A < 0 for q ∈ [−1, q0).
In particular, if A > 0, we can put q = +1 (q̃ = 2−1
√
b2(
√
b1 +
√
b2) in this case)
and r = (2A)−1
√
b2(
√
b1 +
√
b2). One can observe that g(t, x, y) = 0 for y < 0 in this
case, so, if x(0) ≥ 0, then x(t) ≥ 0 for all t ≥ 0. If, additionaly, b1 = b2 = 1, then the
corresponding process is called a slowly reflecting Brownian motion; it was investigated
in [1].
Remark 5. For t > 0, x ∈ R, and a continuous bounded function ϕ on R, we put
u(t, x, ϕ) = Exϕ(x(t)).
This function satisfies the equation
∂u
∂t
=
1
2
bi
∂2u
∂x2
in the region t > 0, x ∈ Di for i = 1, 2, and the initial condition
lim
t↓0
u(t, x, ϕ) = ϕ(x), x ∈ R.
In addition, the relations u(t, 0+, ϕ) = u(t, 0−, ϕ) = u(t, 0, ϕ) and
∂u(t, 0, ϕ)
∂t
=
1
cr
[
q2
√
b2
∂u(t, 0+, ϕ)
∂x
− q1
√
b1
∂u(t, 0−, ϕ)
∂x
]
are valid for t > 0.
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E-mail : portenko@imath.kiev.ua
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