One example of a random change of time that transforms a generalized diffusion process into an ordinary one
We propose a random change of time for a class of generalized diffusion processes such that the corresponding stochastic differential equation (with generalized coefficients) is transformed into an ordinary one (its coefficients are some non-generalized functions). It turns out that the latter stoc...
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| Цитувати: | One example of a random change of time that transforms a generalized diffusion process into an ordinary one / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 12–21. — Бібліогр.: 5 назв.— англ. |
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irk-123456789-45022009-11-20T12:00:39Z One example of a random change of time that transforms a generalized diffusion process into an ordinary one Aryasova, O.V. Portenko, M.I. We propose a random change of time for a class of generalized diffusion processes such that the corresponding stochastic differential equation (with generalized coefficients) is transformed into an ordinary one (its coefficients are some non-generalized functions). It turns out that the latter stochastic differential equation has no property of the (weak) uniqueness of a solution. 2007 Article One example of a random change of time that transforms a generalized diffusion process into an ordinary one / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 12–21. — Бібліогр.: 5 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4502 519.21 en Інститут математики НАН України |
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We propose a random change of time for a class of generalized diffusion processes such
that the corresponding stochastic differential equation (with generalized coefficients) is transformed into an ordinary one (its coefficients are some non-generalized functions). It turns out that the latter stochastic differential equation has no property of the (weak) uniqueness of a solution. |
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Aryasova, O.V. Portenko, M.I. |
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Aryasova, O.V. Portenko, M.I. One example of a random change of time that transforms a generalized diffusion process into an ordinary one |
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Aryasova, O.V. Portenko, M.I. |
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Aryasova, O.V. |
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One example of a random change of time that transforms a generalized diffusion process into an ordinary one |
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One example of a random change of time that transforms a generalized diffusion process into an ordinary one |
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One example of a random change of time that transforms a generalized diffusion process into an ordinary one |
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One example of a random change of time that transforms a generalized diffusion process into an ordinary one |
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One example of a random change of time that transforms a generalized diffusion process into an ordinary one |
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one example of a random change of time that transforms a generalized diffusion process into an ordinary one |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4502 |
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One example of a random change of time that transforms a generalized diffusion process into an ordinary one / O.V. Aryasova, M.I. Portenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 12–21. — Бібліогр.: 5 назв.— англ. |
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AT aryasovaov oneexampleofarandomchangeoftimethattransformsageneralizeddiffusionprocessintoanordinaryone AT portenkomi oneexampleofarandomchangeoftimethattransformsageneralizeddiffusionprocessintoanordinaryone |
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2025-07-02T07:43:54Z |
| last_indexed |
2025-07-02T07:43:54Z |
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1836520292879433728 |
| fulltext |
Theory of Stochastic Processes
Vol. 13 (29), no. 3, 2007, pp. 12–21
UDC 519.21
OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
ONE EXAMPLE OF A RANDOM CHANGE OF
TIME THAT TRANSFORMS A GENERALIZED
DIFFUSION PROCESS INTO AN ORDINARY ONE
We propose a random change of time for a class of generalized diffusion processes such
that the corresponding stochastic differential equation (with generalized coefficients)
is transformed into an ordinary one (its coefficients are some non-generalized func-
tions). It turns out that the latter stochastic differential equation has no property of
the (weak) uniqueness of a solution.
Introduction
Let S denote a closed bounded surface in a d-dimensional Euclidean space Rd such
that it divides the space Rd into two open parts: interior Di and exterior De, so that
Rd = Di ∪De ∪ S; we put D = Di ∪De. The surface S is assumed to be smooth enough
(see below for the precise assumptions), so that there exists a normal vector ν(x) to S
at any point x ∈ S; it will be considered as a unit vector directed into De.
Suppose that an operator-valued function b(x), x ∈ Rd, is given such that for any
x ∈ Rd, the operator b(x) is a symmetric positively definite linear operator on Rd.
The positive square root of b(x) will be denoted by b(x)1/2. For x ∈ S, the vector
N(x) = b(x)ν(x) is called the co-normal vector to S at the point x.
A continuous function q(x), x ∈ S, with its values being in the interval [−1, 1] of a real
line will be supposed to be given.
Under some assumptions on the function (b(x))x∈Rd (see below), a continuous Markov
process (x0(t))t≥0 in Rd exists such that its trajectories satisfy the stochastic differential
equation (see [4], Ch. 3)
(1) dx0(t) = q(x0(t))δS(x0(t))N(x0(t))dt + b(x0(t))1/2dw(t),
where (w(t))t≥0 is a standard Wiener process in Rd and (δS(x))x∈Rd is a generalized
function on Rd that acts on a test function (ϕ(x))x∈Rd according to the rule
(2) 〈δS , ϕ〉 =
∫
S
ϕ(x)dσ
(the integral in this equality is a surface integral).
Let a continuous bounded function r(·) : S → (0, +∞) be now given. For t ≥ 0, we
put
x(t) = x0(ζt),
where
(3) ζt = inf
{
s : s +
∫ s
0
r(x0(τ))δS(x0(τ))dτ ≥ t
}
,
2000 AMS Mathematics Subject Classification. Primary 60J60, 60J35.
Key words and phrases. Diffusion process, random change of time, stochastic differential equation,
uniqueness of solution.
12
ONE EXAMPLE OF A RANDOM CHANGE 13
the functional
(4) ηt =
∫ t
0
r(x0(τ))δS(x0(τ))dτ, t ≥ 0,
of the process (x0(t))t≥0 is well defined as an additive homogeneous continuous functional
(see [4], Ch.3). As is known (see [2], Theorem 10.11), the process (x(t))t≥0 is a continuous
Markov process in Rd as a result of the random change of time for the process (x0(t))t≥0.
The aim of this article is to show that the trajectories of the process (x(t))t≥0 satisfy
the stochastic differential equation
(5) dx(t) =
q(x(t))
r(x(t))
1IS(x(t))N(x(t))dt + 1ID(x(t))b(x(t))1/2dw(t),
where 1IΓ(x), x ∈ Rd, stands for the indicator function of a set Γ ⊂ Rd.
As a consequence of this result, we obtain the fact that Eq. (5) has no property of th
(weak) uniqueness of a solution, since the representation of a given function A(x), x ∈ S,
in the form A(x) = q(x)
r(x) does not determine uniquely any pair of functions q(·) and r(·).
The situation of b(x) ≡ I (I is an identical operator in Rd) and S being a hyperplane
in Rd was considered in [1]. The reader can find there some discussion about the place of
the result in the modern context of the theory of stochastic differential equations. The
main point that draws a distinction between the situation of this article and the one of
[1] is that the transition probability density of the process (x0(t))t≥0 in [1] is given by
an explicit formula, whereas here, on the contrary, we have only some equations for the
corresponding density. This makes the computation of the local characteristics of the
process (x(t))t≥0 more complicated.
Nevertheless, as in [1], there is a very simple and intuitively evident way to explain
why, under the time changing (3), the coefficients of Eq. (1) are transformed into the
ones of Eq. (5). Namely, according to some very general result (see [2], Ch. 10), in order
to obtain the diffusion coefficients of the proccess (x(t))t≥0, we have to divide the ones
of the proccess (x0(t))t≥0 by the function 1 + r(x)δS(x), x ∈ Rd. So, for the coefficient
of dt, we get
(6)
q(x)δS(x)N(x)
1 + r(x)δS(x)
=
q(x)
r(x)
1IS(x)N(x), x ∈ R
d,
and, for the one of dw(t), we have
(7)
b(x)1/2
(1 + r(x)δS(x))1/2
= 1ID(x)b(x)1/2, x ∈ R
d.
Of course, formulae (6) and (7) cannot serve as a proof of our main result. Some
arguments for proving it can be found below in Section 2. Section 1 is devoted to a brief
description of the process (x0(t))t≥0 constructed in [4]. Moreover, some new properties
of this process (as compared with [4]) are established in Section 1. These properties are
then used in the considerations of Section 2.
1. Some properties of the process (x0(t))t≥0
1.1. A brief description of the process. Fix an orthonormal basis in Rd and denote,
by xj for j = 1, 2, . . . , d, the coordinates of a vector x ∈ Rd and, by bij(x) for i =
1, 2, . . . , d and j = 1, 2, . . . , d, the elements of the matrix of the operator b(x) in that
basis. We assume that the functions bij(·) for all i = 1, 2, . . . , d and j = 1, 2, . . . , d are
bounded and Hölder continuous, that is, the inequality
(8) |bij(x) − bij(y)| ≤ C|x − y|α,
14 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
holds true for all x ∈ Rd and y ∈ Rd with some constants C > 0 and α ∈ (0, 1]. In
addition, the condition of uniform non-degeneracy
(9) inf
x∈Rd
min
θ∈Rd:|θ|=1
(b(x)θ, θ) > 0
will be assumed to be held. Under these assumptions, there exists the fundamental
solution of the equation (see [3], Ch.I)
∂u
∂t
=
1
2
d∑
i,j=1
bij(x)
∂2u
∂xi∂xj
.
Denote it by g0(t, x, y), t > 0, x ∈ Rd, and y ∈ Rd. This function is the transition
probability density of a stochastic process (ξ0(t))t≥0 that is the solution to the following
stochastic differential equation:
dξ0(t) = b(ξ0(t))1/2dw(t).
This means that the equality
Exϕ(ξ0(t)) =
∫
Rd
g0(t, x, y)ϕ(y)dy
is valid for all t > 0, x ∈ Rd, and ϕ ∈ B, where B stands for the Banach space of all real-
valued bounded measurable functions ϕ(x), x ∈ Rd, with the norm ‖ϕ‖ = supx∈Rd |ϕ(x)|
(the subspace of this space consisting of all continuous functions is itself a Banach space
denoted by C).
As for the surface S, we will assume the following conditions to be fulfilled (see [4],
Ch.3, § 3):
(i) S belongs to the class H1+κ for some κ ∈ (0, 1);
(ii) each point of S possesses the property of sphericity from the side of both the
interior and exterior domains.
Let a continuous function q(·) : S → [−1, 1] be now given. For ϕ ∈ B, we consider the
integral equation
(10) V (t, x, ϕ) =
∫
Rd
∂g0(t, x, z)
∂N(x)
ϕ(z)dz +
∫ t
0
dτ
∫
S
∂g0(τ, x, z)
∂N(x)
V (t − τ, z, ϕ)q(z)dσz
in the domain t > 0 and x ∈ S. As proved in [4] (see Ch.3, § 3), this equation has the
unique solution that is continuous in the arguments t > 0 and x ∈ S and satisfies the
inequality
(11) |V (t, x, ϕ)| ≤ KT ‖ϕ‖t−1/2
in any domain of the form (0, T ]× S with some positive constant KT .
For t > 0, x ∈ Rd, and ϕ ∈ B, we put
(12) u(t, x, ϕ) =
∫
Rd
g0(t, x, z)ϕ(z)dz +
∫ t
0
dτ
∫
S
g0(τ, x, z)V (t − τ, z, ϕ)q(z)dσz .
It turns out that a continuous Markov process (x0(t))t≥0 in Rd exists such that
(13) Exϕ(x0(t)) = u(t, x, ϕ)
for all t > 0, x ∈ Rd, and ϕ ∈ B. Moreover, the trajectories of this process satisfy the
stochastic differential equation (1). All these assertions are proved in [4] (see Ch.3, §§
3–4).
ONE EXAMPLE OF A RANDOM CHANGE 15
1.2. Transition probability density of the process. We now show that the process
(x0(t))t≥0 possesses a transition probability density. We denote it by G0(t, x, y), t > 0,
x ∈ Rd, and y ∈ Rd. This means that the function u(t, x, ϕ) given by (12) can be written
in the form
(14) u(t, x, ϕ) =
∫
Rd
G0(t, x, y)ϕ(y)dy
for all t > 0, x ∈ Rd, and ϕ ∈ B.
For an arbitrary y ∈ Rd, we formally put ϕ(z) = δy(z), z ∈ Rd, in (12) and (13), where
δy(·) is Dirac’s δ-function concentrated at the point y (this means that 〈δy, ψ〉 = ψ(y)
for any test function ψ on Rd). We then arrive at the relation
(15) G0(t, x, y) = g0(t, x, y) +
∫ t
0
dτ
∫
S
g0(τ, x, z)V (t − τ, z, y)q(z)dσz,
where the notation V (t, x, y) for V (t, x, δy), t > 0, x ∈ S, and y ∈ R
d, is used. Equation
(10) now implies the integral equation
(16) V (t, x, y) =
∂g0(t, x, y)
∂N(x)
+
∫ t
0
dτ
∫
S
∂g0(τ, x, z)
∂N(x)
V (t − τ, z, y)q(z)dσz
that must be held for t > 0, x ∈ S, and y ∈ Rd.
We will prove that representation (15), (16) for the function G0 holds indeed true.
Moreover, one more representation formula for G0 will be given. Namely, we show that
(17) G0(t, x, y) = g0(t, x, y) +
∫ t
0
dτ
∫
S
Ṽ (τ, x, z)
∂g0(t − τ, z, y)
∂N(z)
q(z)dσz
for t > 0, x ∈ Rd, and y ∈ Rd, where Ṽ is the solution to the integral equation
(18) Ṽ (t, x, y) = g0(t, x, y) +
∫ t
0
dτ
∫
S
Ṽ (τ, x, z)
∂g0(t − τ, z, y)
∂N(z)
q(z)dσz,
where t > 0, x ∈ Rd, and y ∈ S.
To prove these statements, consider the function ∂g0(t,x,y)
∂N(x) that is defined for t > 0, x ∈
S, and y ∈ Rd. Its restriction on the set (0, +∞) × S × S will be denoted by Q(t, x, y).
We now put Q(1) = Q and, for k ≥ 1,
Q(k+1)(t, x, y) =
∫ t
0
dτ
∫
S
Q(k)(τ, x, z)Q(t − τ, z, y)q(z)dσz =
=
∫ t
0
dτ
∫
S
Q(τ, x, z)Q(k)(t − τ, z, y)q(z)dσz,
where t > 0, x ∈ S, and y ∈ S.
Proposition 1. The series
R(t, x, y) =
∞∑
k=1
Q(k)(t, x, y)
is convergent uniformly in x ∈ S and y ∈ S and locally uniformly in t > 0. The kernel R
is continuous in the arguments t > 0, x ∈ S, y ∈ S, x
= y and satisfies the inequality
(19) |R(t, x, y)| ≤ K̃T
tβ1 |y − x|β2
16 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
in any domain of the form (0, T ] × S × S with some constants β1 ∈ (0, 1) and β2 ∈
(0, d − 1). In addition, this kernel is the solution to each one of the following pair of
integral equations (t > 0, x ∈ S, y ∈ S)
R(t, x, y) = Q(t, x, y) +
∫ t
0
dτ
∫
S
R(τ, x, z)Q(t − τ, z, y)q(z)dσz,
(20)
R(t, x, y) = Q(t, x, y) +
∫ t
0
dτ
∫
S
Q(τ, x, z)R(t − τ, z, y)q(z)dσz.
Finally, each equation in (20) has no more than one solution satisfying estimate (19).
Proof. Denote, by δ, the minimal one of the numbers α from (8) and κ mentioned in (i)
above. It is clear that δ is a positive number. As proved in [3] (see Ch. V, § 2), for any
β ∈ (
1 − δ
2 , 1
)
, the kernel Q satisfies the inequality
|Q(t, x, y)| ≤ const
tβ |y − x|d+1−δ−2β
in any domain of the form (0, T ]×S ×S. We now put ρ = d + 1− δ− 2β, γ = 2β + δ − 2
and σ = 1− β; then γ > 0 and σ > 0. Making use of Lemma 2 from [3] (see Ch. V, § 2),
one can obtain the estimate
|Q(2)(t, x, y)| ≤ const
tβ−σ|y − x|ρ−γ
valid for (t, x, y) ∈ (0, T ]× S × S. In the same way, we get the estimate
|Q(3)(t, x, y)| ≤ const
tβ−2σ|y − x|ρ−2γ
valid for (t, x, y) ∈ (0, T ] × S × S. Therefore, an integer k0 exists such that
|Q(k0)(t, x, y)| ≤ const in any domain of the form (0, T ] × S × S for T < +∞. By
induction on k, we then obtain the inequality
|Q(k0+n)(t, x, y)| ≤ Cn
T
tn−β
(1 − β)(2 − β) . . . (n − β)
fulfilled for (t, x, y) ∈ [0, T ] × S × S and n = 1, 2, . . . with some constant CT < ∞ for
T < ∞. This inequality guarantees the convergence of the series mentioned above. The
remaining assertions of Proposition 1 are elementary.
Corollary. The solution to Eq. (16) can be given by
(21) V (t, x, y) =
∂g0(t, x, y)
∂N(x)
+
∫ t
0
dτ
∫
S
R(τ, x, z)
∂g0(t − τ, z, y)
∂N(z)
q(z)dσz
for t > 0, x ∈ S, and y ∈ Rd, and the solution to Eq. (18) can be written as
(22) Ṽ (t, x, y) = g0(t, x, y) +
∫ t
0
dτ
∫
S
g0(τ, x, z)R(t − τ, z, y)q(z)dσz
for t > 0, x ∈ Rd, and y ∈ S.
It is now not difficult to verify that the result of substituting (21) into (15) coincides
with that of substituting (22) into (17). So, the right-hand sides of (15) and (17) define
the same function G0. The fact that it is the transition probability density of the process
(x0(t))t≥0 follows now from the comparison of (10) and (12) with (16) and (15). We have
thus proved that there exists the transition probability density for the process (x0(t))t≥0
and it can be represented by (15) and (17).
ONE EXAMPLE OF A RANDOM CHANGE 17
Remark 1. Applying the well-known theorem on the jump of the co-normal derivative of
a single-layer potential (see, for example, [4], Ch. 3, § 3) to (15), we get the relation
(23)
∂G0(t, x±, y)
∂N(x)
= (1 ∓ q(x))V (t, x, y)
valid for t > 0, x ∈ S and y ∈ Rd, where ∂G0(t,x+,y)
∂N(x) means the limit of ∂G0(t,z,y)
∂N(x) as z → x
in such a non-tangent way that (z, ν(x)) > 0; the value of ∂G0(t,x−,y)
∂N(x) is determined in
the same way, but (z, ν(x)) < 0 must be held this time. By analogy, (17) and (18) yeild
the relation
(24) G0(t, x, y±) = (1 ± q(y))Ṽ (t, x, y)
fulfilled for t > 0, x ∈ Rd, and y ∈ S.
Remark 2. Formulae (20) – (24) and the theorem mentioned in Remark 1 imply the
relations
a) R(t, x, y) = 1
2 [V (t, x, y+) + V (t, x, y−)] =
=
1
2
[
∂Ṽ (t, x+, y)
∂N(x)
+
∂Ṽ (t, x−, y)
∂N(x)
]
;
b) R(s + t, x, y) =
∫
Rd
V (s, x, z)Ṽ (t, z, y)dz
valid for s > 0, t > 0, x ∈ S, and y ∈ S.
Remark 3. The function δS in formulae (1) and (4) possesses the property of symmetry
in the sense
(25) 〈δS , ψ〉 =
1
2
∫
S
[ψ(x+) + ψ(x−)]dσ
for any function ψ having, at the points of S, the non-tangent limits from the sides of De
and Di such that they are integrable. Notice that (25) is a natural generalization of (2).
This statement follows from the definition of the functional (ηt)t≥0 (see [4], Ch. 3,
§ 4), according to which
ηt = l.i.m.
h→0
∫ t
0
h−1fh(x0(τ))dτ, t ≥ 0,
where
ft(x) =
∫ t
0
dτ
∫
S
Ṽ (τ, x, y)r(y)dσy , t ≥ 0, x ∈ R
d.
One can easily verify that
lim
h→0
1
h
∫
Rd
fh(x)ψ(x)dx =
1
2
∫
S
[ψ(x+) + ψ(x−)]r(x)dσ
for any test function ψ having the same properties as those in (25).
Remark 4. Equation (1) is analytically equivalent to the two relations (see [4], Ch. 3, §
4)
(26)
∫
Rd
(y, θ)G0(t, x, y)dy = (x, θ) +
∫ t
0
dτ
∫
S
(N(y), θ)q(y)Ṽ (τ, x, y)dσy ,
18 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
∫
Rd
(y, θ)2G0(t, x, y)dy = (x, θ)2 +
∫ t
0
dτ
∫
Rd
G0(τ, x, y)(b(y)θ, θ)dy+
+ 2
∫ t
0
dτ
∫
S
(N(z), θ)(z, θ)Ṽ (τ, x, z)q(z)dσz(27)
valid for t > 0, x ∈ Rd, and θ ∈ Rd. Indeed, equality (26) means that the process
ξ̃0(t) = x0(t) − x0(0) −
∫ t
0
q(x0(τ))δS(x0(τ))N(x0(τ))dτ, t ≥ 0,
is a martingale. Equality (27) implies that its square characteristic is given by (see [4],
Ch. 3, § 4)
〈ξ̃0〉t =
∫ t
0
b(x0(τ))dτ, t ≥ 0.
1.3. The Feynman–Kac formula. For λ ≥ 0, we define a function Gλ of the argu-
ments t > 0, x ∈ Rd, and y ∈ Rd by the relation
(28) Ex(ϕ(x0(t)) exp{−ληt}) =
∫
Rd
ϕ(y)Gλ(t, x, y)dy
that must be fulfilled for all t > 0, x ∈ Rd, and ϕ ∈ B.
Proposition 2. The function Gλ is determined uniquely by (28) and it satisfies each
one of the following pair of equations (in the domain t > 0, x ∈ Rd, and y ∈ Rd):
(29) Gλ(t, x, y) = G0(t, x, y) − λ
∫ t
0
dτ
∫
S
Ṽ (τ, x, z)Gλ(t − τ, z, y)r(z)dσz ,
(30) 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 .
Proof of this assertion is quite similar to that given in [5] for the case of b(x) ≡ I and
S being a hyperplane in Rd. We omit it here.
Notice that Gλ as a function of the third argument has a jump at the points of S (at
those of them, for which the function q does not vanish). Namely, we have
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
from (30) for y ∈ S. In this equality (and in (30) too), we have
Gλ(τ, x, z) =
1
2
[Gλ(τ, x, z+) + Gλ(τ, x, z−)], z ∈ S,
because of the property of symmetry of δS . So, for t > 0, x ∈ Rd, and y ∈ S, Eqs. (29)
and (30) can be rewritten in the following way:
(31) Gλ(t, x, y) = Ṽ (t, x, y) − λ
∫ t
0
dτ
∫
S
Ṽ (τ, x, z)Gλ(t − τ, z, y)r(z)dσz ,
(32) Gλ(t, x, y) = Ṽ (t, x, y) − λ
∫ t
0
dτ
∫
S
Gλ(τ, x, z)Ṽ (t − τ, z, y)r(z)dσz .
We now introduce the kernels
Gλ(x, y) =
∫ ∞
0
e−λtGλ(t, x, y)dt, λ > 0, x ∈ R
d, y ∈ R
d;
ONE EXAMPLE OF A RANDOM CHANGE 19
G̃0(λ, x, y) =
∫ ∞
0
e−λtG0(t, x, y)dt, λ > 0, x ∈ R
d, y ∈ R
d;
Ṽλ(x, y) =
∫ ∞
0
e−λtṼ (t, x, y)dt, λ > 0, x ∈ R
d, y ∈ S.
Then equalities (29) and (30) yield the relations
(33) Gλ(x, y) = G̃0(λ, x, y) − λ
∫
S
Ṽλ(x, z)Gλ(z, y)r(z)dσz,
(34) Gλ(x, y) = G̃0(λ, x, y) − λ
∫
S
Gλ(x, z)G̃0(λ, z, y)r(z)dσz
valid for λ > 0, x ∈ R
d, and y ∈ R
d. If y ∈ S, we have
(35) Gλ(x, y) = Ṽλ(x, y) − λ
∫
S
Gλ(x, z)Ṽλ(z, y)r(z)dσz ,
(36) Gλ(x, y) = Ṽλ(x, y) − λ
∫
S
Ṽλ(x, z)Gλ(z, y)r(z)dσz .
2. A martingale characterization of the process (x(t))t≥0
2.1. The main formula. Supposing that ϕ ∈ C and taking into account (3), we obtain
Ex
∫ ∞
0
e−λtϕ(x(t))dt = Ex
∫ ∞
0
e−λtϕ(x0(ζt))dt =
= Ex
∫ ∞
0
e−λt−ληtϕ(x0(t))dt + Ex
∫ ∞
0
e−λt−ληtϕ(x0(t))r(x0(t))δS(x0(t))dt =
=
∫
Rd
Gλ(x, y)ϕ(y)dy +
∫
S
Gλ(x, y)r(y)ϕ(y)dσy .
By the standard reason, this equality can be extended on all functions ϕ ∈ B. We have
just proved the following statement.
Proposition 3. The formula
Ex
∫ ∞
0
e−λtϕ(x(t))dt =
∫
Rd
Gλ(x, y)ϕ(y)dy+
+
∫
S
Gλ(x, y)r(y)ϕ(y)dσy(37)
holds true for all λ > 0, x ∈ Rd, and ϕ ∈ B.
It is clear that this formula remains to be true for some unbounded measurable func-
tions ϕ.
2.2. Calculating the local characteristics of the process. Equality (26) can be
rewritten as∫
Rd
(y, θ)G̃0(λ, x, y)dy =
1
λ
(x, θ) +
1
λ
∫
S
(N(y), θ)q(y)Ṽλ(x, y)dσy ,
where λ > 0, x ∈ Rd, and θ ∈ Rd. Using this equality and (34), we get∫
Rd
(y, θ)Gλ(x, y)dy =
1
λ
(x, θ) +
1
λ
∫
S
Ṽλ(x, y)(N(y), θ)q(y)dσy−
−
∫
S
Gλ(x, z)(z, θ)r(z)dσz−
20 OLGA V. ARYASOVA AND MYKOLA I. PORTENKO
−
∫
S
Gλ(x, z)r(z)
[ ∫
S
(N(y), θ)q(y)Ṽλ(z, y)dσy
]
dσz .
This equality and (35) together with (37) yield
Ex
∫ ∞
0
e−λt(x(t), θ)dt =
∫
Rd
Gλ(x, y)(y, θ)dy +
∫
S
Gλ(x, y)r(y)(y, θ)dσy =
=
1
λ
(x, θ) +
1
λ
∫
S
Ṽλ(x, y)(N(y), θ)q(y)dσy−
− 1
λ
∫
S
(N(y), θ)q(y)[Ṽλ(x, y) − Gλ(x, y)]dσy =
=
1
λ
(x, θ) +
1
λ
∫
S
(N(y), θ)q(y)Gλ(x, y)dσy =
=
1
λ
(x, θ) +
1
λ
∫
S
(N(y), θ)
q(y)
r(y)
Gλ(x, y)r(y)dσy =
=
1
λ
(x, θ) + Ex
∫ ∞
0
e−λtdt
∫ t
0
(N(x(τ)), θ)
q(x(τ))
r(x(τ))
1IS(x(τ))dτ.
This means that the process
ξ(t) = x(t) − x(0) −
∫ t
0
q(x(τ))
r(x(τ))
1IS(x(τ))N(x(τ))dτ, t ≥ 0,
is a martingale. It remains to prove that the equality
(38) Ex(ξ(t), θ)2 = Ex
∫ t
0
1ID(x(τ))(b(x(τ))θ, θ)dτ
holds true for all t > 0, x ∈ Rd, and θ ∈ Rd.
From (27) and (34), we get∫
Rd
Gλ(x, y)(y, θ)2dy =
1
λ
(x, θ)2 +
1
λ
∫
Rd
G̃0(λ, x, y)(b(y)θ, θ)dy+
+
2
λ
∫
S
(z, θ)(N(z), θ)Ṽλ(x, z)q(z)dσz−
−λ
∫
S
Gλ(x, z)r(z)
[
1
λ
(z, θ)2 +
1
λ
∫
Rd
G̃0(λ, z, y)(b(y)θ, θ)dy+
+
2
λ
∫
S
Ṽλ(z, y)(y, θ)(N(y), θ)q(y)dσy
]
dσz .
Making use of this relation, (34), and (35), we obtain∫
Rd
Gλ(x, y)(y, θ)2dy +
∫
Rd
Gλ(x, z)r(z)(z, θ)2dσz =
(39) =
1
λ
(x, θ)2 +
2
λ
∫
S
Gλ(x, y)(y, θ)(N(y), θ)q(y)dσy +
1
λ
∫
Rd
Gλ(x, y)(b(y)θ, θ)dy.
We now taking into account the relation
Ex(ξ(t), θ)2 = Ex(x(t), θ)2 − (x, θ)2−
−2Ex
∫ t
0
(x(τ), θ)(N(x(τ)), θ)
q(x(τ ))
r(x(τ ))
1IS(x(τ))dτ
ONE EXAMPLE OF A RANDOM CHANGE 21
that is a simple consequence of the definition of the process (ξ(t))t≥0. This equality and
(39) together with the main formula allow us to arrive at the equality
Ex
∫ ∞
0
e−λt(ξ(t), θ)2dt =
1
λ
∫
Rd
Gλ(x, y)(b(y)θ, θ)dy =
=
1
λ
∫
Rd
Gλ(x, y)(b(y)θ, θ)1ID(y)dy,
and it can be rewritten as
Ex
∫ ∞
0
e−λt(ξ(t), θ)2dt = Ex
∫ ∞
0
e−λtdt
∫ t
0
(b(x(τ))θ, θ)1ID(x(τ))dτ.
This relation is equivalent to (38), and the proof of our main result has been completed.
Bibliography
1. O.V. Aryasova and M.I. Portenko, One class of multidimensional stochastic differential equa-
tions having no property of weak uniqueness of a solution, Theory Stochast. Process. 11(27)
(2005), 14–28.
2. E.B. Dynkin, Markov Processes, Fizmatgiz, Moscow, 1963.; English transl., vol. I, II, New York:
Acad.Press., Berlin: Springer, 1965.
3. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice–Hall, Englewood Cliffs,
N.J. (1964).
4. N.I. Portenko, Generalized Diffusion Processes, Naukova Dumka, Kyiv, 1982; English transl.,
Amer. Math. Soc., Providence, RI, 1990.
5. N.I. Portenko, On multidimensional skew Brownian motion and the Feynman–Kac formula,
Theory Stochast. Process. 4 (20) (1998), 60–70.
E-mail : oaryasova@mail.ru
E-mail : portenko@imath.Kyiv.ua
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