The limit stochastic equation changes type
We study the weak convergence of solutions of the Itˆo stochastic equation, whose coeffcients depend on a small parameter. Conditions under which the limit process changes the type and will be a solution of the stochastic equation with a local time are obtained.
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irk-123456789-44312009-11-10T12:00:34Z The limit stochastic equation changes type Makhno, S.Ya. We study the weak convergence of solutions of the Itˆo stochastic equation, whose coeffcients depend on a small parameter. Conditions under which the limit process changes the type and will be a solution of the stochastic equation with a local time are obtained. 2005 Article The limit stochastic equation changes type / S.Ya.Makhno // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 104–109. — Бібліогр.: 11 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4431 519.21 en Інститут математики НАН України |
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We study the weak convergence of solutions of the Itˆo stochastic equation, whose
coeffcients depend on a small parameter. Conditions under which the limit process
changes the type and will be a solution of the stochastic equation with a local time
are obtained. |
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Makhno, S.Ya. |
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Makhno, S.Ya. The limit stochastic equation changes type |
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Makhno, S.Ya. |
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Makhno, S.Ya. |
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The limit stochastic equation changes type |
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The limit stochastic equation changes type |
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The limit stochastic equation changes type |
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The limit stochastic equation changes type |
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The limit stochastic equation changes type |
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limit stochastic equation changes type |
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Інститут математики НАН України |
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2005 |
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The limit stochastic equation changes type / S.Ya.Makhno // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 104–109. — Бібліогр.: 11 назв.— англ. |
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Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 104–109
UDC 519.21
SERGEY YA. MAKHNO
THE LIMIT STOCHASTIC EQUATION CHANGES TYPE
We study the weak convergence of solutions of the Itô stochastic equation, whose
coefficients depend on a small parameter. Conditions under which the limit process
changes the type and will be a solution of the stochastic equation with a local time
are obtained.
We study the solutions of Itô stochastic equations, whose coefficients depend on a
small parameter ε > 0. We investigate their convergence as ε → 0 without assuming
the convergence of the coefficients themselves.Conditions under which these solutions
converge in the weak sense to a solution of the Itô stochastic equation are known [6,7].
In [11], W.Rosenkrants considered the random processes xε(t) as solutions of the Itô
stochastic equations
xε(t) = x +
1
ε
∫ t
0
b
(
xε(s)
ε
)
ds + w(t).
Let we suppose that
∫∞
−∞ b(x)dx �= 0. The results of [11, Theorems 1 and 3] show that
the limit process x(t) does not possess the Itô stochastic integral representation. By Le
Gall [3 , Corollary 3.3], the limit process is a solution of the stochastic equation with a
local time, or it is the skew Brownian motion in the terminology of Walsh. On the other
hand, Portenko [10] considered the class of random processes he called as generalized
diffusion processes. He proved [10, Theorem 3.4] that the random process x(t) described
above is also a generalized diffusion process, and it can be represented in integral form
with the Dirac delta function in the drift coefficient [10, Corollary of Theorem 3.5].
Thus, if we have Itô stochastic equations with coefficients unbounded in the parameter
ε, we may have another type of the equation for the limit processes. For a particular
form of coefficients in the Itô equations, the same situation arises in [4,10], and the limit
process was classified as a generalized diffusion process. In the present paper, we consider
a similar problem in the case where the coefficients are not assumed to be smooth and
depend irregularly on a small parameter. Moreover, the coefficients may not be uniformly
bounded in ε at certain points and tend to infinity as ε → 0 or may not have a limit at all.
Let (Ω, F, Ft, P ) denote the main probability space with filtration Ft, t ∈ [0, T ], R is the
one-dimensional Euclidian space, (w(t), Ft) are one-dimensional Wiener processes, and
(C[0, T ], Ct), t ∈ [0, T ], is the space of continuous functions f(t) on the interval [0, T ]. We
use the following notations: B0(R) is the space of all measurable bounded functions with
compact support on R, and C∞
0 (R) is a subspace of all infinitely differentiated functions
from B0(R). The notations L2([0, T ] × R), L2,loc, and W 1,2
2,loc (the Sobolev space) have
standard sense [5], and || · ||2 is the norm in L2. For the weak convergence in L2,loc, we
use the symbol ⇀. The different positive constants are denoted by �L.
Consider the one-dimensional Itô stochastic equations
(1) ξε(t) = x +
∫ t
0
(bε(ξε(s)) + gε(s, ξε(s)))ds +
∫ t
0
(aε(ξε(s)) + Aε(s, ξε(s)))
1
2 dw(s).
2000 AMS Mathematics Subject Classification. Primary 60H10,60J60.
Key words and phrases. Local time, stochastic equation, limit process.
104
THE LIMIT STOCHASTIC EQUATION CHANGES TYPE 105
The limit process for the processes ξε(t) changes the Itô type and is a solution of the
stochastic equation with a local time
(2) ξ(t) = x + βLξ(t, 0) +
∫ t
0
g(ξ(s))ds +
∫ t
0
σ(ξ(s))dw(s),
where Lξ(t, 0) is the symmetric local time of the process ξ(t) at the level 0. The symmetric
local time for the continuous semimartingale can be defined in the following way. Let
X(t) be a continuous semimartingale with the canonical decomposition X(t) = X(0) +
M(t) + A(t), where M stands for a continuous local martingale, and A is a continuous
process of finite variation. Then its symmetric local time at the level a is given by the
Tanaka formula
LX(t, a) = |X(t) − a| − |X(0) − a| −
∫ t
0
sgn(X(s) − a)dX(s),
where
sgnx =
⎧⎪⎨
⎪⎩
1, for x > 0,
0, for x = 0,
−1, for x < 0.
Under conditions of the theorem proved below, Eq. (2) has unique weak solution by [2,
Theorem 4.35].
Let Du(x) denote the symmetric derivative of the function u(x),
Du(x) = lim
h→0
u(x + h) − u(x − h)
2h
,
and the signed measure nu(dx) on (R,R) is the second derivative of u(x) in the sense of
distributions if, for any χ(x) ∈ C∞
0 ,∫
R
χ′′(x)u(x)dx =
∫
R
χ(x)nu(dx).
For every convex real function u(x), the generalized Itô formula
(3) u(X(t)) = u(X(0)) +
∫ t
0
Du(X(s))dX(s) +
1
2
∫
LX(t, y)nu(dy)
holds. Let
u(x) =
{
u1(x), for x ≤ 0,
u2(x), for x ≥ 0.
Suppose that u1(x) and u2(x) are twice continuously differentiable functions for x ∈ R
such that u1(0) = u2(0). Then
Du(x) =
u
′
2(x) + u
′
1(x)
2
+
u
′
2(x) − u
′
1(x)
2
sgnx.
nu(dx) = (u
′
2(0) − u
′
1(0))δ0(x)dx + Nu(x)dx,
where δ0(x) is the Dirac function at point 0 and
Nu(x) =
u
′′
2 (x) + u
′′
1 (x)
2
+
u
′′
2 (x) − u
′′
1 (x)
2
sgnx.
Let l and �L be constants such that 0 < l ≤ �L < ∞. We say that the couple of functions
(r, a) ∈ L(�L, l), if the functions r(x) and a(x) are measurable functions, and there are
the constants �L, l > 0 such that
|r(x)| + a(x) ≤ �L, a(x) ≥ l.
Let με, μ be the measures on (C[0, T ], Ct) corresponding to the random processes ξε(t)
and ξ(t), respectively. To indicate the weak convergence of measures, we use the symbol
106 SERGEY YA. MAKHNO
=⇒ . With the coefficients of Eq. (1), we connected the functions
F ε(x) = exp
{
−2
∫ x
0
bε(y)
aε(y)
dy
}
, f ε(x) =
∫ x
0
F ε(y)dy.
We introduce the following condition.
Condition (I).
I1. For any ε > 0, the functions bε(x), gε(t, x), aε(x), Aε(t, x) are measurable func-
tions, l ≤ aε(x) ≤ �L, Aε(t, x) ≥ 0, and Eq. (1) has a weak solution.
I2. For any x ∈ R, ∣∣∣∣
∫ x
0
bε(y)
aε(y)
dy
∣∣∣∣≤ �L.
I3. There exist functions rε(x), αε(t), αε(t), and hε(t, x) such that |rε(x)| ≤ �L,
|gε(t, x) − rε(x)| + (|bε(x)| + 1)Aε(t, x) ≤ αε(t) + hε(t, x),
and
lim
ε→0
(
||hε||2 +
∫ T
0
αε(t)dt
)
= 0
I4. There exists
≤ f ε(x) = f(x) =
{
f1(x), for x ≤ 0,
f2(x), for x ≥ 0,
where f1(x) and f2(x) are twice continuously differentiable monotonically in-
creasing functions for x ∈ R such that f1(0) = f2(0).
By φε(x), we denote the function inverse to the function f ε(x). Then
≤ φε(x) = φ(x) =
{
φ1(x), for x ≤ 0,
φ2(x), for x ≥ 0,
where φi(x) is the inverse function for fi(x), i = 1, 2. Set β1 = f
′
1(0), β2 = f
′
2(0).
Theorem. Let condition (I) be satisfied and
i) ≤
∫
R
χ(y)
1
F ε(y)aε(y)
dy =
∫
R
χ(y)a(y)dy for any χ ∈ B0(R)
ii) ≤
∫
R
χ(y)
rε(y)
aε(y)
dy =
∫
R
χ(y)r(y)dy for any χ ∈ B0(R),
iii) the couple of functions (r + Nφ(f), a) ∈ L(�L, l).
Then με =⇒ μ, and
β =
β1 − β2
β1 + β2
, σ(x) =
1√
a(x)
, g(x) =
r(x)
a(x)
+ Nφ(f(x)).
To prove the theorem, we use Theorem 2.1 from [9]. For convenience, we present a
complete formulation of this theorem for d=1.
Lemma (Theorem 2.1 [9]). Let xε(t) and x(t) be solutions of the stochastic equations
xε(t) = xε +
∫ t
0
(γε(s, xε(s)) + Bε(s, xε(s)))ds +
∫ t
0
(qε(s, xε(s)) + Qε(s, xε(s)))
1
2 dw(s)
and
x(t) = x +
∫ t
0
γ(s, x(s))ds +
∫ t
0
q(s, x(s))dw(s).
Suppose that the couple (γε, qε) ∈ L(�L, l) and the functions γε(t, x) and qε(t, x) satisfy
the following conditions (V) and (N).
THE LIMIT STOCHASTIC EQUATION CHANGES TYPE 107
Condition (V): There exists a function V ε(t, x) ∈ W 1,2
2,loc such that
V1) γ̂ε def
= : γε +
1
2
qε ∂2V ε
∂x2
⇀ γ;
V2) lim
ε→0
sup
t∈[0,T ],x∈D
|V ε(t, x)| = 0, for any bounded D ∈ R;
V3) lim
ε→0
∣∣∣∣
∣∣∣∣∂V e
∂t
+ γε ∂V ε
∂x
+ γ̂ε − γ
∣∣∣∣
∣∣∣∣
2,loc
= 0.
Condition (N): There exists a function N ε(t, x) ∈ W 1,2
2,loc such that
N1) q̂e def
= : qε + qε ∂2N ε
∂x2
⇀ q;
N2) lim
ε→0
sup
t∈[0,T ],x∈D
|N ε(t, x)| = 0, for any bounded D ∈ R;
N3) lim
ε→0
∣∣∣∣
∣∣∣∣∂Ne
∂t
+ γε ∂N ε
∂x
+
1
2
(q̂ε − q)
∣∣∣∣
∣∣∣∣
2,loc
= 0.
Let the functions (γ, q) ∈ L(�L, l). In addition, we assume that, for any bounded domain
D ∈ R, the following conditions are satisfied:
V4) lim
ε→0
sup
t∈[0,T ],x∈D
∣∣∣∣∂V ε(t, x)
∂x
∣∣∣∣= 0;
V5) sup
t∈[0,T ],x∈D
∣∣∣∣∂
2V ε(t, x)
∂x2
∣∣∣∣≤ �L;
N4) lim
ε→0
sup
t∈[0,T ],x∈D
∣∣∣∣∂N ε(t, x)
∂x
∣∣∣∣= 0;
N5) sup
t∈[0,T ],x∈D
∣∣∣∣∂
2N ε(t, x)
∂x2
∣∣∣∣≤ �L.
Suppose also that
|Bε(t, x)| + Qε(t, x) ≤ αε(t) + hε(t, x),
lim
ε→0
(
||hε||2 +
∫ T
0
αε(t)dt
)
= 0.
Then xε =⇒ x.
It is known [7] that the limit coefficients in conditions (V), (N) are uniquely deter-
mined.
Proof of the theorem. Denote ηε(t) = f ε(ξε(t)). To use the result of the lemma, we apply
the Itô formula for the process ξε(t) and for the function f ε(x). We have
ηε(t) = f ε(x) +
∫ t
0
(γε(ηε(s)) + Bε(s, ηε(s))ds
+
∫ t
0
(qε(ηε(s)) + Qε(s, ηε(s))
1
2 dw(4)
In (4),
γε(x) = F ε(φε(x))rε(φε(x)),
Bε(t, x) = F ε(φε(x))(gε(t, φε(x)) − rε(φε(x))) +
1
2
(F ε)
′
(φε(x))Aε(t, φε(x)),
qε(x) = (F ε)2(φε(x)aε(φε(x)),
Qε(t, x) = (F ε)2(φε(x))Aε(t, φε(x)).
108 SERGEY YA. MAKHNO
Now we verify that the functions γε, qε satisfy conditions (V), (N) in the lemma and
conditions V4, V5, N4, N5 are valid for the functions V ε(x) and N ε(x). From condition
i) of the theorem, we have∫ x
0
1
qε(y)
dy =
∫ φε(x)
0
1
F ε(y)aε(y)
dy −→
ε→0
∫ φ(x)
0
a(y)dy =
=
∫ x
0
a(φ(y))Dφ(y)dy.(5)
Denote
(6) N ε(x) =
∫ x
0
∫ y
0
[
1
qε(z)a(φ(z))Dφ(z)
− 1
]
dzdy
From (6), we get
qε(x) + qε(x)
d2N ε(x)
dx2
=
1
a(φ(x))Dφ(x)
.
From (6) and (5), we have
≤ sup
x∈D
(
|N ε(x)| +
∣∣∣∣dN ε(x)
dx
∣∣∣∣
)
= 0.
It is obvious that
sup
x∈D
∣∣∣∣d
2N ε(x)
dx2
∣∣∣∣≤ �L.
So, conditions (N1) - (N5) from the lemma are valid, and the limit coefficient equals
q(x) =
1
a(φ(x))Dφ(x)
.
Reasoning similarly and using condition ii) in the theorem, we conclude that the function
V ε(x) = 2
∫ x
0
∫ y
0
[
r(φ(z))
qε(z)a(φ(z))
− γε(z)
qε(z)
]
dzdy
satisfies conditions (V1) - (V5) from the lemma with the limit coefficient
γ(x) =
r(φ(x))
a(φ(x))
.
For the functions Bε(t, x) and Qε(t, x) from condition (I3), we have
|Bε(t, x)| + Qε(t, x) ≤ �L(αε(t) + hε(t, x)).
Thus, ηε(t) =⇒ η(t), where
(7) η(t) = f(x) +
∫ t
0
γ(η(s))ds +
∫ t
0
√
q(η(s))dw(s).
As follows from condition (I2), the limit ≤ fε(x) = f(x) and ≤ φε(x) = φ(x) uniformly
on the compact sets. From Theorem 1.5.5 [1], we conclude that ξε(t) =⇒ ξ(t) = φ(η(t)).
Observing that Dφ(f(x)) = 1 for x �= 0, we can rewrite Eq. (7) as
(8) η(t) = f(x) +
∫ t
0
r(ξ(s))
a(ξ(s))
ds +
∫ t
0
1√
a(ξ(s))
dw(s).
Using formula (3) for the function φ(x) and for the process η(t) from (8), we get the
statement of the theorem by Lemma 1 [8]. The theorem is proved.
Consider the model example. Introduce the functions
αε(t) =
ε3|2t − 1|
[t(t − 1) + ε2]2
, hε(x) =
ε
1
8
(2πε)
1
4
exp
{
−x2
4ε
}
, τ ε(t, x) = αε(t) + hε(x),
THE LIMIT STOCHASTIC EQUATION CHANGES TYPE 109
and study the solutions of the stochastic equations
ξε(t) = x +
1
ε
∫ t
0
b
(
ξε(s)
ε
)
ds +
∫ t
0
[
g
(
ξε(s)
ε
)
+τ ε(s, ξε(s))
]
ds+
+
∫ t
0
σ
(
ξε(s)
ε
)
dw(s).(9)
It is obvious that, for t = 0 or t = 1 or x = 0, the function τ ε(t, x) tends to infinity as
ε → 0, but condition I3 is valid.
Suppose that∣∣∣∣
∫ x
0
b(y)
σ2(y)
dy
∣∣∣∣< const,
∫ 0
−∞
b(y)
σ2(y)
dy = B1,
∫ ∞
0
b(y)
σ2(y)
dy = B2
and that the limits
lim
|x|→∞
1
x
∫ x
0
g(y)
σ2(y)
dy = A1,
lim
|x|→∞
1
x
∫ x
0
dy
σ2(y)
= A2 > 0
exist. In this case, the conditions of the theorem are valid, and
β1 = exp(2B1), β2 = exp(−2B2), β = th(B1 + B2), a(x) =
1
A2
, r(x) =
A1
A2
, Nφ(x) = 0.
Then the limit process for Eq. (9) is
ξ(t) = x + th(B1 + B2)Lξ(t, 0) +
A1
A2
t +
1√
A2
w(t)
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E-mail : makhno@iamm.ac.donetsk.ua
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