Limit theorems for backward stochastic equations
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irk-123456789-45562009-12-07T12:00:29Z Limit theorems for backward stochastic equations Makhno, S.Ya. Yerisova, I.A. 2008 Article Limit theorems for backward stochastic equations / S.Ya. Makhno, I.A. Yerisova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 93–107. — Бібліогр.: 17 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4556 519.21 en Інститут математики НАН України |
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Makhno, S.Ya. Yerisova, I.A. Limit theorems for backward stochastic equations |
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Makhno, S.Ya. Yerisova, I.A. |
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Makhno, S.Ya. |
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Limit theorems for backward stochastic equations |
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Limit theorems for backward stochastic equations |
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Limit theorems for backward stochastic equations |
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Limit theorems for backward stochastic equations |
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Limit theorems for backward stochastic equations |
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limit theorems for backward stochastic equations |
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Інститут математики НАН України |
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2008 |
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Limit theorems for backward stochastic equations / S.Ya. Makhno, I.A. Yerisova // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 2. — С. 93–107. — Бібліогр.: 17 назв.— англ. |
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2025-07-02T07:46:18Z |
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Theory of Stochastic Processes
Vol. 14 (30), no. 2, 2008, pp. 93–107
UDC 519.21
SERGEY YA. MAKHNO AND IRINA A. YERISOVA
LIMIT THEOREMS FOR BACKWARD STOCHASTIC EQUATIONS
Consider a weak convergence in the Meyer–Zheng topology of solutions of a backward
stochastic equation in the form
Y ε(t) = E
�
gε
�
Xε(T )
�
+
� T
t
fε
�
s, Xε(s), Y ε(s)
�
ds
����F Xε
t
�
as ε → 0 for different classes of random processes Xε(t) with the irregular dependence
on the parameter ε. The equations for the limit process are obtained.
1. Introduction.
Backward stochastic differential equations of the form
Y (t) = g(X(T )) +
∫ T
t
f(s,X(s), Y (s))ds−
∫ T
t
Z(s)dW (s),
X(t) = x+
∫ t
0
b(s,X(s))ds+
∫ t
0
σ(s,X(s))dW (s),
where W (s) is a Wiener process, have been introduced by E. Pardoux and S. Peng
[14,15], who proved the existence and uniqueness of a FW adapted solution. That is, the
solutions of the equations are strong solutions. The aim of the authors was to describe
a solution of a second-order quasilinear partial equation in probabilistic terms. Due to
such stochastic representation of the solution of a quasilinear partial equation, the study
of the limit behavior of backward stochastic differential equations allows us to develop a
theory of the limit behavior of the corresponding partial equations. The limit behavior
of stochastic systems when coefficients of the process X(t) and a function f are of the
form h(x
ε ) or 1
εh(x
ε ), where ε is a small parameter and h(x) is a periodic function, was
studied in [3,12,13,16] as ε→ 0.
Here, we continue this investigation in several directions. First, following [2], we write
the equation for the processes Y ε in another form. This will allow us to consider weak
solutions of the processes Xε(t). Second, the coefficients of the processes Xε(t) may
depend on the parameter ε in any way. This dependence can be irregular: we do not
assume that the coefficients have limits as ε → 0, they may tend to infinity as ε → 0 or
may have no limit at all. Third, the coefficients for the processes Y ε(t) will also depend
on a small parameter. The functions gε(x) will converge uniformly on compact sets, and
the functions f ε(t, x, y) will have limit in the space of summable functions.
The paper is organized as follows. In this section, we introduce our notation and
assumptions. The limit result for the processes Y ε(t) is proved in Section 2. The identi-
fication of the limit processes for different classes is realized in Sections 3–5. 5. For this,
we will use the method developed in [12,13,16].
2000 AMS Mathematics Subject Classification. Primary 60H10, 60H20.
Key words and phrases. Backward stochastic equation, weak convergence, Meyer–Zheng topology.
93
94 SERGEY YA. MAKHNO AND IRINA A. YERISOVA
Let Rp be a p-dimensional Euclidean space. By D([0, T ];Rp), we denote the space of
cadlag (right-continuous with left-hand limits) functions x(t), t ∈ [0, T ]. On this space,
we consider the Skorokhod topology [1, part 3] and the Meyer–Zheng topology [11,2].
For weak convergence in these topologies, we use the notation S=⇒ and M−Z=⇒ , respectively.
Let Ω = D([0, T ];Rd) × D([0, T ];Rm), F be the σ-algebra of Borel subsets of this set,
(Ω,F , P ) be a probability space, and E be a symbol of expectation. If ξ(t), t ∈ [0, T ],
is a random process, then Fξ
t is the smallest filtration generated by ξ(s), s ∈ [0, t]. The
notation Lp([0, T ]×K) has a standard sense of the space of p-order integrable functions
on [0, T ] × K with the norm ‖ · ‖Lp([0,T ]×K). In this paper, we denote, by C, different
constants independent of ε, and I(A) is the indicator of an event A.
We consider solutions of backward stochastic equations (BSE) in the form
Y ε(t) = E
[
gε
(
Xε(T )
)
+
∫ T
t
f ε
(
s,Xε(s), Y ε(s)
)
ds
∣∣∣∣FXε
t
]
(1.1)
and investigate their weak convergence as ε → 0. Let the process Xε(t) in (1.1) be
a cadlag process with values in D([0, T ];Rd), and let the process Y ε(t) be a cadlag
process with values in D([0, T ];Rm). The measure corresponding to the process Xε on
D([0, T ];Rm) is denoted by με, and the measure corresponding to the process Y ε on
D([0, T ];Rm) is denoted by Qε. Following [2], we define the strong solution of (1.1) as a
process Y ε(t) such that, for any ε > 0,
E|gε(X(T ))| + E
∫ T
0
∣∣∣∣f ε
(
s,Xε(s), Y ε(s)
)∣∣∣∣ds <∞,
and (1.1) is valid.
For the processes Xε(t), we suppose that the following condition (I) is satisfied:
Condition (I):
I1. μ
ε S=⇒ μ, and let the process X(t) correspond to the measure μ.
I2. Let Xε(0) = x, and let the moment estimates
E sup
t∈[0,T ]
|Xε(t)|2 + E sup
t∈[0,T ]
|X(t)|2 ≤ C
be valid.
I3. Krylov’s estimates for the processes Xε(t) and X(t)
E
∫ T
0
∣∣∣∣h(t,Xε(t)
)∣∣∣∣dt+ E
∫ T
0
∣∣∣∣h(t,X(t)
)∣∣∣∣dt ≤ C‖h‖Ld+1([0,T ]×Rd)
are fulfilled.
We introduce the conditions for the coefficients in Eq. (1.1).
Condition (II). For measurable functions gε(x) and f ε(t, x, y) :
II1. |gε(x)| ≤ C(1 + |x|).
II2. |f ε(t, x, y)| ≤ C(1 + |y|).
II3. |f ε(t, x, y2) − f ε(t, x, y1)| ≤ C|y2 − y1|.
Condition (III): There exist a continuous function g(x) and a measurable function
f(t, x, y) such that
III1. For any compact K ∈ Rd, lim supx∈K |gε(x) − g(x)| = 0.
III2. For fixed y ∈ Rm and any compact K ∈ Rd,
LIMIT THEOREMS FOR BACKWARD STOCHASTIC EQUATIONS 95
limε→0 ‖f ε(·, ·, y) − f(·, ·, y)‖Ld+1([0,T ]×K) = 0.
III3. For the functions g(x) and f(t, x, y), condition (II) is valid, if the symbol ε is
omitted.
It was proved in [2, Proposition 2.1] that, under conditions (II) and I2 for any ε > 0,
there exists a strong solution of Eq. (1.1), and this solution is unique.
2. Limit for Y ε
Consider the strong solutions of Eq. (1.1). The main result of this section is Theorem
1.
Theorem 1. Suppose that conditions (I), (II), and (III) are satisfied. Then there exist
a subsequence Qεk of Qε and a probability law Q on D([0, T ];Rm) such that Qεk
M−Z=⇒ Q,
and, for corresponding process Y (t), we have
Y (t) = E
[
g
(
X(T )
)
+
∫ T
t
f
(
s,X(s), Y (s)
)
ds
∣∣∣∣FX,Y
t
]
. (2.1)
Proof. We will prove the theorem in several steps.
Step 1). As the constants in conditions (I) and (II) do not depend on ε, we can obtain
the following estimate [2, proof of Proposition 2.1] (cf. Corollary 1 below):
E sup
t∈[0,T ]
|Y ε(t)|2 ≤ C. (2.2)
Step 2). We verify that this sequence is relatively compact in the Meyer–Zheng topology.
To prove this, we use a result from [11, Theorem 4]. For a subdivision 0 = t0 < t1 <
... < tn = T , we define
Vn(Y ε) = E|Y ε(T )| +
n−1∑
k=0
E
∣∣∣∣E[(
Y ε(tk+1) − Y ε(tk)
)∣∣∣∣FY ε
tk
]∣∣∣∣.
Using estimates (2.2) and the conditions of the theorem, we get
Vn(Y ε) = E|Y ε(T )| +
n−1∑
k=0
E
∣∣∣∣E[∫ tk+1
tk
f ε(s,Xε(s), Y ε(s))ds
∣∣∣∣FY ε
tk
]∣∣∣∣≤
≤ E|Y ε(T )| + E
∫ T
0
∣∣∣∣f ε(s,Xε(s), Y ε(s))
∣∣∣∣ds ≤ E|Y ε(T )|+
+ C
∫ T
0
E(1 + |Y ε(s)|)ds ≤ C.
Then there exists a subsequence of Qε which converges weakly to the law Q in the Meyer–
Zheng topology on D([0, T ];Rm). We denote this subsequence by Qε again, and let Y (t)
be a process on D([0, T ];Rm) corresponding to the measure Q.
Step 3). Let t ∈ [0, T ], G(t) ∈ D([0, T ];Rm), k ∈ [0,∞), and let Gk(t) = G(t)((k+1−
|G(t)|)+ ∧ 1), where a+ = a ∨ 0 and ∨,∧ are the symbols of max and min, respectively.
It is relevant to remark that
|Gk(t)| ≤ |G(t)| and |G(t) −Gk(t)| = |G(t)|I(|G(t)| > k).
For l ≥ k, we denote Ul,k(t) = |Gl(t)−Gk(t)|. It is not difficult to check that the sequence
Ul,k(t) monotonically increases in l, and
lim
l→∞
Ul,k(t) = |G(t) −Gk(t)|.
96 SERGEY YA. MAKHNO AND IRINA A. YERISOVA
Introduce the functional
Nt,δ(G) = δ−1
∫ T∧(t+δ)
t
G(s)ds.
The functionals Nt,δ(Gk) and Nt,δ(Ul,k) are bounded and continuous in the Meyer–Zheng
topology [2].
Step 4). In virtue of step 3),
E|Y ε(t) − Y ε
k (t)| = E|Y ε(t)|I(|Y ε(t)|>k) ≤
(
E|Y ε(t)|2
) 1
2
P{|Y ε(t)| > k}.
From this, estimate (2.2), and the Chebyshev inequality, we have
sup
ε
sup
t∈[0,T ]
E|Y ε(t) − Y ε
k (t)| ≤ Ck−2 (2.3)
and
lim
k→∞
sup
ε
sup
t∈[0,T ]
E|Y ε(t) − Y ε
k (t)| = 0. (2.4)
Step 5). As the process Y (t) belongs to D(Rm), we use the Fatou’s lemma and obtain
E|Y (t) − Yk(t)| =E lim
δ↓0
1
δ
∫ t+δ
t
|Y (s) − Yk(s)|ds ≤
≤ limδ↓0E
1
δ
∫ t+δ
t
|Y (s) − Yk(s)|ds.
We continue the last inequality by the theorem on monotone convergence and the prop-
erty of continuity of the functional Nt,δ(Ul,k) from step 3). Then
E|Y (t) − Yk(t)| ≤ limδ↓0 lim
l↑∞
E
1
δ
∫ t+δ
t
|Yl(s) − Yk(s)|ds =
= limδ↓0 lim
l↑∞
lim
ε→0
E
1
δ
∫ t+δ
t
|Y ε
l (s) − Y ε
k (s)|ds.
From this and (2.3), we get
lim
k→∞
E|Y (t) − Yk(t)| = 0. (2.5)
Now we show that the limit process Y (t) is uniformly integrable. From the inequality
|Y (t)| ≤ |Y (t) − Yk(t)| + |Yk(t) −Nt,δ(Yk)| + |Nt,δ(Yk)|,
by using the continuity in the Meyer–Zheng topology of the functional Nt,δ(|Zk|), we get
E|Y (t)| ≤ E|Y (t) − Yk(t)| + E|Yk(t) −Nt,δ(Yk)| + lim
ε→0
ENt,δ(|Y ε
k |).
By virtue of (2.2),
E|Y (t)| ≤ E|Y (t) − Yk(t)| + E
1
δ
∣∣∣∣∫ t+δ
t
(Yk(t) − Yk(s))ds
∣∣∣∣+C.
Approaching the limit as δ → 0 in this inequality and taking (2.5) into account, we have
E|Y (t)| ≤ C. (2.6)
Take (2.5) and step 3) into account, we get the uniform integrability of the process Y (t):
lim
k→∞
sup
t∈[0,T ]
E|Y (t)|I(|Y (t)| > k) = lim
k→∞
sup
t∈[0,T ]
E|Y (t) − Yk(t)| = 0. (2.7)
LIMIT THEOREMS FOR BACKWARD STOCHASTIC EQUATIONS 97
We now prove that
E
∣∣∣∣E(
[Y ε
k (t) −Nt,δ(Y ε
k )]
∣∣∣∣FXε
t
)∣∣∣∣≤ C
(
1
k2
+ δ
)
. (2.8)
We have
E
∣∣∣∣E(
[Y ε
k (t) −Nt,δ(Y ε
k )]
∣∣∣∣FXε
t
)∣∣∣∣≤ E|Y ε
k (t) − Y ε(t)| +
1
δ
∫ t+δ
t
E|Y ε(s)−
−Y ε
k (s)|ds+
1
δ
∫ t+δ
t
E
∣∣∣∣E{
[Y ε(t) − Y ε(s)]
∣∣∣∣FXε
t
}∣∣∣∣ds. (2.9)
Next, relation (1.1) yields
E
{
[Y ε(t) − Y ε(s)]
∣∣∣∣FXε
t
}
=
∫ s
t
E
{
f ε(u,Xε(u), Y ε(u))
∣∣∣∣FXε
t
}
du.
From this and (2.2), we get
E
∣∣∣∣E{
[Y ε(t) − Y ε(s)]
∣∣∣∣FXε
t
}∣∣∣∣≤ CE
∫ s
t
E
{
(1 + |Y ε(u)|)
∣∣∣∣FXε
t
}
du ≤
≤ C
∫ s
t
E(1 + |Y ε(u)|)du ≤ C(s− t).
(2.10)
Inequality (2.8) follows from (2.9), (2.3), and (2.10).
Step 6). Let Φt(x, y) be a bounded continuous functional on D([0, t];Rd) × D([0, t];
Rm) equipped with a product of the Skorokhod topology on the first factor and the
Meyer–Zheng topology on the second factor. As follows from (1.1) for any such functional,
EΦt(Xε, Y ε)[Y ε(t) − gε(Xε(T )) −
∫ T
t
f ε(s,Xε(s), Y ε(s))ds] = 0.
We rewrite the left-hand side of the last equality as
3∑
k=1
Jε
k + EΦt(X,Y )[Y (t) − g(X(T )) −
∫ T
t
f(s,X(s), Y (s))ds] = 0, (2.11)
where
Jε
1 = E[Φt(Xε, Y ε)Y ε(t) − Φt(X,Y )Y (t)],
Jε
2 = E[Φt(X,Y )g(X(T )) − Φt(Xε, Y ε)gε(Xε(T ))],
Jε
3 = E[Φt(X,Y )
∫ T
t
f(s,X(s), Y (s))ds−
− Φt(Xε, Y ε)
∫ T
t
f ε(s,Xε(s), Y ε(s))ds],
and estimate each of Jε
k. The expression for Jε
1 can be represented in the form
Jε
1 =
6∑
i=4
Jε
i + J7 + J8, (2.12)
where
Jε
4 = E[Φt(Xε, Y ε)[Y ε(t) − Y ε
k (t)],
Jε
5 = EΦt(Xε, Y ε)[Y ε
k (t) −Nt,δ(Y ε
k )],
Jε
6 = EΦt(Xε, Y ε)Nt,δ(Y ε
k ) − EΦt(X,Y )Nt,δ(Yk),
J7 = EΦt(X,Y )[Nt,δ(Yk) − Yk(t)],
J8 = EΦt(X,Y )[Yk(t) − Y (t)].
98 SERGEY YA. MAKHNO AND IRINA A. YERISOVA
Relations (2.4) and (2.5) yield
lim
k→∞
sup
ε
|Jε
4 | = 0 (2.13)
and
lim
k→∞
|J8| = 0. (2.14)
From (2.8), we get
lim
k→∞
lim
δ↓0
sup
ε
|Jε
5 | = 0. (2.15)
In view of the weak convergence of (Xε, Y ε) to (X,Y ),
lim
ε→0
|Jε
6 | = 0. (2.16)
As with probability one, limδ↓0Nt,δ(Yk) = Yk(t), and Yk(t) are uniformly bounded on t,
we have
lim
δ↓0
|J7| = 0. (2.17)
Approaching the limit in (2.12) firstly as ε → 0, then as δ → 0, and then as k → ∞, we
get, by virtue (2.13)–(2.17):
lim
ε→0
|Jε
1 | = 0. (2.18)
Before the estimation of Jε
2, we introduce the continuous functions rN (x) : 0 ≤ rN (x) ≤ 1,
rN (x) = 1 if |x| ≤ N and rN (x) = 0 if |x| ≥ N + 1 and define gN (x) = g(x)rN (x). The
expression Jε
2 can be presented as the sum
Jε
2 = Jε
9 + Jε
10 + Jε
11 + J12, (2.19)
where
Jε
9 = EΦt(Xε, Y ε)[g(Xε(T )) − gε(Xε(T ))],
Jε
10 = EΦt(Xε, Y ε)[gN (Xε(T )) − g(Xε(T ))],
Jε
11 = E[Φt(X,Y )gN (X(T )) − Φt(Xε, Y ε)gN(Xε(T ))],
J12 = EΦt(X,Y )[g(X(T )) − gN(X(T ))].
Using the estimate from I2, it is not difficult to get that, for any K,
|Jε
9 | ≤ CE|g(Xε(T ) − gε(Xε(T ))|I(|Xε(T )| ≤ N) +
C
N2
.
Taking the limits in the last formula firstly in ε → 0 and then in N → ∞, we have, by
condition III1,
lim
ε→0
|Jε
9 | = 0. (2.20)
As |g(x) − gN(x)| ≤ |g(x)|I(|x| > N), estimate I2 and the Chebyshev inequality yield
sup
ε
|Jε
10| ≤
C
N2
.
From this, we have
lim
N→∞
sup
ε
|Jε
10| = 0. (2.21)
Similarly,
lim
N→∞
|J12| = 0. (2.22)
From the weak convergence (Xε, Y ε) to (X,Y ), we get
lim
ε→0
|Jε
11| = 0. (2.23)
From (2.19)–(2.23), we conclude
lim
ε→0
Jε
2 = 0. (2.24)
LIMIT THEOREMS FOR BACKWARD STOCHASTIC EQUATIONS 99
To estimate Jε
3 , we will do some constructions. Let R(a;A) = {y : |y−a| ≤ A} be a ball of
radius A with its center at the point a. In R(0;N), we introduce a δ-net {y1, y2, ..., yM} :
|yi+1 − yi| ≤ δ and a family of functions qi(y) with the properties:
a) qi(y) ≥ 0 and qi(y) = 0 outside R(yi, δ),
b)
M∑
i=1
qi(y) = 1,
c) qi(y) are differentiable functions on y ∈ R(0;N).
For the function g(t, x, y), we define the function
gN,δ(t, x, y) =
M∑
i=1
qi(y)g(t, x, yi).
From this and condition II3, we get, for any t ∈ [0, T ], x ∈ Rd, and y ∈ Rm,
|f ε(t, x, y) − f ε
N,δ(t, x, y)| ≤
M∑
i=1
qi(y)|f ε(t, x, y) − f ε(t, x, yi)| ≤
≤ Cδ.
(2.25)
Similarly, we have
|f(t, x, y) − fN,δ(t, x, y)| ≤ Cδ. (2.26)
Due to the convolution for the function f(t, x, y), we can define a sequence of continuous
functions f (n)(t, x, y) bounded for each n and such that
lim
n→∞ ||f(·, ·, y) − f (n)(·, ·, y)||Ld+1([0,T ];Rd) = 0
for any y ∈ Rm. From this, we get
lim
n→∞ ||fN,δ(·, ·, y) − f
(n)
N,δ(·, ·, y)||Ld+1([0,T ];Rd) = 0. (2.27)
Then we rewrite the expression for Jε
3 in the form
Jε
3 = Jε
13 + Jε
14 + Jε
15 + Jε
16 + Jε
17 + J18 + J19, (2.28)
where
Jε
13 = EΦt(Xε, Y ε)
∫ T
t
[f ε
N,δ(s,Xε(s), Y ε(s)) − f ε(s,Xε(s), Y ε(s))]ds,
Jε
14 = EΦt(Xε, Y ε)
∫ T
t
[fN,δ(s,Xε(s), Y ε(s)) − f ε
N,δ(s,Xε(s), Y ε(s))]×
× rK(Xε(s))ds,
Jε
15 = EΦt(Xε, Y ε)
∫ T
t
[fN,δ(s,Xε(s), Y ε(s)) − f ε
N,δ(s,Xε(s), Y ε(s))]×
× (1 − rK(Xε(s)))ds,
Jε
16 = EΦt(Xε, Y ε)
∫ T
t
[f (n)
N,δ(s,Xε(s), Y ε(s)) − fN,δ(s,Xε(s), Y ε(s))]ds,
100 SERGEY YA. MAKHNO AND IRINA A. YERISOVA
Jε
17 = EΦt(X,Y )
∫ T
t
[f (n)
N,δ(s,X(s), Y (s))ds−
− EΦt(Xε, Y ε)
∫ T
t
f
(n)
N,δ(s,Xε(s), Y ε(s))]ds,
J18 = EΦt(X,Y )
∫ T
t
[fN,δ(s,X(s), Y (s)) − f
(n)
N,δ(s,X(s), Y (s))]ds
J19 = EΦt(X,Y )
∫ T
t
[f(s,X(s), Y (s)) − fN,δ(s,X(s), Y (s))]ds.
By (2.25) and (2.26,) we have
sup
ε
|Jε
13| + |J19| ≤ Cδ. (2.29)
In view of Krylov’s estimate I3, we get
|Jε
14| ≤ C
M∑
i=1
||f(·, ·, yi) − f ε(·, ·, yi)||Ld+1([0,T ]×K). (2.30)
Similarly,
|Jε
16| ≤ C
M∑
i=1
||f (n)(·, ·, yi) − f(·, ·, yi)||Ld+1([0,T ]×Rd) (2.31)
and
|J18| ≤ C
M∑
i=1
||f(·, ·, yi) − f (n)(·, ·, yi)||Ld+1([0,T ]×Rd). (2.32)
From conditions of the theorem and from estimate I2, we get
|Jε
15| ≤ CM
∫ T
t
E(1 + |Xε(s)|)I(|Xε(s)| > K)ds ≤ CM
K2
. (2.33)
As Xε S=⇒ X , Y ε M−Z=⇒ Y , and the functional Φt(x(·), y(·))
∫ T
t
f
(n)
N,δ(x(s), y(s))ds is con-
tinuous in the corresponding topologies,
lim
ε→0
|Jε
17| = 0. (2.34)
Finally, we have
|Jε
3 | ≤ Cδ + C
M∑
i=1
||f ε(·, ·, yi) − f(·, ·, yi)||Ld+1([0,T ]×K)+
+ C
M∑
i=1
||f(·, ·, yi) − f (n)(·, ·, yi)||Ld+1([0,T ];Rd) +
CM
K2
+ |Jε
17|.
Approaching the limit in this inequality firstly as ε → 0, then as n → ∞, K → ∞, and
δ → 0, we have, by (2.29)–(2.34),
lim
ε→0
Jε
3 = 0. (2.35)
From (2.11), (2.8), (2.24), and (2.35), we obtain
EΦt(X,Y )[Y (t) − g(X(T )) −
∫ T
t
f(s,X(s), Y (s))ds] = 0. (2.36)
From this, we get (2.1). The theorem is proved.
LIMIT THEOREMS FOR BACKWARD STOCHASTIC EQUATIONS 101
Corollary 1. Under the conditions of Theorem 1,
E sup
t∈[0,T ]
|Y (t)|2 ≤ C. (2.37)
Proof. From (2.1), we have
sup
t∈[0,T ]
|Y (t)| ≤
≤ sup
t∈[0,T ]
E
(
|g(X(T ))| +
∫ T
t
|f(s,X(s), Y (s))|ds
∣∣∣∣FX,Y
t
)
≤
≤ sup
t∈[0,T ]
E
(
|g(X(T ))| +
∫ T
0
|f(s,X(s), Y (s))|ds
∣∣∣∣FX,Y
t
)
.
As E|g(X(T ))|+E
∫ T
0 |f(s,X(s), Y (s))|ds ≤ C, we can use Doob’s inequality and obtain
E sup
t∈[0,T ]
|Y (t)|2 ≤ 4E
(
|g(X(T )|2 + |
∫ T
0
f(s,X(s), Y (s))ds|2
)
≤
≤ C(1 +
∫ T
0
E sup
u∈[0,s]
|Y (u)|2ds
)
.
The Gronwall’s lemma yields (2.37).
Corollary 2. Under the conditions of Theorem 1, the process Y (t) admits the decom-
position in the following form
Y (t) = g(X(T )) +
∫ T
t
f(s,X(s), Y (s))ds−M(T ) +M(t), (2.38)
where (M(t),FX,Y
t ) is a square integrable martingale, and EM(t) = 0.
Proof. We define
M(t) = E
[
g
(
X(T )
)
+
∫ T
0
f
(
s,X(s), Y (s)
)
ds
∣∣∣∣FX,Y
t
]
−E
[
g
(
X(T )
)
+
+
∫ T
0
f
(
s,X(s), Y (s)
)
ds
]
.
The required assertion follows from (2.1) and Corollary 1.
3. Itô equation with coefficients bounded on ε.
In this section, we consider one-dimensional processes Xε(t), because the conditions
for weak convergence με S=⇒ μ have a very simple form in this case. The multidimensional
case was considered in [6,7,10]. LetXε(t) in Eq. (1.1) be a solution of the one-dimensional
stochastic equation
Xε(t) = x+
∫ t
0
[bε1(s,Xε(s)) + bε2(Xε(s))]ds +
∫ t
0
σε(s,Xε(s))dw, (3.1)
where w(t) is the standard Wiener process. We consider a weak solution of this equation
and introduce the conditions for the coefficients of (3.1). Let the constants 0 < λ ≤ Λ <
∞ be given. We say that a pair of measurable functions (f(t, x), g(t, x)) ∈ L(λ,Λ) if
|f(t, x)| + |g(t, x)| ≤ Λ, g(t, x) ≥ λ. (3.2)
102 SERGEY YA. MAKHNO AND IRINA A. YERISOVA
For the coefficients of Eq. (3.1), we suppose that
(bε1 + bε2, a
ε) ∈ L(λ,Λ), (3.3)
where aε(x) := (σε(x))2.
Introduce the following condition.
Condition (IV). There exist the measurable functions b1(t, x), b2(x), Gε(x), and
G(x) such that, for any compact K ∈ R1,
IV1. limε→0 ‖bε1 − b1‖L2([0,T ];K) = 0, limε→0 ‖aε −Gε‖L2([0,T ];K) = 0,
IV2. For any x ∈ R1 limε→0
∫ x
0
bε
2(y)
Gε(y)dy =
∫ x
0
b2(y)dy,
limε→0
∫ x
0
1
Gε(y)dy =
∫ x
0 G(y)dy,
IV3. (b1 + b2
G , G) ∈ L(λ,Λ).
In [7, Theorem 4], it was proved that if (3.3) and condition (IV ) are valid, then
με S=⇒ μ, and the limit process X(t) is a solution of the equation
X(t) = x+
∫ t
0
B(s,X(s))ds+
∫ t
0
σ(X(s))dw,
where B(s, x) = b1(s, x) + b2(x)
G(x) , σ(x) = 1√
G(x)
.
Theorem 2. Let the processes Xε(t), Y ε(t) be defined by Eqs. (3.1) and (1.1). Sup-
pose that (3.3) and conditions (II), (III), and (IV ) are fulfilled. Then Qε M−Z=⇒ Q,
and there exist the Wiener process (w̄(t),FX,Y
t ) and a process (Z(t),FX,Y
t ) such that
E
∫ T
0
|Z|2(t)dt <∞ and
X(t) = x+
∫ t
0
B(s,X(s))ds+
∫ t
0
σ(X(s))dw̄, (3.4)
Y (t) = g(X(T )) +
∫ T
t
f(s,X(s), Y (s))ds−
∫ T
t
Z(s)dw̄(s). (3.5)
Proof. To use Theorem 1, we must verify condition (I). The property I1 is valid in view of
the noted above [7, Theorem 4]. Estimate I2 under condition (3.3) is a standard estimate
from the theory of Itô stochastic equations. See, for example, [5. Corollary 2.5.12]. The
estimate from I3 is a result of [5, chapter 2]. Therefore, Theorem 1 may be employed.
After the extraction of a suitable sequence, which we omit as an abuse notation, we have
that με S=⇒ μ and Qε M−Z=⇒ Q. Let Φt(x, y) be a functional as in step 6) in the proof of
Theorem 1. Since Y ε(t) is a strong (FXε
t - measurable) solution of Eq. (1.1), it is proved
in [6, proof of Theorem 1] that, for any infinitely differentiable function with a compact
support φ(x),
lim
ε→0
EΦt(Xε, Y ε)
[
φ(Xε(s)) − φ(Xε(t))−
−
∫ s
t
(
φ
′
(Xε(u))B(Xε(u)) +
1
2
φ
′′
(Xε(u))σ2(X(u))
)
du
]
= 0.
(3.6)
From this,
EΦt(X,Y )
[
φ(X(s)) − φ(X(t)) −
∫ s
t
(
φ
′
(X(u))B(u,X(u))+
+
1
2
φ
′′
(X(u))a(X(u))
)
du
]
= 0.
LIMIT THEOREMS FOR BACKWARD STOCHASTIC EQUATIONS 103
Passing to the limit causes no difficulty if the functions b(x), a(x) are continuous func-
tions. For only measurable functions, the passing to the limit proves by Krylov’s estimate
as for Iε
3 in the proof of Theorem 1. From [17, Theorem 4.5.1], we have that there exists
a Wiener process (w̄(t),FX,Y
t ) such that (3.4) is valid.
Let (Ȳ (t), Z(t)) be a unique solution of BSE [2,13,14]
Ȳ (t) = g(X(T )) +
∫ T
t
f(s,X(s), Ȳ (s))ds−
∫ T
t
Z(s)dw̄(s). (3.7)
We denote M̄(t) =
∫ t
0 Z(s)dw̄(s). From (2.38) and (3.7), we conclude that
Y (t) − Ȳ (t) =
∫ T
t
[f(s,X(s), Y (s)) − f(s,X(s), Ȳ (s))]ds+
+ (M(t) − M̄(t)) + (M̄(T ) −M(T )).
From Itô’s formula for semimartingales,
E(Y (t) − Ȳ (t))2 + E[M − M̄ ]T − E[M − M̄ ]t = E
∫ T
t
[f(s,X(s), Y (s))−
− f(s,X(s), Ȳ (s))](Y (s) − Ȳ (s))ds ≤ C
∫ T
t
E|Y (s) − Ȳ (s)|2ds.
Hence, from Gronwall’s lemma for all t ∈ [0, T ], the processes Y (t) = Ȳ (t) and M(t) =
M̄(t). By the unique solution of Eq. (3.5), we conclude that all sequence Qε M−Z=⇒ Q.
The theorem is proved.
4. Itô equation with drift unbounded on ε
Let Xε(t) in Eq. (1.1) be a solution of the one-dimensional stochastic equation
Xε(t) = x+
∫ t
0
bε(Xε(s))ds+
∫ t
0
σε(Xε(s))dw, (4.1)
where w(t) is a standard Wiener process. We consider a weak solution of this equation
and introduce conditions for the coefficients of (4.1). For the constant λ > 0,
0 < λ ≤ aε(x) := (σε(x))2 ≤ C. (4.2)
And, for any x ∈ R1, ∣∣∣∣∫ x
0
bε(y)
aε(y)
dy
∣∣∣∣≤ C. (4.3)
Under these restrictions, the coefficients bε(x) may not be bounded at certain points
and tend to infinity as ε → 0 or may not have a limit at all. The limit process for the
processes Xε(t) may be also a solution of the Itô stochastic equation or may change its
type and be a solution of the stochastic equation with a local time. In this section, we
consider the case of the Itô equation for the limit process.
Denote
Hε(x) = exp
{
−2
∫ x
0
bε(y)
aε(y)
dt
}
, hε(x) =
∫ x
0
Hε(y)dy.
We set
βε(x) = 2
∫ x
0
Hε(y)
[∫ y
0
b(z) − bε(z)
Hε(z)aε(z)
dz + β
]
dy,
αε(x) =
∫ x
0
a(y) − aε(y)[1 + (βε(y))
′
]2
Hε(y)aε(y)
dy,
where the prime denotes a derivative, and introduce conditions (α) and (β).
104 SERGEY YA. MAKHNO AND IRINA A. YERISOVA
Condition (β). There exist a measurable bounded function b(x) and a constant β
such that, for any x ∈ R1, limε→0 β
ε(x) = 0.
Condition (α). There exists a measurable bounded function a(x) ≥ λ such that, for
any x ∈ R1, limε→0 α
ε(x) = 0.
We note that the limit functions b(x), a(x) and the constant β are uniquely determined
by conditions (β) and (α) [8, Lemma 3.2]. In [8, Theorem 1], it is proved that if (4.2) and
(4.3) are satisfied, then conditions (α) and (β) are necessary and sufficient conditions for
με S=⇒ μ, and X(t) is the unique weak solution of the stochastic equation
X(t) = x+
∫ t
0
b(X(s))ds+
∫ t
0
√
a(X(s))dw. (4.4)
Theorem 3. Let processes Xε(t), Y ε(t) be defined by Eqs. (4.1) and (1.1). Suppose
that (4.2), (4.3) and conditions (II), (III), (α), and (β) are fulfilled. Then Qε M−Z=⇒ Q,
and there exist the Wiener process (w̄(t),FX,Y
t ) and a process (Z(t),FX,Y
t ) such that
E
∫ T
0 |Z|2(t)dt <∞ and
X(t) = x+
∫ t
0
b(X(s))ds+
∫ t
0
√
a(X(s))dw̄, (4.5)
Y (t) = g(X(T )) +
∫ T
t
f(s,X(s), Y (s))ds−
∫ T
t
Z(s)dw̄(s). (4.6)
Proof. The proof of this theorem is completely analogous to that of Theorem 2, and we
only indicate where the corresponding results were proved. Property I1 is valid in view
of the noted above [8, Theorem 1]. The estimate from I2 follows from the same estimate
in [8, Lemma 3.5]. The Krylov’s estimate for the process Xε(t) is a result of [8, Lemma
3.7]. The analog of formula (3.6) is formula (1.3) in [8]. Theorem is proved.
As an example of the use of Theorem 3, we consider the solutions of stochastic equa-
tions
Xε(t) = x+
1
ε
∫ t
0
b
(
Xε(s)
ε
)
ds+
∫ t
0
σ
(
Xε(s)
ε
)
dw. (4.7)
In contrast to [3,12,14], we do not assume that b(x), a(x) are periodic functions. Let the
following limits exist:
lim
|x|→∞
1
x
∫ x
0
exp
{
−2
∫ y
0
b(z)
a(z)
dz
}
dy = B1 > 0, (4.8)
lim
|x|→∞
1
x
∫ x
0
1
a(y)
exp
{
2
∫ y
0
b(z)
a(z)
dz
}
dy = B2 > 0. (4.9)
It can easily be verified that conditions (β) and (α) are satisfied with the function b(x) = 0
and a constant β : 2β + 1 = B−1
1 , a(x) = (B1B2)−1. In this case, (4.5) takes the form
X(t) = x+ (B1B2)−
1
2 w̄(t).
Let us consider Eq.(4.7) under another assumptions,∫ ∞
−∞
∣∣∣∣ b(x)
a(x)
∣∣∣∣dx <∞,
∫ ∞
−∞
b(x)
a(x)
dx = 0, (4.10)
and let there exist
lim
|x|→∞
1
x
∫ x
0
dy
a(y)
= A > 0. (4.11)
LIMIT THEOREMS FOR BACKWARD STOCHASTIC EQUATIONS 105
In this case, conditions (β) and (α) are satisfied with the function b(x) = 0 and a constant
β :
2β + 1 = exp
{
2
∫ ∞
0
b(y)
a(y)
dy
}
,
a(x) = A−1. In this case, (4.5) takes the form
X(t) = x+
1√
A
w̄(t).
5. Limit process changes type
We now consider the stochastic equation (4.1) under conditions (4.2) and (4.3), but
we introduce another suppositions for the coefficients of the equation. In this case, the
limit process is the solution of a stochastic equation with local time. We will use notation
from Section 4. Let hi > 0, i = 1, 2 be some fixed constants. Suppose
lim
ε→0
hε(x) = h(x) =
{
h1x, if x ≤ 0;
h2x, if x ≥ 0.
(5.1)
Under condition (4.3), the limit in (5.1) exists uniformly on compacts. As hε(x) is a mono-
tonically increasing function on x, there exists an inverse function λε(x) : λε(hε(x)) = x.
Then limε→0 λ
ε(x) = λ(x) uniformly on compacts, and λ(x) is an inverse function to the
function h(x). By Dh(x), we denote symmetric derivatives of the function h(x):
Dh(x) = lim
δ→0
h(x+ δ) − h(x− δ)
2δ
and introduce the following condition: for any x ∈ R1,
lim
ε→0
∫ x
0
1
Hε(y)aε(y)
dy =
∫ t
0
1
a(y)Dh(y)
dy. (5.2)
Let X(t) be a weak solution of the equation
X(t) = x+ βLX(t, 0) +
∫ t
0
σ(X(s))dw. (5.3)
It follows from [4, Theorem 4.35] that Eq. (5.3) has the unique weak solution.
In [9, Theorem], it is proved that if (4.2), (4.3,) and (5.1) are satisfied, then conditions
(5.2) and
β =
h1 − h2
h1 + h2
(5.4)
are necessary and sufficient conditions for με S=⇒ μ, and X(t) is the unique weak solution
of the stochastic equation (5.3).
Theorem 4. Let the processes Xε(t), Y ε(t) be defined by Eqs. (4.1) and (1.1). Suppose
that (4.2), (4.3), (5.1), (5.2), (5.4), (II), and (III) are fulfilled. Moreover, the function
f(t, x, y) is continuous on x. Then Qε M−Z=⇒ Q, and there exist the Wiener process
(w̄(t),FX,Y
t ) and a process (Z(t),FX,Y
t ) such that E
∫ T
0
|Z|2(t)dt <∞ and
X(t) = x+ βLX(t, 0) +
∫ t
0
√
a(X(s))dw̄ (5.5)
Y (t) = g(X(T )) +
∫ T
t
f(s,X(s), Y (s))ds−
∫ T
t
Z(s)dw̄(s). (5.6)
106 SERGEY YA. MAKHNO AND IRINA A. YERISOVA
Proof. We use Theorem 3 in the proof of Theorem 4. Let ηε(t) = hε(Xε(t)). Then, by
the Itô formula,
ηε(t) = ηε(0) +
∫ t
0
σ̂ε(ηε(s))dw, (5.7)
where σ̂ε(x) = Hε(λε(x))σε(λε(x)). By using the one-to-one correspondence of the pro-
cesses Xε(t) and ηε(t), we get FXε
t = Fηε
t , and (1.1) yields
Y ε(t) = E
[
ĝε
(
ηε(T )
)
+
∫ T
t
f̂ ε
(
s, ηε(s), Y ε(s)
)
ds
∣∣∣∣Fηε
t
]
. (5.8)
In this formula, ĝε(x) = gε(λε(x)) and f̂ ε(t, x, y) = f ε(t, λε(x), y). The stochastic system
(5.7), (5.8) has the same type as (4.1), (1.1). To prove the theorem, we verify the
conditions of Theorem 3. According to the conditions of Theorem and the properties
of the functions hε(x), Hε(x), λε(x) and λ(x), we have that, uniformly on compacts,
limε→0 g
ε(λε(x)) = g(λ(x)) and, for any compact K,
lim
ε→0
‖f̂ ε(·, ·, y) − f(·, λ(·), y)‖L2([0,T ]×K) = 0.
Hence, condition (III) is valid. For process (5.7), condition (β) is automatically valid
with the limit function b(x) = 0 and the constant β = 0. From (5.2), we get that
condition (α) is valid too with the function a(x) = a(λ(x)). Consequently, by Theorem
3, there exists the Wiener process (w̄(t),Fη,Y
t ), and Fη,Y
t is a measurable process Z(t),
E
∫ T
0
Z2(t)dt <∞, such that
η(t) = h(x) +
∫ t
0
√
a(λ(η(s)))
D(λ(η(s))
dw̄, (5.8)
Y (t) = g(λ(η(T ))) +
∫ T
t
f(s, λ(η(s)), Y (s))ds−
∫ T
t
Z(s)dw̄(s). (5.9)
Observe that Xε S=⇒ X = λ(η). Applying the Tanaka formula to the process η(t) from
(5.8) and using the function λ(x), we get that the process X(t) satisfies Eq. (5.5), and
formula (5.9) coincides with formula (5.6). As above, we have Fη,Y
t = FX,Y
t . The
theorem is proved.
As a model example for this theorem, we again consider the stochastic equation (4.7)
but change the second condition in (4.10). Suppose that∫ ∞
−∞
∣∣∣∣ b(x)
a(x)
∣∣∣∣dx <∞,
∫ 0
−∞
b(x)
a(x)
dx = K1,
∫ ∞
0
b(x)
a(x)
dx = K2,
and condition (4.11) is satisfied. Then the limit process X(t) for the solution to Eq. (4.7)
has the type
X(t) = x+ th(K1 +K2)LX(t, 0) +
1√
A
w̄(t).
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