Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics
Подождите немного. Документ загружается.
206 8 Mean Derivatives in Linear Spaces
E
0
(f(ζ(t)) lim
Δt→+0
E
0ζ
t
ζ(t) − ζ(t − Δt)
Δt
θ(l)
= lim
Δt→+0
E
0
(f(W (t)) −f (W (t − Δt)))
W (t) − W (t − Δt)
Δt
θ(l)
+ lim
Δt→+0
E
0
f(W (t −Δt))
W (t) − W (t −Δt)
Δt
θ(l)
.
As above, the second summand on the right hand side is equal to
E
0
(f(ζ(t))θ(t)a(t)). Let us calculate the first summand. Here we apply The-
orem 6.12,theItˆo formula, and integration by parts. Denoting by the same
symbol the conditional expectation and the corresponding regression (see
Section 6.1.2), we have:
lim
Δt→+0
E
0
(f(W (t) −f (W (t − Δt)))
W (t) − W (t − Δt)
Δt
θ(l)
= lim
Δt→+0
E
0
([f
(W (t − Δt))Δt
+ (gradf(W (t − Δt)))(W (t) −W (t − Δt))]
W (t) − W (t − Δt)
Δt
θ(l))
= E
0
(gradf(W (t))θ(l))
=
[0,l]×R
n
[(gradf(W (t)))θ(l)]ρ
W
dλ
=
[0,l]×R
n
f(W (t))
−
gradρ
W
ρ
W
θ(l)
ρ
W
dλ
+
[0,l]×R
n
f(W (t))
−
gradE
W
t
(θ(l))
θ(l)
θ(l)
ρ
W
dλ
= E
0
f(W (t))
−
gradρ
W
ρ
W
θ(l)
−E
0
f(W (t))
θ(l)
−1
gradE
W
t
(θ(l))
θ(l)
= E
0
f(ζ(t))
W (t)
t
θ(l)
− E
0
f(ζ(t))
θ(l)
−1
gradE
W
t
(θ(l))
θ(l)
,
where −
gradρ
W
ρ
W
=
W (t)
t
by Lemma 8.22.
Lemma 8.35 The following formulae hold:
D
∗
ξ(t)=E
ξ
t
(a(t)) +
ξ(t)
t
− E
ξ
t
(κ(t)), (8.36)
D
ξ
∗
w(t)=
ξ(t)
t
− E
ξ
t
(κ(t)), (8.37)
where κ(t)=θ(l)
−1
gradE
W
t
(θ(l)).
Proof. From the above formulae it follows that
8.3 Calculation of Mean Derivatives for Itˆo Processes 207
D
∗
ξ(t)=E
W
t
(θ(l))
−1
E
W
t
(θ(t)a(t)) + E
W
t
W (t)
t
θ(l)
+ E
W
t
[θ(l)
−1
gradE
W
t
(θ(l))]θ(l)
and having applied (8.31), (8.34)and(8.35) we obtain (8.36). (8.37)isa
consequence of (8.35)and(8.36).
Lemma 8.36 Let g(t) and h(t) be L
1
-stochastic processes with continuous
sample paths in R
n
defined for t ∈ [0,l) on the same probability space. Con-
sider the process E
h
t
g(t).SupposeDh(t) and D
∗
h(t) exist. Then:
(i) D
h
g(t) exists if and only if D
h
E
h
t
g(t) exists and D
h
E
h
t
g(t)=D
h
g(t);
(ii) D
h
∗
g(t) exists if and only if D
h
∗
E
h
t
g(t) exists and D
h
∗
E
h
t
g(t)=D
h
∗
g(t).
Proof. Fix an arbitrary smooth function f : R
n
→ R with compact support.
Using the equality
(E
h
t+Δt
g(t + Δt))f (h(t + Δt)) − (E
h
t
g(t))f(h(t))
= {(E
h
t+Δt
g(t + Δt) − E
h
t
g(t)}f(h(t))
+ E
h
t+Δt
g(t + Δt)){f (h(t + Δt)) − f(h(t))}
we obtain
ED
h
{(E
h
t
g(t))f(h(t))}
= lim
Δt→+0
E
E
h
t
E
h
t+Δt
g(t + Δt))f (h(t + Δt)) − (E
h
t
g(t))f(h(t))
Δt
= E((D
h
E
h
t
g(t))f(h(t))) + E((E
h
t
g(t))D
h
∗
f(h(t))),
if the limit exists (cf. [188, 190]). Note that the existence of the second sum-
mand on the right-hand side follows from the conditions of the Lemma. Thus
the limit exists if and only if D
h
E
h
t
g(t) exists. On the other hand
E(E
h
t
{(E
h
t+Δt
g(t + Δt))f (h(t + Δt)) − (E
h
t
g(t))f(h(t))})
= E(g(t + Δt)f(h(t + Δt)) − g(t)f (h(t)))
and by analogous arguments we obtain
ED
h
{(E
h
t
g(t))f(h(t))} = E((D
h
g(t))f(h(t))) + E(g(t)D
h
∗
f(h(t)))
if and only if D
h
g(t) exists. Clearly,
E((E
h
t
g(t))D
h
∗
f(h(t))) = E(g(t)D
h
∗
f(h(t))),
hence
208 8 Mean Derivatives in Linear Spaces
E((D
h
E
h
t
g(t))f(h(t))) = E((D
h
g(t))f(h(t))).
This proves (i). The proof of (ii) is analogous and is based on the equality
(E
h
t+Δt
g(t + Δt))f (h(t + Δt)) − (E
h
t
g(t))f(h(t))
= {(E
h
t+Δt
g(t + Δt)) − E
h
t
g(t)}f(h(t + Δt))
+ E
h
t
g(t)){f(h(t + Δt)) −f (h(t))}.
Lemma 8.37
(i) D
ξ
ξ(t)
t
= E
ξ
t
a(t)
t
−
ξ(t)
t
2
.
(ii) D
ξ
∗
ξ(t)
t
= E
ξ
t
a(t)
t
−
κ(t)
t
.
Lemma 8.37 is a corollary of Theorem 8.7 and Lemma 8.35. In particu-
lar, (ii) follows immediately from Theorem 8.7 and the construction of the
derivative.
Lemma 8.38 The following equalities hold:
DD
∗
ξ(t)=D
ξ
a(t)+E
ξ
t
a(t)
t
−
ξ(t)
t
2
, (8.38)
D
∗
Dξ(t)=D
ξ
∗
a(t), (8.39)
D
ξ
E
ξ
t
(κ(t)) = 0. (8.40)
Proof. To prove (8.40) we apply Lemma 8.36 and formula (8.31) as follows:
D
ξ
E
ξ
t
(κ(t))
= D
ξ
E
ξ
t
θ(l)
−1
gradE
W
t
(θ(l))
= D
ξ
θ(l)
−1
gradE
W
t
(θ(l))
= lim
Δt→+0
E
ξ
t
θ(l)
−1
grad
E
W
t+Δt
(θ(l)
− E
W
t
(θ(l)))
Δt
= E
W
t
(θ(l))
−1
lim
Δt→+0
E
W
t
θ(l)
−1
grad
E
W
t+Δt
(θ(l)
− E
W
t
(θ(l)))
Δt
θ(l)
= E
W
t
(θ(l))
−1
lim
Δt→+0
E
W
t
grad
E
W
t+Δt
(θ(l)
− E
W
t
(θ(l)))
Δt
= E
W
t
(θ(l))
−1
gradD
W
E
W
t
θ(l)
= E
W
t
(θ(l))
−1
gradD
W
θ(l)
=0.
8.4 First Order Differential Equations and Inclusions with Mean Derivatives 209
Formulae (8.38)and(8.39) follow from Lemmas 8.35–8.37,Theorem8.7 and
formula (8.40).
8.4 First Order Differential Equations and Inclusions
with Mean Derivatives
Let a(t, x)andα(t, x) be Borel mappings from [0,T]×R
n
to R
n
and to
¯
S
+
(n),
respectively. According to Definition 8.21 we call the system of the form
Dξ(t)=a(t, ξ(t)),
D
2
ξ(t)=α(t, ξ(t)),
(8.41)
a first order differential equation with forward mean derivatives.
It is clear that the first equation of (8.41) determines the drift and the
second one determines the diffusion coefficient of the process.
Definition 8.39. We say that (8.41)hasaweak solution on [0,T] with initial
condition ξ(0) = x
0
if there exists a probability space (Ω, F, P) and a process
ξ(t) given on (Ω, F, P) and taking values in R
n
such that for almost all
t ∈ [0,T] equation (8.41) is satisfied P-a.s. by ξ(t)
Later we shall need the following technical statement.
Lemma 8.40 Let α(t, x) be a jointly continuous (measurable, smooth) map-
ping from [0,T] × R
n
to S
+
(n). Then there exists a jointly continuous (mea-
surable, smooth, respectively) mapping A(t, x) from [0,T] ×R
n
to L(R
n
, R
n
)
such that for all t ∈ R, x ∈ R
n
the equality A(t, x)A
∗
(t, x)=α(t, x) holds.
Proof. Since the symmetric matrices α(t, x)fromS
+
(n) are positive definite,
all diagonal minors of α(t, x) are positive (in particular, they are not equal
to zero). Then the Gauss decomposition is valid for α(t, x)(see[239,The-
orem II.9.3]): α = ζδz,whereζ is a lower-triangular matrix with units on
the diagonal, z is an upper-triangular matrix with units on the diagonal and
δ is a diagonal matrix. In addition, the elements of matrices ζ, δ and z are
rationally expressed via the elements of α, hence if the matrices α(t, x)are
continuous (measurable, smooth) jointly in t, x, the matrices ζ, δ and z are
also continuous (measurable, smooth, respectively) jointly in variables t, x.
From the fact that the elements of α are symmetric matrices one can easily
derive that z = ζ
∗
(i.e., z equals the transpose of ζ). One can also easily see
that the elements of the diagonal matrix δ are positive. Thus the diagonal
matrix
√
δ is well-defined: its diagonal contains the square roots of the cor-
responding diagonal elements of δ. Consider the matrix A(t, x)=ζ
√
δ.By
construction A(t, x) is jointly continuous (measurable, smooth, respectively)
in t, x and A(t, x)A
∗
(t, x)=ζ(t, x)δ(t, x)z(t, x)=α(t, x).
210 8 Mean Derivatives in Linear Spaces
If α(t, x) takes values in
¯
S
+
(n) it is possible to construct continuous A(t, x)
under some stronger assumptions.
Lemma 8.41 If α(t, x) is a C
2
-smooth mapping from [0,T] ×R
n
to
¯
S
+
(n),
there exists a mapping A(t, x) from [0,T]×R
n
to the space L(R
n
, R
n
) of n×n
matrices, jointly continuous in t, x, such that for all t ∈ R and x ∈ R
n
the
equality A(t, x)A
∗
(t, x)=α(t, x) holds.
Lemma 8.41 is derived from [75, Theorem 1].
We can now prove several simple existence of solution theorems for (8.41).
Theorem 8.42 Let α(t, x) in the system (8.41) be jointly continuous in t, x
and positive definite (i.e., for all t ∈ [0,T] and x ∈ R
n
, α(t, x) belongs to
S
+
(n)). In addition, let the estimate
trα(t, x) <K(1 + x)
2
(8.42)
hold for some K>0.Leta(t, x) be Borel measurable and satisfy the estimate
a(t, x) <K(1 + x) (8.43)
for some K>0. Then for every initial condition ξ(0) = x
0
∈ R
n
equation
(8.41) has a weak solution that is well-defined on the entire interval [0,T].
Proof. Since α(t, x) is continuous and positive-definite, by Lemma 8.40 there
exists a continuous A(t, x) such that A(t, x)A
∗
(t, x)=α(t, x). Directly from
the definition of trace we obtain in this case that tr α(t, x) equals the sum of
the squares of the elements of the matrix A(t, x), i.e., it is the square of the
Euclidean norm in the space of n × n matrices. Since in a finite-dimensional
space S(n) of symmetric matrices all norms are equivalent, from condition
(8.42) it immediately follows that A(t, x) <K(1 + x)forsomeK>0.
Since α(t, x) is positive-definite, the matrix A(t, x) is invertible for all t, x.
Since a(t, x) is Borel measurable and satisfies (8.43), the pair a(t, x)and
A(t, x) satisfies [83, Theorem III.3.3] and so there exists a weak solution of
the stochastic differential equation
ξ(t)=ξ
0
+
t
0
a(s, ξ(s))ds +
t
0
A(s, ξ(s))dw(s), (8.44)
that is well-defined on the entire interval [0,T]. From Theorems 8.7 and 8.12
it follows that the solution ξ(t)of(8.44) P-a.s. satisfies (8.41).
Theorem 8.43 Let α(t, x) be C
2
-smooth, positive semi-definite (i.e., for all
t ∈ [0,T] and x ∈ R
n
, α(t, x) belongs to
¯
S
+
(n)) and satisfy (8.42).Leta(t, x)
be continuous and satisfy (8.43). Then for every initial condition ξ(0) = x
0
∈
R
n
equation (8.41) has a weak solution well-defined on the entire interval
[0,T].
8.4 First Order Differential Equations and Inclusions with Mean Derivatives 211
Proof. By Lemma 8.41 there exists a continuous mapping A(t, x)toL(R
n
, R
n
)
such that α(t, x)=A(t, x)A
∗
(t, x). As in the proof of Theorem 8.42, one can
obtain the estimate A(t, x) <K(1+x)forsomeK>0. Since a(t, x)and
A(t, x) are continuous and (8.43) holds, equation (8.44) satisfies the condi-
tions of [83, Theorem III.2.4], i.e., it has a weak solution well-defined on the
entire interval [0,T]. It is obvious that P-a.s. the solution satisfies (8.41).
Now we turn to differential inclusions with mean derivatives. We refer the
reader to Section 4.1 for the definitions and main results in the theory of
set-valued mappings that we use here.
Let a(t, x)andα(t, x) be set-valued mappings from [0,T] ×R
n
to R
n
and
to
¯
S
+
(n), respectively. The system
Dξ(t) ∈ a(t, ξ(t)),
D
2
ξ(t) ∈ α(t, ξ(t)).
(8.45)
is called the first order differential inclusion with forward mean derivatives.
Definition 8.44. We say that (8.45)hasaweak solution on [0,T] with initial
condition ξ(0) = x
0
if there exist a probability space (Ω,F, P) and a process
ξ(t) given on (Ω, F, P) and taking values in R
n
such that P-a.s. and for almost
all t (8.45) is satisfied.
Analogous definitions are also valid for inclusions with backward deriva-
tives and with current velocities.
In this section we will mainly look for weak solutions in the class of diffu-
sion type processes.
In the simplest cases the problem of the existence of weak solutions for
(8.45) can be reduced to that for (8.41). We present some examples of such
statements.
Everywhere below for the set B in R
n
or in L(R
n
, R
n
)weusethenorm
defined by the formula B =sup
y∈B
y.
Theorem 8.45 Assume that α(t, x) takes values in positive definite matrices
S
+
(n), has closed convex values, is lower semicontinuous and for every α ∈
α(t, x) the estimate
trα(t, x) <K(1 + x)
2
holds for a certain K>0.Leta(t, x) be a Borel measurable set-valued map-
ping that satisfies the estimate
a(t, x) <K(1 + x) (8.46)
for some K>0. Then for every initial condition ξ(0) = ξ
0
there exists a
weak solution of (8.45) that is well-defined on the entire interval [0,T].
Proof. Under the hypothesis, by Michael’s Theorem (Theorem 4.7)theset-
valued mapping α(t, x) has a continuous single-valued selector α(t, x). The
212 8 Mean Derivatives in Linear Spaces
Borel measurable set-valued mapping a(t, x) has a Borel measurable single-
valued selector a(t, x). Then the system
Dξ(t)=a(t, ξ(t)),
D
2
ξ(t)=α(t, ξ(t))
satisfies the conditions of Theorem 8.42 and so it has a weak solution that is
evidently a solution of (8.45).
Assume that α(t, x)anda(t, x) are lower semicontinuous, have closed con-
vex values in
¯
S
+
and satisfy the estimates from the hypothesis of Theo-
rem 8.45. Suppose in addition that it is known that a continuous selector
α(t, x)ofα(t, x) (that exists by Michael’s Theorem) is represented in the
form α(t, x)=A(t, x)A
∗
(t, x) with continuous A(t, x). Then one can easily
prove the existence of a weak solution of (8.45) by reducing the problem to
Theorem 8.43.
Theorem 8.46 Let a(t, x) be an upper semicontinuous set-valued mapping
with closed convex values from [0,T]×R
n
to R
n
and let it satisfy the estimate
a(t, x) <K(1 + x) (8.47)
for some K>0.
Let α(t, x) be an upper semicontinuous set-valued mapping with closed
convex values from [0,T] × R
n
to
¯
S
+
(n) such that for each α(t, x) ∈ α(t, x)
the estimate
tr α(t, x) <K(1 + x)
2
(8.48)
holds for some K>0.
Then for any initial condition ξ(0) = ξ
0
∈ R
n
inclusion (8.45) has a weak
solution ξ(t), well-defined on the entire interval t ∈ [0,T], that is a semi-
martingale.
Proof. For the norm in S(n) we take the restriction to S(n) of the Euclidean
norm (i.e., the square root of the sum of the squares of the elements of
a matrix) in the space L(R
n
, R
n
) isomorphic to R
n
2
. Since all norms in the
finite-dimensional space S(n) are equivalent to each other, for this norm (8.48)
is also valid, perhaps with another constant, for which we keep the notation
K.
Since a(t, x) is an upper semicontinuous set-valued mapping with closed
convex values, for any ε>0 it has an ε-approximation (see Section 4.1,in
particular Definition 4.10). We shall use the ε-approximations from Theo-
rem 4.11, i.e., for ε
i
→ 0theε
i
-approximations point-wise converge to a
Borel measurable selector of the set-valued mapping.
Choose a positive sequence ε
i
→ 0. Denote by a
i
(t, x) the continuous ε
i
-
approximations of a(t, x)inR
n
from Theorem 4.11 and by a(t, x)theBorel
measurable selector of a(t, x)towhicha
i
(t, x) converge point-wise. It is clear
8.4 First Order Differential Equations and Inclusions with Mean Derivatives 213
that all a
i
(t, x)anda(t, x) satisfy (8.47) for some constant that is bigger
than the constnant K from the condition of the Theorem. Nevertheless, for
simplicity, we shall retain the notation K for this constant.
Like a(t, x), α(t, x) has in S(n)anε-approximation from Theorem 4.11
for any ε>0sinceα(t, x) is an upper semicontinuous set-valued mapping
with closed convex values. Note that
¯
S
+
(n) is a convex set in S(n)andso
by Theorem 4.11 those approximations also take values in
¯
S
+
(n). For the se-
quence (ε
i
)(seeabove)denoteby¯α
i
(t, x)an
ε
i
2
-approximation of α(t, x). Let
α
i
(t, x)=¯α
i
(t, x)+
ε
i
4
I where I is the unit matrix. Immediately from the con-
struction it follows that α
i
(t, x), for any i, is a continuous ε
i
-approximation
of α(t, x) and that at any (t, x) it belongs to S
+
(n), i.e., it is strictly positive
definite. In addition, α
i
(t, x) satisfies (8.48) where the constant K>0is
bigger than the constant from the hypothesis of the Theorem. Also, by con-
struction, the sequence α
i
(t, x) point-wise converges to a Borel measurable
selector α(t, x)ofα(t, x).
By Lemma 8.40 for any i there exists a continuous A
i
(t, x) such that
A
i
(t, x)A
∗
i
(t, x)=α
i
(t, x). Directly from the definition of trace we obtain
that tr α
i
(t, x) is equal to the sum of the squares of the elements of A
i
(t, x),
i.e., it is the square of the Euclidean norm in L(R
n
, R
n
). Hence from (8.48)
it follows that A
i
(t, x) <K(1 + x)forsomeK>0.
Thus the stochastic differential equation
ξ(t)=ξ
0
+
t
0
a
i
(s, ξ(s))ds +
t
0
A
i
(s, ξ(s))dw(s) (8.49)
satisfies the hypothesis of Theorem 6.26 and so it has a weak solution that is
well-defined on the entire interval [0,T]. Denote this solution by ξ
i
(t).
Below in this section we use the measure space (
˜
Ω,
˜
F) and the family
˜
P
t
of σ-subalgebras of
˜
F introduced in Section 6.1.1. On the measure space
([0,T], B), where B is the Borel σ-algebra, we denote the Lebesgue measure
by λ
1
.
The process ξ
i
(t) determines a measure μ
i
on (
˜
Ω,
˜
F). On the probability
space (
˜
Ω,
˜
F,μ
i
) the process ξ
i
(t) is the coordinate process, i.e., ξ
i
(t, x(·)) =
x(t), x(·) ∈
˜
Ω. In addition it is clear that Lemma 6.28 is valid for measures
μ
i
and so the set of measures {μ
i
} is weakly compact, i.e., it is possible to
select a subsequence weakly convergent to some measure μ. Denote by ξ(t)
the coordinate process on the probability space (
˜
Ω,
˜
F,μ).
Define the measures ν
i
on (
˜
Ω,
˜
F) by the relations dν
i
=(1+x(·))dμ
i
.
By Lemma 6.29 these measures weakly converge to the measure ν given by
the relation dν =(1+x(·))dμ.
Since the sequence a
i
(t, x(·)) converges to a(t, x(·)) point-wise, it converges
almost surely with respect to all λ×μ
k
and so the functions
a
i
(t,x(·))
1+x(·)
converge
to
a(t,x(·))
1+x(·)
almost surely with respect to all λ × ν
k
.
Let δ>0. By Egorov’s theorem (see, e.g., [235]) for every k there exists
a subset
˜
K
k
δ
⊂ [0; T ] ×
˜
Ω such that (λ × ν
k
)(
˜
K
k
δ
) > 1 − δ and the sequence
214 8 Mean Derivatives in Linear Spaces
a
i
(t,x(·))
1+x(·)
converges to
a(t,x(·))
1+x(·)
on
˜
K
k
δ
uniformly. Let
˜
K
δ
=
∞
k=0
˜
K
k
δ
. Then the
sequence
a
i
(t,x(·))
1+x(·)
converges on
˜
K
δ
to
a(t,x(·))
1+x(·)
uniformly and (λ ×ν
k
)(
˜
K
δ
) >
(λ ×ν
k
)([0; T ] ×
˜
Ω) − δ for all k =0,...,∞ where ν
∞
= ν.
Note that a(t, x(·)) is continuous on a set of full measure λ×ν on [0; T ]×
˜
Ω.
Indeed, consider a sequence δ
k
→ 0 and the corresponding sequence
˜
K
δ
k
from Egorov’s theorem. From the above arguments we see that a(t, x(·)) is
a uniform limit of continuous functions on every
˜
K
δ
k
. Thus this mapping is
continuous on every
˜
K
δ
k
and so on each finite unit
n
k=1
˜
K
δ
i
. We have lim
n→∞
(λ×
ν)(
n
k=1
˜
K
δ
k
)=(λ ×ν)([0; T ] ×
˜
Ω). Thus
a(t,x(·))
1+x(·)
is continuous on a set of full
measure λ ×ν on [0; T ] ×
˜
Ω.
Let g
t
(x(·)) be an arbitrary continuous bounded function on
˜
Ω that is
˜
P
t
measurable. In particular let |g
t
(x(·))| <Ξ for all x(·)from
˜
Ω.
From the above-mentioned uniform convergence of
a
i
(t,x(·))
1+x(·)
to
a(t,x(·))
1+x(·)
on
˜
K
δ
for all k and from the boundedness of g
t
we obtain that for i large enough
˜
K
δ
t+Δt
t
(a
i
(τ,x(·)) − a(τ,x(·)))dτ
g
t
(x(·))dμ
k
=
˜
K
δ
t+Δt
t
a
i
(τ,x(·)) − a(τ,x(·))
1+x(·)
dτ
g
t
(x(·))dν
k
<δ
uniformly for all k. Since (λ × μ
k
)(
˜
K
δ
) > 1 − δ for all k,
a
i
(t,x(·))
1+x(·)
<K
by (8.47) for all i =0, 1,...,∞ (where i = ∞ corresponds to a) and since
|g
t
(x(·))| <Ξ, we obtain
˜
Ω\
˜
K
δ
t+Δt
t
(a
i
(τ,x(·)) − a(τ,x(·)))dτ
g
t
(x(·))dμ
k
=
˜
Ω
\
˜
K
δ
t+Δt
t
a
i
(τ,x(·)) − a(τ,x(·))
1+x(·)
dτ
g
t
(x(·))dν
k
< 2QΞδ.
From the last two formulae it follows that for k large enough
˜
Ω
t+Δt
t
(a
k
(τ,x(·)) − a(τ,x(·)))dτ
g
t
(x(·))dμ
k
<δ(2QΞ +1)
From the fact that δ is an arbitrary number it follows that
8.4 First Order Differential Equations and Inclusions with Mean Derivatives 215
lim
k→∞
˜
Ω
t+Δt
t
a
k
(τ,x(·))dτ −
t+Δt
t
a(τ,x(·))dτ
g
t
(x(·))dμ
k
=0.
(8.50)
The function
a(t,x(·))
1+x(·)
is continuous λ ×ν-a.s. (see above) and bounded on
[0; T ] ×
˜
Ω. Hence by a lemma from [82, Section VI.4] we derive from the weak
convergence of the measures ν
k
to ν that
lim
k→∞
˜
Ω
t+Δt
t
a(τ,x(·))dτ
g
t
(x(·))dμ
k
= lim
k→∞
˜
Ω
t+Δt
t
a(τ,x(·))
1+x(·)
dτ
g
t
(x(·))dν
k
=
˜
Ω
t+Δt
t
a(τ,x(·))
1+x(·)
dτ
g
t
(x(·))dν
=
˜
Ω
t+Δt
t
a(τ,x(·))dτ
g
t
(x(·))dμ. (8.51)
Using the same arguments as above, we obtain
lim
i→∞
˜
Ω
(x(t + Δt) − x(t))g
t
(x(·))dμ
i
= lim
i→∞
˜
Ω
x(t + Δt) − x(t)
1+x(·)
g
t
(x(·))dν
i
=
˜
Ω
x(t + Δt) − x(t)
1+x(·)
g
t
(x(·))dν
=
˜
Ω
(x(t + Δt) − x(t))g
t
(x(·))dμ. (8.52)
Recall that
˜
Ω
x(t + Δt) − x(t) −
t+Δt
t
a
i
(τ,x(·))dτ
g
t
(x(·))dμ
i
= E
ξ
i
(t + Δt) − ξ
i
(t) −
t+Δt
t
a
k
(τ,ξ
i
(τ))dτ
g
t
(ξ
i
(·))
!
= 0 (8.53)
since ξ
i
(t) is a solution of (8.49)andg
t
(ξ
i
(·)) is independent from ξ
i
(t+Δt)−
ξ
i
(t) −
t+Δt
t
a
k
(τ,ξ
i
(τ))dτ .