1. Introduction
The paper firstly demonstrates convergence theorem that set-valued Superpramart is in the sense of weak convergence under the X* separable condition. On this basis, using support function and results about real- valued Superpramart, we give a class of Riesz decomposition of set-valued Superpramart.
2. Method
Assume (X,‖・‖) as a separable Banach space, D1 is X-fan subset of the columns that can be condensed. X* is the dual space X. X* is separable. D* = is X*-fan subset of the columns that can be condensed, remember
Any, define
If for any
, we call An weak convergence in A, denote as
, or
.
If, then we call
Kuratowski-Mosco significance of convergence in A, denote as (K-M)
, or
.
Assume (W,G,P) is a complete probability space. {Gn, n ≥ 1} is the G’s rise, and G = ÚGn,
indicates stop (bounded stopping),
.
, said the set value is mapped
to a random set (or on G measurable). If for any open set G, it has
. We define {Fn, Gn, n ≥1}
as adapted random set columns. If, Fn can be measured by Gn. If
, F is bounded integrable.
represents that the value of
is all the integrable bounded random set.
In order to write simply, often eliminating the almost certainly established under the meaning of the equations, inequalities and tag contain relations sense “a.s.”, {Fn, Gn, n ≥ 1}. {xn, Gn, n ≥ 1} is often referred as, {xn, n ≥ 1}.
Definition 1 Supposing is a real-valued integrable adapted column
1) If, call
as Subpramart.
2) If, call
as Superpramart.
Definition 2 Supposing is a
valued adapted random set column
If, call
as set-valued Superpramart.
Definition 3 Supposing we call A and B are homothetic, if
exists, then it has A = B + C.
Definition 4 We call set-valued Superpramart has Riesz decomposition, if set-valued martingale
and set-valued Superpramart
exist,
lets
.
Lemma 1 [11] If is set-valued Superpramart, then
1),
is real-valued Superpramart.
2),
is real-valued Subpramart.
Lemma 2 Supposing is real-valued Superpramart and
, then
exist and is integrable.
Proof: is the real-valued Subpramart, and reference [12] theorem 5 (corollary 1) is known.
Lemma 3 [7] Supposing, if
1);
2) are limited existing, if it has
let
.
Lemma 4 Supposing is set-valued Superpramart and
, then it has ran-
dom set, let
.
Proof:, we know
is a real-valued Superpramart from reference [11] theorem 3.1,
and because, we know
exists and is limited from refer-
ence [12] theorem 5, and through the list of D*, we know exists and is limited in little-known set of N,
,
, by the maximum inequality and Lemma 3, the existence of F lets
, then by reference [2] corollary 2.1.1 and theorem 2.1.19, we know F is a random set, the conclusion is proved.
Lemma 5 Supposing is a real-valued Superpramart, and
, then it has an unique fac-
torization, where
is a real-valued martingale,
is a real-valued Superpramart, and
.
Proof: From Lemma 2, we know, noting
, it’s easy to find taht
is a real-valued martingale, making
, it’s easy to know
is a real-valued Superpramart,
and, it also has
.
The uniqueness is proved by the following: Supposing, so
, because
(i = 1, 2) is a real-valued martingale,
is a real-valued martingale from the above equation, then
, (from reference [12] theorem 7), so
, the uniqueness is proved.
Lemma 6 Supposing is a set-valued Superpramart, and
, if
, then
.
Proof: From Lemma 4, we know the random set exists, then it has
, and
noting, from reference [11] theorem 3.1, we know
is a real-valued
consistent Subpramart, then from reference [2] Lemma 4.4.2, we know the little-known set N exists,
,
, From inequality
and the usual density method, we known
,
, it indicates
, then
.
Lemma 7 Supposing is a set-valued Superpramart, and
, the followings are equivalent:
1) can be the Riesz decomposition.
2), Fn and E(F|Gn)(n ≥ 1) are homothetic, where
Proof: We prove firstly, because
, it’s easy to know
, by the lemma Fatou, we know
, then
.
1) 2) because
,
,
Then, S(x*,Fn) = S(x*,Gn) + S(x*,Zn).
From Lemma 1, Lemma 6 and reference [2] lemma 4.1.3, it’s easy to know the above equation is the Riesz decomposition of real-valued Superpramart, and from Lemma 5 and its proof process, we notice
and know
, by the separability of X* and the continuity of X* support function, we know from reference [2] corollary 1.4.1 that:
, namely, Fn and E(F|Gn) are homothetic.
2) 1) Noting
E(F|Gn), it’s easy to know {Gn, n ≥ 1} is the value martingale of
, and
, making
, the following is the proof that
is the value Super-
pramart of, because
S(x*,Fn) = S(x*,Gn) + S(x*,Zn)
S(x*,Zn) = S(x*,Fn) – S(x*,Gn)
It’s easy to prove, so
.
E(Ft|Gs) = Gs + E(Zt|Gs), sÎT, tÎT (s), from reference [2] lemma 5.3.6, we know
Then, we know is set-valued Superpramart the proof is set below
, because
S(xi*,Zn) = S(xi*,Fn) – S(xi*,Gn), x*iÎD*
=S(xi*,Fn) – S(xi*,E(F|Gn))
=S(xi*,Fn) – E(S (xi*,F)|Gn), and from the list of D*, we know the little-known set N1, and,
,
, using Lemma 3, we know
,
Noting, from reference [11] lemma 3.2, we know
is a real-valued
consistent Subpramart, and from reference [2] lemma 4.4.2, we know the little-known set N2 exists,
,
, from inequality
and the usual density method, we know
,
, then from reference [2] lemma 4.5.4,
.
3. Conclusion
The paper proves the convergence theorem of Superpramart in the sense of weak convergence. And on the basis of this certificate, through the support function and the results of real-valued Superpramart, we give the one of Riesz decomposition forms of set-valued Superpramart. It provides new ideas for the research of Riesz decomposition.
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