Applied Mathematics E-Notes, 14(2014), 135-143 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
On The Solution Sets Of Semicontinuous Quantum
Stochastic Di¤erential Inclusions
Michael Oluniyi Ogundirany, Victor Folarin Paynez
Received 1 June 2014
Abstract
The aim of this paper is to provide a uni…ed treatment of the existence of
solution of both upper and lower semicontinuous quantum stochastic di¤erential
inclusions. The quantum stochastic di¤erential inclusion is driven by operatorvalued stochastic processes lying in certain metrizable locally convex space. The
uni…cation of solution sets to these two discontinuous non-commutative stochastic
di¤erential inclusions is established via the existence of directionally continuous
selections.
1
Introduction
Existence results for the solutions of quantum stochastic di¤erential inclusions of Hudson and Parthasarathy quantum stochastic calculus was established in [9]. The Topological properties of solution sets for this Lipschitzian quantum stochastic di¤erential
inclusions were established in [2]. The cases of coe¢ cients that are discontinuous multivalued stochastic processes were established in [14, 15, 16]. In [15] the existence of
solutions for upper semicontinuous was established via Fixed point theorem while in
[14] the multivlaued stochastic processes possess minimal selections. The existence of
solution for the case of Lower semicontinuous multivlaued stochastic processes were
established in [16] via continuous selection of some prede…ned integral operators. The
extension of quantum stochastic di¤erential inclusions to discontinuous cases was essentially to enhance further applications of the rich quantum stochastic calculus to
quantum stochastic control theory and evolutions. The quantum stochastic di¤erential inclusions considered in [9] have Lipschitzian coe¢ cients de…ned on certain locally
convex space and in [10] more locally convex spaces were considered. By employing
one of the locally convex spaces de…ned in [10], a uni…ed treatment of upper and lower
semicontinuous cases in this work was established.
For classical di¤erential inclusions, the solution sets of upper and lower semicontinuous di¤erential inclusions were considered via a directionally continuous selection in
[4]. This directionally continuous selection which is a non-convex analogue of Michael
selection was …rst considered in [3] for …nite dimensional case and for an arbitrary Banach space in [6]. A more general case was established in [5], which shall be employed
Mathematics Subject Classi…cations: 81S25, 34A37.
of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria
z Department of Mathematics, University of Ibadan, Ibadan, Nigeria
y Department
135
136
On the Solution Sets of Di¤erential Inclusions
in our work. The paracompact property of the locally convex space which is the domain of our multivalued stochastic processes in this work guaranteed the existence of
directional continuous selection which provides a link between upper and lower semicontinuous multifunctions. In the sequel, the work shall be as follows; in section 2,
preliminaries on quantum stochastic di¤erential inclusions shall be given. In section 3,
our main result shall be proved.
2
Preliminaries
In this section, we shall adopt the notations in [10]. Let D be some pre-Hilbert space
whose completion is R; is a …xed Hilbert space and L2 (R+ ) is the space of square
integrable -valued maps on R+ . The inner product of the Hilbert space R (L2 (R+ ))
will be denoted by h:; :i and k : k the norm induced by h:; :i. Let E be linear space
generated by the exponential vectors in Fock space (L2 (R+ )) and (D E)1 be the
set of all sequences = f n g1
= f n g1
n=1 and
n=1 of members of D E, such that
1
+
j
h
;
x
i
j<
1,
8x
2
A,
where
A
L
((D
E)1 ; R
(L2 (R+ ))). Then the
n
n
w
n=1
family of seminorms fk : k ; ; 2 (D E)1 g, where
kxk
=
1
X
n=1
jh
n ; x n ij
for x 2 A;
generates a -weak topology, denoted by
w [10]. The completion of (A;
w ) is
e
e
denoted by A: The underlying elements of A consist of linear maps from (D E)1 into
R
(L2 (R+ )) having domains of their adjoints containing (D E)1 .
REMARK 1. By Theorem V.5 [18], we remark that the -weak topology w is
metrizable since (D E)1 has a countable base, hence Ae is a paracompact space [13].
e L1 (R+ ) and Lp (I A)
e are
For a …xed Hilbert space , the spaces Lploc (A),
;loc
loc
adopted as in [10]. For a topological space N , let clos(N ) be the collection of all
e as
nonempty closed subsets of N ; we shall employ the Hausdor¤ topology on clos(A)
de…ned in [9]. Moreover, for A; B 2 clos(C) and x 2 C, a complex number, we de…ne
the Hausdor¤ distance, (A; B) as
d(x; B)
Then
inf jx
y2B
yj ,
(A; B)
sup d(x; B), and
(A; B)
max( (A; B); (B; A)):
x2A
is a metric on clos(C) and induces a metric topology on the space.
DEFINITION 1.
(a) By a multivalued stochastic process indexed by I = [0; T ]
e
multifunction on I with values in clos(A).
R+ , we mean a
(b) If is a multivalued stochastic process indexed by I R+ , then a selection of
is a stochastic process X : I ! Ae with the property that X(t) 2 (t) for almost
all t 2 I.
M. O. Ogundiran and V. F. Payne
137
(c) A multivalued stochastic process
(i) adapted if
will be called
Aet for each t 2 R+ ;
(t)
e
(ii) measurable if t 7! d (x; (t)) is measurable for arbitrary x 2 A,
D E;
(iii) locally absolutely p-integrable if t 7!k
arbitrary ; 2 D E.
;
2
e for
(t) k , t 2 R+ , lies in Lploc (A)
(d) The set of all absolutely p-integrable multivalued stochastic processes will be
e mvs and for p 2 (0; 1), Lp (I A)
e mvs is the set of maps
denoted by Lploc (A)
loc
e such that t 7! (t; X(t)), t 2 I lies in Lp (A)
e mvs for every
: I Ae ! clos(A)
loc
p
e
X 2 L (A):
loc
e and (0; x0 ) be a …xed point
Consider stochastic processes E; F; G; H 2 L2loc (I A)
e
in [0; T ] A. Then, a relation of the form
Z t
X(t) 2 x0 +
(E(s; X(s))d (s) + F (s; X(s))dAf (s)
0
+ G(s; X(s))dA+
g (s) + H(s; X(s))ds for t 2 [0; T ]
will be called a stochastic integral inclusion with coe¢ cients E; F; and G and H:
The stochastic di¤erential inclusion corresponding to the integral inclusion above is
dX(t) 2 E(t; X(t))d
+
(t) + F (t; X(t))dAf (t)
G(t; X(t))dA+
g (t)
+ H(t; X(t))dt;
(1)
X(0) = x0 for almost all t 2 [0; T ]:
2
Let P : [0; T ] Ae ! 2sesq(D E) be sesquilinear form valued stochastic process de…ned
in [9] in terms of E; F; G; H by using the matrix elements in Hudson and Parthasarathy
quantum stochastic calculus [12], it was established that problem (1) is equivalent to
d
h ; X(t) i 2 P(t; X(t))( ; );
dt
X(0) = x0 for almost all t 2 [0; T ]:
(2)
As explained in [9], the map P is such that:
e h ; x i)
P(t; x)( ; ) 6= P(t;
e de…ned on I C for t 2 I, x 2 A,
e ; 2 D E:
for some complex-valued multifunction P
The notion of solution of (1) or equivalently (2) is de…ned as follows:
DEFINITION 2. By a solution of (1) or equivalently (2), we mean a stochastic
e wac \ L2 (A)
e such that
process ' 2 Ad(A)
loc
d'(t) 2 E(t; '(t))d
+
(t) + F (t; '(t))dAf (t)
G(t; '(t))dA+
g (t)
'(t0 ) = '0 ;
+ H(t; '(t))dt for almost all t 2 I;
138
On the Solution Sets of Di¤erential Inclusions
or equivalently
d
h ; '(t) i 2 P(t; '(t))( ; ):
dt
'(t0 ) = '0 ;
for arbitrary ; 2 (D E)1 , almost all t 2 I. A multivalued stochastic process
e
1
e
: I Ae ! 2A is said to be lower semicontinuous if for every open set V
A,
(V )
e
A
e
is open. Also, : I A ! 2 is said to be upper semicontinuous if, for every x 2 Ae
and > 0, there exists > 0 such that
d ((t1 ; x); (t2 ; y)) <
where
and
=)
(t2 ; y)
n
d ((t1 ; x); (t2 ; y)) = max jt1
n
B( (t1 ; x); ) = (t; z) 2 I
Ae : jt
!0
meas(J \ [t
2
t2 j ; kx
t1 j <
Let meas(J) be the Lebesgue measure of a set J
lim
B( (t1 ; x); );
yk
and kz
o
xk
<
o
:
R, t is a point of density for J if
; t + ])
= 1:
e
It follows from the previous works; [15], [16] and [14], that if E; F; G; H 2 L2loc (I A)
are upper semicontinuous (resp. lower semicontinuous) then the equivalent sesquilinear
form valued stochastic process P is upper semicontinuous (resp. lower semicontinuous).
e
We consider a topology + on I Ae stronger than the usual metric topology of I A:
+
A topology
is said to satisfy a property (P ):
(P ) For every pair of sets A B with A closed and B open (in the original topology)
there exists a set C closed-open with respect to the topology + such that A
C B
Let I = [a; b] and
I
neighbourhoods for a topology
property P: For every (t; x) 2
n
V (t; x; ) = (s; y) 2
e the following set de…ned in [7] is a basis of open
A,
+
on
stronger than the metric one, and satis…es
and > 0,
o
: t s < t + and ky xk
M (s t) :
Moreover, each set V (t; x; ) is closed-open in the topology
3
+
:
Main Results
The following Lemma shall be employed in the proof of the main result.
M. O. Ogundiran and V. F. Payne
139
LEMMA 1. Let X(:) be a Caratheodory solution of upper (lower) semicontinuous
quantum stochastic di¤erential inclusion
d
h ; X(t) i 2 (t; X(t))( ; ) on [a; b]:
dt
Assume that J is the set of times t 2 [a; b] such that
(i)
d
h ; X(t) i 2 (t; X(t))( ; ):
dt
(ii) If there exists a sequence tk ; strictly decreasing to t, with
d
d
d
h ; X(tk ) i 2 h ; X(t) i and h ; X(tk ) i 2 (tk ; X(tk ))( ; )
dt
dt
dt
for any k and ; 2 (D E)1 .
Then meas(J) = b
a:
PROOF. Let J1 be the set of times where (i) holds. Since X is a caratheodory
d
h ; X(t) i is measurable,
solution, then meas(J1 ) = b a: Fix any > 0, since dt
by Lusin’s theorem there exists a weakly continuous stochastic process u such that
d
h ; u(t) i = dt
h ; X(t) i for every t in a set J2
J1 with meas(J2 ) > b a
:
Clearly (ii) holds at every t 2 J2 which is a point of density for J2 : Hence meas(J)
meas(J2 ) > b a
, since was arbitrary, the lemma is proved.
The following is an adaptation of Theorem 1 in [5] to our non commutative setting.
THEOREM 1. Suppose the following hold:
(i) For almost all t 2 I and ; 2 D E, the maps X ! (t; X)( ; ),
2
f E; F; G; Hg are non-empty lower semicontinuous multivalued stochastic processes.
(ii) For almost all t 2 I and ; 2 D E, the maps t !
(iii)
+
is a topology on I
(t; X)( ; ) are closed.
Ae with property (P).
Then the sesquilinear form valued multifunction, (t; X(t)) ! P(t; X(t))( ; )
P(t; X(t))( ; ) = ( E)(t; X(t))( ; ) + ( F )(t; X(t))( ; )
+ ( G)(t; X(t))( ; ) + H(t; X(t))( ; )
admits a
+
-continuous selection.
PROOF. P is non-empty since each of 2 f E; F; G; Hg is non-empty then P
is a non-empty lower semicontinuous sesquilinear form-valued multifunction. We shall
employ a similar procedure as in the proof of Theorem 3.2 in [5] to construct a + continuous -approximate selections P of P, hence by inductive hypothesis we obtain
a + -continuous selection P of P: Let > 0 be …xed, since X ! P(t; X)( ; ) is lower
140
On the Solution Sets of Di¤erential Inclusions
e we choose point y
semicontinuous, for every X(t) 2 A,
neighbourhood UX of X(t) such that
y
inf
0
;P (t)2P(t;X(t ))(
; )
jy
;X (t)
y
;P (t)j
;X (t)
2 P(t; X(t))( ; ) and
for X(t0 ) 2 UX :
<
(3)
Now, let (V ) 2 be a local …nite open re…nement of (UX )X(t)2Ae, with V
UX ,
and let (W ) 2 be another open re…nement such that cl(W ) V for all 2 .
By property (P), for each , we can choose a set Z , clopen w.r.t. + , such that
cl(W )
int(Z )
cl(Z )
V :
e Let
Then (Z ) is a local …nite + clopen covering of A.
set , de…ne for each 2 ,
[
=Z n
Z :
(4)
be a well-ordering of the
<
Set O = ( ); 2 . By well-ordering, every x 2 Ae belongs to exactly one set
e Moreover, since Z
where = minf 2
: x 2 Z g. Hence, O is a partition
of A.
S
+
+
is locally …nite(wrt and therefore wrt
), the sets
clopen. Hence
< Z are
O is a + clopen disjoint covering of Ae such that, fcl( )g re…nes (V ) . By setting
y ; = y ;X and P (t; X(t))( ; ) = y ;X , 8 2 ; we have + continuous function
P , which by (3), satis…es
y
inf
;P (t)2P(t;X(t))(
; )
jP (t; X(t))( ; )
y
;P (t)j
< :
Therefore, there exists an -approximate selection P of P. Since was arbitrarily
chosen,thus we have a + -continuous selection P of P.
Let P : I Ae ! sesq(D E)21 be sesquilinear form -valued directionally continuous
map as de…ned above. The upper semicontinuous, convex valued regularization of
P , corresponding to a given ; 2 (D E)1 is de…ned as
o
\ n
R(t; x)( ; ) =
co P (s; y)( ; ) : jt sj < and kx yk <
:
(5)
>0
e and let P : I Ae ! 2sesq(D E)1 )
THEOREM 2. Let be a closed subset of I A,
be a bounded, lower semicontinuous multifunction. Then there exists an upper semicontinuous map R :
! 2sesq(D E)1 ) with compact convex values such that every
Caratheodory solution of
is also a solution of
d
h ; X(t) i 2 P(t; X(t))( ; )
dt
(6)
d
h ; X(t) i 2 R(t; X(t))( ; ):
dt
(7)
d
PROOF. Let X(t) be a Caratheodory solution of dt
h ; X(t) i 2 R(t; X(t))( ; ) on
[a; b]: De…ne J [a; b] to be the set of times t such that
M. O. Ogundiran and V. F. Payne
(i)
d
dt h
141
; X(t) i 2 R(t; X(t))( ; ).
(ii) There exists a sequence of times tk strictly decreasing to t such that
d
d
R(tk ; X(tk ))( ; ) and dt
h ; X(tk ) i ! dt
h ; X(t) i.
d
dt h
; X(tk ) i 2
By Lemma 1 above, J has a full measure in [a; b]: We claim that
P (t; X(t))( ; ) for every t 2 J: Assume on the contrary that t 2 J but
d
dt h
; X(t) i =
d
h ; X(t) i
dt
P (t; X(t))( ; ) = > 0:
Using the directional continuity of P at the point (t; X), choose
jP (s; y)( ; )
P (t; X(t))( ; )j <
(8)
> 0 such that
(9)
2
whenever t
s < t + , k y X(t) k
M (s t): Let tk ! t be a sequence with
properties stated in (ii), then there exists k large enough so that 0 < tk t < and
d
h ; X(tk ) i
dt
d
h ; X(t) i < :
dt
2
(10)
The boundedness assumption j P (t; X)( ; ) j< L implies that R(t; X)( ; ) B(0; L)
for all (t; X): Our solution X(t) is therefore Lipschitz continuous with constant L: In
particular,
kX(tk ) X(t)k
L(tk t) < M (tk t):
Then we conclude that
R(tk ; X(tk ))
B P (t; X(t))( ; );
2
:
(11)
:
(12)
Hence
d
h ; X(tk ) i
dt
P (t; X(t))( ; )
2
Comparing we obtain a contradiction, which proves that the caratheodory solutions of
d
h ; X(t) i = P (t; X(t))( ; )
dt
and (7) coincide. Now since P is bounded we can assume P (t; X)( ; )
B(0; L)
for some constant L and all (t; X) 2 : Choose M > L and let P be + -continuous
selection of P; by Theorem 1 above, such a P exists. Then if R is the regularization
multivalued stochastic process as de…ned above, R is upper semicontinuous compact
convex-valued [1]. Let now X(:) be a Caratheodory solution of (7) on [a; b] since P is
a selection of P, then X(:) is also a solution of (6).
142
On the Solution Sets of Di¤erential Inclusions
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