Riccati Equations, and Applications
Michael Kohlmann y
Shanjian Tang z
September 7, 2000
Abstract
Multi-dimensional backward stochastic Riccati dierential equations (BSRDEs
in short) are studied. A closed property for solutions of BSRDEs with respect to
theircoeÆcientsisstatedandisprovedforgeneralBSRDEs,whichisusedtoobtain
theexistence of a global adapted solution to some BSRDEs. The global existence
and uniqueness results are obtained for two classes of BSRDEs, whose generators
containaquadratictermof L(thesecond unknowncomponent). Morespecically,
thetwo classesofBSRDEs are(for theregularcase N >0)
(
dK = [A
K+KA+Q LD(N +D
KD) 1
D
L]dt+Ldw;
K(T) = M
and (forthesingularcase)
8
>
<
>
:
dK = [A
K+KA+C
KC+Q+C
L+LC
(KB+C
KD+LD)(D
KD) 1
(KB+C
KD+LD)
]dt+Ldw;
K(T) = M:
This partially solves Bismut-Peng's problem which was initiallyproposed by Bis-
mut (1978) in the Springer yellow book LNM 649. The arguments given in this
paperarecompletely new,and they consist of some simpletechniques of algebraic
transformations and direct applications of the closed property mentioned above.
We makefulluseofthe specialstructure(thenonnegativityof thequadraticterm,
forexample)oftheunderlyingRiccatiequation. Applicationsinoptimalstochastic
controlareexposed.
Key words: backward stochasticRiccatiequation,stochasticlinear-quadraticcon-
trolproblem,algebraictransformation, Feynman-Kac formula
AMS Subject Classications. 90A09, 90A46, 93E20, 60G48
Abbreviated title: Multi-dimensionalbackward stochastic Riccatiequation
Bothauthors gratefullyacknowledge the support bythe CenterofFinance and Econo-
metrics,University of Konstanz.
y
DepartmentofMathematicsandStatistics,UniversityofKonstanz,D-78457,Konstanz,Germany
z
DepartmentofMathematics,FudanUniversity,Shanghai200433,China. Thisauthorissupported
byaResearchFellowshipfromtheAlexandervonHumboldtFoundationandbytheNational
Natural ScienceFoundation ofChina underGrantNo. 79790130.
Let(;F;P;fF
t g
t0
)beaxedcompleteprobabilityspaceonwhichisdenedastandard
d-dimensional F
t
-adapted Brownian motion w(t) (w
1
(t);;w
d (t))
. Assume that
F
t
is the completion, by the totality N of all null sets of F, of the natural ltration
fF w
t
g generated by w. Denote by fF 2
t
;0 t Tg the P-augmented natural ltration
generated by the (d d
0
)-dimensional Brownian motion (w
d
0 +1
;:::;w
d
). Assume that
all the coeÆcients A;B;C
i
;D
i
are F
t
-progressively measurable bounded matrix-valued
processes, dened on [0;T]; of dimensions nn;nm;nn;n m respectively.
AlsoassumethatM isanF
T
-measurablenonnegativebounded nn randommatrix,and
Qand N are F
t
-progressively measurable,bounded, nonnegativeand uniformlypositive,
nn and mm matrix processes, respectively.
Consider the following backward stochastic Riccati dierential equation
(BSRDE inshort):
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
dK = [A
K+KA+ d
X
i=1 C
i KC
i
+Q+ d
X
i=1 (C
i L
i +L
i C
i )
(KB+ d
X
i=1 C
i KD
i +
d
X
i=1 L
i D
i )(N +
d
X
i=1 D
i KD
i )
1
(KB+
d
X
i=1 C
i KD
i +
d
X
i=1 L
i D
i )
]dt+ d
X
i=1 L
i dw
i
; 0t<T;
K(T) = M:
(1)
ItwillbecalledtheBSRDE(A;B;C
i
;D
i
;i=1;:::;d;Q;N;M)inthefollowingforconve-
nienceofindicatingtheassociatedcoeÆcients. WhenthecoeÆcientsA;B;C
i
;D
i
;Q;N;M
arealldeterministic,thenL
1
==L
d
=0and theBSRDE (1)reduces tothe following
nonlinear matrix ordinary dierentialequation:
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:
dK = [A
K+KA+ d
X
i=1 C
i KC
i
+Q (KB+ d
X
i=1 C
i KD
i )
(N + d
X
i=1 D
i KD
i )
1
(KB+ d
X
i=1 C
i KD
i )
]dt;
0t<T;
K(T) = M;
(2)
which was completely solved by Wonham [28] by applying Bellman's principle of quasi-
linearization and a monotone convergence approach. Bismut [2, 3] initially studied the
case ofrandom coeÆcients,but he could solve onlysome special simplecases. He always
assumedthattherandomnessofthecoeÆcientsonlycomesfromasmallerltrationfF 2
t g,
which leads to L
1
==L
d
0
=0. He further assumed in hispaper [2]that
C
d
0 +1
==C
d
=0; D
d
0 +1
==D
d
=0; (3)
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
dK = [A
K+KA+ d
0
X
i=1 C
i KC
i +Q
(KB+ d
0
X
i=1 C
i KD
i )(N +
d
0
X
i=1 D
i KD
i )
1
(KB+ d
0
X
i=1 C
i KD
i )
]dt
+ d
X
i=d0+1 L
i dw
i
; 0t <T;
K(T) = M;
(4)
and the generator does not involve L atall. In his work [3]he assumed that
D
d0+1
==D
d
=0; (5)
underwhich the BSRDE (1) becomesthe following one
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
dK = [A
K+KA+ d
X
i=1 C
i KC
i
+Q+ d
X
i=d
0 +1
(C
i L
i +L
i C
i )
(KB+ d
0
X
i=1 C
i KD
i )(N +
d
0
X
i=1 D
i KD
i )
1
(KB+ d
0
X
i=1 C
i KD
i )
]dt
+ d
X
i=d
0 +1
L
i dw
i
; 0t <T;
K(T) = M;
(6)
and the generator depends on the second unknown variable (L
d
0 +1
;:::;L
d )
in a linear
way. Moreover his method was rather complicated. Later, Peng [18] gave a nice treat-
ment on the proof of existence and uniqueness for the BSRDE (6), by using Bellman's
principleof quasi-linearizationand amethod of monotoneconvergence|a generalization
of Wonham's approachto the randomsituation.
As earlyasin1978, Bismut[3]commentedonpage220 that:"Nous ne pourronspas
demontrer l'existence de solution pour l'equation (2.49)dans le cas general." (We could
not prove the existence of solution for equation (2.49) for the general case.) On page
238,he pointed outthat the essentialdiÆcultyforsolutionof thegeneralBSRDE (1)lies
in the integrand of the martingale term which appears in the generator in a quadratic
way. Two decades later in 1998, Peng [19] included the aboveproblem inhislist ofopen
problems onBSDEs. Recently, Kohlmannand Tang [13] solved the onedimensional case
of the above Bismut-Peng's problem.
In this paper, we prove the global existence and uniqueness result for BSRDE (1)
for the following special multi-dimensionalcase:
d=1; B =C =0:
That is,we solve the followingBSRDE
8
>
<
>
:
dK = [A
K+KA+Q LD(N +D
KD) 1
D
L]dt+Ldw;
0t<T;
K(T) = M:
(7)
This result isstated as Theorem 2.3.
ConsiderthenthecasewherethecontrolweightmatrixN reducestozero. Kohlmann
and Zhou [14] discussed such a case. However, their context is rather restricted, asthey
make the following assumptions: (a) all the coeÆcients involved are deterministic; (b)
C
1
= =C
d
= 0;D
1
==D
d
=I
mm
; and M =I;(c) A+A
BB
. Their argu-
ments are based onapplyinga resultof Chen, Liand Zhou [4]. Kohlmann and Tang [12]
consideredageneral frameworkalong thoseanalogues of Bismut[3]and Peng [18], which
has the followingfeatures: (a) the coeÆcients A;B;C;D;N;Q;M are allowed to be ran-
dom,but are onlyF 2
t
-progressivelymeasurable processesorF 2
T
-measurable randomvari-
able; (b) the assumptions in Kohlmann and Zhou [14] are dispensed with orgeneralised;
(c) the condition (5) is assumed to be satised. Kohlmann and Tang [12] obtained a
general result and generalised Bismut's previous result on existence and uniqueness of a
solution of BSRDE (6) to the singular case under the following additionaltwo assump-
tions:
M "I
nn
; d
X
i=1 D
i D
i
(t)"I
mm
for some deterministicconstant ">0: (8)
KohlmannandTang[13]provedtheexistenceanduniquenessresultfortheone-dimensional
singular case N =0 under the assumption (8), but for a more generalframework of the
following features: the coeÆcientsA;B;C;D;N;Q;M are allowed tobe F
t
-progressively
measurableprocessesorF
T
-measurablerandomvariable,andthecoeÆcientDisnotnec-
essarilyzero. Inthispaperweobtaintheglobalexistenceanduniquenessforthefollowing
multi-dimensionalsingularcase:
d=1; D
D"I
mm
; M "I
nn
for some deterministicconstant ">0:
That is,we solve the followingBSRDE:
8
>
>
>
<
>
>
>
:
dK = [A
K+KA+C
KC+Q+C
L+LC
(KB+C
KD+LD)(D
KD) 1
(KB+C
KD+LD)
]dt+Ldw;
0t<T;
K(T) = M:
(9)
This result isstated as Theorem 2.2.
The BSRDE (1) arises fromsolution of the optimalcontrol problem
inf
u()2L 2
F (0;T;R
m
)
J(u;0;x) (10)
where fort 2[0;T] and x2R n
,
J(u;t;x):=E F
t
[ Z
T
t
[(Nu;u)+(QX t;x;u
;X t;x;u
)]ds+(MX t;x;u
(T);X t;x;u
(T))] (11)
and X t;x;u
()solvesthe following stochastic dierential equation
8
>
>
<
>
>
:
dX = (AX+Bu)ds+ d
X
i=1 (C
i
X+D
i u)dw
i
; tsT;
X(t) = x:
(12)
solution for the above linear-quadratic optimal control problem (LQ problem in
short) has the followingclosed form (also calledthe feedback form):
u(t)= (N + d
X
i=1 D
i KD
i )
1
[B
K+ d
X
i=1 D
i KC
i +
d
X
i=1 D
i L
i
]X(t) (13)
and the associated value function V isthe following quadraticform
V(t;x):= inf
u2L 2
F (t;T;R
m
)
J(u;t;x)=(K(t)x;x); 0tT;x2R n
: (14)
In this way, on one hand, solution of the above LQ problem is reduced to solving the
BSRDE (1). On the other hand, the formula(14) actually provides arepresentation|of
Feynman-Kac type| for the solution of BSRDE (1). The reader will see that this kind
of representation plays an important role in the proofs given here for Theorems 2.1, 2.2
and 2.3.
The arguments given in this paper are completely new. They results from two
observations. The rst one is that inthe following simple case
A=B =C =0;d=1;m=n;
Dis nonsingular, and D and N are constant matrices,
(15)
thediÆcultquadratictermofL canberemovedbydoingsome simplealgebraictransfor-
mation,and the resulting BSRDE is globally solvable in viewof the result of Bismut[3]
andPeng[18]. Asaconsequence,theabovesimplecaseisgloballysolvable. However, this
caseistoorestricted. Thencomes outthe secondobservation: byusingsome othertricks
and by applying the closedness theorem 2.1, some more general cases can be attacked.
Specically, the followingrestrictions
A=0;m =n; and D isnonsingular (16)
are allremoved, and the restricted assumption
D and N are constant matrices (17)
is improved. For the singular case, we only have the one restriction d = 1 remained.
Theorem 2.1 providesaway toobtainthe solvability of moregeneral BSRDEs fromthat
of simple ones. We hope that Bismut-Peng's problem will be completely solved in the
near future, by using the above-mentioned methodology.
The rest of the paper is organized as follows. Section 2 contains a list of notation
and two preliminarypropositions,and the statementof the main resultswhichconsist of
Theorems 2.1-2.3. The proofs of these three theorems are given in Sections 3-5, respec-
tively. Finally, in Section 6, application of Theorems 2.2 and 2.3 is given to the regular
and singularstochastic LQproblems, both with and withoutconstraints.
Let(;F;P;fF
t g
t0
)beaxedcompleteprobabilityspaceonwhichisdenedastandard
d-dimensional F
t
-adapted Brownian motion w(t) (w
1
(t);;w
d (t))
. Assume that
F
t
is the completion, by the totality N of all null sets of F, of the natural ltration
fF w
t
g generated by w. Denote by fF 2
t
;0 t Tg the P-augmented natural ltration
generated by the (d d
0
)-dimensional Brownian motion (w
d
0 +1
;:::;w
d
). Assume that
all the coeÆcients A;B;C
i
;D
i
are F
t
-progressively measurable bounded matrix-valued
processes, dened on [0;T]; of dimensions nn;nm;nn;n m respectively.
Also assume that M is an F
T
-measurable, nonnegative, and bounded n n random
matrix. Assume that Q and N are F
t
-progressively measurable, bounded, nonnegative
and uniformlypositive, nn and mm matrix processes, respectively.
Notation. Throughoutthis paper, the followingadditionalnotationwill beused:
M
: the transpose of any vector or matrix M;
jMj : =
q
P
ij m
2
ij
for any vector or matrix M =(m
ij );
(M
1
;M
2
) : the innerproduct of the two vectors M
1
and M
2
;
R n
: the n-dimensionalEuclidean space;
R
+
: the set of allnonnegative real numbers;
S n
: the Euclidean space of allnn symmetricmatrices;
S n
+
: the set of allnn nonnegative denitematrices;
C([0;T];H) : the Banachspace of H-valued continuous functions on[0;T],
endowed with the maximum norm fora given Hilbert space H;
L 2
F
(0;T;H) : the Banachspace of H-valued F
t
-adapted square-integrable
stochastic processes f on[0;T],endowed with the norm
(E R
T
0
jf(t)j 2
dt) 1=2
fora given Euclidean space H;
L 1
F
(0;T;H) : the Banachspace of H-valued, F
t
-adapted, essentially
bounded stochastic processes f on [0;T], endowed with the
norm esssup
t;!
jf(t)j for agiven Euclidean space H;
L 2
(;F;P;H) : the Banachspace of H-valued norm-square-integrable random
variableson the probabilityspace (;F;P) for agiven
Banach space H;
and L 1
(;F;P;C([0;T];R n
)) is the Banach space of C([0;T];R n
)-valued, essentially
maximum-norm-boundedrandomvariablesf ontheprobabilityspace(;F;P),endowed
with the norm esssup
!2 max
0tT
jf(t;!)j.
Proposition2.1. AssumethatallthecoeÆcientsA;B;C
i
;D
i areF
2
t
-progressively
measurableboundedmatrix-valuedprocesses, denedon[0;T];ofdimensionsnn;n
m;nn;nm respectively. Alsoassume thatM isan F 2
T
-measurable, nonnegative,and
bounded nn random matrix. Assume that Q and N are F 2
t
-progressively measurable,
bounded, nonnegative and uniformly positive, nn and mm matrix processes, respec-
tively. Then, the BSRDE (6)has a unique F 2
t
-adapted global solution (K;L)with
K 2L 1
F 2
(0;T;S n
+ )\L
1
(;F 2
T
;P;C([0;T];S n
+
)); L2L 2
F 2
(0;T;S n
):
Proposition 2.1 is due to Bismut [3] and Peng [18], and the reader is referred to
Consider the optimalcontrolproblem
inf
u()2L 2
F (0;T;R
m
)
J(u;0;x) (18)
where fort 2[0;T] and x2R n
,
J(u;t;x):=E Ft
[ Z
T
t
[(Nu;u)+(QX t;x;u
;X t;x;u
)]ds+(MX t;x;u
(T);X t;x;u
(T))] (19)
and X t;x;u
()solvesthe following stochastic dierential equation
8
>
>
<
>
>
:
dX = (AX+Bu)ds+ d
X
i=1 (C
i
X+D
i u)dw
i
; tsT;
X(t) = x:
(20)
Proposition 2.2. Let (K;L) be an F
t
-adapted solution of the BSRDE (1) with
K 2L 1
F
(0;T;S n
)\L 1
(;F
T
;P;C([0;T];S n
)); L2L 2
F
(0;T;S n
);
and N(t)+ P
d
i=1 D
i KD
i
(t) being uniformly positive. Then,
(K(t)x;x)=V(t;x):= inf
u2L 2
F (t;T;R
m
)
J(u;t;x); 8x2R n
:
This proposition is a special case of Theorem 6.1, and the reader is referred to
Section6 for the proof.
The main results of this paperare stated by the followingthree theorems.
Theorem2.1. Assumethat8 0thecoeÆcientsA
;B
;C
i
;D
i
;Q
;andN
are
F
t
-progressively measurable matrix-valued processes, dened on [0;T]; of dimensions
nn;nm;nn;nm;nn; and mm; respectively. Assume that M
is an F
T -
measurableandnonnegativennrandom matrix. Assume thatQ
isa:s:a:e: nonnegative.
Assumethattherearetwodeterministicpositiveconstants"
1 and"
2
whichareindependent
of the parameter , such that
jA
(t)j;jB
(t)j;jC
i
(t)j;jD
i
(t)j;jQ
(t)j;jN
(t)j;jM
j"
1
and
N
"
2 I
mm :
Assume that as ! 0, A
(t);B
(t);C
i (t);D
i (t);Q
(t), and N
(t) converge uniformly
in (t;!)to A 0
(t);B 0
(t);C 0
i (t);D
0
i (t);Q
0
(t) andN 0
(t), respectively. Assume thatM
uni-
formlyin!convergestoM 0
as !0. Assumethat8 >0theBSRDE(A
;B
;C
i
;D
i
;i=
1;:::;d;Q
;N
;M
) has a unique F
t
-adapted global solution (K
;L
) with
K
2L 1
F
(0;T;S n
+ )\L
1
(;F
T
;P;C([0;T];S n
+
)); L
2L 2
F
(0;T;S n
):
K 2L 1
F
(0;T;S n
+ )\L
1
(;F
T
;P;C([0;T];S n
+
)); L2L 2
F
(0;T;S n
);
such that
lim
!0 K
=K strongly in L 1
F
(0;T;S n
+ )\L
1
(;F
T
;P;C([0;T];S n
+ ));
lim
!0 L
=L strongly in L 2
F
(0;T;S n
);
(21)
and such that (K;L) is a unique F
t
-adapted solution of the
BSRDE (A 0
;B 0
;C 0
;D 0
;Q 0
;N 0
;M 0
).
If the aboveassumption of uniform convergenceof (A
;C
;Q
;M
) isreplacedwith
the followingone:
lim
!0
esssup
!2 Z
T
0 (jA
A 0
j+jC
C 0
j 2
+jQ
Q 0
j)ds+jM
M 0
j !0: (22)
then the above assertions stillhold.
Remark2.1. Whentheassumptionofuniformpositivityonthecontrolweightma-
trixN is relaxed to nonnegativity, Theorem 2.1 stillholds withthe additional assumption
that there is a deterministic positive constant "
3
such that
d
X
i=1 (D
i )
D
i "
3 I
mm
; M
"
3 I
nn :
Theorem 2.2. (the singular case) Assume that d = 1 and Q(t) 0. Also
assume that there is a deterministic positive constant "
3
such that
M "
3 I
nn
(23)
and
D
D(t)"
3 I
mm
: (24)
Then, the BSRDE (9) has a unique F
t
-adapted global solution (K;L) with
K 2L 1
F
(0;T;S n
+ )\L
1
(;F
T
;P;C([0;T];S n
+
)); L2L 2
F
(0;T;S n
);
and K(t;!) being uniformly positive w.r.t. (t;!):
Theorem 2.3. (the regular case) Assume that d = 1;M 0;Q(t) 0 and
N(t) "
3 I
mm
for some positive constant "
3
>0: Further assume that B = C =0, and
D and N satisfy the following
lim
h!0+
esssup
!2
max
t
1
;t
2 2[0;T];jt
1 t
2 jh
jD(t
1
) D(t
2
)j = 0;
lim
h!0+
esssup
!2
max
t1;t22[0;T];jt1 t2jh jN(t
1
) N(t
2
)j = 0:
(25)
Then, the BSRDE (7) has a unique F
t
-adapted global solution (K;L) with
K 2L 1
F
(0;T;S n
+ )\L
1
(;F
T
;P;C([0;T];S n
+
)); L2L 2
F
(0;T;S n
):
TheproofsoftheabovethreetheoremsaregiveninSections3,4,and5,respectively.
For8(t;K;L)2[0;T]S n
+ (S
n
) d
, write
F
(t;K;L):= [KB
(t)+ d
X
i=1 C
i (t)
KD
i (t)+
d
X
i=1 L
i D
i (t)]
[N
(t)+ d
X
i=1 D
i (t)
KD
i (t)]
1
[KB
(t)+ d
X
i=1 C
i (t)
KD
i (t)+
d
X
i=1 L
i D
i (t)]
:
(26)
The generator of the BSRDE (A
;B
;C
i
;D
i
;i=1;:::;d;Q
;N
;M
) is
G
(t;K;L) := (A
)
K+KA
+ d
X
i=1 (C
i )
KC
i +Q
+ d
X
i=1 ((C
i )
L
i +L
i C
i )+F
(t;K;L):
(27)
Wehave the followinga prioriestimates.
Lemma 3.1. Let the set of coeÆcients (A
;B
;C
i
;D
i
;i = 1;:::;d;Q
;N
;M
)
satisfytheassumptionsmadeinTheorem2.1,andlet (K
;L
)beaglobaladaptedsolution
to the BSRDE (A
;B
;C
i
;D
i
;i=1;:::;d;Q
;N
;M
) with
K
2L 1
F
(0;T;S n
)\L 1
(;F
T
;P;C([0;T];S n
)); L
2L 2
F
(0;T;S n
);
andN(t) + P
d
i=1 D
i KD
i
(t)beinguniformlypositive. Then,thereisadeterministicpositive
constant "
0
which isindependent of ; such that8 0; the following estimates hold:
0K
(t)"
0 I
nn
; E Ft
Z
T
t jL
j 2
ds
!
p
"
0
; 8p1: (28)
Proof of Lemma 3.1. From Proposition 2.2, we see that K
0. Note that
(K
;L
)satises the BSRDE:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
: dK
=
(A
)
K
+K
A
+ d
X
i=1 (C
i )
K
C
i +Q
+ d
X
i=1 ((C
i )
L
i +L
i C
i )
+F
(t;K
;L
)
dt+ d
X
i=1 L
i dw
i
; 0t <T;
K
(T) = M
:
(29)
UsingIt^o's formula, we get
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
djK
j 2
=
4tr h
(K
) 2
A
i
+ d
X
i=1
2tr [K
(C
i )
K
C
i
]+2tr (K
Q
)
+ d
X
i=1
4tr (K
L
i C
i
)+2tr [K
F
(t;K
;L
)] jL
j 2
dt
+ d
X
i=1
2tr (K
L
i )dw
i
; 0t<T;
jK
j 2
(T) = jM
j 2
:
(30)
F
(t;K
;L
)0; K
0;
we have
2tr [K
F
(t;K
;L
)]=2tr h
(K
) 1
2
F
(t;K
;L
)(K
) 1
2 i
0: (31)
Hence,
jK
j 2
(t)+ Z
T
t jL
j 2
ds jM
j 2
+ Z
T
t
4tr h
(K
) 2
A
i
+ d
X
i=1
2tr [K
(C
i )
K
C
i ]
+2tr (K
Q
)+ d
X
i=1
4tr (K
L
i C
i )
ds
Z
T
t d
X
i=1
2tr (K
L
i )dw
i
; 0t<T:
(32)
Usingthe elementaryinequality
2aba 2
+b 2
and taking the expectation onboth sides with respect toF
r
for rt, we obtain that
E Fr
jK
j 2
(t)+ 1
2 E
Fr Z
T
t jL
j 2
ds "
4 +"
4 Z
T
t E
Fr
jK
j 2
(s)ds; 0rt<T: (33)
Using Gronwall'sinequality, We derive from the lastinequality the rst one of the esti-
mates (28). In return, we derivefrom the second lastinequality that
Z
T
t jL
j 2
ds"
5 +"
5 Z
T
0 jL
jds Z
T
t d
X
i=1
2tr (K
L
i ) dw
i
: (34)
Therefore,
E Ft
Z
T
t jL
j 2
ds
!
p
3 p
"
"
p
5 +"
p
5 E
Ft Z
T
t jL
jds
!
p
+E Ft
Z
T
t d
X
i=1 2trK
L
i dw
i
p
#
:(35)
We have fromthe Burkholder-Davis-Gundy inequality the following
E Ft
Z
T
t d
X
i=1
2tr (K
L
I ) dw
i
p
2 p
E Ft
Z
T
t jK
j 2
jL
j 2
ds
p=2
;
while fromthe Cauchy-Schwarz inequality, we have
E Ft
Z
T
t jL
jds
!
p
T p=2
E Ft
Z
T
t jL
j 2
ds
!
p=2
:
Finally,we get
E F
t Z
T
t jL
j 2
ds
!
p
3 p
"
p
5 +[3
p
T p=2
"
p
5 +6
p
n p=2
"
p
0 ]E
F
t Z
T
t jL
j 2
ds
!
p=2
; (36)
Now, consider the optimalcontrolproblem
Problem P
inf
u()2L 2
F (0;T;R
m
) J
(u;0;x) (37)
where fort 2[0;T] and x2R n
,
J
(u;t;x):=E F
t
[ Z
T
t [(N
u;u)+(Q
X t;x;u
;X t;x;u
)]ds+(M
X t;x;u
(T);X t;x;u
(T))] (38)
and X t;x;u
()solvesthe following stochastic dierential equation
8
>
>
<
>
>
:
dX = (A
X+B
u)ds+ d
X
i=1 (C
i
X+D
i u)dw
i
; tsT;
X(t) = x:
(39)
The associatedvalue function V
is dened as
V
(t;x):= inf
u()2L 2
F (t;T;R
m
) J
(u;t;x): (40)
Then, fromProposition2.2, wehave
(K
(t)x;x)=V
(t;x); 8(t;x)2[0;T]R n
:
From the a priori estimates result Lemma 3.1, we have
V
(t;x)"
0 jxj
2
; 8(t;x)2[0;T]R n
:
So, the optimal control b
u
for the problemP
satises
"
2 E
Ft Z
T
t jb
u
j 2
ds=E Ft
Z
T
t (N
b
u
; b
u
)ds "
0 jxj
2
:
Set
U x
ad
(t;T):=
(
u2L 2
F
(t;T;R m
):"
2 E
Ft Z
T
t juj
2
ds "
0 jxj
2 )
; 8x2R n
: (41)
Then, wehave
V
(t;x):= inf
u()2U x
ad (t;T)
J
(u;t;x): (42)
Dene
K
:=K
K
; L
i
:=L
i L
i
; X t;x;u
:=X t;x;u
X t;x;u
;
A
:=A
A
; B
:=B
B
; C
i
:=C
i C
i
;
D
:=D
D
; Q
:=Q
Q
; N
:=N
N
;
M
:=M
M
:
(43)
deterministicpositiveconstants"
6
;"
7
, and "
8
, which are independentof the parameters
and such thatthe following three estimates hold. (i) For each x2R n
,
E Ft
max
tsT jX
t;x;u
(s)j
2
"
6 jxj
2
+"
6 E
Ft Z
T
t juj
2
ds: (44)
(ii) For each (t;x)2[0;T]R n
,
E Ft
max
tsT jX
t;x;u
(s)j
2
"
7 E
Ft Z
T
t (jA
j+jC
j 2
)jX t;x;u
(s)j
2
ds
+"
7 E
F
t Z
T
t (jB
j+jD
j 2
)juj 2
(s)ds:
(45)
(iii) For each (t;x)2[0;T]R n
,
jJ
(u;t;x) J
(u;t;x)j
"
8 E
Ft
[jM
jjX t;x;u
(T)j 2
+jX t;x;u
(T)j(jX t;x;u
(T)j+jX t;x;u
(T)j)]
+"
8 E
Ft Z
T
t jX
t;x;u
(s)j[jX t;x;u
(s)j+jX t;x;u
(s)j]ds
+"
8 E
F
t Z
T
t jQ
jjX t;x;u
(s)j
2
ds+"
8 E
F
t Z
T
t jN
jjuj 2
(s)ds:
(46)
Proof of Lemma 3.2. NotethatX t;x;u
satisesthe followingstochastic dieren-
tialequation:
8
>
>
<
>
>
: dX
= (A
X
+A
X
+B
u)ds+ d
X
i=1 (C
i X
+C
i X
+D
i
u)dw
i
;
X
(t) = 0:
So, inviewofthe assumptionsof Theorem 2.1,the rst two estimates areactually acon-
sequenceofthecontinuousdependenceupontheparametersofthesolutionofastochastic
dierentialequation, and the proof isstandard. The last estimateresults fromanimme-
diate applicationof the mean-value formulafor a dierentialfunction.
Lemma 3.3. Let the assumptions of Theorem2.1 be satised. Then, we have the
followingthree inequalities. (i) For each x2R n
;8u2U x
ad (t;T);
E F
t
max
tsT jX
t;x;u
(s)j
2
"
6 (1+"
1
2
"
0 )jxj
2
: (47)
(ii) For each (t;x)2[0;T]R n
;8u2U x
ad (t;T);
E Ft
max
tsT jX
t;x;u
(s)j
2
"
7
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! Z
T
0 (jA
j+jC
j 2
)ds
+"
7
"
1
2
"
0 jxj
2
esssup
s;!
(jB
j+jD
j 2
)(s):
(48)
(iii) For each (t;x)2[0;T]R ;8u2U
ad (t;T);
jJ
(u;t;x) J
(u;t;x)j
"
8
esssup
! jM
j E F
t
jX t;x;u
(T)j 2
+"
8 h
E Ft
jX t;x;u
(T)j
2 i
1=2 h
E Ft
(2jX t;x;u
(T)j 2
+2jX t;x;u
(T)j 2
) i
1=2
+"
8 T
"
E Ft
sup
tsT jX
t;x;u
(s)j
2
#
1=2
"
E Ft
sup
tsT [2jX
t;x;u
(s)j
2
+2jX t;x;u
(s)j
2
]
#
1=2
+"
8
esssup
! Z
T
0 jQ
jds E Ft
sup
tsT jX
t;x;u
(s)j
2
+"
8
"
1
2
"
0 jxj
2
esssup
s;!
jN
j(s):
(49)
Proof of Lemma 3.3. Sinceu2U x
ad
(t;T), we have
E Ft
Z
T
t juj
2
ds"
1
2
"
0 jxj
2
: (50)
Putting(50)intotherstestimateofLemma3.2,wegettherstinequalityofLemma3.3.
Putting (50) and the rst inequality of Lemma 3.3 into the second estimate of Lemma
3.2, we get the second one. The last one is a combination of (50) and applying the
Cauchy-Schwarz inequality inthe thirdestimate of Lemma3.2.
Combining the rst and the last inequalities of Lemma 3.3, we conclude that for
each(t;x)2[0;T]R n
;8u2U x
ad (t;T);
jJ
(u;t;x) J
(u;t;x)j
"
8
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! jM
j
+2jxj"
8
(T +1) q
"
6 (1+"
1
2
"
0 )
"
E F
t
sup
tsT jX
t;x;u
(s)j
2
#
1=2
+"
8
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! Z
T
0 jQ
jds+"
8
"
1
2
"
0 jxj
2
esssup
s;!
jN
j(s):
(51)
Puttingthe secondinequality of Lemma3.3 intothis, wehave that
jJ
(u;t;x) J
(u;t;x)j
"
8
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! jM
j+2jxj"
8
(T +1) q
"
6 (1+"
1
2
"
0 )
"
7
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! Z
T
0 (jA
j+jC
j 2
)ds
+"
7
"
1
2
"
0 jxj
2
esssup
s;!
(jB
j+jD
j 2
)(s)
1=2
+"
8
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! Z
T
0 jQ
jds+"
8
"
1
2
"
0 jxj
2
esssup
s;!
jN
j(s)
(52)
hold for each (t;x)2[0;T]R ;8u2U
ad
(t;T):Therefore, we have
jV
(t;x) V
(t;x)j
"
8
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! jM
j+2jxj"
8
(T +1) q
"
6 (1+"
1
2
"
0 )
"
7
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! Z
T
0 (jA
j+jC
j 2
)ds
+"
7
"
1
2
"
0 jxj
2
esssup
s;!
(jB
j+jD
j 2
)(s)
1=2
+"
8
"
6 (1+"
1
2
"
0 )jxj
2
esssup
! Z
T
0 jQ
jds+"
8
"
1
2
"
0 jxj
2
esssup
s;!
jN
j(s)
(53)
hold for each (t;x)2[0;T]R n
:
In view of the assumptions of Theorem 2.1, (53) implies that for each (t;x) 2
[0;T]R n
,V
(t;x)convergestoV 0
(t;x)as! 0. Moreover, thisconvergenceisuniform
in(t;!). Hence, K
converges tosome K 0
inthe Banach space
L 1
F
(0;T;S n
+ )\L
1
(;F
T
;P;C([0;T];S n
+ )):
Inthefollowing,weshowthestrongconvergenceofL
. Notethat(K
;L
)satises
the BSDE
8
>
>
<
>
>
: dK
(t) = [G
(t;K
;L
) G
(t;K
;L
)] dt+ d
X
i=1 L
i dw
i
;
K
(T) = M
:
(54)
UsingIt^o's formula, we have
EjK
j 2
(t)+E Z
T
t jL
j 2
(s)ds
= EjM
j 2
+E Z
T
t K
[G
(s;K
;L
) G
(t;K
;L
)]ds:
(55)
Since
jG
(s;K
;L
) G
(t;K
;L
)j"(1+jL
j 2
+jL
j 2
) (56)
for somedeterministic constant " which is independent of and , we have
E Z
T
t jL
j 2
(s)dsEjM
j 2
+"esssup
s;!
jK
(s)jE Z
T
t
(1+jL
j 2
+jL
j 2
)ds: (57)
From the seconda priori estimateof Lemma2.1, weconclude that L
converges tosome
L 0
strongly in L 2
F
(0;T;S n
). By passing to the limitin the BSRDE (A
;B
;C
i
;D
i
;i=
1;:::;d;Q
;N
;M
), we show that (K 0
;L 0
) solves the BSRDE (A 0
;B 0
;C 0
i
;D 0
i
;
i=1;:::;d;Q 0
;N 0
;M 0
).
This sectiongivesthe proof of Theorem 2.2. The main idea isto dothe inverse transfor-
mation:
f
K :=K 1
; (58)
which turnsout tosatisfy aRiccati equationwhose generatordepends onthe martingale
term ina linear way.
First, since D isinversable, we can rewritethe BSRDE (9) as
8
>
<
>
:
dK = [
e
A
K K
e
A+Q K e
BK 1
e
B
K LK
1
L
+K e
BK 1
L+LK 1
e
B
K]dt+Ldw;
K(T) = M;
(59)
where
e
A := A+BD 1
C;
e
B := BD 1
:
Notethatwehavethefollowingrulefortherstandthe seconddierentialsoftheinverse
of apositivematrix as a matrix-valued function:
d
K 1
= K
1
(dK)K 1
; d 2
K 1
=2K 1
(dK)K 1
(dK)K 1
: (60)
UsingIt^o's formula, we can write the equation for the inverse f
K of K:
(
d f
K = [
f
K e
A
+ e
A f
K f
KQ f
K+ e
B f
K e
B
+ e
B e
L+ e
L e
B
]dt+ e
Ldw;
f
K(T) = M 1
;
(61)
where
e
L:= K 1
LK 1
:
FromProposition2.1,theaboveBSRDE
e
A;Q 1=2
; e
B;0;0;I
mm
;M 1
hasauniqueglobal
adapted solution ( f
K; e
L) with
f
K 2L 1
F
(0;T;S n
+ )\L
1
(;F
T
;P;C([0;T];S n
+ ));
e
L2L 2
F
(0;T;S n
);
which implies that f
K 1
(t) is uniformly positive in (t;!). Moreover, from the fact that
f
K(T)=M 1
"
1
1 I
nn
,wederive that f
K isuniformlypositive. Thisshows that f
K 1
(t)
is uniformly bounded. Therefore (K;L) is a global adapted solution to the BSRDE (9)
with
K :=
f
K 1
2L 1
F
(0;T;S n
+ )\L
1
(;F
T
;P;C([0;T];S n
+ ));
L:=
f
K 1
e
L f
K 1
2L 2
F
(0;T;S n
):
TheuniquenessresultsfromtheFeynman-KacrepresentationresultProposition2.2.
In fact, assume that ( c
K;
b
L) also solves the BSRDE (9). Then, from Proposition 2.2, we
see that
(K(t)x;x)=V(t;x)=( c
K(t)x;x); a:s:; 8(t;x)2[0;T]R n
:
So, we have K(t)= c
K(t) almost surely for 8(t;x)2[0;T]R :Set
ÆK :=K c
K; ÆL
i :=L
i b
L
i
; ÆG:=G(t;K;L) G(t;
c
K; b
L ):
Then, wehave ÆK =0. Notethat (ÆK;ÆL) satisesthe following BSDE:
8
>
>
<
>
>
:
dÆK(t) = ÆGdt+ d
X
i=1 ÆL
i (t)dw
i
(t); 0t <T;
ÆK(T) = 0:
(62)
From this, proceedingidenticallyas inthe last paragraph of Section3, wehave
E Z
T
t jÆLj
2
(s)dsEjÆK(T)j 2
+"esssup
s;!
jÆK(s)jE Z
T
t
(1+jLj 2
+j b
L j 2
)ds=0: (63)
Hence, ÆL=L b
L=0.
5 The Proof of Theorem 2.3
Fortheregularcase, thesituationisalittlecomplex: weeasilysee thatthe aboveinverse
transformationonthe rstunknown variablecan not eliminatethe quadratictermof the
second unknown variable. However, we can still solve some classes of BSRDEs with the
help of doing some appropriatetransformation.
Proposition 5.1. Assume that Q A
(D 1
)
ND 1
+(D 1
)
ND 1
A;m = n;
and D and N are positive constantmatrices. Then, Theorem 2:3 holds.
Proof of Proposition 5.1. Write
c
N :=(D 1
)
ND 1
: (64)
Then, the BSRDE (7)reads
8
>
<
>
:
dK = [A
K +KA+Q L(
c
N +K) 1
L]dt+Ldw;
0t<T;
K(T) = M:
(65)
The equation for c
K :=
c
N +K is
8
>
<
>
: d
c
K = [A
c
K + c
KA+Q A
c
N c
NA b
L c
K 1
b
L]dt+ b
Ldw;
0t<T;
c
K(T) = c
N +M:
(66)
Notethat c
N+M is uniformlypositive. From Theorem 2.2,wesee that the BSRDE (66)
has a unique global adapted solution ( c
K; b
L ). Therefore ( c
K c
N; b
L ) is a global adapted
solutionto the BSRDE (7).
Proposition 5.2. Assume thatA =0 andD andN areconstant matrices. Then,
Theorem 2:3 holds.
imatingBSRDEs:
(
dK = [Q LD
(N +D
KD
)
1
D
L]dt+Ldw;
K(T) = M
(67)
where
D
:=D+ I
mm
>0;>0:
From Proposition 5.1, we see that the BSRDE (67) has a unique globaladapted solution
(K
;L
) for every >0. From Proposition 2.2, K
can be represented as
(K
(t)x;x)=V
(t;x); 8(t;x)2[0;T]R n
: (68)
From Theorem 2.1, we see that K
uniformly converges to some K 2 L 1
F
(0;T;S n
+ )\
L 1
(;F
T
;P;C([0;T];S n
+
)) and L
strongly converges to some L 2 L 2
F
(0;T;S n
), and
that (K;L)is anadapted solutionof the BSRDE (7)when A=0.
Considerthecasen >m. Thenconsiderthennmatrices f
Dwhoserstmcolumns
are D and whose last (n m) columns are zero column vectors, and f
N whichis dened
as
f
N :=
R 0
0 I
!
:
The BSRDE (7) when A =0 isrewritten as
(
dK = [Q L
f
D(
f
N + f
D
K f
D) 1
f
D
L]dt+Ldw;
K(T) = M
From the preceding result, we obtainthe desiredexistence result.
Consider thecase n<m. Then,thereisamm orthogonaltransformationmatrix
T such that
D=[ c
D;0]T;
c
D2R nn
and is non-singular.
Write
f
N :=(T 1
)
NT 1
:=
c
N
11 c
N
12
c
N
12 c
N
22
!
>0:
Then, c
N
11
>0:The BSRDE (7)when A=0 is rewritten as
(
dK = [Q L
c
D(
f
N
11 +
c
D
K c
D) 1
c
D
L]dt+Ldw;
K(T) = M
From the preceding result, we obtainthe desiredexistence result.
Proposition 5.3. Assume that A = 0; and D and N are piece-wisely constant
F
t
-adapted bounded matrix processes. Then, Theorem 2:3 holds.
Proof of Proposition 5.3. Since Dand N are piece-wisely constantF
t
-adapted
bounded matrix processes, there is a nitepartion:
0=:t
0
<t
1
<<t
J :=T
i i+1 t
i
randommatrices. From Proposition 5.2, the BSRDE
8
>
<
>
:
dK = [Q LD(N +D
KD) 1
D
L]dt+Ldw;
t
J 1
t<T;
K(T) = M
(69)
has a unique F
t
-adapted solution (K J
;L J
) with
K J
2L 1
F (t
J 1
;T;S n
+ )\L
1
(;F
T
;P;C([t
J 1
;T];S n
+
)); L J
2L 2
F (t
J 1
;T;S n
):
Assume that for some i=2;:::;J; the BSRDE
8
>
<
>
:
dK = [Q LD(N+D
KD) 1
D
L]dt+Ldw;
t
i 1
t<t
i
;
K(t
i
) = K i+1
(t
i )
(70)
has a unique F
t
-adapted solution (K i
;L i
)with
K i
2L 1
F (t
i 1
;t
i
;S n
+ )\L
1
(;F
t
i
;P;C([t
i 1
;t
i ];S
n
+
)); L i
2L 2
F (t
i 1
;t
i
;S n
):
Note that when i = J, we use the convention K J+1
(t
J
) :=M. Then, we conclude from
Proposition5.2 that the BSRDE
8
>
<
>
:
dK = [Q LD(N +D
KD) 1
D
L]dt+Ldw;
t
i 2
t<t
i 1
;
K(t
i 1
) = K i
(t
i 1 )
(71)
has a unique F
t
-adapted solution (K i 1
;L i 1
) with
K i 1
2L 1
F (t
i 2
;t
i 1
;S n
+ )\L
1
(;F
t
i 1
;P;C([t
i 2
;t
i 1 ];S
n
+
)); L i 1
2L 2
F (t
i 2
;t
i 1
;S n
):
Inthisbothinductiveandbackwardway,wemaydeneJpairesofprocessesf(K i
;L i
)g J
i=1 .
Deneonthewholetimeinterval[0;T]the pairofF
t
-adaptedprocesses(K;L)asfollows:
K(t):=
J
X
i=1 K
i
(t)
[t
i 1
;t
i )
(t); L(t):=
J
X
i=1 L
i
(t)
[t
i 1
;t
i )
(t):
We see that(K;L) satisesthe BSRDE (7). We thenobtain the desiredexistenceresult.
Proposition 5.4. Assume that A=0:Then, Theorem2:3 holds.
Proof of Proposition 5.4. For an arbitrary positive integer k, consider the
2 k
-partionof the time interval. Dene
D k
(t)=D
i 1
2 k
T
; 8t2
i 1
2 k
T; i
2 k
T
;i=1;2;:::;2 k
;
and
N k
(t)=N
i 1
2 k
T
; 8t2
i 1
2 k
T; i
2 k
T
;i=1;2;:::;2 k
:
For each k, D and N are are piece-wisely constant, F
t
-adapted, bounded matrix pro-
cesses. Further, in view of (25), D k
(t) and N k
(t) converge respectively to D and N,
uniformlyin(t;!):That is, we have
lim
k!1
esssup
!2 max
t2[0;T]
jD k
(t) D(t)j=0; lim
k!1
esssup
!2 max
t2[0;T]
jN k
(t) N(t)j=0:
FromProposition5.3,weseethattheBSRDE(0;0;0;D k
;Q;N k
;M)has aglobaladapted
solution(K k
;L k
), and then from Theorem 2.1, we see that Theorem 2.3 holds.
Proof of Theorem 2.3. The case A = 0 is solved by Proposition 5.4. For the
case A6=0,consider the following transformation
f
K :=
K;
e
L:=
L
where solves the dierentialequation
8
<
: d
dt
(t) = A(t)(t); t2(0;T];
(0) = I
nn :
UsingIt^o's formula, we get the BSDE for ( f
K; e
L):
(
d f
K(t) = [ e
Q e
L f
D(N + f
D f
K f
D) 1f
D e
L ]dt+ e
Ldw(t); t 2(0;T];
f
K(T) = f
M
where e
Q:=
Q;
f
M :=(T)
M(T);
f
D :=
1
D. Note that the trajectories of f
D are
still uniformly continuous like D. From Proposition 5.4, we see that the
BSRDE (0;0;0;
f
D;
e
Q;N;
f
M) has a global adapted solution( f
K;
e
L ), and thus the pair
((
) 1
f
K 1
;(
) 1
e
L 1
)
solves the originalBSRDE (A;0;0;D;Q;N;M).
The uniqueness can be proved in the same way as inthe proof of Theorem 2.2.
6 Application to Stochastic LQ Problems
6.1 The unconstrainted case
Assume that
2L 2
(;F
T
;P;R n
); q;f;g
i 2L
2
F
(0;T;R n
): (72)
Consider the following optimalcontrolproblem (denoted by P
0 ):
min
u2L 2
F (0;T;R
m
)
J(u;0;x) (73)
J(u;t;x)= E F
t
(M(X t;x;u
(T) );X t;x;u
(T) )
+E F
t Z
T
t
[(Q(X t;x;u
q);X t;x;u
q)+(Nu;u)]ds
(74)
and X t;x;u
solving the equation
8
>
>
<
>
>
:
dX = (AX+Bu+f)ds+ d
X
i=1 (C
i
X+D
i u+g
i )dw
i
; t<sT;
X(t) = x; u2L 2
F
(t;T;R m
):
(75)
The value function V is dened as
V(t;x):= min
u2L 2
F (t;T;R
m
)
J(u;t;x); (t;x)2[0;T]R n
: (76)
Dene :[0;T]S n
+ R
nd
!R mn
by
(;S;L)= (N + d
X
i=1 D
i SD
i )
1
(B
S+ d
X
i=1 D
i SC
i +
d
X
i=1 D
i L
i
): (77)
and
b
A:=A+B (;K;L);
b
C
i :=C
i +D
i
(;K;L); i=1;:::;d: (78)
Let( ;) be the F
t
-adapted solutionof the followingBSDE
8
>
>
<
>
>
:
d (t) = [ b
A
+ d
X
i=1 b
C
i (
i Kg
i
) Kf d
X
i=1 L
i g
i
+Qq]dt+ d
X
i=1
i dw
i
;
(T) = M
(79)
where (K;L)is the unique F
t
-adapted solution of the BSRDE (1). The following can be
veried by apure completion of squares.
Theorem 6.1 Suppose that the assumptions of Theorem 2.2 or Theorem 2.3 are
satised. Let (K;L) be the unique F
t
-adapted solution of BSRDE (1). Then, the optimal
control b
u for the non-homogeneous stochastic LQ problem P
0
exists uniquely and has the
followingfeedback law
b
u = (N + d
X
i=1 D
i KD
i )
1
[(B
K + d
X
i=1 D
i KC
i +
d
X
i=1 D
i L
i )
c
X
B
+ d
X
i=1 D
i (Kg
i
i )]:
(80)
The value function V(t;x);(t;x)2[0;T]R n
has the following explicit formula
V(t;x)=(K(t)x;x) 2( (t);x)+V 0
(t); (t;x)2[0;T]R n
(81)
V 0
(t):= E F
t
(M;)+E F
t Z
T
t
(Qq;q)ds 2E F
t Z
T
t
( ;f)ds
+E F
t Z
T
t d
X
i=1 [(Kg
i
;g
i
) 2(
i g
i )]ds
E Ft
Z
T
t
((N+ d
X
i=1 D
i KD
i )u
0
;u 0
)ds
(82)
and
u 0
:=(N + d
X
i=1 D
i KD
i )
1
[B
+ d
X
i=1 D
i (
i Kg
i
)]; tsT: (83)
Proof Set
e
u=u (;K;L)X: (84)
Then the system (75) reads
8
>
>
<
>
>
:
dX = ( b
AX+B e
u+f)ds+ d
X
i=1 (
b
C
i
X+D
i e
u+g
i )dw
i
; t<sT;
X(t) = x; u2L 2
F
(t;T;R m
):
(85)
Applying It^o's formula, we have the equation for X =:XX
:
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
dX = [ b
AX +X b
A
+X(B e
u+f)
+(B e
u+f)X
]ds
+ d
X
i=1 [
b
C
i X
b
C
i +
b
C
i X(D
i e
u+g
i )
+ b
C
i X(D
i e
u+g
i )X
b
C
i
+(D
i e
u+g
i )(D
i e
u+g
i )
]ds
+ d
X
i=1 [
b
C
i
X +X b
C
i
+X(D
i e
u+g
i )
+(D
i e
u+g
i )X
]dw
i
; t<sT;
X(t) = xx
; u2L 2
F
(t;T;R m
):
(86)
Notethat the BSRDE (1) can be rewritten as
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
dK =
b
A
K +K b
A+ d
X
i=1 b
C
i K
b
C
i +
d
X
i=1 (
b
C
i L
i +L
i b
C
i )+Q
+ (t;K;L)
N (t;K;L)
dt d
X
i=1 L
i dw
i
;
K(T) = M:
(87)
E Ft
(MX(T);X(T))+E Ft
Z
T
t
([Q+ (s;K;L)
N (s;K;L)]X;X)ds
= (K(t)X(t);X(t))+2E F
t Z
T
t
(K(B e
u+f);X)ds
+E F
t Z
T
t d
X
i=1
2(K(D
i e
u+g
i );
b
C
i X)ds
+E Ft
Z
T
t d
X
i=1 (K(D
i e
u+g
i );D
i e
u+g
i )ds
+2E F
t Z
T
t d
X
i=1 (L
i (D
i e
u+g
i
);X)ds;
and
E Ft
"
(M;X(T))+ Z
T
t
(Qq;X)ds
#
= E Ft
"
(T)X(T)+ Z
T
t
QqXds
#
= ( (t);X(t))+E F
t Z
T
t
( ;B e
u+f)ds
+E F
t Z
T
t d
X
i=1 (
i
;D
i e
u+g
i )ds
+E Ft
Z
T
t (
d
X
i=1 b
C
i Kg
i
+Kf+ d
X
i=1 L
i g
i
;X)ds:
J(u;t;x)
= E Ft
"
(M(X(T) );X(T) )+ Z
T
t
(Q(X q);X q)ds+ Z
T
t
(Nu;u)ds
#
= E Ft
"
(MX(T);X(T))+ Z
T
t
([Q+ (s;K;L)
N (s;K;L)]X;X)ds
#
2E F
t
"
(M;X(T))+ Z
T
t
(Qq;X)ds
#
+E F
t
"
(M;)+ Z
T
t
(Qq;q)ds
#
+E Ft
Z
T
t [(N
e
u;
e
u)+2(N (s;K;L)X;
e
u)]ds
= (KX(t);X(t)) 2( (t);X(t))+E Ft
"
(M;)+ Z
T
t
(Qq;q)ds
#
+E Ft
Z
T
t d
X
i=1 (K(D
i e
u+g
i );D
i e
u+g
i
)ds+E Ft
Z
T
t (N
e
u;
e
u)ds
2E F
t Z
T
t
( ;B e
u+f)ds 2E F
t Z
T
t d
X
i=1 (
i
;D
i e
u+g
i )ds
= (K(t)x;x) 2( (t);x)+E Ft
"
(M;)+ Z
T
t
(Qq;q)ds
#
2E Ft
Z
T
t
( ;f)ds+E Ft
Z
T
t d
X
i=1 [(Kg
i
;g
i
) 2(
i
;g
i )]ds
+E F
t Z
T
t
((N+ d
X
i=1 D
i KD
i )(
e
u u
0
);
e
u u
0
)ds
E Ft
Z
T
t
((N + d
X
i=1 D
i KD
i )u
0
;u 0
)ds:
This completes the proof.
6.2 The constrainted case
Fixx
T 2R
n
. Dene
U
ad
(t;x):=fu2L 2
F
(t;T;R m
):EX t;x;u
(T)=x
T
g; 8(t;x)2[0;T]R n
; (88)
where X t;x;u
solving the equation (75). Then, consider the following constrainted LQ
problem(denoted by P t;x
c ):
inf
u2U
ad (0;x)
J(u;0;x) (89)
where the cost functional J(u;t;x) is dened by (74). Note that the set of admissible
controls U
ad
(t;x)contains the terminalexpected constraint.
Let (;t) be the unique solution of the SDE:
8
>
>
<
>
>
: dY
s
= A(s)Y
s ds+
d
X
i=1 C
i (s)Y
s dw
i
(s); t sT;
Y
t
= I
nn :
(90)
ad
:=E Z
T
0 E
Fs
(T;s)B(s)B
(s)E Fs
(T;s)ds (91)
isnonsingular. Then, 8x2R n
,the following control
u(s):=B
(s)E Fs
(T;s) 1
[x
T E
Z
T
t
(T;s)f(s)ds]; s2(t;T]; (92)
belongs toU
ad (t;x).
Wehave the followingexistence result.
Theorem 6.2. Let the assumptions of Theorem 2.2 or Theorem 2.3 be satised.
Assume that U
ad
(0;x) isnot empty. Then, the problemP 0;x
c
has a unique optimal contol.
Proof of Theorem6.2. The proofissimilartothat ofKohlmannand Tang [12].
The main idea isto choose asequence fu k
;k=1;2;:::g such that
u k
2U
ad
(0;x); lim
k!1 J(u
k
;0;x)= inf
u2U
ad (0;x)
J(u;0;x):
Then, we prove that this sequence is a Cauchy sequence by using the uniform convexity
of the cost functional J(u;0;x) in the control u. This uniform convexity is obvious for
the regular case, and has been proved for the singular case by Kohlmann and Tang [12].
The details are left tothe reader.
Due to the limitationof space, we will inwhat follows just sketch how tosolve the
uniqueoptimalcontrolofTheorem 6.2interms ofthe solutionofthe associatedBSRDE.
Using the stochastic maximum principle (see Peng [20], and Tang and Li [27], for
example), we have the following. Let e
u be the optimal control, and f
X :=X 0;x;eu
. Then,
there issome 2R n
, and a pair of processes (e
p;
e
q),such that
8
>
>
<
>
>
: d
e
p = [A
e
p+Q(
f
X q)+ d
X
i=1 C
i e
q
i ]ds+
d
X
i=1 e
q
i dw
i
; 0<s T;
e
p(T) = M( f
X(T) )
(93)
and
B
e
p+ d
X
i=1 D
i e
q
i +N
e
u=0: (94)
Using It^o'sformula and the equality (94), we get the equation for e
:=K f
X e
p:
8
>
>
<
>
>
: d
e
(t) = [ b
A
e
+ d
X
i=1 b
C
i (
e
i Kg
i
) Kf d
X
i=1 L
i g
i
+Qq]dt+ d
X
i=1 e
i dw
i
;
e
(T) = M+
(95)
t
of the optimalcontrol:
e
u = (N + d
X
i=1 D
i KD
i )
1
[(B
K + d
X
i=1 D
i KC
i +
d
X
i=1 D
i L
i )
c
X
B
e
+ d
X
i=1 D
i (Kg
i e
i )]
(96)
wheretheLagrangemultipleisdeterminedsuchthat theterminalconstraintE f
X(T)=
x
T
issatised.
6.3 A comment onapplication of theLQ theoryin mathematical
nance
One-dimensional singular LQ problems arise from mathematical nance. The mean-
variancehedgingproblemandthedynamicversionofMarkowitz'smean-varianceportfolio
selectionproblem, are one-dimensional singular LQproblems.
The mean-variance hedging problem was initiallyintroduced by Follmer and Son-
dermann [7], and later was widely studied among others by DuÆe and Richardson [5],
Follmer and Schweizer [8], Schweizer [23, 24, 25], Hipp [11], Monat and Stricker [16],
Pham, Rheinlander and Schweizer [21], Gourieroux, Laurent and Pham [10], and Lau-
rent and Pham [15]. All of these works are based on a projection argument. Recently,
KohlmannandZhou[14]usedanaturalLQtheoryapproachtosolvethecaseofdetermin-
isticmarketconditions. Kohlmann and Tang [12,13]used anaturalLQ theoryapproach
to solve the case of stochastic market conditions, and the optimalhedging portfolio and
thevariance-optimalmartingalemeasure are characterizedinterms ofthe solutionof the
associatedBSRDE.
The continuoustime mean-varianceportfolioselectionproblemwasinitiallyconsid-
eredbyRichardson[22]. ThereaderisreferredtoZhouandLi[29]forrecentdevelopments
onthis problem.
Acknowledgement The second author would like to thank the hospitality of
DepartmentofMathematicsand Statistics, andthe Center of FinanceandEconometrics,
UniversitatKonstanz, Germany.
References
[1] Bismut, J. M., Conjugate convex functions in optimal stochastic control, J. Math.
Anal.Appl., 44 (1973),384{404
[2] Bismut, J. M., Linear quadratic optimal stochastic control with random coeÆcients,
SIAM J. Control Optim.,14 (1976),419-444
[3] Bismut, J.M.,Controledes systems linearesquadratiques: applications de l'integrale
stochastique, Seminaire de ProbabilitesXII, eds : C. Dellacherie, P. A. Meyer etM.
Weil, LNM 649, Springer-Verlag, Berlin1978
