Riccati Equations and Their Applications
Michael Kohlmann y
Shanjian Tang z
October 25, 2000
Abstract
ThefollowingbackwardstochasticRiccatidierentialequation(BSRDEinshort)
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
dK = [A
0
K+KA+ d
X
i=1 C
0
i KC
i
+Q+ d
X
i=1 (C
0
i L
i +L
i C
i )
(KB+ d
X
i=1 C
0
i KD
i +
d
X
i=1 L
i D
i )(N+
d
X
i=1 D
0
i KD
i )
1
(KB+ d
X
i=1 C
0
i KD
i +
d
X
i=1 L
i D
i )
0
]dt+ d
X
i=1 L
i dw
i
; 0t<T;
K(T) = M:
is motivated, and is then studied. Some properties are presented. The existence
anduniquenessofaglobaladaptedsolutionto aBSRDEhasbeenopenforthecase
D
i
6=0formorethantwodecades. Ourrecentresultsonthistopic aresummarized.
Finally,applicationsare addressed,bothinnanceand control.
Key words: backward stochasticRiccatiequation,stochasticlinear-quadraticcon-
trolproblem,mean-variance hedging, variance-optimalmartingale measure
AMS Subject Classications. 93E20, 60H10, 91B28
Abbreviated title: Backward stochastic Riccati equations and applications
1 Introduction
Let(;F;P;fF
t g
t0
)beaxedcompleteprobabilityspaceonwhichisdenedastandard
d-dimensionalF
t
-adaptedBrownian motionw(t)(w
1
(t);;w
d (t))
0
. Denoteby F
t the
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.
completion,bythetotalityN ofallnullsetsofF,ofthenaturalltrationfF
t
ggenerated
by w.
Consider the following BSRDE:
8
>
>
<
>
>
:
dK = G(t;K;L)dt+ d
X
i=1 L
i dw
i
; 0t <T;
K(T) = M
(1)
where the generator G isgiven by
G(t;K;L) := A 0
K+KA+ d
X
i=1 C
0
i KC
i
+Q+ d
X
i=1 (C
0
i L
i +L
i C
i
)+F(t;K;L) (2)
with
F(t;K;L):= [KB(t)+ d
X
i=1 C
i (t)
0
KD
i (t)+
d
X
i=1 L
i D
i
(t)][N(t)+ d
X
i=1 D
i (t)
0
KD
i (t)]
1
[KB(t)+ d
X
i=1 C
i (t)
0
KD
i (t)+
d
X
i=1 L
i D
i (t)]
0
;
8(t;K;L)2[0;T]S n
+ (S
n
) d
:
(3)
It will be called the BSRDE (A;B;C
i
;D
i
;i = 1;:::;d;Q;N;M) in the following for
convenience of indicatingthe associated coeÆcients. The coeÆcients appearing here will
be dened in Section2.
BSRDEs have atleast the following two motivations.
(1) The control-theoretic motivation. The BSRDE (1) arises from solution
of the optimalcontrolproblem
inf
u2L 2
F (0;T;R
m
)
J(u;0;x) (4)
where fort 2[0;T] and x2R n
,
J(u;t;x):=E F
t
[ Z
T
t [(Nu
s
;u
s
)+(QX t;x;u
s
;X t;x;u
s
)]ds+(MX t;x;u
T
;X t;x;u
T
)] (5)
and X t;x;u
solves the following stochastic dierentialequation
8
>
>
<
>
>
: dX
s
= (AX
s +Bu
s )ds+
d
X
i=1 (C
i X
s +D
i u
s )dw
i
; tsT;
X
t
= x:
(6)
We have the following connection: if the BSRDE (1) has a solution (K;L), the solution
for the above linear-quadratic optimal control problem (LQ problem in short) has the
following closed form(also called the feedback form):
u
s
= (N+ d
X
i=1 D
0
i KD
i )
1
[B 0
K + d
X
i=1 D
0
i KC
i +
d
X
i=1 D
0
i L
i ]X
s
(7)
V(t;x):= inf
u2L 2
F (t;T;R
m
)
J(u;t;x)=(K(t)x;x); 0tT;x2R n
: (8)
Inthis way, solutionof theabove LQproblemisreduced tosolving the BSRDE (1). The
LQ problemwith a terminalexpected constraint (EX(T)= x
T
for some xed x
T 2R
n
,
for example),is alsoreduced to solutionof a BSRDE.
(2) The nancial motivation. A mean-variance hedging problem is a one-
dimensional, nonhomogeneous, singular stochastic LQ problem. A mean-variance port-
folio selection problem is a one-dimensional, nonhomogeneous, singular stochastic LQ
problem with an expected terminal state constraint. Solution of these two classes of
mathematicalnancialproblemsis reduced tosolutionof the associatedone-dimensional
BSRDEs.
The rest of the paper is organized as follows. Preliminaries are done in Section
2 where the notation is listed and a solution of a BSRDE is dened. In Section 3, a
historicalreviewisgivenonBSRDEs,and theknown existenceanduniqueness resultdue
to Bismut [5] and Peng [29] is stated. Section 4 collects various properties of BSRDEs.
Section 5 summarizes our recent results on the existence and uniqueness of a global
adapted solutionofBSRDE (1). Finallyinsection6, BSRDEsare appliedtocontroland
nance.
2 Preliminaries
Notation. Throughout this paper, the following additionalnotationwillbe used:
M 0
: 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
with the norm esssup
!2 max
0tT
jf(t;!)j.
Wemakethe following two basic assumptions.
(A1)ThecoeÆcientsA;B;C
i
, andD
i areF
t
-progressivelymeasurableboundedmatrix-
valuedprocesses,denedon[0;T];of dimensionsnn;nm;nn;nm respectively.
M isanF
T
-measurable,nonnegative,andboundednnrandommatrix,andQandN are
F
t
-progressively measurable, bounded, and nonnegative nn and mm matrixprocesses,
respectively.
(A2) N is uniformly positive. Or
(A3) M and d
X
i=1 D
0
i D
i
are uniformly positive.
Denition 2.1. A solution of the BSRDE (1) is a pair (K;L) of processes such
that
(i) K 2L 1
F
(0;T;S n
)\L 1
(;F
T
;P;C([0;T];S n
)); L2
L 2
F
(0;T;S n
)
d
;
(ii) N(t)+ d
X
i=1 D
i (t)
0
KD
i
(t) is uniformlypositive with respect to(t;!),
(iii)K(t)=M + Z
T
t
G(s;K(s);L(s))ds Z
T
t
L(s)dw(s); 0tT:
When the pair (K;L)is asolutionof the BSRDE (1), we alsosay that itsolves the
BSRDE (1).
3 A Historical Review
First, consider the regular case, i.e., N is assumed to be uniformly positive. When the
coeÆcients A;B;C
i
;D
i
;Q;N;M are all deterministic, then L
1
= = L
d
= 0 and the
BSRDE (1)reduces tothe following nonlinear matrix ordinary dierentialequation:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
dK = [A
0
K+KA+ d
X
i=1 C
0
i KC
i
+Q (KB + d
X
i=1 C
0
i KD
i )
(N + d
X
i=1 D
0
i KD
i )
1
(KB+ d
X
i=1 C
0
i KD
i )
0
]dt; 0t<T;
K(T) = M;
(9)
which was completely solved by Wonham [41] by applying Bellman's principle of quasi-
linearizationand a monotone convergence approach.
The attention to the randomness of the coeÆcients A;B;C;D;Q;N;M is due to
Bismut. Bismut [4, 5] initially studied the case of random coeÆcients, but he could
solve only some special simple cases at that time. Letthe integer d
0
0,and denoteby
fF 2
t
;0tTgtheP-augmentednaturalltrationgeneratedbythe (d d
0
)-dimensional
Brownianmotion(w
d0+1
;:::;w
d
). HeassumedthattherandomnessofthecoeÆcientsonly
comes from the smaller ltration fF 2
t
g, which leads to L
1
= = L
d
0
= 0. He further
assumed in the paper[4] that
C
d
0 +1
==C
d
=0; D
d
0 +1
==D
d
=0; (10)
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
dK = [A
0
K+KA+ d
0
X
i=1 C
0
i KC
i +Q
(KB + d
0
X
i=1 C
0
i KD
i )(N +
d
0
X
i=1 D
0
i KD
i )
1
(KB+ d
0
X
i=1 C
0
i KD
i )
0
]dt
+ d
X
i=d0+1 L
i dw
i
; 0t<T;
K(T) = M;
(11)
and the generator does not involve L atall. In the work [5], he assumed that
D
d0+1
==D
d
=0; (12)
underwhich the BSRDE (1) becomesthe following one
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
dK = [A
0
K+KA+ d
X
i=1 C
0
i KC
i
+Q+ d
X
i=d
0 +1
(C 0
i L
i +L
i C
i )
(KB + d
0
X
i=1 C
0
i KD
i )(N +
d
0
X
i=1 D
0
i KD
i )
1
(KB+ d
0
X
i=1 C
0
i KD
i )
0
]dt
+ d
X
i=d0+1 L
i dw
i
; 0t<T;
K(T) = M;
(13)
and the generator depends on the second unknown variable (L
d0+1
;:::;L
d )
0
in a linear
way. Moreover, hismethod was rathercomplicated.
Later, Peng [29] gavea nice treatment onthe proof of existence and uniqueness for
the BSRDE (13), by using Bellman's quasilinear principle and a method of monotone
convergence|a generalizationof Wonham'sapproach tothe random situation.
Thefollowingpropositionstatestheabove-mentionedresultonexistenceandunique-
ness of a globaladapted solutionto the BSRDE (1), due to Bismut [5]and Peng [29].
Theorem 3.1. Let the assumptions (A1) and (A2) be satised. Assume that
all the coeÆcients A;B;C
i
;D
i
;Q and N are F 2
t
-progressively measurable and that M is
F 2
T
-measurable. Then, the BSRDE (13) has a unique F 2
t
-adapted global solution (K;L)
with K(t)0;t2[0;T]:
ThegeneralcasewherethegeneratorofBSRDE(1)isallowedtocontainaquadratic
term of L,turns out tobecomea long-standingproblem. As earlyas in1978, Bismut[5]
commented on page 220 that:"Nous ne pourrons pas demontrer l'existence de solution
pour l'equation(2.49)dans lecas general." (Wecouldnot provethe existenceof solution
for equation (2.49) for the general case.) On page 238, he pointed out that the essential
diÆculty for solution of the general BSRDE (1) lies in the integrand of the martingale
term which appears in the generator in a quadratic way. Two decades later in 1998,
Peng [32] included the above probleminhis listof open problems onBSDEs.
Second, consider the singular case N = 0. Kohlmann and Zhou [20] studied the
following case:
C =0;M =I
nn
;D
i
=I
mm
; i=1;;d:
and (b) A+A 0
BB
0
. Under the above-described framework, they obtain the existence
and uniqueness of a global solution. Their method is based on an existence criterion of
Chen, Liand Zhou [6].
4 Fundamentals of BSRDEs
This sectioncollects various properties of BSRDEs, most of which turn out tobe helpful
to the proof of the existence and uniqueness of a global adapted solution. They are
consequences of the special structure of BSRDE (1)from dierent points of view.
Forconvenience of followingexposition,we introducethefollowingnotation. Dene
:[0;T]S n
R nd
!R mn
by
(;K;L)= (N + d
X
i=1 D
0
i KD
i )
1
(KB+ d
X
i=1 C
0
i KD
i +
d
X
i=1 L
i D
i )
0
(14)
and
b
A:=A+B (;K;L);
b
C
i :=C
i +D
i
(;K;L); i=1;:::;d: (15)
Consider the following SDE
8
>
>
<
>
>
: dY
s
= b
A(s)Y
s ds+
d
X
i=1 b
C
i Y
s dw
i
(s); t sT;
Y
t
= I
nn :
(16)
In view of Gal'chuk [11], this equation has a unique strong solution (denoted by (;t))
when L2(L 2
F
(0;T;S n
)) d
.
4.1 Regularity
Theorem 4.1. Let (K;L) solve the BSRDE (1). Then,
(;K;L)(;t)2L 2
F
(t;T;R mn
);
:= max
tsT
j(s;t)j2L 2
(;F;P): (17)
Consider the optimalcontrolproblem
inf
u2L 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
s
;u
s
)+(QX t;x;u
s
;X t;x;u
s
)]ds+(MX t;x;u
T
;X t;x;u
T
)] (19)
and X t;x;u
solves the SDE
8
>
>
<
>
>
: dX
s
= (AX
s +Bu
s )ds+
d
X
i=1 (C
i X
s +D
i u
s )dw
i
; tsT;
X
t
= x:
(20)
can show that the assumption (A2) or (A3) implies the existence and uniqueness of the
optimalcontrol b
u2L 2
F
(0;T;R m
). From the optimality conditions, wehave
b
u= (;K;L)(;t)x; X t;x;bu
=(;t)x:
Then, Theorem 4.1follows. The reader is referred toSubsection 6.1for more details.
4.2 The Feynman-Kac Representation
Theorem 4.2. Let (K;L) solve the BSRDE (1). Then,
(K(t)x;x)=V(t;x):= inf
u2L 2
F (t;T;R
m
)
J(u;t;x);8t2[0;T]; x2R n
: (21)
In viewof Theorem 4.1, the proof is straightforward. This theorem plays a crucial
roleinthe proof ofuniqueness: itimmediatelygivesthe uniqueness of K asthe rst part
of the solution. It alsoplays animportant role inthe proof of the closed property of the
solutions(See Theorem 4.9).
4.3 Nonnegativity
Theorem 4.3. If (K;L) solves the BSRDE (1), then K(t)0; 0t T.
Noticing the nonnegativity of M;Q;N, we haveK(t)0from Theorem 4.2.
4.4 Monotonicity
FromTheorem4.2,weimmediatelyobtainthefollowingmonotonepropertyofthesolution
tothe BSRDE (1).
Theorem4.4. Let(K;L)and( f
K;
e
L )bethesolutionstotheBSRDEs(A;B;C
i
;D
i
;i=
1;;d;Q;N;M) and (A;B;C
i
;D
i
;i=1;;d;
e
Q;
f
N;
f
M), respectively. If
e
QQ0;
f
N N >0;
f
M M 0;
then f
K K;a:s:a:e::
4.5 Minimality
Theorem 4.5. Let e
2L 1
F
(0;T;R mn
) and let ( f
K;
e
L ) be the solution of the following
linear BSDE
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
dK = [
e
A 0
K +K e
A+ d
X
i=1 e
C 0
i K
e
C
i
+Q+ d
X
i=1 (
e
C 0
i L
i +L
i e
C
i )
+ e
N e
]dt+ d
X
i=1 L
i dw
i
; 0t <T;
K(T) = M
(22)
e
A:=A+B e
; e
C
i :=C
i +D
i e
; I =1;:::;d: (23)
If (K;L) is the solution to the BSRDE (1), then K f
K;a:s:a:e::
Note that adeterministic version of Theorem4.5 was given by Wonham [41].
For the one-dimensional case, inview of the minimalityof the generator:
G(t;K;L)[ e
A 0
K+K e
A+ d
X
i=1 e
C 0
i K
e
C
i
+Q+ d
X
i=1 (
e
C 0
i L
i +L
i e
C
i )+
e
N e
]; 8 e
2R mn
;
this theorem is an immediate consequence of the existing comparison theory for one-
dimensional BSDEs. For the general multi-dimensional case, it is an immediate conse-
quence of the above monotone theorem 4.4for the BSRDE (A;0;C;0;Q;N;M).
4.6 A priori estimates and the BMO-property.
We have the following a prioriestimate of boundedness.
Theorem 4.6. Let (K;L) solve the BSRDE (A;B;C
i
;D
i
;i = 1;:::;d;Q;N;M):
Then, there is a deterministic positive constant"
0
such thatthe followingestimates hold:
0K(t)"
0 I
nn
; E Ft
Z
T
t jLj
2
ds
!
p
"
0
; 8p1: (24)
Here "
0
depends on the uniform upper bound of all the coeÆcients.
Proof of Theorem 4.6. From Theorem 4.3, we have K 0. Note that (K;L)
satisesthe BSRDE:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
dK =
A 0
K+KA+ d
X
i=1 C
0
i KC
i
+Q+ d
X
i=1 (C
0
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:
(25)
UsingIt^o's formula, we get
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
djKj 2
=
4tr
K 2
A
+ d
X
i=1
2tr (KC 0
i KC
i
)+2tr(KQ)
+ d
X
i=1
4tr (KL
i C
i
)+2tr [KF(t;K;L)] jLj 2
dt
+ d
X
i=1
2tr (KL
i ) dw
i
; 0t<T;
jKj 2
(T) = jMj 2
:
(26)
We observethat since
F(t;K;L)0; K 0;
2tr [KF(t;K;L)]=2tr h
K 1
2
F(t;K;L)K 1
2 i
0: (27)
Hence,
jKj 2
(t)+ Z
T
t jLj
2
ds jMj 2
+ Z
T
t
4tr
K 2
A
+ d
X
i=1
2tr (KC 0
i KC
i )
+2tr(KQ)+ d
X
i=1
4tr (KL
i C
i )
ds
Z
T
t d
X
i=1
2tr (KL
i ) dw
i
; 0t <T:
(28)
Usingthe elementaryinequality
2aba 2
+b 2
and taking the expectation onboth sides with respect toF
r
for rt, we obtain that
E F
r
jKj 2
(t)+ 1
2 E
F
r Z
T
t jLj
2
ds"
1 +"
1 Z
T
t E
F
r
jKj 2
(s)ds; 0rt<T: (29)
Using Gronwall's inequality, we derive from the last inequality the rst one of the esti-
mates (24). In return, we derivefrom the second lastinequality that
Z
T
t jLj
2
ds"
2 +"
2 Z
T
t
jLjds Z
T
t d
X
i=1
2tr (KL
i ) dw
i
: (30)
Therefore,
E F
t Z
T
t jLj
2
ds
!
p
3 p
"
"
p
2 +"
p
2 E
F
t Z
T
t
jLjds
!
p
+E F
t
Z
T
t d
X
i=1
2trKL
i dw
i
p
#
: (31)
We have fromthe Burkholder-Davis-Gundy inequality the following
E Ft
Z
T
t d
X
i=1
2tr (KL
i ) dw
i
p
2 p
E Ft
Z
T
t jKj
2
jLj 2
ds
p=2
;
while fromthe Cauchy-Schwarz inequality, we have
E Ft
Z
T
t
jLjds
!
p
T p=2
E Ft
Z
T
t jLj
2
ds
!
p=2
:
Finally,we get
E Ft
Z
T
t jLj
2
ds
!
p
3 p
"
p
2 +[3
p
T p=2
"
p
2 +6
p
n p=2
"
p
0 ]E
Ft Z
T
t jLj
2
ds
!
p=2
; (32)
which implies the lastestimateof the lemma.
Theorem4.7. Let(K;L)solvetheBSRDE (1). Then,
0
L(s)dw(s)isaBMO(P)
martingale.
Proof of Theorem 4.7 From the inequality (30), we get that for any stopping
time T,
Z
T
jLj
2
ds"
5 +"
5 Z
T
jLjds Z
T
d
X
i=1
2tr (KL
i ) dw
i
: (33)
From this it follows that
E F
Z
T
jL(s)j 2
ds":
Then, Theorem 4.7follows.
Notethatasimplecase (where thegeneratorof BSRDE(1)dependsonthe martin-
gale term L in a linear way) of Theorem 4.7 has been obtained by Bismut [5]by aquite
dierent argument. This theorem plays animportantrole in solving a generalnonhomo-
geneous stochastic LQproblemand in solvinga generalmean-variance hedgingproblem.
The readeris referred toSection 6 fordetails.
We have the following a priori estimate of the uniform positivity for the rst part
K of the solution.
Theorem4.8. Lettheassumption(A3)besatised,and(K;L)solvetheBSRDE(1).
Then, there isa deterministic positive constant "
0
such that
K(t)"
0 I
nn
: (34)
The proof is an immediate consequence of a combination of Theorem 4.2 and the
following estimate.
Lemma 4.1. Let the assumption (A3) be satised. Let X t;x;u
solve the SDE (20).
Then, there is a deterministicpositiveconstant"
0
, which is independent of the control u,
such that
jxj 2
+ 1
2 E
F
t Z
T
t ju
s j
2
dsexp("
0
(T t))E F
t
jX t;x;u
T j
2
; 8u2L 2
F
(t;T;R m
): (35)
4.7 Closedness of solution.
Theorem 4.9. Assume that 8 0 the coeÆcients A
;B
;C
i
;D
i
;Q
; and N
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-
formly converges to M 0
as ! 0. Assume that 8 >0 the BSRDE (A
;B
;C
i
;D
i
;i=
1;:::;d;Q
;N
;M
) has a unique solution (K
;L
) with K
(t) 0;t 2 [0;T]: Then,
there isa pair of processes (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
);
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
);
(36)
and such that (K;L) solves 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: (37)
then the above assertions stillhold.
The proof is referred toKohlmann and Tang [19].
Remark2.1. Whentheassumptionofuniformpositivityonthecontrolweightma-
trixN is relaxed to nonnegativity, Theorem 4.9 stillholds withthe additional assumption
that there is a deterministic positive constant "
3
such that
d
X
i=1 (D
i )
0
D
i "
3 I
mm
; M
"
3 I
nn :
4.8 Transformation of BSRDEs
Let solve the dierentialequation:
8
<
: d
dt
(t) = A(t)(t); t2(0;T];
(0) = I
nn :
Consider the followingtransformation
f
K :=
0
K;
e
L:=
0
L (38)
of the BSRDE (1).
Using It^o's formula, we verify that the BSRDE (A;B;C
i
;D
i
;i =1;:::;d;Q;N;M)
becomesthenewBSRDE(0;
e
B;
e
C
i
; f
D
i
;i=1;;d;
e
Q ;N;
f
M),whichissatisedby( f
K;
e
L).
Here,
e
B :=
1
B;
e
C :=
1
C;
f
D:=
1
D
e
Q:=
0
Q;
f
M :=(T) 0
M(T):
(39)
Since A isbounded, the resultingtransformation isalsobouned.
It isinterestingtodiscussthemore generaltransformation whichsolvesthe SDE:
8
>
>
<
>
>
:
d (t) = A(t) (t)dt+ d
X
i=1 C
i
(t) (t)dw
i
(t); t2(0;T];
(0) = I
nn :
(40)
This transformation can anneal both A and C in the BSRDE (1). Note that it is not
necessarilybounded thoughA and C=(C
1
;:::;C
d
)are assumed to be bounded.
4.9 The Markov case : a PDE characterization
Sinceoptimal controls or optimalhedging/investing strategies are characterized interms
of the solutions of associated BSRDEs, it is important to characterize the solutions of
BSRDEs. When the coeÆcientsof BSRDEs are Markovian functions of anIt^o's process,
itis naturaltoconnect the solution ofa BSRDE with asystem of parabolicPDEs. Note
thatingeneralthe associated systemof parabolicPDEs contains aquadraticterm of the
gradient. We are not going into the details due to the limitation of space. The reader
is referred to among others, Peng [30, 31], Pardoux and Peng [27], and Pardoux and
Tang [28] for this direction.
5 Recent Advances on Existence and Uniqueness.
Recently, Kohlmann and Tang [17] extended Theorem 3.1to the singular case under the
assumption (A3).
Theorem5.1. Lettheassumption(A3)be satised. Then, Theorem3:1stillholds
even if N =0. Moreover, K(t) is uniformly positive.
KohlmannandTang[18]solvedtheonedimensionalcaseoftheaboveBismut-Peng's
problem.
Theorem 5.2. Let the assumptions (A1) and (A2) be satised, and n = 1:
Then, BSRDE (1) has a unique solution. Further, assume that (A3) is satised. Then,
BSRDE (1) still has a unique solution (K;L) even if N = 0, and moreover, K(t) is
uniformly positive.
This theoremmeetsthe needsforBSRDEs tobeappliedtonance. Anapproxima-
tiontechnique isused,whichismotivatedby theworksof Kobylansky [16],and Lepeltier
and San Martin [22, 23].
Kohlmann and Tang [19] proved the global existence and uniqueness result for
BSRDE(1)forsomemulti-dimensionalcases: They arespecial buttypical, forthegener-
atorcontainsaquadratictermonL. Theresultsarestatedbythefollowingtwotheorems.
Theorem 5.3. (the singular case) Let the assumptions (A1) and (A2) be satis-
ed, and d =1. Assume that there isa deterministic positive constant " such that
M "I
nn
(41)
D 0
D(t)"I
mm
: (42)
Then, the BSRDE:
8
>
>
>
<
>
>
>
:
dK = [A
0
K+KA+C 0
KC+Q+C 0
L+LC
(KB+C 0
KD+LD)(D 0
KD) 1
(KB+C 0
KD+LD) 0
]dt+Ldw;
0t<T;
K(T) = M:
(43)
has a unique solution (K;L) with K(t;!) beinguniformly positive w.r.t. (t;!):
The main idea for the proof of Theorem 5.3 isto dothe inverse transformation:
f
K :=K 1
; (44)
which turnsout tosatisfy aRiccati equationwhose generatordepends onthe martingale
term ina linear way.
First, since D isinversable, we can rewritethe BSRDE (43) as
8
>
<
>
:
dK = [
e
A 0
K K
e
A+Q K e
BK 1
e
B 0
K LK
1
L
+K e
BK 1
L+LK 1
e
B 0
K]dt+Ldw;
K(T) = M;
(45)
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
: (46)
UsingIt^o's formula, we can write the equation for the inverse f
K of K:
(
d f
K = [
f
K e
A 0
+ e
A f
K f
KQ f
K+ e
B f
K e
B 0
+ e
B e
L+ e
L e
B 0
]dt+ e
Ldw;
f
K(T) = M 1
;
(47)
where
e
L:= K 1
LK 1
:
FromTheorem3.1,the aboveBSRDE
e
A;Q 1=2
; e
B;0;0;I
mm
;M 1
has aunique solution
( f
K;
e
L) with f
K 0; which implies that f
K 1
(t) is uniformly positive in (t;!). More-
over, from the fact that f
K(T) = M 1
"
1
1 I
nn
, we derive that f
K is uniformly posi-
tive. This shows that f
K 1
(t) isuniformlybounded. Therefore ( f
K 1
; f
K 1
e
L f
K 1
) solves
BSRDE (43).
The uniqueness results from the Feynman-Kac representation result Theorem 4.2.
Infact,assumethat( c
K;
b
L)alsosolvesthe BSRDE(43). Then,fromTheorem 4.2,wesee
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:
(48)
From this, proceedingidenticallyas inthe proof of Theorem 4.6, 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: (49)
Hence, ÆL=L b
L=0.
Theorem5.4. (theregularcase) Lettheassumptions(A1)and(A2)besatised.
Further assume that d=1;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
t
1
;t
2 2[0;T];jt
1 t
2 jh
jN(t
1
) N(t
2
)j = 0:
(50)
Then, the BSRDE:
8
>
<
>
:
dK = [A
0
K+KA+Q LD(N +D 0
KD) 1
D 0
L]dt+Ldw;
0t <T;
K(T) = M:
(51)
has a unique solution (K;L) with K(t)0;8t2[0;T]:
For the regular case, the situation is alittle complex: we easilysee that the above
inversetransformationonthe rstunknown variablecannot eliminatethe quadraticterm
ofthesecondunknown variable. However, wecanstillsolvesomeclassesofBSRDEswith
thehelpofdoingsomeappropriatetransformation. Thewholeproofisdividedintoseveral
propositions.
Proposition 5.1. Assume that QA 0
(D 1
) 0
ND 1
+(D 1
) 0
ND 1
A;m=n; and
D and N are positive constantmatrices. Then, Theorem 5:4 holds.
Proof of Proposition 5.1. Write
c
N :=(D 1
) 0
ND 1
: (52)
Then, the BSRDE (51) reads
8
>
<
>
:
dK = [A
0
K+KA+Q L(
c
N +K) 1
L]dt+Ldw;
0t <T;
K(T) = M:
(53)
The equation for K :=N+K is
8
>
<
>
: d
c
K = [A
0
c
K + c
KA+Q A 0
c
N c
NA b
L c
K 1
b
L]dt+ b
Ldw;
0t<T;
c
K(T) = c
N +M:
(54)
Notethat c
N+M isuniformlypositive. FromTheorem5.3,itfollowsthattheBSRDE(54)
has a solution( c
K;
b
L). Therefore ( c
K c
N; b
L)solvesBSRDE (51).
Proposition 5.2. Assume thatA =0 andD andN areconstant matrices. Then,
Theorem 5:4 holds.
Proof of Proposition 5.2. First assume m=n. Consider the followingapprox-
imatingBSRDEs:
(
dK = [Q LD
(N +D 0
KD
)
1
D 0
L]dt+Ldw;
K(T) = M
(55)
where
D
:=D+ I
mm
>0;>0:
FromProposition5.1,weseethattheBSRDE(55)hasasolution(K
;L
)forevery>0.
From Theorem 4.9, it follows that K
uniformlyconverges to some K 2 L 1
F
(0;T;S n
+ )\
L 1
(;F
T
;P;C([0;T];S n
+
))andL
stronglyconvergestosomeL2L 2
F
(0;T;S n
),andthat
(K;L)solvesBSRDE (51) 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 (51) when A=0is rewritten as
(
dK = [Q L
f
D(
f
N + f
D 0
K f
D) 1
f
D 0
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
) 0
NT 1
:=
c
N
11 c
N
12
c
N 0
12 c
N
22
!
>0:
Then, c
N
11
>0:The BSRDE (51) whenA =0 isrewritten as
(
dK = [Q L
c
D(
f
N
11 +
c
D 0
K c
D) 1
c
D 0
L]dt+Ldw;
K(T) = M
From the preceding result, we obtainthe desiredexistence result.
F
t
-adapted bounded matrix processes. Then, Theorem 5:4 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
suchthatoneachinterval[t
i
;t
i+1
][0;T],DandN areconstantF
ti
-measurablebounded
random matrices. Then, we can solve the BSRDE one interval by one, in an inductive
and backward way.
Proposition 5.4. Assume that A=0:Then, Theorem5:4 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 k
and N k
are are piece-wisely constant, F
t
-adapted, bounded matrix pro-
cesses. Further, since the trajectories of D and N are uniformly continuous in !, D k
(t)
and N k
(t)converge respectively toD and N, uniformlyin(t;!): Thatis, 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,wesee thattheBSRDE(0;0;0;D k
;Q;N k
;M)hasaglobalsolution
(K k
;L k
), and then fromTheorem 4.9, it follows that Theorem 5.4holds.
Proof of Theorem 5.4. 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
((
0
) 1
f
K 1
;( 0
) 1
e
L 1
)
solves the originalBSRDE (A;0;0;D;Q;N;M). Here, f
D;
e
Q; and f
M are dened by (39).
The uniqueness can be proved in the same way as inthe proof of Theorem 5.3.
6 Applications to Control and Finance
6.1 Control-theoretic application
Assume that
2L 2
(;F
T
;P;R n
); q;f;g
i 2L
2
F
(0;T;R n
): (56)
Consider the following optimalcontrolproblem (denoted by P
0 ):
min
u2L 2
F (0;T;R
m
)
J(u;0;x) (57)
J(u;t;x)= E F
t
(M(X t;x;u
T
);X t;x;u
T
)
+E Ft
Z
T
t
[(Q(X t;x;u
q);X t;x;u
q)+(Nu;u)]ds
(58)
and X t;x;u
solving the followinglinear SDE
8
>
>
<
>
>
: dX
s
= (AX
s +Bu
s
+f(s))ds+ d
X
i=1 (C
i X
s +D
i u
s +g
i
(s))dw
i
; t<sT;
X
t
= x; u2L 2
F
(t;T;R m
):
(59)
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
: (60)
Theorem 6.1 Let the two assumptions (A1) and (A2), or (A1) and (A3) be
satised. Let (K;L) solve the BSRDE (1). Then, the BSDE
8
>
>
<
>
>
:
d (t) = [ b
A 0
+ d
X
i=1 b
C 0
i (
i Kg
i
) Kf d
X
i=1 L
i g
i
+Qq]dt+ d
X
i=1
i dw
i
;
(T) = M
(61)
where b
A and b
C
i
are dened by (15),has a unique F
t
-adapted solution ( ;) with
2L 2
F
(0;T;R n
)\L 2
(;F
T
;P;C([0;T];R n
)); 2
L 2
F
(0;T;R n
)
d
: (62)
Moreover, theoptimalcontrol b
u forthenon-homogeneousstochasticLQ problemP
0 exists
uniquely and has the followingfeedback law
b
u = (N+ d
X
i=1 D
0
i KD
i )
1
[(B 0
K+ d
X
i=1 D
0
i KC
i +
d
X
i=1 D
0
i L
i )
c
X
B 0
+ d
X
i=1 D
0
i (Kg
i
i )]
(63)
where c
X :=X 0;x;bu
:
Remark 6.1. Notethat b
A and b
C
i
dependon Lin general, andthus theymightnot
beuniformlybounded. Inthis case,wehavenoavailable|totheauthors'bestknowledge|
theoremtoguaranteethe existence andtheuniquenessof aglobaladaptedsolution,though
the BSDE (61) is linear.
Proof of Theorem 6.1. Our assumptions guarantee that there is a unique
optimalcontrol b
u2L 2
F
(0;T;R m
): The optimality condition implies
B 0
e
p+ X
i=1 D
0
i e
q
i +N
b
u=0
where (p;q) solvesthe BSDE (calledthe adjoint equation)
8
>
>
<
>
>
: d
e
p(t) = [A 0
e
p(t)+ d
X
i=1 C
0
i e
q
i
(t)+Q(
c
X
t
q(t))]dt+ d
X
i=1 e
q
i (t)dw
i (t);
e
p(T) = M( c
X
T )
(64)
with
e
p2L 2
F
(0;T;R n
)\L 2
(;F
T
;P;C([0;T];R n
));
e
q2
L 2
F
(0;T;R n
)
d
: (65)
ViaIt^o's formula, we check out that the pair ( ;) dened by the following
(t):=K(t) c
X
t e
p(t);
i
(t):=K(t)[C
i c
X
t +D
i b
u
t +g
i
(t)]+L
i c
X
t e
q
i (t)
solves the BSDE (61). It is obvious that 2L 2
F
(0;T;R n
)\L 2
(;F
T
;P;C([0;T];R n
)).
Since R
0
L(s)dw(s) is a BMO(P)-martingale, it follows from Theorem 1.1 (i) and (iii) of
Ba~nuelos and Bennett [1] that R
T
0 L
i (s)
c
X
s dw
i
(s) is square-integrable. Therefore, L
i c
X 2
L 2
F
(0;T;R n
), and
i 2L
2
F
(0;T;R n
):
It is standard to get the explicit formula (63) of b
u from the optimality condition.
The proof iscomplete.
The following can be veried by a pure completion of squares.
Theorem 6.2 Suppose thatthetwo assumptions (A1)and(A2), or (A1)and(A3)
are satised. Let (K;L) solve BSRDE (1). Then, 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
(66)
with
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
0
i KD
i )u
0
;u 0
)ds
(67)
and
u 0
:=(N + d
X
i=1 D
0
i KD
i )
1
[B 0
+ d
X
i=1 D
0
i (
i Kg
i
)]; tsT: (68)
6.2 Financial application.
As anapplication of the above results, the mean-variance hedgingproblem withrandom
marketconditions is considered. The mean-variancehedging problemwas initiallyintro-
ducedbyFollmerandSondermann[9],andlater widelystudiedby DuÆeandRichardson
Pham, Rheinlander and Schweizer [34], Gourieroux, Laurent and Pham [12], and Lau-
rent and Pham [21]. All of these works are based on a projection argument. Recently,
KohlmannandZhou[20]usedanaturalLQtheoryapproachtosolvethecaseofdetermin-
istic market conditions. Kohlmann and Tang [17] used a natural LQtheory approach to
solvethecase ofstochasticmarketconditions,but themarketconditionsare onlyallowed
toinvolvea smallerltrationfF 2
t
g. Kohlmannand Tang [18] completelysolved the case
of random market conditions by using Theorems 6.1 and 6.2, and the optimal hedging
portfolioand the variance-optimalmartingale measure are characterized by the solution
of the associated BSRDE.
Considerthenancialmarketinwhichtherearem+1primitiveassets: onenonrisky
asset (the bond) of price process
S
0
(t)=exp( Z
t
0
r(s)ds); 0tT; (69)
and m risky assets (the stocks)
dS(t)=diag(S(t))((t)dt+(t)dw(t)); 0tT: (70)
Here w =(w
1
;:::;w
d )
0
is a d-dimensional standard Brownian motion dened on a com-
pleteprobabilityspace(;F;P),andfF
t
;0tTgistheP-augmentationofthenatu-
ralltrationgeneratedbythed-dimensionalBrownianmotionw. Assumethattheinstan-
taneousinterestrater,them-dimensionalappreciationvectorprocessandthevolatility
md matrix process are progressively measurable with respect to fF
t
;0 t Tg.
For simplicity of exposing the main ideas, assume that they are uniformly bounded and
there exists a positive constant" such that
0
(t)"I
mm
; 0tT;a:s: (71)
The risk premiumprocess is given by
(t)= 0
( 0
) 1
e
(t); 0t T (72)
where e
m
=(1;:::;1) 0
2R m
; and e
:= re
m :
Foranyx2Rand 2L 2
F
(0;T;R m
),denetheself-nanced wealthprocessX with
initialcapitalx and with quantity invested inthe risky asset S by
(
dX
t
= [rX
t +(
e
;)]dt+ 0
dw; 0<t T;
X
0
= x; 2L 2
F
(0;T;R m
):
(73)
Given a random variable 2 L 2
(;F
T
;P), consider the quadratic optimal control
problem:
Problem P
0;x
() min
2L 2
F (0;T;R
m
) EjX
0;x;
T
j 2
(74)
whereX 0;x;
isthe solutiontothe wealth equation(73). The associated value functionis
denotedbyV(t;x);(t;x)2[0;T]R : TheminimumpointofV(t;x)overx2R forgiven
time t is dened to bethe approximate price for the contingent claim at time t.
0;x
nance. It is aone-dimensional singular stochastic LQproblem P
0 .
Denote by
i
the i-th column of the volatility matrix . The associated Riccati
equationis a non-linear singularBSDE:
dK = [2rK (e
0
K+ P
d
i=1 L
i
0
i
)(K 0
) 1
(K e
+ P
d
i=1 L
i
i )]dt+
P
d
i=1 L
i dw
i
= [(2r jj 2
)K 2(;L) K 1
L 0
0
( 0
) 1
L]dt+(L; dw); 0t<T
K(T)=1:
(75)
Let( ;) is the F
t
-adapted solution of the BSDE
d = f[r jj 2
(;K 1
L)]
P
d
i=1 [
i +K
1
0
i (
0
) 1
L]
i gdt+
P
d
i=1
i dw
i
;
= f[r jj 2
(;K 1
L)] (+K 1
0
( 0
) 1
L;)gdt+(; dw);
(T)=
(76)
Theorem 6.1 providesanexplicit formulafor the optimalhedging portfolio:
= (
d
X
i=1
i K
0
i )
1
[(e
K+
d
X
i=1
i L
i )X
e d
X
i=1
i
i ]
= (K
0
) 1
[(
e
K+L)X e
]
= (
0
) 1
[(
e
+K 1
L)X e
K 1
K 1
]
(77)
where(K;L) isthe F
t
-adapted solution tothe Riccati equation(75). The value function
V isalso given by
V(t;x)=K(t)x 2
2 (t)x+E Ft
2
E Ft
Z
T
t (
e
+)
0
(K 0
)(e
+)ds (78)
where:=(
1
;:::;
n )
0
:So, theapproximatepricep(t)attimetforthe contingentclaim
is given by
p(t)=K 1
(t) (t): (79)
The abovesolutionneednot introducetheadditionalconcepts oftheso-calledhedg-
ing numeraire and variance-optimal martingale measure, and therefore is simpler than
that ofGourierouxet al[12], and Laurentand Pham [21]. To beconnected tothe latter,
the optimalhedging portfolio(77) is rewritten as
= (
0
) 1
[(
e
+
e
L )(X e
)
e
]: (80)
Here,
e
L:=LK 1
; e
:= K 1
; e
:=K 1
L K 2
: (81)
and the pair ( e
; e
) solves the BSDE:
(
d e
= fr e
+( e
; e
)gdt+( e
; dw); 0t<T;
e
(T) =
(82)
e
:= [I
0
( 0
) 1
]LK 1
: (83)
Theprocess e
isjusttheapproximatepriceprocess,andtheBSDE(82)istheapproximate
pricing equation.
In view of Theorem 6.1, it follows from Theorem 1.1 (i) and (iii) of Ba~nuelos and
Bennett [1]that
e
2L 2
F
(0;T;R n
)\L 2
(;F
T
;P;C([0;T];R ));
e
2L 2
F
(0;T;R d
): (84)
Note that the optimalhedging portfolio(77) consists of the followingtwo parts:
1
:= (
0
) 1
( e
+
e
L)X (85)
and
0
:=( 0
) 1
[(
e
+
e
L) e
+ e
]; (86)
and satises
=
1
+ 0
: (87)
Therst part 1
istheoptimalsolutionof thehomogeneousmean-variancehedgingprob-
lemP
0;x
(0)(thatisthecase of=0forthe problemP
0;x
()). Thecorrespondingoptimal
wealth process X 0;1;
1
is the solution tothe followingoptimal closed system
(
dX
t
= X
t
[(r jj 2
(;
e
L ))dt (+ 0
( 0
) 1
e
L; dw)]; 0<t T;
X
0
= 1;
(88)
andisjustthehedgingnumeraire. So,thehedgingnumeraireisjustthestate(wealth)tran-
sitionprocess of the optimalclosedsystem (88) fromtime 0,oritis justthe fundamental
solution of the optimalclosed system (88).
To understand the quantity e
, consider the BSDEsatised by (K ;L)
(
dK = f(2r jj 2
)K+2(;L)+K 1
L 0
[I
0
( 0
) 1
]Lgdt+(L; dw);
K (T) = 1
(89)
with K:=K 1
and L := LK 2
. It is the BSRDE for the following singular stochastic
LQproblem (denoted by P 0
0;x ):
Problem P 0
0;x
min
2L 2
F (0;T;R
d
) EjX
0;x;
T j
2
(90)
where X 0;x;
is the solution tothe followingstochastic dierentialequation
(
dX
t
= X
t
[ rdt (; dw)]+([I 0
( 0
) 1
];dw); 0t T;
X
0
= x; 2L 2
F
(0;T;R d
):
(91)
Its optimalcontrol has the followingfeedback form
b
= K
1
LX =LK 1
X: (92)
TheproblemP 0
0;1
isjustthe so-calleddualproblemof theproblemP
0;1
(0)in[12,21], and
sothe variance-optimalmartingalemeasure is P
dened as
dP
:=exp
Z
T
0 (
e
;dW) 1
2 Z
T
0 j
e
j 2
dt
dP: (93)
P
is anequivalent martingalemeasure.
Note that e
has the followingexplicit formula:
e
(t)=E Ft
exp( Z
T
t
r(s)ds); 0tT: (94)
Here, thenotationE Ft
stands forthe expectationoperatorconditioningonthe -algebra
F
t
with respect to the probability P
. The discounted e
is just the integrand of the
stochastic-integral-representation of the P
-martingale fE Ft
exp( R
T
0
r(s)ds);0 t
Tg(w.r.t. the P
-martingaleW + R
0 e
dt).
As inKohlmann and Zhou[20], again, the formula(80) has aninteresting interpre-
tationintermsof mathematicalnance. The optimalhedgingportfolio in(80)consists
of the two components: (a) ( 0
) 1
~
|it may be interpreted as the perfect hedging
portfoliofor the contingent claim with the risk premiumprocess
~
(that is, under the
variance-optimal martingale measure), (b) ( 0
) 1
(~+
~
L )(
~
X)|it is a generalized
Merton-type portfolio for a terminal utility function c(x) = x 2
(see Merton [24]), which
invests the capital (
~
X) leftover after fulllingthe obligationfromthe perfect hedge
underthe variance-optimalmartingale measure.
Acknowledgement The second author would like to thank the hospitality of
DepartmentofMathematicsand Statistics, andthe Center of FinanceandEconometrics,
UniversitatKonstanz, Germany.
References
[1] Ba~nuelos, R. and Bennett, A., Paraproducts and commutators of martingale trans-
forms. Proceedings of the American Mathematical Society, 103 (1988), 1226{1234
[2] Bellman, R., Functional equations in the theory of dynamic programming,positivity
and quasilinearity, Proc.Nat. Acad. Sci. U.S.A., 41 (1955),743{746
[3] Bismut, J. M., Conjugate convex functions in optimal stochastic control, J. Math.
Anal.Appl., 44 (1973),384{404
[4] Bismut, J. M., Linear quadratic optimal stochastic control with random coeÆcients,
SIAM J. Control Optim.,14 (1976),419-444
[5] 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
control weight costs, SIAM J. ControlOptim. 36 (1998), 1685-1702
[7] DuÆe, D. and Richardson, H. R., Mean-variance hedging in continuous time, Ann.
Appl. Prob., 1 (1991),1-15
[8] Follmer, H. and Leukert, P., EÆcient hedging: cost versus shortfall risk, Finance &
Stochastics, 4(2000), 117{146
[9] Follmer, H. and Sonderman, D., Hedging of non-redundant contingent claims. In:
Mas-Colell, A., Hildebrand, W. (eds.) Contributions to Mathematical Economics.
Amsterdam: North Holland 1986, pp. 205-223
[10] Follmer,H.and Schweizer,M.,Hedgingof contingentclaimsunderincompleteinfor-
mation.In: Davis,M.H.A.,Elliott,R.J.(eds.)AppliedStochasticAnalysis.(Stochas-
tics Monographs vol.5) London-New York: Gordon &Breach, 1991, pp. 389-414
[11] Gal'chuk, L. I., Existence and uniqueness of a solution for stochastic equations with
respect to semimartingales, Theory of Probability and Its Applications, 23 (1978),
751{763
[12] Gourieroux,C., Laurent,J.P.andPham,H.,Mean-variance hedgingand numeraire,
Mathematical Finance,8 (1998), 179-200.
[13] Heston, S., A closed-form solution for option with stochastic volatility with applica-
tions to bond and currency options. Rev.Financial Studies, 6(1993), 327{343
[14] Hipp, C., Hedging general claims, Proceedings of the 3rd AFIR colloqium, Rome,
Vol. 2,pp. 603-613(1993)
[15] Hull, J. and White, A., The pricing of options on assets with stochastic volatilities.
J. Finance,42 (1987),281-300
[16] Kobylanski, M., Resultats d'existence et d'unicite pour des equations dierentielles
stochastiques retrogrades avec des generateurs a croissance quadratique, C. R. Acad.
Sci. Paris, 324 (1997),Serie I, 81-86
[17] Kohlmann,M.and Tang,S., Optimalcontrolof linear stochasticsystems with singu-
lar costs, and the mean-variance hedgingproblem with stochastic market conditions.
submitted
[18] Kohlmann, M. and Tang, S., Global adapted solution of one-dimensional backward
stochastic Riccati equations, with application to the mean-variance hedging. submit-
ted
[19] Kohlmann,M.andTang,S.,Multi-dimensionalbackwardstochasticRiccatiequations
and applications. submitted
equations and stochastic controls: a linear-quadratic approach, SIAM J. Control &
Optim.,38 (2000),1392-1407
[21] Laurent, J. P. and Pham, H., Dynamic programming and mean-variance hedging,
Finance and Stochastics, 3 (1999),83-110
[22] Lepeltier, J. P. and San Martin, J., Backward stochastic dierential equations with
continuous coeÆcient, Statistics& Probability Letters,32 (1997),425-430
[23] Lepeltier, J. P. and San Martin, J., Existence for BSDE with superlinear-quadratic
coeÆcient, Stochastics& Stochastics Reports, 63(1998), 227-240
[24] Merton, R., Optimum consumption and portfolio rules in a continuous time model,
J. Econ. Theory, 3 (1971),pp. 373-413;Erratum 6 (1973),213-214.
[25] Monat, P. and Stricker, C., Follmer-Schweizer decomposition and mean-variance
hedging of general claims, Ann. Probab. 23,605{628
[26] Pardoux,E.and Peng, S.,Adapted solutionof backwardstochasticequation, Systems
Control Lett., 14(1990), 55{61
[27] Pardoux,E.andPeng, S.,Backwardstochasticdierentialequationsand quasi-linear
parabolic partial dierential equations, in: Rozovskii, B. L., Sowers, R. S. (eds.)
Stochastic Partial Dierential Equations and Their Applications, Lecture Notes in
Control and Information Science 176, Springer, Berlin,Heidelberg, New York 1992,
pp. 200{217
[28] Pardoux, E. and Tang, S., Forward-backward stochastic dierential equations with
application to quasi-linear partial dierential equations of second-order, Probability
Theory and Related Fields,114 (1999), 123-150
[29] Peng, S., StochasticHamilton-Jacobi-Bellmanequations,SIAM J.Controland Opti-
mization30 (1992) ,284-304
[30] Peng, S., Probabilistic interpretation for systems of semilinear parabolic PDEs,
Stochastics &Stochastic Reports, 37 (1991),61-74
[31] Peng, S., A generalized dynamic programming principle and Hamilton-Jacobi-
Bellman equation, Stochastics &Stochastics Reports, 38 (1992),119{134
[32] Peng, S., Some open problems on backward stochastic dierential equations, Control
of distributed parameter and stochastic systems, proceedings of the IFIP WG 7.2
international conference, June 19-22, 1998, Hangzhou, China
[33] Peng,S.,Ageneralstochasticmaximumprincipleforoptimalcontrolproblems,SIAM
J. ControlOptim., 28(1990), 966-979
[34] Pham,H.,Rheinlander,T.,andSchweizer,M.,Mean-variancehedgingforcontinuous
processes: New proofs and examples, Finance and Stochastics, 2 (1998),173-198
agement Science, 35(1989), 1045-1055
[36] Schweizer,M.,Mean-variancehedgingforgeneralclaims,Ann.Appl.Prob.,2(1992),
171-179
[37] Schweizer, M., Approaching random variables by stochastic integrals, Ann. Probab.,
22(1994), 1536{1575
[38] Schweizer, M., Approximation pricing and the variance-optimal martingalemeasure,
Ann. Probab., 24(1996), 206{236
[39] Stein, E. M. and Stein, J. C., Stock price distributions with stochastic volatility: an
analytic approach,Rev. Financial Studies, 4 (1991), 727-752
[40] Tang, S. and Li, X., Necessary conditions for optimal control of stochastic systems
with random jumps, SIAM J. Control Optim.,32(1994), 1447{1475
[41] Wonham,W.M.,On amatrixRiccatiequationofstochasticcontrol,SIAMJ.Control
Optim.,6 (1968),312-326
[42] Zhou,X. and Li,D., The continuous time mean-variance selection problem, Applied
Mathematics and Optimization,(2000),
