JOHANNES LEITNER
JOHANNES.LEITNER@UNI-KONSTANZ.DE
CENTER OF FINANCE AND ECONOMETRICS (COFE)
UNIVERSITY OF KONSTANZ
Abstract. Inanarbitragefreeincompletemarketweconsiderthe
problemofmaximizingterminalisoelasticutility. Therelationship
betweentheoptimalportfolio,theoptimalmartingalemeasure in
thedualproblemandtheoptimalvaluefunctionoftheproblemis
describedbyanBSDE.Foratotally unhedgeable pricefor instan-
taneousrisk,isoelasticutilityofterminalwealthcanbemaximized
usingaportfolioconsisting of thelocally risk-freebond andalo-
cally eÆcient fund only. Inamarkovianmarketmodel wend a
non-linearPDEforthelogarithmofthevaluefunction. Fromthe
solutionwecanconstructtheoptimalportfolioandthesolutionof
thedual problem.
Keywords: Utility,OptimalPortfolios,DualityTheory.
AMS91Classications: 90A09,90A10
JELClassications: G11
IwouldliketothankProfessorM.Kohlmannforhissuggestionsandsupport. I
amalsothankfultoProfessorM.SchweizerandProfessorS.Tang.
Introduction
We study the problem of maximizing expected isoelastic utility of
terminalwealth inanincomplete continuous time marketwith contin-
uouspriceprocess. Theisoelasticutilityofexponentp6=0;1isdened
asu (p)
(x):=sgn(1 p) jxj
p
p
andforp=0byu (0)
(x):=ln(jxj). The two
cases p < 1 and p > 1 are very dierent in there economic interpre-
tation, but can be treated to some extend by the same mathematical
methods. Solving theoptimizationproblemforp<1isaplausibleap-
proachtondportfoliosofoptimalexpectedgrowth. Thereareseveral
papers on this topic: See, e.g. Merton (1990), Pliska (1986), He and
Pearson (1991), Karatzas, Lehoczky, Shreve and Xu (1991), Karatzas
and Shreve (1999),Kramkov and Schachermayer (1999).
For p = 2 the problem is related to the mean-variance hedging
problem,see Gourieroux, Laurent and Pham (1998),(GLP98), Pham,
Rheinlander and Schweizer (1998) and Laurentand Pham (1999).
The theory of stochastic duality, which goes back to Bismut (1973,
1975), is the central tool for solving these problems. This theory al-
lows to formulate an optimization problem over a set of martingale
measures, beingdualtothe originaloptimizationproblemoveraset of
self-nancing hedging-strategies. Under quite general conditions, the
solution of one of the problems can be transformed into a solution of
the corresponding dual problem.
Anotherimportantapproach,istotrytosolvetheoptimizationprob-
lemlocally,i.e. by so-calledmyopic strategies whichmaximizeinsome
sensetheexpectedgrowthrateoftheportfolioateveryinstantoftime.
In some important cases these strategies turn out to be globally op-
timal too. See, e.g., Mossin (1968), Leland (1972), Aase (1984, 1986,
1987, 1988), Foldes (1991), Goll and Kallsen (2000). This approach
isrelated tothe risk-sensitive stochastic controlapproach, see Bielecki
and Pliska (1999,2000).
We consider an arbitrage-free (ina sense to be specied later) con-
tinuous time market model with unrestricted trading. We use the
modern equivalent martingale measure approach, see Harrison and
Pliska (1981), Delbaen and Schachermayer (1994). After some techni-
calpreparations inSection 1and specication of the model inSection
2, we formulate the optimization problem and its corresponding dual
Problem in Section 3. We show a representation property (formula
(3.8)), relating the terminal value V opt
T
of a portfolio to a martingale
Research supported by the Center of Finance and Econometrics, ProjectMathe-
maticalFinance.
measure Z opt
T
, to be suÆcient for the optimality of V opt
T
for the utility
maximizationproblemandthe optimalityof Z opt
T
forthe dualproblem.
Theoptimalvaluesofthetwoproblemarerelatedbyasimpleformula.
In Section 4 weintroduce the notion of a totally unhedgeable price for
instantaneous risk. In this situation we can explicitlysolve the utility
optimizationproblem. TheoptimalportfolioisalocallyeÆcientportfo-
lio,anotionweintroduceinSection5. InSection6wegiveanexistence
result for the solutions of the two optimization problems. In Section
7 we derive abackward stochastic dierentialequation, (BSDE), such
thatfromthesolutiontheoptimalportfolio,theoptimalvaluefunction
and the solutionof the dualoptimization problemcan beconstructed.
See Yong and Zhou (1999) for an introduction to BSDEs. In Section
8 we consider a markovian market model. We transform the BSDE
into a non-linear PDE for the logarithm of the value function. From
the partial derivatives of the solution, we can construct under addi-
tional assumptions the optimal portfolio and the solution of the dual
optimization problem.
1. Self-financing Hedging Strategies
Let a ltered probability space
1
:= (;F;(F
s )
s0
;P), satisfying
theusualconditionsbegiven. ForsimplicityweassumeF
0
tobetrivial
up to sets of measure 0 with respect to P and F
1
= F
1
:=F. For
an adapted process X set X
0
:=X
0 , X
t
:=lim
h&0 X
t h
for t >0 if
thelimitexists anddenethe processesX :=(X
t )
0t<1
andX :=
X X if X
t
exists for allt >0. The components of X are denoted
as X i
; 1 id. Fora process X and a map :!
R
+
, denote the
stopped process attime by X
. Wewilloftenrestrict asemimartin-
gale X on
1
to an interval [t;T]; 0 t T < 1, resp. to [t;1).
Therefore we introduce the following ltered probability space (again
satisfying the usual conditions),
[t;T]
:=
;F
T
;
F [t;T]
s
s0
;P
jF
T
for all 0 t T 1, t <1, where F [t;T]
s
:=F
t_s^T
for 0 s <1.
The process X [t;T]
s
:=X
t_s^T
isthen a semimartingaleon
[t;T]
. How-
ever, on[t;T] we oftenwrite X insteadof X [t;T]
. Set
T :=
[0;T] .
For q >1 dene L q
(
[t;T]
), respectively L q
t (
[t;T]
), asthe set of F
T -
measurablerandomvariablesX,suchthatE[X]<1a.s.,respectively
E
t
[X] < 1 a.s., where E
t
[] :=E[jF
t
] denotes the generalized condi-
tional expectation. Denote the conditional variance by Var
t
(). The
stochastic exponential of a semimartingale X is denoted as E(X) and
we set E(X):=E(1 X). Asa generalreferences wecite Jacod and
Shiryaev (1987), (J&S 87), and Jacod (1979). Denote the set of pre-
dictableprocesses which are locallyintegrable, resp. locally Riemann-
Stieltjes integrable, with respect to a local martingale M, resp. with
respectto aprocessA ofnitevariation,byL 1
l oc
(M),resp. by L 1
l oc (A).
If the semimartingale X admits a decomposition X = X
0
+A+M,
where M is a local martingale and A is a process of nite variation
then L 1
l oc
(X):=L 1
l oc
(M)\L 1
l oc (A).
We can now dene the market model: Let S = (S
t )
0t<1
be a R d
-
valued semimartingale. M := (
1
;S) = ((;F;(F
s )
s0
;P);S) is a
model for a market, where S describesthe price processes of d assets.
We will often consider such a market on an interval [t;T]; 0 t <
T < 1. This is equivalent to work with the following market model
M
[t;T]
denedbyM
[t;T]
:=
[t;T]
;S [t;T]
. Set M
T
:=M
[0;T]
. Wewant
to model the economic activity of investing money into a portfolio of
assetsand changingthenumberof assetsheldovertimeaccording toa
certainhedgingstrategy. Thisisachieved withthefollowingdenition:
Denition 1.1. AhedgingstrategyinthemarketMisaH 2L 1
l oc (S).
The corresponding value process V H
of H is dened as V H
:= HS.
The gains process of H is dened as the semimartingaleG H
:=HS.
H is called self-nancing if V H
= V H
0 +G
H
, i.e. H
t S
t
= H
0 S
0 +
R
t
0 H
s dS
s
;8t0. Denote thespaceofallself-nancinghedgingstrate-
gies in Mby SF(M).
Note that for H 2SF(M), we have H [t;T]
2SF(M
[t;T]
). The idea
of a self-nancing hedging strategy is that the changes over time of
thecorrespondingvalue processaresolelycausedbythechangesofthe
value ofthe assetsheldintheportfolioandnot bywithdrawing money
fromor adding money tothe portfolio.
Denition 1.2. A semimartingaleB such that B and B are strictly
positiveiscalledanumeraireforthemarketM. Themarketdiscounted
withrespecttoB isthendened asM B
:=
1
;S B
,whereS B
:=
S
.
For 0 t T < 1, the market restricted to the interval [t;T] is
dened asM B
[t;T]
:= M B
[t;T]
=
[t;T]
; S B
[t;T]
.
Note that for a numeraire B, B 1
is a numeraire too and S B
is a
semimartingale.
Usually there is in addition to the market M a numeraire B given
and the market
M := (
1
;
S);
S := (S;B) is considered. Often B
is the price process of a locally risk-free bond. If the numeraire is
traded, i.e. the value process of a portfolio in M, one can try to
extend ahedgingstrategyinMtoaself-nancing hedgingstrategyin
M. Dene the discounted market
M B
=(
1
;(S B
;1)). The ideaisto
extend H toaself-nancing hedgingstrategy
H =(H;
^
H)2SF(
M B
)
by dening the process
^
H :=H
0 S
B
0
+HS B
HS B
andthen toshow
that
H is a self-nancing hedging strategy in
M too, see Geman, El
Karui and Rochet (1995) and Goll and Kallsen (2000). (Note that
HS B
HS B
=(HS B
) +(HS B
) HS B
=(HS B
) +HS B
HS B
=(H S B
) HS
B
ispredictable, hence
H aswell.)
Proposition 1.3. Let B be a numeraire for the market M. Then
SF(M B
)=SF(M) holds.
Proof. Let H 2 SF(M B
). Set V B
= HS B
. First, we have to show
H 2 L 1
l oc
(S). Since S = S B
B =S
0 +S
B
B +B S B
+[S B
;B] this
followsifweshowthatH 2L 1
l oc (S
B
B)\L 1
l oc
(B S B
)\L 1
l oc ([S
B
;B]).
Note that HS B
= H(S B
S B
) = V B
(H S B
) = V B
0
+H
S B
(H S B
) = V B
0
+(H S B
) = V B
, which is locally bounded.
Since[S B
B;S B
B]=(S B
S B
)[B;B]and H(S B
S B
)H =(V B
) 2
is locally integrable with respect to [B;B], we nd H 2 L 1
(S B
B).
That H 2L 1
l oc
(B S B
)\L 1
l oc ([S
B
;B]) iseasy tosee. We calculate
HS = H(S B
B)=H(S B
B+B S B
+[S B
;B])
= (HS B
)B+(B H)S B
+[HS B
;B]
= V B
B+B (H S B
)+[V B
;B]
= V B
B V
B
0 B
0
=HS B
B H
0 S
B
0 B
0
= HS H
0 S
0 :
This implies SF(M B
) SF(M). Now observe that (M B
) B
1
= M,
since B 1
is anumeraire. This impliesthe reverse inclusion.
There is an alternative way to construct self-nancing hedgingstrate-
gies:
Lemma 1.4. Let H 2 SF(M) be such that V H
6= 0 and V H
6= 0
almost surely. Set
~
H :=
H
V H
. Then
~
H2L 1
l oc (S),
~
HS =1 and
V H
=V H
0 +V
H
(
~
HS)=V H
0 E(
~
HS);
(1.1)
holds. Conversely, let
~
H 2 L 1
l oc
(S) with
~
HS = 1 be given and set
H := v
0 E(
~
H S)
~
H for a F
0
-measurable random variable v
0
. Then
H 2 SF(M) and V H
= v
0 E(
~
HS). We call
~
H a generator for the
self-nancing strategy H and dene V (
~
H)
:=V H
.
Proof. Since V H
1
islocallybounded wehave
~
H 2L 1
l oc
(S). Wehave
V H
V H
0
=G H
=HS=(V H
~
H)S =V H
(
~
HS). Thesecondidentity
Dolean-DadeSDEdeningthestochasticexponential,seeJ&S87,I.4f.
Conversely, we calculate
HS = v
0 E(
~
HS)
~
HS =v
0 E(
~
HS) (
~
HS +
~
HS)
= v
0 E(
~
HS) (1+(
~
HS))
= v
0
E(
~
HS) +E(
~
HS) (
~
HS)
= v
0
E(
~
HS) + E(
~
HS) (
~
HS)
= v
0
E(
~
HS) + E(
~
HS) 1
= v
0 E(
~
HS)=v
0 +v
0 E(
~
HS) (
~
HS)
= v
0 +v
0 E(
~
HS)
~
HS=V H
0 +G
H
:
2. Arbitrage-free Markets
Sofarwedid not worryaboutarbitrage. Weconsider inthissection
themarket
M:=(
1
;
S),where
S :=(S;B)isR d
R-valuedandB is
a numeraire, with B
0
=1, which we assume to be uniformlybounded
and uniformly bounded away from 0 on nite intervals. For 0 t
T 1;t < 1, denote the set of uniformly integrable, resp. local
martingales, living on
[t;T] by L
u
(
[t;T]
), resp. by L(
[t;T]
). Dene
the followingsets of localmartingale measures:
D(
M
[t;T] ):=
Z 2L(
[t;T] )jZ1
[0;t]
=1;Z 0;(S B
) [t;T]
Z 2L(
[t;T] ) ; (2.1)
D(
M
[t;T] ):=
Z 2L(
[t;T] )jZ1
[0;t]
=1;Z >0;(S B
) [t;T]
Z 2L(
[t;T] ) ; (2.2)
D abs
(
M
[t;T] ):=
Z 2
D(
M
[t;T]
)jZ uniformlyintegrablemartingale ; (2.3)
and
D e
(
M
[t;T]
):=
Z 2D(
M
[t;T]
)jZ uniformlyintegrable martingale : (2.4)
Wewillwork with the following No-Arbitragecondition:
D e
(
M
T
)6=;; 8T <1:
(2.5)
This condition isknown tobeequivalenttothe NFLVR-condition,see
Delbaen and Schachermayer (1994). It implies that
D e
(
M
[t;T]
)6=;; 80t T <1:
(2.6)
We will often work with the following sets of equivalent, resp. abso-
lutelycontinuous, localmartingalemeasures, for q >1:
D q
(
M
[0;T]
):=
Z 2D(
M
[0;T] )jZ
T 2L
q
(
[0;T] ) ; (2.7)
D q
t (
M
[t;T]
):=
Z
t_
Z
t
jZ 2D q
(
M
[0;T] )
; (2.8)
D q
(
M
[0;T] ):=
Z 2D abs
(
M
[0;T]
)jZ
T 2L
q
(
[0;T] ) ; (2.9)
D q
t (
M
[t;T] ):=
Z
t_
Z
t jZ 2
D q
(
M
[0;T] );Z
t
>0
: (2.10)
Note that Z 2 D q
t (
M
[t;T]
) implies Z
T 2 L
p
t (
[t;T]
). For q < 1 set
D q
t (
M
[t;T]
) := D(
M
[t;T] ) and
D q
(
M
[t;T] ) :=
D q
t (
M
[t;T] ) :=
D(
M
[t;T] ).
ForZ 2D q
t (
M
[t;T]
)andt t 0
T 0
T,wehave Z
[t 0
;T 0
]
Z
t 0
2D q
t 0
(
M
[t 0
;T 0
] ).
Note also that D q
0 (
M
[0;T]
) = D(
M
[0;T]
), since F
0
was assumed to be
trivial.
p willalways denote a real number dierentfrom 1. We dene q:=
p
p 1
, such that forp6=0;1,p 1
+q 1
=1 holds, but forp=0 wehave
q=0.
Let B SF(
M
[t;T]
). We call a H 2 B an B-arbitrage, if V H
0
= 0,
V H
T
0 and V H
T
6= 0 almost surely. If there exists no B-arbitrage,
then B is called arbitrage-free. In all probabilistictheories of nancial
markets allowingto tradeat aninnitely large number of instances of
time one has to exclude certain self-nancing hedging strategies, e.g.
doublingstrategies, inorder to avoidarbitrage opportunities. We will
dene several arbitrage-free subsets of SF(
M ):
1. For p > 1 and D q
t (
M
[t;T]
) 6= ;, (see Delbaen and Schachermayer
(1996), (DS96)):
SF p
(
M
[t;T] ) :=
H2SF(
M
[t;T]
)jV H
T 2L
p
(
[t;T] );
V H
B [t;T]
Z 2L u
(
[t;T]
);8Z 2D q
(
M
[t;T]
)
; (2.11)
resp.
SF p
t (
M
[t;T] ) :=
H2SF(
M
[t;T]
)jV H
T 2L
p
t (
[t;T] );
V H
B [t;T]
Z 2L u
(
[t;T]
);8Z 2D q
t (
M
[t;T]
)
: (2.12)
Notethat
SF p
t (
M
[t;T] ) =
H 2SF(
M
[t;T] )jV
H
T 2L
p
t (
[t;T] );
V H
B [t;T]
Z 2L u
(
[t;T]
);8Z 2
D q
t (
M
[t;T] )
; (2.13)
since for Z 2
D q
t (
M
[t;T]
)we can nd a
Z 2
D q
(
M
[0;T]
) with Z =
Zt_
Z
t
and for Z 0
2 D q
(
M
[0;T]
), we have
~
Z :=
Z+Z 0
2
2 D q
(
M
[0;T]
),
which implies
^
Z :=
~
Z
t_
~
Zt 2 D
q
t (
M
[t;T]
) and for H 2 SF p
t (
M
[t;T] )
that V
H
B [t;T]
Z = V
H
B [t;T]
((
Z
t +Z
0
t )
^
Z Z
0
t_
Z 0
t
) is a uniformlyintegrable
martingale.
2. Forp<1
SF p
(
M
[t;T]
):=SF p
t (
M
[t;T] ):=
H 2SF(
M
[t;T] )jV
H
0 : (2.14)
3. Forp>1 and
S 2S p
l oc (
[t;T] )
G p
(
M
[t;T]
):=
H 2SF(
M
[t;T]
)jV H
2S p
(
[t;T] ) ; (2.15)
where S p
(
[t;T]
) denotes the space of L p
-integrable semimartin-
gales,seeDelbaen,Monat,Schachermayer, SchweizerandStricker
(1997)(DMSSS97)forthecasep=2and GranditsandKrawczyk
(1998), (GK98),for the general case p>1.
Lemma 2.1. For p > 1 assume D q
t (
M
[t;T]
) 6= ; and
S to be contin-
uous. Then G p
(
M
[t;T]
) SF p
t (
M
[t;T]
). In particular G p
(
M
[t;T] ) is
arbitrage-free.
Proof. For H 2 G p
(
M
[t;T] ) set
n
:= inf n
s0
V
H
s
B
s
n
o
, H n
:= H
on [0;
n ) and
0;
V H
n
B
n
2 R d
R on [
n
;T]. Then H n
2 SF p
t (
M
[t;T] ),
since
V
H n
B [t;T]
n. It follows E h
V H
n
T
B
T Z
T jF
s i
= V
H n
s
B
s Z
s
for all t
s T and all Z 2 D q
t (
M
[t;T] ). V
H n
s
converges almost surely to V H
s
and
V
H n
T
B
T Z
T
sup
tsT (
V H
s )
B
T
Z
T
2 L
1
(
[t;T]
), since sup
tsT V
H
s
2
L p
(
[t;T]
)byDoob'smaximalinequality,hencewendE h
V H
T
B
T Z
T jF
s i
=
V H
s
Bs Z
s
for alltsT.
Dene for F
t
-measurable v
A p
v (
M
[t;T] ):=
V H
T
B
T
H 2SF p
(
M
[t;T] );
V H
t
B
t
=v
; (2.16)
and
G p
v (
M
[t;T] ):=
V H
T
B
T
H2G p
(
M
[t;T] );
V H
t
B
t
=v
: (2.17)
For p > 1 and D q
(
M
[t;T]
) 6= ;, SF p
(
M
[t;T]
) has an important prop-
erty: A p
1 (M
[t;T]
)isknowntobeclosed,if S
B
[t;T]
islocallyinL p
(
[t;T] )
in the sense, that there exists a sequence U
n
;n 2 N of localizing
stopping times increasing to innity such that for each n, the fam-
ily fS [t;T]
j stoppingtime; U
n
g is bounded in L p
(
[t;T]
), see DS96.
This condition certainly holds if
S is continuous. To work with the
spaces G p
(
M
[t;T]
)isinsome sense morenatural,since itsdenition in-
volvesonlytheobjectiveprobabilitymeasureP andnoequivalentmar-
tingale measures. Furthermore G p
(
M
[t;T]
) is stable under stopping, a
desirablepropertyfromaneconomicpointofview. However, thisspace
has in generalweaker properties than SF p
(
M
[t;T]
),see DMSSS97 and
GK98.
WewilloftenworkwithacontinuouspriceprocessS,resp.
S. Inthis
caseL 1
l oc
(S)=L 2
l oc
(S)holds. Thepriceprocessadmitsarepresentation
S =S
0
++M;
(2.18)
where =( i
)
1id
is predictable, is a predictable, increasing, con-
tinuous,locallyintegrableprocesssuchthatislocallyintegrablewith
respectto . Furthermore,thereexistsasymmetricnon-negativedd-
matrix-valued predictable process C = (C ij
) , locally integrable
withrespect to ,suchthat[S i
;S j
]=[M i
;M j
]=< M i
;M j
>=C ij
.
can bechosen such that B =E(r ) for a predictableprocess r.
In the continuous case, D e
(
M
[0;T]
) 6= ; implies = rS C, d -
almost surely for a predictable process 2 L 2
l oc
(M) and every Z 2
D e
(
M
[t;T]
) is of the form Z = E
t
( M + N) T
, where N is a not
necessarilycontinuouslocalmartingaleorthogonaltoM with[M;N]=
0, see Ansel and Stricker(1992).
3. Optimal Portfolios
Consider the problem of maximizing expected utility from termi-
nal wealth. We follow a stochastic duality approach, which goes back
to Bismut (1973, 1975), see also Karatzas, Lehoczky, Shreve and Xu
(1991), (KLSX91), and Karatzas and Shreve (1999), Kramkov and
Schachermayer(1999) and Schachermayer (2000)for general results.
We have already dened the so-called isoelastic utility functions
u (p)
;p 6= 1, with constant index of relative risk-aversion, see Pratt
(1964) and Arrow (1976). For optimization multiplication of the util-
ity functionwith a constant factor oradding a constant has no eect.
Wechoose tonormalizethe utility functionsuch that jU (p)
(1)j=1 for
allp6=0;1and dene for p<1;p6=0
U (p)
(x):=sgn(p)x p
; 8x0;
(3.1)
U (p)
(x)= 1 for x<0 and
U (0)
(x):=ln(x); 8x>0;
(3.2)
U (0)
(x)= 1 for x0. Forp>1 set
U (p)
(x):= jxj p
; 8x2R;
(3.3)
Wehave for p<1;p6=0
dU (p)
dx
(x)=jpjx p 1
; 8x>0;
(3.4)
and
dU (0)
dx
(x)= 1
x
; 8x>0;
(3.5)
and set dU
(p)
dx
(0):=1for p<1.
We want to solve the following optimization problem for xed 0
tT <1and p6=1:
V(p;t;T;B):=esssup H2B
H
S=1 E
t
U (p)
V H
T
(3.6)
where B 2 fSF p
t (
M
[t;T] );fSF
p
(
M
[t;T] );G
p
(
M
[t;T]
)g for p > 1, resp.
B=SF p
t (
M
[t;T]
) for p<1, and the dual problem
W
(q;t;T;C):=essinf
Z2C E
t
U (q)
B
t Z
T
B
T
; (3.7)
where C 2 fD q
t (
M
[t;T] );
D q
t (
M
[t;T]
)g. ( U (q)
equals the convex dual
to U (p)
up toa constant factor, see Rockafellar (1970)). See Karatzas
and Shreve (1999)forthe denitionofesssup andessinf. IfforH 2B
withV H
0
=1andV(p;t;T;B)=E
t
U (p)
V H
T
,thenwesayV H
solves
Problem (3.6) forB. Iffor Z 2C, W
(q;t;T;C)=E
t h
U (q)
BtZ
T
B
T i
,
then we say Z solves the dual Problem (3.7) for C. For the moment
we are interested in the Problem (3.6) for B = SF p
t (
M
[t;T]
) and set
V(p;t;T) := V(p;t;T;SF p
t (
M
[t;T]
)). For p > 1, we set W
(q;t;T) :=
W
(q;t;T;
D q
t (
M
[t;T]
)),respectivelyfor p<1,we deneW
(q;t;T):=
W
(q;t;T;D q
t (
M
[t;T]
)). (It will turn out later, that W
(q;t;T) =
W
(q;t;T;
D q
t (M
[t;T]
))for p>1and for p<1if V(p;0;T)<1.)
Thefollowingpropositionshows thecloserelationbetween thesetwo
problems and gives the key idea how to handle the incompleteness of
the market.
Proposition 3.1. Assume that there exists an H 2SF p
t (
M
[t;T] ) with
V H
T
0andaZ
opt;q;t;T
2
D
t (
M
[t;T]
)suchthatforsomeF
t
-measurable
random variable c>0
Z
opt;q;t;T
T
=cB
T
sgn(1 p) dU
(p)
dx V
H
T
; (3.8)
andsuchthat V
H
B [t;T]
Z
opt;q;t;T
isauniformlyintegrablemartingale. Then
V opt;p;t;T
:=
V H
V H
0
solves Problem (3.6) for SF p
t (
M
[t;T]
) and Z
opt;q;t;T
solves for p > 1, resp. p < 1, the dual Problem (3.7) for
D
t (
M
[t;T] ),
resp. for D
t (
M
[t;T] ) and
D
t (
M
[t;T]
). There exists at most one such
opt;p;t;T opt;q;t;T
corresponding optimal values satisfy
jV(p;t;T)j p
1
jW
(q;t;T)j q
1
=1:
(3.9)
Proof. Notethatforp<1(3.8)impliesV H
T
>0. ForH 2SF p
t (
M
[t;T] )
with V H
0
=1and since U (p)
is concave we have
U (p)
V H
T
U (p)
V opt;p;t;T
T
+ dU
(p)
dx
V opt;p;t;T
T
(V H
T V
opt;p;t;T
T
):
(3.10)
Taking conditionalexpectations we nd
E
t
U (p)
V H
T
E
t
"
U (p)
V opt;p;t;T
T
+Z
opt;q;t;T V
H
T V
opt;p;t;T
T
sgn (1 p)cB
T
#
E
t h
U (p)
V opt;p;t;T
T
i
;
since E
t h
V H
T
B
T Z
opt;q;t;T i
V
H
t
Bt
= V
opt;p;t;T
t
Bt
for p < 1, respectively since
Z
opt;q;t;T
2
D q
t (
M
[t;T]
) for p>1. LetQ2
D q
t (
M
[t;T]
)and calculate
U (q)
dQ
T
dP
T
B
T B
1
t
!
= U
(q) 0
@ Z
opt;q;t;T
T
+
dQ
T
dP
T Z
opt;q;t;T
T
B
T B
1
t
1
A
U
(q) Z
opt;q;t;T
T
B
T B
1
t
!
dU (q)
dx B
t Z
opt;q;t;T
T
B
T B
1
t
!
dQ
T
dP
T Z
opt;q;t;T
T
B
T B
1
t
= U
(q) Z
opt;q;t;T
T
B
T B
1
t
!
sgn(1 p)k V
opt;p;t;T
T
B
T
dQ
T
dP
T Z
opt;q;t;T
T
;
for some F
t
-measurable random variable k > 0. Taking conditional
expectations wend
E
t
"
U (q)
B
t Z
opt;q;t;T
T
B
T
!#
E
t
"
U (q)
B
t dQ
T
dP
T
B
T
!#
; (3.11)
since E
t h
V opt;p;t;T
T
B
T dQ
T
dP
T i
V
opt;p;t;T
t
B
t
for p < 1, resp. since for p > 1
Z
opt;q;t;T
2
D q
t (
M
[t;T]
). Theuniquenessofthepair(V opt;p;t;T
T
;Z
opt;q;t;T
T
)
follows from the strict concavity of U (p)
. Since V
opt;p;t;T
B [t;T]
Z
opt;q;t;T
is a
uniformlyintegrable martingale, itis determined by V
opt;p;t;T
T
B
T Z
opt;q;t;T
T
.
LetH 0
2SF p
t (
M
[t;T]
) with V H
0
0
=1and Z 0
2D
t (
M
[t;T]
) such that
Z 0
T
=c 0
B
T
sgn (1 p) dU
(p)
dx
V H
0
T
; (3.12)
holds for a F
t
-measurable random variable c 0
> 0. Then V opt;p;t;T
T
=
V H
0
T , Z
opt;q;t;T
T
= Z 0
T and
V opt;p;t;T
B [t;T]
Z
opt;q;t;T
= V
H 0
B [t;T]
Z 0
. Assume that
there exists a t s T with A := fV H
0
s
> V opt;p;t;T
s
g 6= ;. In
this case we can change the self-nancing hedging strategy H 0
on
A [s;T] to a H 00
2 SF p
t (
M
[t;T]
) such that V H
0 0
T
V
opt;p;t;T
T
and
V H
00
T
> V opt;p;t;T
T
on A. From this we conclude the uniqueness of the
pair (V opt;p;t;T
;Z
opt;q;t;T
). Forp6=0wend
U (p)
(V opt;p;t;T
T
) = sgn (q)
V opt;p;t;T
T
p
= sgn (q)
V opt;p;t;T
T
p 1
V opt;p;t;T
T
= Z
opt;p;t;T
T
cp sgn(1 p) V
opt;p;t;T
T
B
T
;
hence
V(p;t;T)=E
t h
U (p)
(V opt;p;t;T
T
) i
=
1
cp sgn(1 p)B : (3.13)
Forq 6=0 we nd
U (q)
B
t Z
opt;p;t;T
T
B
T
!
= sgn (p)B q
t Z
opt;p;t;T
T
B
T
!
q
= sgn (p)B q
t Z
opt;p;t;T
T
B
T
!
q 1
Z
opt;p;t;T
T
B
T
= sgn (p)B q
t
cjpj V opt;p;t;T
T
p 1
1
p 1 Z
opt;p;t;T
T
B
T
= sgn (p)B q
t (cjpj)
1
p 1
Z
opt;p;t;T
T
V opt;p;t;T
T
B
T
;
hence
W
(q;t;T)=E
t
"
U (q)
B
t Z
opt;p;t;T
T
B
T
!#
=sgn (p)(B
t cjpj)
1
p 1
: (3.14)
(3.13) and (3.14) together imply(3.9).
Wecall(V opt;p;t;T
;Z
opt;q;t;T
)theoptimalpairforthemarket
M
[t;T]
with
respecttooptimizationinSF p
t (
M
[t;T]
). Wehavethefollowingstability
property for optimalpairs:
Proposition 3.2. If the pair (V opt;p;t;T
;Z
opt;q;t;T
) admits a represen-
tation (3.8) with V opt;p;t;T
=V H
for a H 2SF p
t (
M
[t;T]
) with V H
0
=1
andZ
opt;q;t;T
2D q
t (
M
[t;T]
),thentheoptimalpairforthemarket
M
[t 0
;T]
exists and is given by
V opt;p;t
0
;T
;Z opt;q;t
0
;T
= V
opt;p;t;T
t 0
_
V opt;p;t;T
t 0
; Z
opt;q;t;T
t 0
_
Z
opt;q;t;T
t 0
!
: (3.15)
Proof. Noterst,thatbyJ&S87,LemmaIII.3.6,wehaveZ
opt;q;t;T
>0.
ForZ 2
D q
t 0
(
M
[t 0
;T]
) set A:=
E
t 0
[Z q
T ]<E
t 0
Z opt;q;t;T
T
Z opt;q;t;T
t 0
q
. Dene
~
Z :=Z
opt;q;t;T
on[t;t 0
)[A c
[t 0
;T],and
~
Z :=Z
opt;q;t;T
0
Z onA[t 0
;T].
Wecalculate
E
t h
~
Z q
T i
= E
t h
E
t 0
h
~
Z q
T ii
= E
t h
1
A
Z
opt;q;t;T
t 0
q
E
t 0
[Z q
T ]+1
A c
E
t 0
h
Z
opt;q;t;T
T
q ii
E
t h
E
t 0
h
Z
opt;q;t;T
T
q ii
=E
t h
Z
opt;q;t;T
T
q i
;
hence
~
Z 2
D q
t (
M
[t;T] ),
~
Z
T
= Z
opt;q;t;T
T
and A = ; and we conclude
Z opt;q;t
0
;T
= Z
opt;q;t;T
t 0
_
Z opt;q;t;T
t 0
. By assumption we have V
opt;p;t;T
T
B
T Z
opt;q;t;T
T
> 0
and since V
opt;p;t;T
B [t;T]
Z
opt;q;t;T
is a non-negative supermartingale we have
V opt;p;t;T
>0. Hence V
opt;p;t;T
T
V opt;p;t;T
t 0
2L p
t 0
(
[t 0
;T]
)and V
opt;p;t;T
t 0
_
V opt;p;t;T
t 0
Z is auniformly
integrablemartingale for allZ 2D q
t 0 (
M
[t 0
;T]
). Since
Z opt;q;t
0
;T
T
=c
t 0
B
T
sgn(1 p) dU
(p)
dx V
opt;p;t;T
T
V opt;p;t;T
t 0
!
; (3.16)
where
c
t 0
:=
c dU
(p)
dx
V opt;p;t;T
t 0
Z
opt;q;t;T
t 0
; (3.17)
we can apply Proposition 3.1.
Lemma 3.3. For p >1, H 2 SF 0
t (
M
[t;T]
) with V H
0
=1, Z
opt;q;t;T
2
D
t (
M
[t;T]
)andassume(V H
;Z
opt;q;t;T
)toadmitarepresentation(3.8).
If V H
T
2 L p+
t (
[t;T]
), (or equivalently Z
opt;q;t;T
T
2 L q+
t (
[t;T]
)), for
some >0, then H 2SF p
t (
M
[t;T]
) and (V H
;Z
opt;q;t;T
) is the optimal
pair for the market
M
[t 0
;T] .
Proof. Observe that V
H
T
B
T Z
T 2 L
1+ ~
t (
[t;T]
) for some ~> 0 for all Z 2
D q
t (
M
[t;T]
). Hence V
H
T
B
T
Z is a uniformly integrable martingale. Now
apply Proposition 3.1.
In the next two sections we look at an example and postpone an
existence result for the optimalpair (V opt;p;t;T
;Z
opt;q;t;T
) untilSection
6.
4. Totally Unhedgeable Price for Instantaneous Risk
Assume
S to be continuous such that we have a representation
(2.18). In this section we seek a suÆcient condition ensuring certain
self-nancing hedging strategies to be optimal for problem 3.6. See
Karatzas andShreve(1999),Example6.7.4forasimilarresultand the
notionof totally unhedgeablecoeÆcients. This notiondescribes amar-
ketmodelwhere the uncertainty in the coeÆcients dening the model
is ina certainsense orthogonal tothe uncertainty of the localmartin-
gale M driving the price process, such that we can not hedge against
this risk. Set :=
p
C2L 2
l oc ( ).
Denition 4.1. For 0t T <1, [t;T]
is called the instantaneous
price for risk process, or instantaneous Sharpe-ratio process, for the
market
M
[t;T] .
Denition 4.2. For 0 t T < 1, let an F
t
-measurable random
variable c > 0 and a not necessarily continuous local martingale N
orthogonal to M, (or equivalently with [N;M] = 0), be given such
that
1.
E
t
pr q
2
2
=cE
t (N)
T : (4.1)
2. Forp<1,E
t
(qM +N) T
isa uniformlyintegrablemartingale.
Wethencalltheinstantaneousprice forrisk [t;T]
inthemarket
M
[t;T]
totally p-unhedgeable. resp. strongly totally p-unhedgeableif E
t (N)
T
is
a uniformlyintegrablemartingale.
Remark 4.3. Forp =0 we have q = 0 and we nd a unique represen-
tation (4.1) with c= 1, N = 0 for any 0 t T < 1, thus [t;T]
is
totally 0-unhedgeable in
M
[t;T]
. For p =0, the optimization problem
(3.6)isalsoknown asmaximizingthe Kelly-criterion,see Kelly(1956),
Breiman (1960) and Karatzas and Shreve (1999). For general results
see Aase (1986) and Golland Kallsen (2000).
Lemma 4.4. If [t;T]
is totally p-unhedgeable in
M
[t;T]
, then [t
0
;T]
is
totally p-unhedgeable in
M
[t 0
;T]
for allt t 0
T.
Proof. For t t 0
T set c 0
:= cE
t
pr q
2
2
+N
t 0
. This
gives us a representation (4.1) and for p < 1, E
t 0
(qM +N) T
is a
uniformlyintegrable martingale.
Proposition 4.5. Assume [t;T]
to be totally p-unhedgeable in
M
[t;T]
witharepresentation(4.1),thentheoptimalpairforthemarket
M
[t 0
;T]
for tt 0
T is givenby
V opt;p;t
0
;T
;Z opt;q;t
0
;T
= V
(H p
)
t 0
_
V (H
p
)
0
;E
t
0(M +N) T
!
; (4.2)
where H p
:=
p 1
; 1
S
p 1
B
[t;T]
generates the value process
V (H
p
)
:=E
t
r
2
p 1
+
p 1 M
T
: (4.3)
Furthermore, for p6=0
V(p;t 0
;T)= sgn (q) E
t 0
h
E
t 0
pr q
2
2
T i
E
t 0[E
t 0(N)
T ]
; (4.4)
resp. if [t;T]
is strongly totally p-unhedgeable in
M
[t;T] ,
V(p;t 0
;T)= sgn (q)E
t 0
E
t 0
pr q
2
2
T
; (4.5)
and
W
(p;t 0
;T)=sgn(p)(V(p;t 0
;T)) 1
1 p
: (4.6)
Proof. Forp6=0calculate
dU (p)
dx
V (H
p
)
T
= p sgn(q) V (H
p
)
T
p 1
= p sgn(q)E
t
r
2
p 1
+
p 1 M
p 1
T
= p sgn(q)E
t
(p 1)r q
2
2
+M
T
=
sgn (p 1)jpjcB
t
B
E
t
(M +N)
T
;
resp. for p=0
dU (0)
dx
V (H
p
)
T
= V
(H p
)
T
1
= E
t
r+ 2
M
1
T
= E
t
( r+M)
T
= B
t
B
T E
t
(M)
T
;
and nd a representation (3.8),since E
t
(M +N)
T
>0. Set
Z
opt;q;t;T
:=E
t
(M +N) T
: (4.7)
V (H
p
)
B [t;T]
Z
opt;q;t;T
= 1
B
t E
t
2
p 1 +
p 1 M
T
E
t
(M +N) T
= 1
B
t E
t
(qM +N) T
;
which is a uniformly integrable martingale on [t;T] for p < 1 by as-
sumption. Forp>1and >1 observe
V (H
p
)
T
p
= E
t
r
2
p 1
+
p 1 M
p
T
= E
t
pr q
2
+
2
2
2
p 2
p
(p 1) 2
+qM
T
= E
t
pr q
2
2
p(2 ) 1
p 1
+qM
T
= E
t
pr q
2
2
(1 q( 1))
+qM
T
;
hencewendV (H
p
)
T
2L p+ ~
t (
[t;T]
)forsome~>0,since(1 q( 1))>0
for close to1. By Lemma 3.3we nd (V (H
p
)
) T
;E(M +N) T
to
be the optimal pair. Forp6=0 we calculate
U (p)
V (H
p
)
T
= sgn (q)
V (H
p
)
T
p
= sgn (q)E
t
r
2
p 1
+
p 1 M
p
T
= sgn (q)E
t
pr q
2
2
+qM
T
= sgn (q)cE
t
(qM +N)
T :
Since E
t
(qM +N) T
is auniformlyintegrablemartingale wend
E
t h
U (p)
V (H
p
)
T i
= sgn (q)c = sgn (q) E
t h
E
t
pr q
2
2
T i
E
t [E
t (N)
T ]
: (4.8)
The lastequation follows from(3.9).
Inthenextsectionwegiveaninterpretationoftheportfoliosgenerated
by H p
.
5. Locally Efficient Portfolios
From Lemma 1.4 and = rS C, d -a.s., we immediately nd
V (
^
H)
:=E
t
(H S) T
=E
t
((r HC)+HM) T
for a process
^
H =
H;
1 HS
B
2L 1
l oc
(S). From Cauchy-Schwarz inequality it follows that
jHCj
p
C p
HCH = p
HCH. We have
[V (
^
H)
;V (
^
H)
]=
V (
^
H)
2
HCH
[t;T]
:
Weinterpret p
HCH asameasurefortherelativeinstantaneousriskof
the portfoliogeneratedby
^
H and
^
H=r HC asameasure for the
instantaneouslyexpectedrelativereturnrate. For 6=0and p
HCH6=
0, we nd for the instantaneous Sharpe-ratio
^
H r
p
HCH
= HC
p
HCH
of in-
stantaneouslyexpected relative excess returnover the instantaneously
risk-free return rate and relative instantaneous risk, HC
p
HCH
and HC
p
HCH
=i H2k+Ker(C)forapredictable,strictly negative,
process k 2 L 2
l oc
(), resp.
HC
p
HCH
= i H 2 k+Ker (C) for a
predictable, strictly positive, process k 2L 2
l oc
(). We call these hedg-
the caseof atotallyp-unhedgeableprice forriskthe optimalportfolios
generated by H p
are locally eÆcient. See Markowitz (1952, 1987) and
Sharpe (1964, 2000).
Dene the following quantities for p6=0;1:
R (p;t;T)
:=
1
p
dln(V(p;t;T))
dT
; (5.1)
R (p)
:= lim
T!1 1
pT
ln(V(p;0;T)) (5.2)
and
R (0;t;T)
:=
dV(0;t;T)
dT (5.3)
R (0)
:= lim
T!1 1
T
V(0;0;T):
(5.4)
Under some regularity conditions, these quantities exist. By Theorem
4.5we nd immediately
Proposition 5.1. Under the assumptions,
t
= t and pr q
2
2 con-
stant for p6=0;1, resp. r+
2
2
constant forp=0, we have
V(p;t;T) = exp pr q
2
2
(T t)
; (5.5)
R (p;t;T)
= R
(p)
=r+
2
2(1 p)
; (5.6)
resp.
V(0;t;T) =
r+
2
2
(T t);
(5.7)
R (0;t;T)
= R
(0)
=r+
2
2 : (5.8)
SeeBieleckiandPliska(1999,2000)foraninterpretationofthequan-
tities R (p)
and the risk-sensitive stochastic control approach. R (p;t;T)
can be interpreted as an implied forward growth rate of the expected
utility of wealth under the optimalself-nancing hedging strategy. As
we will see in Section 7, another related quantity is the right one to
look at: We dene
Y(p;t;T):=ln(jV(p;t;T)j): (5.9)
6. Existence of Optimal Portfolios
Let 0 T < 1 be xed. In this section we will assume
S to be
continuousandD q
(
M
[0;T]
)6=;forp>1,resp. forp<1,D e
(
M
[0;T]
)6=
; and V(p;0;T;SF p
(
M
[0;T]
)) < 1. We assume for simplicity in this
section that B =1. The results can be generalized to the case of a B
such that B and B 1
are uniformlybounded on[0;T].
Theorem 6.1. Under the above assumptions, the optimal pair
(V
opt;p;0;T
;Z
opt;q;0;T
),satisfying(3.8)andZ
opt;q;0;T
2D(
M
[0;T]
),exists
for the market
M
[0;T]
with respect to optimization in SF p
(
M
[0;T] ).
Proof. We rst prove the case p > 1. Since A p
1 (
M
[0;T]
) is closed and
convex andsinceL p
(
[0;T]
)isreexivethereexists anelementV
opt;p;0;T
with minimal norm. As inGLP98, Lemma 4.1 and Theorem 4.1, it is
easily shown that V
opt;p;0;T
0. Since U (p)
is concave we have for all
Y 2A p
0 (
M
[0;T] )
U (p)
V
opt;p;0;T
T
+Y
U (p)
V
opt;p;0;T
T
+ dU
(p)
dx
V
opt;p;0;T
T
Y:
(6.1)
It follows fromthe optimalityof V
opt;p;0;T
T
and since dU
(p)
dx
V
opt;p;0;T
T
2
L q
(
[0;T] ) that
E
B
T dU
(p)
dx
V
opt;p;0;T
T
Y
B
T
=0;
(6.2)
for all Y 2 A p
0 (
M
[0;T]
). From V
opt;p;0;T
T
2 L p
(
[0;T]
) it follows that
dU (p)
V
opt;p;0;T
T
2 L q
(
[0;T]
). Since V
opt;p;0;T
T
2 A p
1 (
M
[0;T]
) we have
E
V
opt;p;0;T
T
p 1
>0. We thereforend
Z
optp;0;T
:=
E h
dU (p)
dx
V
opt;p;0;T
T
F
i
E h
dU (p)
dx
V
opt;p;0;T
T
i
2
D q
(
M
[0;T] ):
(6.3)
Optimality follows now from Proposition 3.1. It was shown in GK98,
Lemma 4.4, that Z
opt ;q;0;T
2D q
(
M
[0;T] ).
Forp<1theresultsofKramkovandSchachermayer(1999),(KS99),
canbeapplied. There,existenceanduniquenessof anoptimalsolution
V
opt;p;0;T
with V
opt;p;0;T
T
> 0 for problem (3.6) is proved. Furthermore,
the existence and uniqueness of a strictly positive process Z opt
, such
that E
Z opt
T
q
= inf
Z2D(
M
[0;T]
) E
h
Z
T
B
T
q i
, and with the following
properties is shown: Z opt
T
= sgn (q)U (p)
(V
opt;p;0;T
T
), V
opt;p;0;T
Z opt
is
a uniformly integrable martingale and for an arbitrary self-nancing
hedgingstrategywithnon-negativevalueprocess V,theprocess VZ opt
is asupermartingale. Wewillshow in the next lemma,that for a con-
tinuous price process Z opt
2D(
M
[0;T] ).
TheworryingfactisofcoursethatZ opt
isingeneralonlyasupermartin-
gale. However,thegivenexample(Example5.1'inKS99),showingthat
Z opt
isingeneralnotalocalmartingale,involvesanon-continuousprice
process. Wedenethe following set ofsemimartingaleslivingon
[0;T]
for 0<1, see KS99:
Y(
M
[0;T] ):=
Y 0jY
0
=1;
V H
B [0;T]
Y is a supermartingale
forall H 2SF 0
(
M
[0;T] )
:
Lemma 6.2. Assume
S to be continuous and let Y 2Y(
M
[0;T] ) with
Y
T
>0 be given. If there exists a H 0
2SF 0
(
M
[0;T]
) with V H
0
0
=1 and
V H
0
T
>0andsuchthatV H
0
Y isauniformlyintegrablemartingale,then
Y 2D(
M
[0;T] ).
Proof. Since Y is a non-negative supermartingale,we have by J&S87,
Lemma III.3.6, that Y > 0 and Y >0 almost surely and hence Y =
E(Z) for Z :=
1
Y
Y. Since Y is a supermartingale it is a special
semimartingale and therefore Z too. Z admits a representation Z =
A+L,whereA=A T
isapredictableprocessofnitevariation,L=L T
is a local martingale and A
0
= L
0
=0. By J&S87, Theorem III.4.11,
we nd a predictable process K 2 L 2
l oc
(M) and a local martingale
N orthogonal to all components of M, with [M;N] = 0, such that
L=KM +N and the representation Y =E(A+K M+N).
Since V H
0
0 is a local martingale with respect to any equiva-
lent martingale measure, V H
0
T
> 0 implies V H
0
> 0. By Lemma 1.4
there exists a
~
H 0
2 L 2
l oc S
T
such that
~
H 0
; 1
~
H 0
S
B
T
generates V H
0
.
By assumption and since M and
~
H 0
C are continuous, V
H 0
B T
Y =
V (
~
H 0
)
B T
Y = E(
~
H 0
C(K ) +A + (K +
~
H 0
) M + N) T
is a uni-
formly integrable martingale. The Doleon-Dade SDE implies that
V H
0
B T
Y
(
~
H 0
C(K )+A) = V
H 0
B T
Y
V H
0
B T
Y
((K+
~
H 0
)
T
[0;T], hence constant on [0;T] almost surely, see J&S87, Corollary
I.3.16. We therefore nd E(
~
H 0
C(K ) + A) T
= 1. Now let
H 2L 2
l oc S
T
,set
H := H;
1 HS
B
T
andconsider thediscountedvalue
processV
:=
V (
H)
B T
=E( HC+HM) T
generatedby
H. Wehave
V
Y =E(HC(K )+A+(K+H)M+N) T
=E((H
~
H 0
)C(K )
+(K+H)M+N) T
isasupermartingaleforallH 2L 2
l oc S
T
byas-
sumption. ForH :=K +
~
H 0
wend(V
Y) ((K )C(K ) )=
V
Y (V
Y) ((K+H)M +N) 1 to be a non-decreasing local
supermartingale on [0;T]. Therefore (K )C(K ) = 0, d -a.s.
and from 1 = E(
~
H 0
C(K )+A) T
= E(A) T
we conclude A = 0
and Y =E(M +N) T
2D(
M
[0;T] ).
In DMSSS97, Theorem A-C,(for p=2), and GK98,Theorem 3.1 and
Theorem 4.1, (for p>1), necessary and suÆcient conditions are given
ensuring G p
0 (
M
[0;T]
) to beclosed. These results imply
Proposition 6.3. If G p
0 (
M
[0;T]
) is closed, then
V(p;0;T;SF p
(
M
[0;T]
))=V(p;0;T;G p
(
M
[0;T] ));
(6.4)
and V
opt;p;0;T
can be obtained by a self-nancing hedging strategy in
G p
(
M
[0;T]
). Furthermore, for 0 t T, the optimal pair for the
market
M
[t;T]
is givenby
V
opt;p;0;T
t_
V
opt;p;0;T
t
; Z
opt;q;0;T
t_
Z
opt;q;0;T
t
!
2G p
(
M
[t;T] )D
q
(
M
[t;T] ):
(6.5)
7. The BSDE Approach
InthissectionwewillputtouseProposition3.1inageneralsetting.
Assumethe existenceofacontinuouslocalmartingaleN orthogonalto
M such that (M;N) has the localmartingale representation property
and [N;N]=
~
C . Since thecase p=0isalready solved (see Remark
4.3) we assume inthis section p6=0;1. Let0tT bexed.
Consider the following formal calculation for the optimal solution
V opt;p;t;T
for a maximizationproblemof terminalutility in the market
M
[t;T]
and anarbitrary attainable Y
B
T 2A
p
0 (M
[t;T] ):
U(V opt;p;t;T
T
+Y)U(V opt;p;t;T
T
)+U 0
(V opt;p;t;T
T
)Y;
(7.1)
implies
E
t
B
T U
0
(V opt;p;t;T
T
) Y
B
T
=0;
(7.2)
sincekY isattainableforallF
t
-measurablerandomvariablesk. Hence
B
T U
0
(V opt;p;t;T
T
)shoulddeneanabsolutelycontinuousmartingalemea-
sure up to normalization. In general this argument breaks down be-
cause of integrability problems. However, for isoelastic utility with
exponent p > 1 this approach works. None the less, we can try the
followingansatz:
c
t B
T U
0
(V opt;p;t;T
T
)=E
t
(M+N)
T
; (7.3)
respectively
ln
c
t B
T U
0
V opt;p;t;T
T
1
E
t
(M+N)
T
=0;
(7.4)
where V opt;p;t;T
=
V (
^
H)
T
for
^
H = H;
1 HS
B
[t;T]
2 L 2
l oc (
S) and 2
L 2
l oc
(N). Ansatz (7.3) leads to a FBSDE. For the isoelastic utility
functions ansatz (7.4) will lead toa BSDE, where (H;) formpart of
the solution. FortsT dene the adapted process
Y p;t;T
s
:=ln
c
t B
s dU
(p)
dx V
opt;p;t;T
s
1
E
t
(M +N)
s
!
: (7.5)
ApplyingIt^o'sformulaandbythedenitionofthe stochasticexponen-
tialwe nd
Y p;t;T
T
= Y p;t;T
t +
Z
T
t dY
p;t;T
s
= Y p;t;T
t +
Z
T
t
(p 1)H
s C
s H
s
s
~
C
s
s
s C
s
s
2
d
s
+ Z
T
t
((p 1)H
s C
s
s pr
s )d
s
+ Z
T
t (
s
(p 1)H
s )dM
s +
Z
T
t
s dN
s :
BecauseofProposition3.2andtheformulas(3.16)and(3.17)weexpect
Y p;t;T
to be independent of t, hence we arrive at the following BSDE
for tt 0
T:
Y (p;T)
t 0
= Z
T
t 0
(p 1)H
s C
s H
s
s
~
C
s
s
s C
s
s
2
d
s
Z
T
t 0
((p 1)H
s C
s
s pr
s )d
s (7.6)
Z
T
t 0
(
s
(p 1)H
s )dM
s Z
T
t 0
s dN
s :
Conversely, givenan adapted solution(Y (p;T)
;H;)tothe BSDE(7.6)
on [t;T], we can dene a self-nancing hedging strategy in
M
[t;T]
by
using
^
H :=
H;
1 HS
B
[t;T]
(7.7)
asa generator for
V (
^
H)
:=E
t
((r HC)+HM) T
2SF 0
(
M
[t;T]
):
(7.8)
Wealsohave
Z
:=E
t
(M+N) T
2D(
M
[t;T] ):
(7.9)
Lemma 7.1. V (
^
H)
T
and Z
T
satisfy (3.8):
Z
T
= exp
Y (p;T)
t
B jpj B
T
sgn(1 p) dU
(p)
dx
V (
^
H)
T
; (7.10)
Proof. Observe
1 = exp Z
T
t
(p 1)H
s C
s H
s
s
~
C
s
s
s C
s
s
2
d
s
!
exp Z
T
t
(p 1)H
s C
s
s pr
s d
s
exp
Y (p;T)
t +
Z
T
t
s
(p 1)H
s dM
s +
Z
T
t
s dN
s
;
whichimplies
E
t
(M +N)
T
= exp Z
T
t
(1 p)H
s C
s H
s
2
+(1 p)H
s C
s
s +pr
s d
s
exp
Y (p;T)
t +
Z
T
t
(p 1)H
s dM
s
=
exp Z
T
t r
s H
s C
s
s H
s C
s H
s
2 d
s
p 1
E
t (r )
T exp
Y (p;T)
t
exp Z
T
t H
s dM
s
p 1
= exp
Y (p;T)
t
B
t
B
T E
t
((r HC)+HM) p 1
T
= exp
Y (p;T)
t
B
t jpj
B
T
sgn(1 p) dU
(p)
dx
V (
^
H)
T
:
Proposition 7.2. Assume(Y (p;T)
;(H;))tobeasolutiontotheBSDE
(7.6)on[t;T]. Dene
^
H, resp. V (
^
H)
,Z
by(7.7),resp. (7.8),(7.9). If
for p< 1, E
t
((H +)M+N) T
is a uniformly integrable martin-
gale, respectively if for p>1, V (
^
H)
^
H 2SF p
(
M
[t;T]
), then
V (
^
H)
;Z
is the optimal pair for the market
M
[t;T]
with respect to optimization
in SF p
t (
M
[t;T]
). Furthermore we have
V(p;t;T)= sgn(q)exp (Y (p;T)
t
)= sgn (q)exp(Y(p;t;T)):
(7.11)
Proof. The rst assertion follows from Proposition 3.1, (7.11) follows
from(3.13).
Conversely, the existence of an optimal pair for the market
M
[t;T] to-
gether with the local martingale representation property of (M;N),
implies the existence of a solution (Y (p;T)
;(H;)) for the BSDE (7.6)
on[t;T] satisfyingthe assumption of Proposition 7.2.
8. Markovian Market Model
As anexample,we willtransforminthis sectionthe BSDE(7.6) for
a (for simplicity time-homogeneous) markovian market model into a
non-linear partial dierentialequation with boundary condition.
Consider the following market model: Assume the existence of a
(m+m 0
)-dimensional Brownian motion W = (W 1
;W 2
) on
1 and
assume F to be generated by W. For simplicity, let ^ = (;
0
) :
R d+d
0
! R d+d
0
and : R d+d
0
! R (d+d
0
)(m+m 0
)
be smooth uniformly
bounded functions with uniformly bounded derivatives of all orders,
such that for all x 2 R d+d
0
,
(x) : R d +d
0
! R d+d
0
is invertible with
uniformlyinxboundedinverse. Furthermore,assume
(x)=C(x)
C 0
(x):R d
R d
0
!R d
R d
0
. Thenthereexists a R d +d
0
-valuedMarkov
process X =(S;S 0
) solving the SDE for x
0 2R
m+m 0
dX
t
=(X^
t
)dt+(X
t )dW
t
; X
0
=x
0 : (8.1)
Denote by M the martingale part of S, and by N the martingale part
of S 0
. Note that M and N are orthogonal. Assume the interest rate
r to be a bounded function of X, and dene B
t
:= exp
R
t
0 r(X
s )ds
for allt 0. Set :=C 1
( rS) and 2
:=C. We nowinterpret
S :=(S;B) asa price process and S 0
as (non-traded)state variables.
Considerthefollowingnon-linearPDEforY :R d
R d
0
[0;1)!R,
(p6=1):
@Y
+L
1 Y +L
2 Y =L
3 Y +L
(p)
Y +q
2
pr;
with boundary condition Y(s;s 0
;0)=0; 8s;s 0
, where
L
1 :=
d
X
i=1
i
@
@s
i +
1
2 d
X
i;j=1 C
i;j
@ 2
@s
i
@s
j
L
2 :=
d 0
X
i=1
0
i
@
@s 0
i +
1
2 d
0
X
i;j=1 C
0
i;j
@ 2
@s 0
i
@s 0
j
;
and for f 2C 1;1
(R d
R d
0
[0;1)),
L
3 f :=
1
2 d
0
X
i;j=1
@f
@s 0
i C
0
i;j
@f
@s 0
j
L (p)
f :=
1
2(p 1) d
X
i;j=1
@f
@s
i C
i;j
@f
@s
j q
d
X
i;j=1
@f
@s
i C
i;j
j :
Assume Y (p)
2 C 2;2;1
(R d
R d
0
[0;1)) to be a solution of the PDE
(8.2),satisfyingtheboundaryconditionY (p)
(;;0)=0. ApplyingIt^o's
formulato the process Y (p;T)
t
:=Y (p)
(S
t
;S 0
t
;T t) we nd
Y (p;T)
; H opt;p;T
; opt;q;T
:=
Y (p;T)
;
(S
;S 0
)
@Y (p)
@s (S
;S 0
;T )
p 1
;
@Y (p)
@s 0
(S
;S 0
;T )
; (8.2)
tobeasolution forthe BSDE(7.6). Wegive(admittedlyquitestrong
and not easy to check) conditions, ensuring Y (p;T)
;(H;)
to be a
useful solution:
Theorem 8.1. If forp>1, E(M+ opt;q;T
N)
T 2L
q+
(
[0;T]
)for
an>0,resp. ifforp<1;p6=0,E (+H opt;p;T
)M + opt;p;T
N
T
is a uniformly integrable martingale, then for all 0tT
V opt;p;t;T
=E
t
r H
opt;p;T
C
+H opt;p;T
M
T
; (8.3)
and
Z
opt;q;t;T
=E
t
(M+ opt;q;T
N) T
: (8.4)
