Statistics & Decisions 23, 1–14 (2005)c© R. Oldenbourg Verlag, Munchen 2005

Optimal consumption strategies under modeluncertainty

Christian Burgert, Ludger Ruschendorf

Received: March 9, 2005; Accepted: April 21, 2005

Summary: In this paper we consider the problem of finding optimal consumption strategies in anincomplete semimartingale market model under model uncertainty. The quality of a consumptionstrategy is measured by not only one probability measure but as common in risk theory by a classof scenario measures. We formulate a dual version of the optimization problem and prove theexistence of a saddle point and give a characterization of an optimal consumption strategy in termsof solutions of the dual problem. This generalizes results of Karatzas and Zitkovic (2003) for theoptimal consumption problem under a fixed probability measure.

1 IntroductionOne of the main problems in mathematical finance is how to invest in assets S =(S1, . . . , Sd) in an optimal way, where optimality is measured with an utility functionU : R −→ R. Admissible investment or trading strategies are predictable processesH = (H1, . . . , Hd) for which the stochastic integral H · S is well defined and uniformlybounded below. Then the problem of optimal investment can be formulated as follows:For a given initial capital x ∈ R+ find an admissible trading strategy H∗ such that themean utility at maturity T ∈ R+ is maximal, that is, H∗ maximizes the expression

EU(x + H · ST ) (1.1)

with respect to all admissible trading strategies H . There are essentially two approachesin the literature for solving problem (1.1). The first one establishes a deterministic partialdifferential equation and then shows that its solution is an optimal strategy. For thisapproach one needs the assumption that S is a Markov process. A more general approachis the so called martingale method which uses the duality theory of convex analysis.Therefore the starting point is an equivalent description of admissible trading strategiesby supermartingale measures. This allows the formulation of a dual problem which canbe solved more easily. By a connection with the primal problem (1.1) one gets an optimaltrading strategy (for a comprehensive overview see Schachermayer (2004)).

AMS 2000 subject classification: Primary: 91B28; Secondary: 91B30Key words and phrases: optimal consumption strategy, risk measure, model uncertainty

2 Burgert -- Ruschendorf

In general one does not only want to invest but also to consume some of the gainedcapital. This is described by consumption strategies c which are optional nonnegativeprocesses. The goal is the determination of an optimal consumption strategy for whichthe mean utility

E

(∫ T

0U(ct) dt

)(1.2)

is maximal with respect to an admissible class of consumption strategies. Under a suit-able definition of admissibility the consumption problem (1.2) can be solved with themartingale method similar to the investment problem (1.1) (see Karatzas and Zitkovic(2003)).

In the following we consider a generalization of problem (1.2): Let ρ be a continuouscoherent risk measure and let L be a loss function. We are searching for an admissibleconsumption strategy c∗ which minimizes the risk of a loss, that is

ρ

(−

∫ T

0L(c∗

t ) dt

)= inf

c(1.3)

with respect to all admissible consumption strategies c. Continuous coherent risk measurescan be represented by sets of probability measures P in the form

ρ(X) = supQ∈P

EQ(−X)

(see Delbaen (2002, Theorem 3.2)). Thus, problem (1.3) is equivalent to the minimizationof

supQ∈P

EQ

(∫ T

0L(ct) dt

)(1.4)

with respect to all admissible consumption strategies c. Therefore, the problem of min-imizing the risk of a loss in (1.3) can equivalently be seen as a problem of utilitymaximization. In contrast to problem (1.2) there is not only one underlying probabilitymeasure but a whole class of these. Therefore, one speaks of model uncertainty. Thecorrect model is unknown, so one chooses a class of models which, from a risk managerspoint of view, describes the reality in a sufficient precise way.

The related generalized version of problem (1.1) of maximizing the robust utilityfunction inf

Q∈P EQ(u(X)) over all admissible claims for incomplete market models has

been considered in a recent paper by Gundel (2003) extending the nonrobust version ofthis problem from Goll and Ruschendorf (2001). A related variational problem was alsoconsidered in Schied (2004) where for law invariant risk measures explicit solutions couldbe given by Neyman-Pearson theory for robust testing problems. In a recent paper theportfolio optimization problem (1.1) has been studied in the robust case by Quenez (2004)extending a result from Cvitanic and Karatzas (1999) and the nonrobust version of thisproblem by Kramkov and Schachermayer (1999) where the solutions are characterizedby duality based methods.

Optimal consumption strategies under model uncertainty 3

The outline of the paper is as follows: In section 2 we specify our market model andformulate the problem of optimal consumption under model uncertainty. In section 3 wegive some conditions under which problem (1.4) has a saddle point. As a result we obtainsimilar as in the nonrobust version of this problem in Karatzas and Zitkovic (2003) anoptimal consumption strategy.

2 The model and the consumption problemIn the following let S = (S1, . . . , Sd) be a semimartingale on a filtered probability space(�,F, (Ft)t∈[0,T ], P), where T ∈ R+. S represents a discounted asset price process.

Trading strategies H = (H1, . . . , Hd) are predictable S-integrable processes, whereHi denotes the amount invested in Si . Let X be the set of all stochastic integrals H · Swhich are bounded from below by some constant. We assume that the set S of probabilitymeasures Q equivalent with respect to P such that each process X ∈ X is a supermartin-gale with respect to Q is not empty. In the following we will show that the optimalconsumption problem under model uncertainty in (1.4) is closely related to a dual op-timization problem on the class S of equivalent supermartingale measures. In order toobtain existence of a solution of the dual problem one needs to consider the weak∗-closureD of S in the space of finitely additive measures ba(P) absolutely continuous with re-spect to P. Apart from trading strategies let an optional nonnegative endowment process(et)t∈[0,T ] be given, which describes additional income. We assume that (et)t∈[0,T ] isessentially bounded. In addition there is the possibility to consume gained capital. Thisis modeled by optional nonnegative processes c. We call such processes c consumptionstrategies.

The following definitions and properties are taken from Karatzas and Zitkovic (2003).

Definition 2.1 A consumption strategy (ct)t∈[0,T ] is called admissible with respect to theinitial capital x ∈ R+ if there exists a trading strategy H such that

x +∫ T

0Ht dSt +

∫ T

0(et − ct) dt ≥ 0.

The set of all admissible trading strategies with respect to the initial capital x ∈ R+ isdenoted by A(x).

For a finitely additive measure Q ∈ D we define a density process as follows: LetQr be the regular part of Q, i.e. the σ-additive part in the Hewitt-Yosida decompositionof Q (see Rao and Rao (1983, Theorem 10.2.1)). Let L Q

t := d(Q|Ft)r

d(P|Ft). Then the density

process Y Q of Q with respect to P is defined by

Y Qt := lim inf

q↓tq∈Q

L Qq .

This implies the following characterization of admissible consumption strategies (seeKaratzas and Zitkovic (2003, Proposition 2.15)).

4 Burgert -- Ruschendorf

Lemma 2.2 c is an admissible consumption strategy with initial capital x, that is c ∈A(x), if and only if

E

(∫ T

0ctY

Qt dt

)≤ x +

⟨Q,

∫ T

0et dt

⟩for all Q ∈ D.

This implies that the set A(x) of admissible consumption strategies is convex andclosed under convergence in probability with respect to P ⊗ λ.

Our goal is to find optimal admissible consumption strategies, that is, those strategiesfor which the risk of a loss is minimal. Losses are measured by some strictly decreasing,strictly convex, continuously differentiable loss function L : R+ −→ Rwhich fulfills theInada-conditions

L ′(0) = −∞ and L ′(∞) = 0.

The risk of a loss is determined by some relevant coherent L1-risk measure ρ : L1(P)→Rhaving the Fatou-property. These risk measures are given by the following axioms: LetX, Y ∈ L1(P), m ∈ R and λ ∈ R+ then

1) If X ≥ Y then ρ(X) ≤ ρ(Y ) (monotonicity).

2) ρ(X + m) = ρ(X) − m (translation invariance).

3) ρ(λX) = λρ(X) and ρ(X + Y ) ≤ ρ(X) + ρ(Y ) (coherence).

4) For all A ∈ F with P(A) > 0 holds ρ(−11A) > 0 (relevance).

5) Let (Xn)n∈N be an uniformly bounded sequence in L1(P) with XnL1(P)−→ X then

ρ(X) ≤ lim infn→∞ ρ(Xn) (Fatou-property).

By a modification of Delbaens (2002) representation theorem, by Nakano (2004) thereexists a weak∗-closed convex set P of probability measures Q equivalent with respect toP with the properties

supQ∈P

∥∥∥∥ dQ

dP

∥∥∥∥∞< ∞

and

ρ(X) = supQ∈P

EQ(−X) for all X ∈ L1(P). (2.1)

Examples for coherent L1-risk measures are given by

a) P = { Q }, where Q ∼ P has a bounded P-density.

b) P ={

Q ∼ P : dQdP ≤ 1

λ

}for some λ ∈ (0, 1].

Optimal consumption strategies under model uncertainty 5

The set P in b) generates the coherent risk measure

AVaRλ( f ) := 1

λ

∫ λ

0VaRα( f ) dα,

where VaRα denotes the Value at Risk at level α. Under appropriate assumptions (seeFollmer and Schied (2002)) AVaR is equal to the worst conditional expectation WCEλ

which is given by

WCEλ( f ) := sup{ E(− f | A) : A ∈ F, P(A) > λ }.AVaRλ is in some sense the smallest coherent risk measure which dominates VaRλ. Inaddition it can be shown that AVaRλ generates the class of all law invariant risk measures(see Inoue (2003)).

For x ∈ R+ let

A1L(x) :=

{c ∈ A(x) :

∫ T

0L(ct) dt ∈ L1(P)

}.

We are searching for a solution of the optimization problem

ρ

(−

∫ T

0L(ct) dt

)= inf

c∈A1L (x)

. (2.2)

By the representation (2.1) of ρ problem (2.2) has the form

infc∈A1

L (x)ρ

(−

∫ T

0L(ct) dt

)= inf

c∈A1L (x)

supQ∈P

EQ

(∫ T

0L(ct) dt

).

One can formulate the problem equivalently by utility functions. Let U := −L. Then weare searching for a solution of

supc∈A1

L (x)

infQ∈P EQ

(∫ T

0U(ct) dt

). (2.3)

In this case our problem is a generalization of Karatzas and Zitkovic (2003) whereP = { P }.

In the following we give some conditions under which (2.3) has a saddle point whichallows to construct an optimal consumption strategy analogously to Karatzas and Zitkovic(2003).

3 A solution of the consumption problemThe optimal consumption problem in (2.3) is defined by the functional

u∗(x) := supc∈A1

L (x)

infQ∈P EQ

(∫ T

0U(ct) dt

), x > 0. (3.1)

6 Burgert -- Ruschendorf

Let u∗ denote the corresponding functional

u∗(x) := infQ∈P sup

c∈A1L (x)

EQ

(∫ T

0U(ct) dt

), x > 0. (3.2)

The dual problem to (3.1) is given by

v(y) := infQ∈D

infQ∈P

{EQ

(∫ T

0V

(y

Y Qt

Z Qt

)dt

)+ y

⟨Q,

∫ T

0et dt

⟩},

y > 0, where Z Qt := E

(dQdP

∣∣∣Ft

)is the density process of Q ∈ P with respect to P and

V is the conjugate of U , that is

V(y) := supx∈R+

(U(x) − xy).

For Q ∈ P let uQ the mean value of the optimal consumption with respect to Q,

uQ(x) := supc∈A1

L (x)

EQ

(∫ T

0U(ct) dt

)(3.3)

and let vQ denote the related dual functional

vQ(y) := infQ∈D

{EQ

(∫ T

0V

(y

Y Qt

Z Qt

)dt

)+ y

⟨Q,

∫ T

0et dt

⟩ }.

We assume the following conditions:

A1)∫ T

0 U(et) dt is P-a.s. bounded below.

A2) The utility function U fulfills the asymptotic elasticity condition, that is

lim supx→∞

xU ′(x)

U(x)< 1.

Assumption A1) is fulfilled if (et)t∈[0,T ] is bounded below by a strictly positive constant.Assumption A2) was introduced by Kramkov and Schachermayer (1999). It implies

Lemma 3.1 For all y ∈ R+ and all Q ∈ P the function y �−→ vQ(y) is finite andcontinuously differentiable.

The proof is analogous to the proof of Lemma 3.10 in Kramkov and Schachermayer(1999).

In the following we identify probability measures Q ∈ P with their P-densities. Inthe following theorem existence of a saddle point is stated and an optimal consumptionstrategy is constructed in terms of the dual problem.

Optimal consumption strategies under model uncertainty 7

Theorem 3.2 1) Let x ∈ R+ and let the assumptions A1) and A2) be satisfied. Thenthere exists a saddle point (Z∗, c∗) ∈ P ×A1

L(x) for (2.3). As consequence

u∗(x) = u∗(x) = E

(Z∗

∫ T

0U(c∗

t ) dt

)(3.4)

and c∗ is an optimal consumption strategy.

2) The following duality relations hold:

u∗(x) = infy∈R+(v(y) + xy),

v(y) = supx∈R+

(u∗(x) − xy).(3.5)

3) u∗ is strictly concave and continuously differentiable.

4) For x ∈ R+ let y := u′∗(x) and let Q∗ ∈ D be the solution of the dual problem v(y).Then the optimal consumption strategy c∗ is given by

c∗t = I

(y

Y Q∗t

Z∗t

), t ∈ [0, T ], (3.6)

where I := (U ′)−1 and Z∗t = E(Z∗ | Ft).

In the following we assume without loss of generality that P ∈ P. Otherwise wereplace the a priori measure P by some equivalent Q ∈ P. The proof of the theorem isbased on some lemmata.

Lemma 3.3 For all Q ∈ P holds limx→∞

uQ(x)x = 0.

Proof: Because of assumption A1) there exists a constant K > 0 such that uQ(x) ≥ −Kfor all x ∈ R+. By definition of vQ holds

uQ(x) ≤ vQ(y) + xy for all y ∈ R+.

Hence uQ(x)x ≤ vQ(y)

x + y. By assumption A2) holds vQ(y) < ∞ for all y ∈ R+ and so

lim supx→∞

uQ(x)

x≤ y for all y ∈ R+,

that is limx→∞

uQ(x)x = 0. �

In the next step we show the existence of a saddle point. Therefore we choosea sequence (cn)n∈N in A1

L(x) such that

8 Burgert -- Ruschendorf

infQ∈P

EQ

(∫ T

0U(cn

t ) dt

)−→ sup

c∈A1L (x)

infQ∈P EQ

(∫ T

0U(ct) dt

). (3.7)

Since cn is nonnegative for all n ∈ N, there exists a subsequence (c n)n∈N of convex combi-nations and some c∗ ≥ 0 such that c n −→ c∗ P⊗λ-a.s. (see Delbaen and Schachermayer(1994, Lemma A1.1)). SinceA(x) is convex and closed under convergence in probability,we have c∗ ∈ A(x).

Lemma 3.4 The consumption strategy c∗ is maximin-optimal, that is c∗ ∈ A1L(x) and

infQ∈P EQ

(∫ T

0U(c∗

t ) dt

)= sup

c∈A1L (x)

infQ∈P EQ

(∫ T

0U(ct) dt

). (3.8)

Proof: With the concavity of U follows

limn→∞ inf

Q∈PEQ

(∫ T

0U(cn

t ) dt

)≤ lim

n→∞ infQ∈P

EQ

(∫ T

0U(c n

t ) dt

)≤ inf

Q∈Plim

n→∞ EQ

(∫ T

0U(c n

t ) dt

).

We have to show that

EQ

(∫ T

0U+(c n

t ) dt

)−→ EQ

(∫ T

0U+(c ∗

t ) dt

).

If (U+(c n))n∈N is uniformly integrable with respect to µ := Q ⊗ λ, then convergenceholds. The proof of this uniform integrability is analogous to that in Kramkov andSchachermayer (2003). We assume without loss of generality that U+(∞) > 0. Assumethat (U+(c n))n∈N is not uniformly µ-integrable. Then there exists a measurable partition(An)n∈N of � × [0, T ] and a constant K ≥ 0 such that∫

U+(c n)11An dµ ≥ K for all n ∈ N.

Let

x0 := inf{ x > 0 : U(x) ≥ 0 },

dn := x0 +n∑

k=1

c k11Ak .

By concavity of U+ holds∫U+(dn) dµ ≥

n∑k=1

∫U+(c k)11Ak dµ ≥ nK.

Optimal consumption strategies under model uncertainty 9

Further, for Q ∈ D we have

E∫ T

0Y Q

t dnt dt ≤ x0 +

n∑k=1

E∫ T

0Y Q

t c kt dt

≤ x0 + nx + n⟨

Q,

∫ T

0et dt

⟩≤ x0 + nx + (n − 1)M +

⟨Q,

∫ T

0et dt

⟩,

where M is chosen such that∥∥ ∫ T

0 et dt∥∥∞ ≤ M. Hence dn ∈ A1

L(x0 + nx + (n − 1)M)

and therefore

lim supy→∞

uQ(y)

y= lim sup

n→∞uQ(x0 + nx + (n − 1)M)

x0 + nx + (n − 1)M

≥ lim supn→∞

∫U+(dn) dµ

x0 + nx + (n − 1)M

≥ lim supn→∞

nK

x0 + nx + (n − 1)M

> 0.

But this is a contradiction to limy→∞

uQ(y)y = 0. So we have shown that (U+(c n))n∈N

is uniformly µ-integrable; in particular∫ T

0 U+(c∗t ) dt ∈ L1(P). The optimality of c∗

implies by assumption A1)

infQ∈P EQ

(∫ T

0U(c∗

t ) dt

)≥ inf

Q∈P EQ

(∫ T

0U(et) dt

)> −∞.

Therefore(∫ T

0 U(c∗t ) dt

)− ∈ L1(P) and thus∫ T

0 U(c∗t ) dt ∈ L1(P). �

Lemma 3.5 There exists some Z∗ ∈ P such that

supc∈A1

L (x)

E

(Z∗

∫ T

0U(ct) dt

)= inf

Q∈P supc∈A1

L (x)

EQ

(∫ T

0U(ct) dt

).

Proof: Let (Zn)n∈N be a sequence in P such that

limn→∞ sup

c∈A1L (x)

E

(Zn

∫ T

0U(ct) dt

)= inf

Q∈P supc∈A1

L (x)

EQ

(∫ T

0U(ct) dt

).

Since P is sequentially weak∗-compact, there exists a subsequence of (Zn)n∈N denotedby (Zm) and some Z∗ ∈ P such that lim

m→∞ E(Zm f ) = E(Z∗ f ) for all f ∈ L1(P).

10 Burgert -- Ruschendorf

Therefore

supc∈A1

L (x)

E

(Z∗

∫ T

0U(ct) dt

)≤ lim

m→∞ supc∈A1

L(x)

E

(Zm

∫ T

0U(ct) dt

)

= infQ∈P sup

c∈A1L (x)

EQ

(∫ T

0U(ct) dt

).

Hence supc∈A1

L (x)

E(

Z∗ ∫ T0 U(ct) dt

)= inf

Q∈P supc∈A1

L (x)

EQ

(∫ T0 U(ct) dt

). �

For showing that (Z∗, c∗) is a saddle point, we use the Minimax-theorem in thefollowing form: Let A, B be nonempty sets and let f : A × B −→ R be a mapping. Thetriple � = (A, B, f ) is called two-person zero-sum game. � is of concave-convex type if

1) For b1, b2 ∈ B and α ∈ [0, 1] exists some b ∈ B, such that for all a ∈ A holds:

f(a, b) ≤ (1 − α) f(a, b1) + α f(a, b2).

2) For a1, a2 ∈ A and α ∈ [0, 1] exists some a ∈ A, such that for all b ∈ B holds:

f(a, b) ≥ (1 − α) f(a1, b) + α f(a2, b).

Minimax-Theorem: Let � = (A, B, f ) be a concave-convex two-person zero-sum gameand let f < ∞. If there exists a topology τ on A with the properties

1) A is τ-compact,

2) For all b ∈ B the mapping f( · , b) : A −→ R is upper-semicontinuous, that is, fora0 ∈ A holds

lim supa→a0

f(a, b) ≤ f(a0, b),

theninfb∈B

supa∈A

f(a, b) = supa∈A

infb∈B

f(a, b).

The Minimax-theorem implies

Lemma 3.6 (Z∗, c∗) ∈ P ×A1L(x) is a saddle point and

u∗(x) = u∗(x) = E

(Z∗

∫ T

0U(c∗

t ) dt

).

Proof: Apply the Minimax-theorem to A := P and B := A1L(x). Then

u∗(x) = u∗(x) = E

(Z∗

∫ T

0U(c∗

t ) dt

).

Optimal consumption strategies under model uncertainty 11

Since Z∗ and c∗ are by construction minimax strategies, it follows that (Z∗, c∗) is a saddlepoint. �

Similar to Karatzas and Zitkovic (2003) holds for Q ∈ P that

uQ(x) = supc∈A1

L (x)

EQ

(∫ T

0u(ct) dt

),

vQ(y) = infQ∈D

{EQ

(∫ T

0V

(y

Y Qt

Z Qt

)dt

)+ y

⟨Q,

∫ T

0et dt

⟩ }are conjugate functions. For showing that this holds true also for u∗ and v, we use againthe Minimax-theorem.

Let wQ(x) := uQ(x) − xy. Then the mapping Q �−→ wQ(x) is convex and upper-semicontinuous above on R+, that is, for every sequence (Qn)n∈N in P which convergesin the weak∗-topology to some Q ∈ P holds

wQ(x) ≤ lim infn→∞ wQn (x).

Further, the mapping x �−→ wQ(x) is concave. An application of the Minimax-theoremto A := P, B := R+ and f(Q, x) := −wQ(x) results in

v(y) = infQ∈P

vQ(y) = infQ∈P

supx∈R+

wQ(x)

= supx∈R+

infQ∈P

wQ(x)

= supx∈R+

( infQ∈P

uQ(x) − xy)

= supx∈R+

(u∗(x) − xy)

= supx∈R+

(u∗(x) − xy).

By standard arguments from convex analysis (see Rockafellar (1970)) follows

u∗(x) = infy∈R+(v(y) + xy).

In the last step we show the representation of the optimal consumption strategy. Let(Z∗, c∗) ∈ P ×A1

L(x) be a saddle point. Then the dual problem v(y) is given by

v(y) = infQ∈D

{E

(Z∗

∫ T

0V

(y

Y Qt

Z∗t

)dt

)+ y

⟨Q,

∫ T

0et dt

⟩ }. (3.9)

This can be shown similar to Karatzas and Zitkovic (2003) by an application of theMinimax-theorem.By the weak∗-closedness ofD there exists a minimal element Q∗ ∈ D.This implies the differentiability of u∗ and v. The mapping v′ is given by

v′(y) = −E

(Z∗

∫ T

0I(

yY Q∗

t

Z∗t

)Y Q∗t

Z∗t

dt

)+

⟨Q∗,

∫ T

0et dt

⟩. (3.10)

12 Burgert -- Ruschendorf

Let y be given by the equation u′∗(x) = y. Since V(y) = U(I(y)) + yI(y), we get

u∗(x) = E

(Z∗

∫ T

0U(c∗

t ) dt

)≤ E

(Z∗

∫ T

0V

(y

Y Q∗t

Z∗t

)dt

)+ E

(Z∗

∫ T

0y

Y Q∗t

Z∗t

I(

yY Q∗

t

Z∗t

)dt

)

= E

(Z∗

∫ T

0V

(y

Y Q∗t

Z∗t

)dt

)+ y

(−v′(y) +

⟨Q∗,

∫ T

0et dt

⟩)= v(y) + xy

= u∗(x).

Because of

E

(∫ T

0

(Y Q

t − Y Q∗t

)I(

yY Q∗

t

Z∗t

)dt

)≤

⟨Q − Q∗,

∫ T

0et dt

⟩for all Q ∈ D and all y ∈ R+ (see Karatzas and Zitkovic (2003)), we have I

(y Y Q∗

Z∗)

∈A1

L(x). This implies that the optimal consumption process c∗ has the representation

c∗t = I

(y

Y Q∗t

Z∗t

)with I = (U ′)−1,

where y is given by the equation u′∗(x) = y.

Remark 3.7 The arguments in this paper are also valid for the more general problem

supc∈A1

L (x)

infQ∈P EQ

(∫ T

0U(ct) µ(dt)

)with some weighting measure µ(dt) which allows to consider the problem of maximizingthe robust expected utility from consumption and terminal wealth

sup infQ∈P EQ

(∫ T

0U1(ct) dt + U2(x + H · ST )

)where the sup is over all admissible investment and consumption strategies (c, H ) (seeKaratzas and Zitkovic (2003), Ex. 3.15). In particular for the robust optimization ofterminal wealth the optimal wealth is of the form

x + H∗ · ST = I

(y

Y Q∗T

Z∗

)

with Q∗ ∈ D, Z∗ ∈ P and y = u′∗(x) as in Theorem 3.2.

Optimal consumption strategies under model uncertainty 13

Acknowledgement. The authors would like to thank A. Schied for pointing out thereference to the related paper of Quenez (2004) on the portfolio optimization problemafter having finished this paper.

References[1] Cvitanic, J., I. Karatzas (1999). On dynamic measures of risk. Finance and Stochas-

tics 4, 451–482.

[2] Delbaen F. (2002). Coherent risk measures on general probability spaces. Essaysin Honour of Dieter Sondermann, Springer-Verlag.

[3] Delbaen, F., W. Schachermayer (1994). A general version of the fundamental theo-rem of asset pricing. Mathematische Annalen 300, 463–520.

[4] Follmer, H., A. Schied (2002). Stochastic Finance, An Introduction in Discrete Time.de Gruyter Studies in Mathematics 27.

[5] Goll, T., L. Ruschendorf (2001). Minimax and minimal distance martingale mea-sures and their relationship to portfolio optimization. Finance and Stochastics 5,557–581.

[6] Gundel, A. (2003). Robust utility maximization for complete and incomplete marketmodels. To appear in Finance and Stochastics.

[7] Inoue, A. (2003). On the worst conditional expectation. Journal of MathematicalAnalysis and Applications 286, 237–247.

[8] Karatzas, I., G. Zitkovic (2003). Optimal consumption from investment and randomendowment in incomplete semimartingale markets. Annals of Probability 31, no. 4,1821–1858.

[9] Kramkov, D., W. Schachermayer (1999). The condition on the asymptotic elasticityof utility functions and optimal investment in incomplete markets. Annals of AppliedProbability 9, no. 3, 904–950.

[10] Kramkov, D., W. Schachermayer (2003). Necessary and sufficient conditions in theproblem of optimal investment in incomplete markets. Annals of Applied Probability13, no. 4, 1504–1516.

[11] Nakano, Y. (2004). Efficient hedging with coherent risk measure. Journal of Math-ematical Analysis and Applications 293, 345–354.

[12] Quenez, M.-C. (2004). Optimal portfolio in a multiple-priors model. Seminar onStochastic Analysis, Random Fields and Applications IV, pp. 291–321. Progr. Prob.58, Birkhauser, Basel.

[13] Rao, K., M. Rao (1983). Theory of Charges. Academic Press.

[14] Rockafellar, R. (1970). Convex Analysis. Princeton University Press.

14 Burgert -- Ruschendorf

[15] Schachermayer, W. (2004). Utility maximization in incomplete markets. In: Sto-chastic Methods in Finance, Lectures given at the CIME-EMS Summer Schoolin Bressanone/Brixen, Italy, July 6–12, 2003 (M. Fritelli, W. Runggaldier, eds.).Springer Lecture Notes in Mathematics, Vol. 1856, pp. 225–288.

[16] Schied, A. (2004). On the Neyman-Pearson problem for law-invariant risk measuresand robust utility functionals. Annals of Applied Probability 14, 1398–1423.

Christian BurgertAbteilung fur Mathematische StochastikUniversitat FreiburgEckerstr. 1D–79104 Freiburgburgert@stochastik.uni-freiburg.de

Ludger RuschendorfAbteilung fur Mathematische StochastikUniversitat FreiburgEckerstr. 1D–79104 Freiburgruschen@stochastik.uni-freiburg.de

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