Robust Utility Maximization for Complete and Incomplete Market Models

Transcription

1 Robust Utility Maximization for Complete and Incomplete Market Models Anne Gundel 1 Humboldt-Universität zu Berlin Institut für Mathematik Unter den Linden Berlin, Germany agundel@math.hu-berlin.de December 1, 23. This version: April 21, 25 Abstract We investigate the problem of maximizing the robust utility functional inf E Q ux. We give the dual characterization for its solution for both a complete and an incomplete market model. To this end, we introduce the new notion of reverse f-projections and use techniques developed for f-divergences. This is a suitable tool to reduce the robust problem to the classical problem of utility maximization under a certain measure: the reverse f-projection. Furthermore, we give the dual characterization for a closely related problem, the minimization of expenditures given a minimum level of expected utility in a robust setting and for an incomplete market. Keywords: f-divergences, utility maximization, robust utility functionals, model uncertainty, incomplete markets, duality theory JEL Classification: D81, G11 AMS 2 Subject Classification: 62C2, 62O5, 91B16, 91B28 1 I thank Hans Föllmer for his help when writing this paper. Furthermore, I thank Alexander Schied for discussing the topic with me and Michael Kupper and the referees for their helpful remarks.

2 2 Anne Gundel 1 Introduction A standard problem in the theory of incomplete financial markets consists in maximizing the utility of payoffs. But how do we measure utility? The usual approach goes back to von Neumann-Morgenstern and Savage. It provides conditions on an investor s preferences which guarantee that the utility of a contingent claim is given by the expectation E Q ux for some measure Q and a utility function u. The problem of utility maximization with the initial endowment x can then be formulated as Maximize E Q [ux] over all X with sup P P Q E P X x 1 for some set of reasonable equivalent local martingale measures P Q. This problem is well understood, in particular due to articles by Kramkov and Schachermayer [13] and Goll and Rüschendorf [1]. However, both from a normative and a descriptive point of view, there are good reasons to consider alternative utility functionals. In 1989, Gilboa and Schmeidler [9] proposed a more flexible set of axioms for preference orders on payoff profiles. representation by a robust utility functional of the form for some set of subjective measures Q. UX := inf E QuX It led to a numerical This approach covers the uncertainty about the probabilities of market events: The agent has in mind a whole set of possible probability distributions and takes a worst case approach in evaluating the expected utility of a payoff. For an overview and more details on such robust representations of preferences, see Föllmer and Schied [6]. In this article we deal with the robust utility maximization problem Maximize inf E Q[uX] over all X with supe P X x 2 P P for some convex set P of reasonable equivalent local martingale measures. We do not require the payoffs to be obtainable from dynamic trading in the underlying assets. However, Goll and Rüschendorf [1] showed that the optimal claim for 1 is replicable by trading in these assets. Furthermore they show that the solution to 1 coincides with the solution to the problem of maximizing portfolios that can be obtained from dynamic trading in the underlying assets see

3 Robust Utility Maximization 3 [1], Theorem 5.1. It follows immediately from the form of the optimal payoff in our Theorem 2 that these results are carried forward to the problem of robust utility maximization. In 22, Baudoin [2] solved Problem 2 for a complete market model of weak information, which means that Q P is the set of measures under which some given random variable has a specific law. In 23, Schied [19] solved it with methods from robust statistics, again for a complete market model and under the condition that there exists a least favorable measure which is for example the case if Q is the core of a convex capacity. Using a similar approach as Goll and Rüschendorf [1] we give a dual characterization for the general solution to 2 for both a complete and an incomplete market model. Furthermore, we give a dual characterization for a closely related problem, the minimization of expenditures given a miminum level w of robust expected utility: Minimize supe P [X] over all X with inf E QuX w. 3 P P The main idea is to identify a measure Q such that the robust utility maximization problem 2 is equivalent to the standard problem 1 corresponding to Q. Goll and Rüschendorf [1] solve the standard problem by means of its dual problem, the minimization of the f- divergence fp Q := E Q f dp over the set of equivalent local martingale measures. In our approach we turn things round: For a given equivalent local martingale measure P, we minimize the f-divergence fp Q over the set Q. This minimizing measure Q P, which we call the reverse f-projection, has the property that for a complete market, problem 1 with Q = Q P is equivalent to our problem 2. With the aid of a characterization of the reverse f-projections Q P it is then possible to give the dual characterization of the solution to the robust utility maximization problem for both a complete and an incomplete market model. The paper is organized as follows. After giving some definitions in Section 2 we introduce reverse f-projections in Section 3 and identify those measures Q P under which for a complete market, problem 1 is equivalent to 2. In Sections 4 and 5 we give the dual characterizations for a complete and for an incomplete market model. Here, we combine the idea of reverse f- projections with techniques developed in [1] by Goll and Rüschendorf. In Section 6 we discuss uniqueness and existence of the robust f-projection. In order to illustrate our approach we

4 4 Anne Gundel discuss a specific diffusion model in Section 7. In Section 8 we show how the introduction of reverse f-projections allows us to easily solve a related problem: the minimization of expenditures given a minimum level of robust expected utility. 2 Preliminaries Let Ω, F, F t t T, Q be a filtered probability space where Ω is Polish and F = F T is the Borel σ-field. For a R d -valued semimartingale S, let P be the set of all equivalent local martingale measures for S. Let us also fix a set Q, the set of subjective measures, which are equivalent to Q. We assume that Q is convex and compact with respect to the weak topology for measures. Definition 1 Let P P, Q Q, and let f :, R be strictly convex. The f-divergence of P with respect to Q is defined as fp Q := f dp, ]. P Q P is called the f-projection of Q on P if it minimizes the f-divergence: fp Q Q = fp Q := inf fp Q. P P P P is called the robust f-projection of Q on P if it minimizes the robust f-divergence inf fp Q: inf fp Q = fp Q := inf P P inf fp Q. [ Example 1 For fx = x p p < or p > 1, we obtain the p-distance E dp Q [ ] fx = x log x, the relative entropy E dp Q log dp = E P [log dp ] log x, the reverse relative entropy E Q [ log dp. The following basic result about f-projections was proved by Rüschendorf [18]: p ], for ], and for fx = Theorem 1 [18], Thm. 5 Let f be differentiable, Q Q, and P Q P with fp Q Q <. Then P Q is the f-projection of Q on P if and only if f dpq dp Q f dpq dp for all P P with fp Q <.

5 Robust Utility Maximization 5 As Goll and Rüschendorf remark in [1] the condition f dp Q L1 P Q assumed in [18] is not used in the proof of this theorem. In this article, a utility function is defined as a function u : R R } which is strictly increasing, strictly concave, continuously differentiable in domu := x R : ux > }, and satisfies for x := infx R : ux > }. u = lim x u x =, u x = lim x x u x = U1 U2 Let I := u 1. The convex conjugate function v : R + R of a utility function u is defined by vy := supux xy = uiy yiy. x R Let us first formulate the problem of robust utility maximization for a complete market model. For an incomplete market model, this formulation will be part of our results. For P P, we want to maximize UX := inf E Q ux P where Q P Q is some set of reasonable subjective measures. We will consider the set X P x := X : X L 1 P, E P X x, and E Q [ux ] < Q Q P } of contingent claims that are affordable under P. Hence, the problem of utility maximization in a complete market can be formulated as Maximize inf E Q ux over all X X P x. P Our aim is to find a characterization of the solution to this problem for complete and incomplete market models. 3 Reverse f-projections In this section we fix an equivalent local martingale measure P P. We then want to characterize the measure Q P that minimizes the f-divergence fp Q over all Q Q. For a strictly convex, differentiable function f :, R, we define 1 ˆfx := xf. 4 x

6 6 Anne Gundel Lemma 1 ˆf :, R is also a strictly convex, differentiable function, and we have ˆf = f. Proof Differentiability and the relation ˆf = f are obvious. For α, 1 and x, y >, x y, we define αx γ := αx + 1 αy. Then we have < γ < 1, and 1 ˆfαx + 1 αy = αx + 1 αyf αx + 1 αy = αx + 1 αyf γ 1 x + 1 γ1 y 1 < αx + 1 αy γf + 1 γf x 1 1 = αxf + 1 αyf x y = α ˆfx + 1 α ˆfy. 1 y Thus, ˆf is strictly convex. For Q Q, we have the following relation between f and ˆf-divergences: ] [ ] dp dp fp Q := E Q [f = E P dp f = E P [ ˆf ] =: dp ˆfQ P. Hence, the f-divergence of P with respect to Q is equal to the ˆf-divergence of Q with respect to P. This symmetry was already observed by Liese and Vaja [14] in Theorem Definition 2 Let Q P Q satisfy fp Q P = fp Q := inf fp Q. Then Q P is called the reverse f-projection of P on Q. Of course, the reverse f-projection coincides with the ˆf-projection. Since Q is weakly compact, by Liese and Vajda [14], Proposition 8.4, the reverse f-projection always exists. Example 2 For fx = x p p < or p > 1, we have ˆfx = x 1 p. Hence the class of p- distances is invariant under this transformation. For fx = x log x, we have ˆfx = log x. Thus, the relative entropy is transformed into the reverse relative entropy and vice versa.

7 Robust Utility Maximization 7 Applying Theorem 1 to the ˆf-projection we see that a measure Q P Q with ˆfQ P P < is the reverse f-projection of P on Q if and only if ˆf P dp P ˆf P dp Q Q with ˆfQ P <. 5 Let u be a utility function as defined in Section 2, and let v denote its convex conjugate function. For λ >, we define v λ and ˆv λ by v λ x := vλx, resp. ˆv λ x := xv λ 1/x. We want to characterize the reverse v λ -projection Q P. To this end, observe that ˆv λ x = v λ λ λ x x v = v x Applying 5 to ˆf = ˆv λ now leads to λ + λ λ x x I = u I x λ. x Proposition 1 Let λ > and Q P λ Q P λ := Q Q : v λ P Q < }. Then Q P λ is the reverse v λ -projection if and only if the following holds: E QP λu I λ dp = inf P λ E Qu I λ dp. 6 P λ P λ With this result we can now address the problem of robust utility maximization. 4 Duality Results for a Complete Market Model In this section let us consider a market with a unique equivalent local martingale measure P which means that the market is complete. Denote by Q P λ the reverse v λ -projection of P on Q and Q P λ := Q Q : v λ P Q < }. We will need the following two assumptions: v µ P Q P λ < for all λ, µ >, A1 and E Q u I λ dp < for all Q Q P λ for all λ >. A2 P λ The following lemma provides some technicalities. Lemma 2 Assume that A1 holds.

8 8 Anne Gundel i For λ >, Hλ := inf v λ P Q is a finite, convex function. ii Let Q Q. If v λ P Q < for all λ >, then I λ dp L 1 P for all λ >. Proof i The convexity of Hλ can be shown with exactly the same argument as in the proof of Theorem 2. Finiteness follows from A1. ii Let us define the function gx := vx v1 v 1x 1 which is convex and positive due to the convexity of v. Furthermore, we have Ix = v x = g x v 1, and hence I L 1 P if and only if g L 1 P. λ dp λ dp Since g is convex we have for < y < x, Therefore, If we set x := λ dp gx gx y yg x gx + y gx. y g x maxgx + y, gx y} gx gx + y + gx y gx µe P g λ dp and y := µ dp E Q g λ + µ dp for < µ < λ, then we get + E Q g λ µ dp E Q g λ dp. Since E Q g λ dp dp dp > for all λ >, and since E Q g λ = E Q v λ v1 v 1 λ 1 < for all λ > by assumption, we have proved ii. For x > x, we define V P x := inf inf E Qv λ dp } + λx. λ> V P as the concave conjugate to H is a concave function of x, and we denote by V P x the superdifferential of V P in x. Now we are ready to prove the main result of this section. Proposition 2 Assume that A1 and A2 hold and let x > x. i We have the following equivalence: λ V P x x = E P I λ dp. P λ

9 Robust Utility Maximization 9 ii Let λ P x V P x, Q P := Q P λ P x, and denote by Q P the reverse v λp x-projection of P on Q. Then sup X X P x inf E Q ux = P inf P sup E Q ux X X P x = sup E QP ux X X P x = inf λ> inf v λp Q + λx } = inf v λ P xp Q + λ P xx = E QP u I λ P x dp P = inf E Q u I λ P x dp P P. Remark 1 The proposition shows that I that is affordable under P. Under Q P λ P x dp P can be interpreted as the optimal claim the expected utility of this claim is minimal. It may therefore be considered as a worst case measure for the robust utility maximization. Furthermore, it follows from the second equality that the maximization of the robust utility functional is equivalent to the standard problem of utility maximization under Q P. However, in general Q P differs for different P. Proof First step. By Lemma 2i the function Hλ := inf E Q v λ dp is convex. By Rockafellar [17], Theorem 23.5, inf λ> Hλ + λx achieves its infimum in λ = λ P x if and only if x Hλ P x which is by [17], Theorem 7.4 and Corollary , equivalent to λ P x V P x. In this case, we have V P x = inf E Qv λ P x dp = E QP v λ P x dp P = inf E QP v λ dp λ> P + λ P xx + λ P xx } + λx.

10 1 Anne Gundel Second step. For all X L 1 P with E P X x and for all λ >, we have E QP ux E QP ux + λx E P X E QP v λ dp + λx P = E QP u I λ dp P + λ x E P I λ dp P 7 where the second inequality and the equality follow from the definition of v. Due to Assumptions U1 and U2 the function I is decreasing with range x,. By Lemma 2ii and Assumption A1, I λ dp P L 1 P for all λ >. Therefore, the function gλ := E P I λ dp P is continuous and decreasing with range x,. Hence, for every x > x, there exists λ P x > such that x = gλ P x. Thus, the above inequalities hold as equalities with λ = λ P x if and only if x = E P I λ P x dp P and X = I λ P x dp P. In this case, we have sup E QP ux = inf E QP v λ dp X X P x λ> P = E QP v λ P x dp P = E QP u I λ P x dp P } + λx + λ P xx. 8 Thus, by our results from the first step we have proved i. Third step. We know from Proposition 1 that E QP u I λ P x dp = inf E Q u I λ P x dp. P P P

11 Robust Utility Maximization 11 Putting everything together we now have sup X X P x inf E Q ux P Now Assumption A2 guarantees that I inf P sup E Q ux X X P x sup E QP ux X X P x = inf λ> = inf λ> inf E Qv E QP v = inf E Qv = E QP u = inf λ dp λ dp P λ P x dp I λ P x dp P E Q u P I } + λx + λx } + λ P xx λ P x dp P. λ P x dp P X P x, which completes the proof. Remark 2 Using methods from robust statistics Schied [19] obtains the corresponding result in the form of Kramkov and Schachermayer [13] for the complete market case under the condition that there exists a measure Q P which is the reverse f-projection for every convex function f. For obtaining a nicer interpretation of our results, we state the following Lemma 3 Assume that I λ dp L 1 P for all Q Q and all λ >. Then we have for λ > and x > x, A3 and Q P λ is independent of λ and x. Q P λ : = Q Q : v λ P Q < } = Q Q : sup E Q ux < }, X X P x Proof Under Assumption A3, 7 and 8 in the proof of the last proposition hold with the corresponding λ P x for all Q Q instead of Q P. implications for Q Q: Therefore, we get the following

12 12 Anne Gundel 1. If sup X XP x E Q ux < for some x > x, then by 8 v λ P Q < for some λ >. 2. If v λ P Q < for some λ >, then by 7 sup X XP x E Q ux < for all x > x. 3. If sup X XP x E Q ux < for all x > x, then, since for all λ >, we have E P I = x x, and hence, λ = λ P x for some x > x, we now get from 8 that v λ P Q < for all λ >. 4. From v λ P Q < for all λ > follows, of course, that sup X XP x E Q ux < for some x > x. λ dp Thus, we have proved the lemma. Remark 3 i If A3 holds, then the representation of the set Q P λ in Lemma 3 leads to the following interpretation of the utility maximization problem: The agent considers only those subjective measures Q that generate a finite maximum expected utility sup X XP x E Q ux and therefore wants to maximize inf P E Q ux and not inf E Q ux. ii Condition A3 is always satisfied for the logarithmic utility function. ux = log x, we have Ix = 1/x and hence, E P I = 1/λ. λ dp Indeed, for For the exponential utility function, ux = 1 e x, we have Ix = log x and vx := 1 x + x log x and hence, E P I = log λ vp Q. Thus, A3 is λ dp satisfied if and only if vp Q < for all Q Q. For the power utility function with ux = xp 1 p, p, 1, we have x =, Ix = x p 1, and vx = 1 p p x p p 1. Hence, E P I λ dp = λ 1 1 p p 1 pvp Q, and we see that A3 is satisfied if and only if vp Q < for all Q Q. So in the latter two cases, we could replace Q by the convex subset Q Q : vp Q < }. Then, for the common utility functions, we may always interpret the set Q P λ as in Lemma 3. iii As Goll and Rüschendorf [1] remark, the v λ -projections are independent of λ for the latter three utility functions. reverse v λ -projections are independent of λ in these cases. It is obvious from the expressions in ii that also the 5 Duality Results for an Incomplete Market Model In this section we assume that the market model is incomplete. This means that, instead of a single measure P, we have a whole set P of equivalent local martingale measures. We start

13 Robust Utility Maximization 13 with a definition of a minimax measure in our robust setting. We define U P x := sup X X P x where Q P := Q P λ P x as in Proposition 2. inf E Q ux P Definition 3 A measure P = P x P is called a robust minimax measure for x if U P x = Ux := inf P P sup X X P x inf E Q ux. P Theorem 2 Let x > x and assume that A1 and A2 hold for all P P. i If λx Ux, then Ux := inf P P sup X X P x inf E Q ux = v λx P Q + λxx. P ii P P is a robust v λx -projection on P for some λx Ux if and only if P is a robust minimax measure. iii If a robust minimax measure P exists, then we have E P I solution to the robust utility maximization problem is X := I λx dp where Q := Q P is the reverse v λx -projection of P. λx dp = x, and the iv If a robust minimax measure P exists, then inf P P sup X X P x inf E Q ux = sup P inf E Q ux : sup E P X x P P P Q } where and P Q := P P : v λx P Q < } Q P := Q Q : v λx P Q < }. If furthermore, Assumption A3 holds for all P P, then P Q = P P : sup E Q ux < } X X P x

14 14 Anne Gundel and Remark 4 Q P = Q Q : and the two sets are independent of λ and x. sup E Q ux < }, X X P x i Q is the measure that minimizes inf P P v λx P Q over all Q Q. Maximizing the utility functional E Q ux is equivalent to maximizing inf P E Q ux. By ii the incomplete market case is reducible to the complete market case under the measure P. Hence, the robust utility maximization problem in an incomplete market can be reduced to the standard problem under Q in a complete market with the equivalent local martingale measure P. ii The representation of the utility maximization problem in iv shows that under Condition A3, if we price contingent claims under the robust v λx -projection P we make sure that the optimal claim is affordable under all measures that are contained in the set P Q, i.e., that generate a finite maximum expected utility under Q. Proof i From Proposition 2 we get Ux := inf P P = inf P P sup X X P x inf λ> = inf inf λ> P P inf E Q X P inf E Qv λ dp } + λx inf E Qv λ dp } + λx Let us define = inf λ> v λp Q + λx}. Hλ := v λ P Q. We now want to show that H is convex. To this end, let ɛ > be fixed and choose λ 1, λ 2 >, P 1, P 2 P, and Q 1, Q 2 Q such that Hλ i + ɛ E Qi v dp i λ i i for i = 1, 2. For γ, 1, define Q := γq γq 2 Q. Then 1 d Q = γ + 1 γ 1 2, 1 2 d Q = 1 γ + γ 1 1, 2

15 Robust Utility Maximization 15 and Therefore, γ 1 d Q + 1 γ 2 = 1. d Q γhλ γhλ 2 + 2ɛ dp 1 γe Q1 v λ 1 1 = E Q v convex = E Qv P convex inf + 1 γe Q2 v dp 2 λ 2 2 [ γ 1 d Q v dp 1 λ γ ] 2 1 d Q v dp 2 λ 2 2 E Qv λ 1 γ 1 dp 1 d Q + λ 2 1 γ 2 dp 2 1 d Q 2 λ 1 γ dp 1 d Q + λ 21 γ dp 2 d Q inf E P P Qv λ 1 γ + λ 2 1 γ dp d Q inf E Qv λ 1 γ + λ 2 1 γ dp P P = Hγλ γλ 2. Statement i now follows from results due to Rockafellar: By [17], Theorem 23.5, inf λ> Hλ+ λx} achieves its infimum in λ = λx if and only if x Hλx which is by [17], Theorem 7.4 and Corollary , equivalent to λx Ux. ii By Proposition 2 we have for all P P, inf inf E Qv λ dp } + λx = sup λ> X X P x inf E Q ux. P Thus, the robust minimax measure coincides with the robust v λx -projection where λx Ux minimizes Hλ + λx = v λ P Q + λx. Statement iii now follows from Proposition 2. To prove iv we apply Theorem 1: P is an inf v λx -projection if and only if I λx dp dp dp P P with v λx P Q <.

16 16 Anne Gundel Recall that v x = Ix. By ii and Proposition 2i E P I λx dp = x. Thus, we have E P I λx dp x P P with v λx P Q <. Since I λx dp is the optimal claim it follows that where inf P P sup X X P x inf E Q ux = sup P inf E Q ux : sup E P X x P P P Q } P Q : = P P : v λx P Q < } and Q P := Q Q : v λx P Q < }. The interpretation of the set P Q follows from the inequalities in the proof of Lemma 2 in exactly the same way as the interpretation of the set Q P. Remark 5 We did not use the fact that P is the set of equivalent local martingale measures. Hence, P could be any set of equivalent measures. 6 Existence and Uniqueness of the Robust Minimax Measure In this section we want to discuss the existence of the robust minimax measure or v-projection and show that there is at most one such measure. Let us start with the problem of uniqueness. In order to keep notations simpler, we will replace f by v and consider v-projections in this section. If v is strictly convex, then so is ˆv. Hence, for every Q Q there is at most one v-projection P Q of Q on P, and for every P P there is at most one reverse v-projection Q P of P on Q. For the robust v-projection, this does not necessarily hold true. However, in our setting where Q is weakly compact by Liese and Vajda [14], Proposition 8.4, there exists a reverse v-projection Q P Q to every P P, and we get Proposition 3 If a robust v-projection P P exists, let Q be its reverse v-projection. Then the density dp is Q -almost surely unique.

17 Robust Utility Maximization 17 Proof Assume that P 1 and P 2 P are two robust v-projections with reverse v-projections Q P1 and Q P2. Then we have γ inf vp 1 Q + 1 γ inf vp 2 Q = inf vp 1 Q inf vγp γp 2 Q. 9 On the other hand we get with the same arguments and definition of Q as in the proof of Theorem 2 γ inf vp 1 Q + 1 γ inf vp 2 Q = γvp 1 Q P1 + 1 γvp 2 Q P2 E Qv γ dp 1 d Q + 1 γdp 2 d Q inf vγp γp 2 Q. The first inequality holds as equality if and only if dp 1 P1 = dp 2 P2 surely. Thus, by 9 the density is Q -almost surely unique. Q- and hence also Q -almost Let us now turn to the problem of existence. As shown by Kramkov and Schachermayer [13], the robust minimax measure P does not necessarily exist even in the classical setting with Q = Q}. In one of their counterexamples, they show that even for a bounded price process the infimum of vp Q may not be attained in P if the utility function is logarithmic [13], Example 5.1 BIS. On the other hand, they show in the classical setting Q = Q} that even if there is no minimax measure, a solution of the utility maximization problem may still exist. Let Y := Y : Y = 1 and XY is a supermartingale for all X that are stochastic integrals of the underlying semimartingale} and let x =. Kramkov and Schachermayer [13] prove that a solution XQ to the utility maximization problem exists if the asymptotic elasticity of the utility function, lim sup x xu x/ux, is strictly less than one. Furthermore, they show that in this case we have V λ := inf Y Y E Q vλy = inf P P E Q v, but there does not necessarily exist a v λ -projection. Schied and Wu [2] extend these results to the robust setting and obtain existence of the solution to the robust utility maximization problem under the condition that the asymptotic elasticity is strictly less than one. In our paper however, the focus is on the representation of the optimal claim in terms of martingale measures. λ dp In this section we will specify conditions which guarantee that the infimum of inf vp Q over P is indeed attained by some measure P P. For a certain class of utility functions, existence results for the classical case Q = Q} can

18 18 Anne Gundel be found in the literature. In 1975, Csiszár [1] showed existence for the case where the set of local equivalent martingale measures P is closed in variation and vx = x log x which is equivalent to ux = e x. In 1987, Liese and Vajda [14] considered the case where P is closed in variation and lim x vx/x =. In 2, Frittelli [8] proved existence for the case where the semimartingale is locally bounded and ux = e x. As he remarks, if the semimartingale is locally bounded, then the set P is closed in variation. In 22, Bellini and Frittelli [3] showed existence under the assumption that the semimartingale is locally bounded and the domain of the utility function is R. The last condition means that x =, and it is easily checked that this is equivalent to the condition lim x vx/x =. Here we will consider the existence problem for robust v-projections. Let M 1 Ω be the set of all probability measures on Ω, F. In the general setting where Q does not only consist of one measure, we obtain the following result: Proposition 4 Let g : [, R satisfy lim x gx/x =. Assume that P is closed in variation and that there exist Q Q and constants c, c 1 > such that for P P, Then there exists a robust v-projection P on P. inf vp Q c = gp Q c 1. 1 Proof We will first show that under Condition 1, the closure of the set } P := P P : inf vp Q c 11 is compact in the weak topology for measures. Since E Q g dp c 1 if inf vp Q c, } the set K := dp : P P is uniformly integrable due to the de la Vallée-Poussin criterion. Hence, by Dunford and Schwartz [4], Corollary IV.8.11, K is weakly sequentially compact on L 1 Q. By [4], Theorem V.6.1, the weak closure of the set K is weakly compact. By [4], Theorem V.3.13, it is contained in the convex and strongly closed set of densities of measures is P with respect to Q. Hence, the closure P of P is weakly compact in the topology on M 1 Ω that corresponds to the weak topology on L 1 Q, and it is contained in P. Since this topology is stronger than the weak topology for measures cf. Liese and Vajda [14], Lemma 1.46, P is compact with respect to the weak topology for measures. Now, by [14], Theorem 1.47, v is lower semicontinuous on the space M 1 Ω M 1 Ω endowed with the weak product topology. Since by Tychonov s theorem P Q is weakly compact, v achieves its infimum P, Q on P Q P Q.

19 Robust Utility Maximization 19 Let us illustrate the application of Proposition 4 in the case where Q is a weakly compact set such that for some β > 1, and where [ ] β sup E Q < 12 vx lim x x p = 13 for some p > 1. The following lemma shows that Condition 1 is satisfied with gx = x α for some α > 1. Recall that the reverse v-projection exists for every P P due to Liese and Vajda [14] because Q is weakly compact. Lemma 4 Under Assumptions 12 and 13, there is an α > 1 such that for any constant c >. Proof Define α := [ dp α ] } sup E Q : vp Q P = inf vp Q c < βp p 1+β α 1p. Then α > 1 and β = 1 + p α. Due to Assumption 13 there Let p := p/α and are constants a R, a 1 > such that vx a + a 1 x p for x. q = p/p α. Then 1/ p + 1/ q = 1, and by Hölder s inequality we have [ dp α ] [ dp α ] α 1 P E Q = E QP P [ ] dp α p 1/ p [ P ] α 1 q 1/ q E QP E QP P [ dp p ] [ α/p P ] β p α/p = E QP E Q P 1 vp Q P a [ α/p P ] β p α/p E Q a 1 a 1 c α/p a c p α/p a 1 1 if vp Q P c and c 1 := sup E Q [ β ].

20 2 Anne Gundel Remark 6 Similar to the equivalence of the two conditions x = and lim x vx/x = above, 13 is equivalent to the condition that the utility function u satisfies ux c x p p 1 ɛ for some ɛ > and for x <. In fact, sufficiency of the condition on u follows from the estimate lim inf x vx x p = lim inf x = lim inf x sup y R uy sup y R sup y R lim inf x = sup y =. y R x p yx 1 p uyx x p/p 1 y uyx x p/p 1 y The necessity can be shown in a similar way. Hence, the utility function must decrease slower than some power function. An example for such a utility function is 1 q ux = x + cq cq q for x qc q x 1/q + q c for x < for some constant c > and < q < 1. The following general result is shown in Föllmer and Gundel [5]. It includes all utility functions that are defined on the whole real line, thus also the case of exponential utility functions. The proof is more involved; instead of Hölder s inequality it uses Young s inequality for certain Orlicz spaces. Theorem 3 Assume that P is closed in variation, that and that vx lim x x =, } : Q Q is weakly compact in L 1 Q for some measure Q Q. Then there exists a robust v- projection P on P. Remark 7 In the classical setting with Q = Q}, Condition 1 is trivially satisfied for g = v and hence by Lemma 4, the v-projection exists if lim x vx/x =. This is also shown by Liese and Vajda [14] in Proposition 8.5. Theorem 3 provides the natural extension to the robust setting.

21 Robust Utility Maximization 21 7 An Example There are, in certain situations, means of determining the f-projection P Q explicitly. This suggests the following method for finding the robust f-projection P : First calculate P Q for each Q and then find the pair P Q, Q that has the smallest f-divergence. Here, we want to give an example for this approach. In diffusion models for financial markets it is feasible to estimate the volatility of assets using historical data. However, estimations of the drift are much less reliable. Let us consider an example of a model in which the volatility and the structure of the drift are known, but there is uncertainty about the size of the drift. Let Ω, F, F t t T, Q be a two-dimensional Wiener space on which we are given two independent Brownian motions B = B t t T and W = W t t T with B = W =. We assume that F = F T and that F t t T is the smallest filtration that contains the filtration which is generated by the two Brownian motions and that satisfies the usual condition see [12]. The price process of an asset is modelled by ds t = S t σ t db t + µ t dt t T. Finding equivalent local martingale measures for this model is equivalent to determining them for the model S t := B t + t α s ds t T with α = µ/σ. We assume that the process α = α t t T is B-integrable and predictable with respect to the filtration F W t t T that is generated by W. For some interval [b 1, b 2 ] R +, we define Q as the set of measures under which S has a drift of bµ, or S has a drift of bα for some b [b 1, b 2 ], i.e., Q := Q b : T b = E b 1 α s db s } for some b [b 1, b 2 ], where E is the Itô exponential T T b T 12 E b 1 α s db s = exp b 1 α s db s α 2 2 sds. We are considering the three utility functions log x, e x, and x p for < p < 1. To solve the dual problems we have to deal with the f-divergences log x, x log x, and x q where q := p p 1 <. We assume that suitable integrability conditions are satisfied for each of the

22 22 Anne Gundel utility functions such that the densities in the following define equivalent measures and hence, the f-projections exist. i fx = log x. For each Q b Q, the f-projection P Qb i.e., P Qb dp Qb b = E T bα s db s b, has the density coincides with the minimal martingale measure see Föllmer and Schweizer [7]. B b is the Brownian motion under the measure Q b. This result was proved by Schweizer [21] for general α. The f-divergence becomes fp Qb Q b = E Qb [ b 2 2 T = E Q [ E Q [E = b2 2 E Q [ T ] αsds 2 T b 1 α s db s FT W ] αsds 2. ] b 2 2 T ] αsds 2 The second equality holds due to the F W T -measurability of T α2 sds and the last equality holds because E[Eb 1 T α2 sdb s F W T ] = 1 due to the independence of B from F W T. ii fx = x log x. For each Q b Q, the f-projection P Qb with C b := E Qb [exp b2 2 [11]. We have dp Qb b T = C b exp bα s ds s has the density T α2 sds] 1. This result is due to Grandits and Rheinländer [ fp Qb Q b = log E Qb exp b2 2 [ = log E Q exp b2 2 T T The second equality holds for the same reasons as above. iii fx = x q. For each Q b Q, the f-projection P Qb dp Qb b ] αsds 2 ] αsds 2. has the density T = C b exp bα s ds s q 1 T 2 b2 αsds 2

23 Robust Utility Maximization 23 with C b := E Qb [exp q b2 2 fp Qb Q b = T α2 sds] 1. This result was also shown in [11]. We have = [ E Qb exp q b2 2 [ E Q exp q b2 2 T T ] 1 q αsds 2 α 2 sds] 1 q. Now we see that in any case, the pair of measures that generates the smallest f-divergence is the one with b = b 1. In this model, Q := Q b1 is the measure that is closest to a martingale measure in the sense that it has the smallest drift. Further interesting examples can be found in the paper [19] by Schied. 8 Expenditure Minimization A problem that is closely related to the one of utility maximization is the minimization of expenditures given the agent has a minimum level w of expected utility. That is, given her subjective probability measure Q and the equivalent local martingale measure P, she wants to solve the problem Minimize E P Y under the constraint E Q uy w. 14 The key idea for solving this is to define the reverse utility function û by the concave conjugate of ˆv: ûx := inf ˆvy + xy} 15 y> for x U := inf x ux, sup x ux and to apply Theorem 2 to these transforms. We have Î := ˆv = û 1 = uix, x x x ûx = u 1 x, and 16 û Î 1 = Ix. x We can replace Y in 14 by ûx and hence uy by X to see that, by interchanging the roles of the sets Q and P, we can apply Theorem 2 to the transforms û and ˆv to solve the expenditure minimization problem.

24 24 Anne Gundel Note that û is also a utility function as defined in Section 2 with and û x as x inf x R ux û x as x sup ux. x R Remark 8 It might not be possible to define the function û for arbitrarily large x. This is, for example, the case if ux = x p for < p < 1, where U =,. But as long as the latter two conditions on û are satisfied, the range of Î is U. This is sufficient to guarantee the existence of λ P x in the second step of the proof of Proposition 2, the only point where Conditions U1 and U2 were used. So this proof still works and hence, Theorem 2 is valid for the utility function û. We need to assume that A1, A2, and A3 hold for the transforms û, Î, and ˆv with the roles of P and Q interchanged. Define P Q λ := P P : v λ P Q < } and assume that the v λ -projection P Q λ of Q on P exists for every Q Q and λ >. Replacing Î and û Î by the terms in 16 and ˆv by xv1/x this leads to the following assumptions for all Q Q, v µ P Q λ Q < for all λ, µ >, A4 E P I λ dp + Qλ < for all P P Q λ for all λ >, A5 and u I λ dp L 1 Q for all P P and all λ >. A6 The last assumption is, as A3, only needed for reasons of economical interpretation in the following. But since this is our aim in this section, we will assume that A6 holds. We set ˆV Q w := inf λ> inf E P ˆv P P λ } λw. dp For ˆλ Q w ˆV Q w, we define P Q := P Q ˆλ Q w, Y Q w := Y : uy L 1 Q, E Q uy w, E P Y + < P P Q }, and Ûw := sup inf Y Y Q w sup E P Y. P P Q Our aim is to find a contingent claim Y that achieves this infimum.

25 Robust Utility Maximization 25 Proposition 5 Let Assumptions A4, A5, and A6 hold and let w U and ˆλw Ûw. We assume the existence of a measure ˆP P that minimizes inf vˆλw P Q over all P P. We denote by ˆQ Q the measure that minimizes vˆλw ˆP Q over all Q Q. i We have the following representation of the expenditure minimization problem in an incomplete market: where and Ûw : = sup = inf = inf P P inf Y Y Q w sup E P [Y ] : P P ˆQ Q ˆP = Q Q : P ˆQ = P P : inf E P ˆv sup E P Y P P Q inf E Q uy w ˆP ˆλw dp } } ˆλww sup E ˆP Y > } E Q uy w sup E P Y > }. E ˆQ uy w ii The solution to the expenditure minimization problem in an incomplete market is given by iii Let I λx dp Y 1 d = I ˆP ˆλw d ˆQ. be the solution to the robust utility maximization problem as in Theorem 2 where λx Ux, P is the robust v λx -projection, and Q is the reverse v λx -projection of P. If w = E Q u I λx dp, then I λx dp is also the solution to the expenditure minimization problem. d ˆP ˆλw d ˆQ iv Let I 1 be the solution to the expenditure minimization problem as in ii. If d x = E ˆP û Î ˆλw ˆQ = E 1 d ˆP I ˆP, then I 1 d ˆP is also the solution Remark 9 d ˆP to the robust utility maximization problem. ˆλw d ˆQ ˆλw d ˆQ i If ˆP exists, then the measure ˆQ always exists since Q is weakly compact. ii The middle term in i has a nice interpretation: The agent wants to minimize her costs sup P P ˆQ E P Y under the condition that her utility measured by the robust utility functional is w at least.

26 26 Anne Gundel iii The last two statements of this proposition that describe the relationship between the problems of utility maximization and expenditure minimization are a well-known result for the case of non-random payoffs see, e.g., Mas-Colell et al. [15], Prop. 3.E.1. Proof i Since we want to apply Theorem 2 we set y := w and define Ũy := Û y = inf sup Y Y Q y inf E P [ Y ] P P Q Then Ûw = Ũy. Hence, instead of minimizing E P Y we will consider the equivalent problem of maximizing E P [ Y ]. In order to avoid a too complicated description we write the constraint simply as E Q uy w instead of Y Y Q w. Define X := uy and replace Y by u 1 X = ûx in the equation above. We get Ũy = inf Now we can apply Theorem 2 to obtain Ũy = inf inf λ> P P = inf P P sup E Q X y inf E P ûx. P P Q inf E P ˆv λ } + λy dp inf E P ˆv ˆλ y dp + ˆλ yy } = sup inf E P ûx : sup E Q X y P P ˆQ ˆP where Q ˆP : = Q Q : ˆvˆλ y Q ˆP < } = Q Q : sup E ˆP ûx < } E Q X y and P ˆQ : = P P : ˆvˆλ y ˆQ P < } = P P : sup E P ûx < } E ˆQ X y and ˆλ y Ũy. Replacing y by w, X by uy, and ûx by Y completes the proof of i.

27 Robust Utility Maximization 27 ii We get from Theorem 2 that for X := Î ˆλw d ˆQ d ˆP, we have sup inf E P ûx : sup E Q X y P P ˆQ ˆP } = inf E P ûx. P P ˆQ Hence, it follows from the proof of i that Y := u 1 X = ûx 1 d = I ˆP ˆλw d ˆQ is the solution to the expenditure minimization problem. The last equality follows from ûî1/y = Iy. iii Let now I λx dp be the solution to the robust utility maximization problem. We want to show that for w = y = E QP u I λx dp, we have P inf inf inf E P ˆv λ } 1 + λy = inf λ> P P dp inf E P ˆv P P λx Then, according to i and ii, I λx dp expenditure minimization with the minimum utility level w. We define the two convex functions and Then Hλ = λĥ1/λ. Hλ := inf P P Ĥλ := inf P P dp + y λx. 17 would also be the solution to the problem of inf E Qv λ dp inf E P ˆv λ. dp By Rockafellar [17], Theorem 23.5, Equation 17 is equivalent to y Ĥ1/λx which is equivalent to y 1 λx + Ĥ y λx λ + Ĥ 1 λ λ >. 18 So we will now show that 18 holds. With the definition of U as in Section 5 we have by [17], Theorem 7.4 and Corollary , that λx Ux if and only if x Hλx. This is equivalent to xλx + H λx xλ + H λ λ >. 19

28 28 Anne Gundel By Theorem 2iii we have x = E P I y = w = E Q u I λx dp. Hence λx dp = E Q v λx dp λxe P I λx dp = inf P P inf E Qv λx dp λxx = Hλx λxx. Replacing x in 19 by Hλx y/λx leads to 18 which completes the proof. iv now follows from iii by interchanging u, I, and v and its transforms and Q and P. References [1] Csiszár, I.: I-Divergence Geometry of Probability Distributions and Minimization Problems. Annals of Probability, Vol.3, No.1, [2] Baudoin, F.: Conditioned Stochastic Differential Equations: Theory, Examples and Application to Finance. Stochastic Processes and their Applications 1, [3] Bellini, F., Frittelli, M.: On the Existence of Minimax Martingale Measures. Mathematical Finance 12, No. 1, [4] Dunford, N., Schwartz, J. T.: Linear Operators. Part 1: General Theory. New York: Interscience Publishers [5] Föllmer, H., Gundel, A.: On the Existence of Robust Projections in the Class of Martingale Measures. Preprint, Humboldt-Universtiät zu Berlin 24. [6] Föllmer, H., Schied, A.: Stochastic Finance Studies in Mathematics 27. Berlin, New York: De Gruyter 22. [7] Föllmer, H., Schweizer, M.: Hedging of Contingent Claims Under Incomplete Information. In: Davis, M., Elliott, R. eds.: Applied Stochastic Analysis. London: Stochastic Monographs 5,

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