Robust Utility Maximization with Unbounded Random Endowment Keita Owari

Received: date / Accepted: date

Abstract This paper studies the problem of robust utility maximization with random endowment. When the endowment is possibly unbounded, but satisfies certain integrability conditions, we first prove the fundamental duality relation between the utility maximization and the dual problem, and the existence of a solution to the dual problem. Then the existence of an optimal strategy in a certain choice of admissible class is discussed. As an application, we introduce a robust version of utility indifference prices. Keywords Robust Utility Maximization Convex Duality Method Utility Indifference Valuation Mathematics Subject Classification (2010) 91G80 60H30 46N10 60G44 JEL Classification C61 G11 Published in: Adv. Math. Econ. 14, pp.147-181, 2011. The final publication is available at Springer. DOI:10.1007/978-4-431-53883-7_7 1 Introduction This paper addresses the convex duality theory for robust utility maximization, in the presence of random endowment. Suppose we are given a semimartingale S describing The author deeply thanks an anonymous referee for a number of valuable suggestions. Also, the financial support from the Global Center of Excellence (COE) program “the Research Unit for Statistical and Empirical Analysis in Social Sciences (G-COE Hi-Stat)” of Hitotsubashi University is greatly acknowledged. K. Owari Graduate School of Economics, Hitotsubashi University 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan Current Address: Graduate School of Economics, The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan E-mail: [email protected]

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Keita Owari

the evolution of prices of tradable assets, a class of admissible integrands for S , a utility function U , a set P of probability measures, and a random variable B. Then the problem is to: maximize

inf E P ŒU. ST C B/;

P 2P

among 2 :

(1.1)

The set P is a mathematical expression of model uncertainty (also called Knightian uncertainty, in Economics). Each element P 2 P is considered a candidate model of financial market, and the case where P is a singleton corresponds to the classical situation without uncertainty. See Föllmer et al. [13] for more background information about model uncertainty and robust utility maximization. In the present paper, we consider (1.1) in the case where the utility function U is defined on the whole real line, and the endowment B is possibly unbounded but satisfies a certain uniform integrability condition in the split of [28]. We will give (1) the duality between the problem (1.1) and a minimization problem of a robust divergence functional over a class of local martingale measures for S (the dual problem), (2) the existence and characterizations of a solution to the dual problem, and (3) the existence of optimal strategy for (1.1) with a certain choice of . One of our primal motivation for studying the problem (1.1) is to give a way of pricing contingent claims in incomplete financial market with model uncertainty. To this aim, we will define and compute the robust utility indifference price of a claim B. This is mathematically a direct corollary of our duality result presented below, but possesses some reasonable properties as fair price, even under model uncertainty. We close the introduction with a brief literature review. When P is a singleton (i.e., no model uncertainty), the convex duality theory for utility maximization is studied by many authors, e.g., the case of no endowment (B 0) by Kramkov and Schachermayer [22, 23] and Schachermayer [32, 33], the case of bounded claim by Bellini and Frittelli [4], the case of exponential utility with integrability conditions on B by Delbaen et al. [10], Kabanov and Stricker [21] and Becherer [3]. Also, Owen and Žitkovi´c [29], Biagini et al. [6] and Owari [27] deal with the case of general utility functions defined on R with unbounded endowment. There is also a vast literature on robust utility maximization without endowment. When the utility function is defined on RC , Quenez [30] considered the problem under rather stronger assumptions including the equivalence of all P 2 P, examining the case of Brownian filtration with logarithmic and power utilities by means of BSDE. More general duality theory is developed by Schied and Wu [35] and Schied [34], followed by Hernández-Hernández and Schied [16, 17] with explicit examples in the setting of a stochastic factor model. For recent developments in this direction, see Föllmer et al. [13] and references there-in. Also, Föllmer and Gundel [12] gives a general existence result for the so called robust f -projection problem, which is the dual of robust utility maximization. For the problem of the type (2.14), Owari [28] extends the duality theory of [10] to the case with model uncertainty (see also [26] for a solvable example), and this paper is an extension of [28] to a slightly general class of utility functions. Müller [25] also studies the robust exponential utility maximization with bounded endowment in a Brownian setting. [25] solves the problem with a direct BSDE argument without

Robust Utility Maximization with Endowment

3

duality. More recently, Wittmüss [37] studies the robust utility maximization from consumption with bounded random endowment for general utility functions defined on the positive half line. Some more detailed information will be given after the proofs. 2 Main Results 2.1 Setup Suppose we are given a complete probability space .˝; F; P / equipped with a filtration F D .F t / t2Œ0;T satisfying the usual conditions of right-continuity and P completeness, where T 2 .0; 1/ is a fixed time horizon. We assume F D FT for simplicity of notation. The expectation of a random variable X under P is denoted simply by EŒX , while we explicitly write E P ŒX for expectation under other probability P . Let S be a d -dimensional càdlàg locally bounded semimartingale on .˝; F; F ; P /. For the set of admissible strategies, we will basically consider: bb WD f 2 L.S / W 0 D 0; S is uniformly bounded from belowg;

(2.1)

where L.S / WD L.S; P / denotes the set of all d -dimensional predictable .S; P /integrable processes. For the precise definitions and basic properties of stochastic integral and the space L.S; P / (or more generally, L.S; P / with P P ), we refer the reader to Jacod [20] and [19, Ch. II, IV]. 2.1.1 Utility Functions and Conjugate In this paper we consider utility functions U defined on the whole real line, i.e., U.x/ is finite for all x 2 R. More precisely, we assume: (A1) U W R ! R is continuously differentiable, increasing, and strictly concave function satisfying the Inada condition: lim U 0 .x/ D C1;

x! 1

and

lim U 0 .x/ D 0:

x!C1

(2.2)

(A2) U satisfies the condition of reasonable asymptotic elasticity: AE

1 .U /

xU 0 .x/ xU 0 .x/ > 1; AEC1 .U / WD lim sup < 1: (2.3) 1 U.x/ x%C1 U.x/

WD lim inf x&

We often need a stronger assumption than (A1): (A10 ) U satisfies (A1) and U.C1/ WD supx2R U.x/ < 1. A typical example of utility function satisfying (A10 ) and (A2) is the exponential utility: U.x/ D e ˛x , where ˛ > 0 is a risk aversion parameter. For a given utility function U , its conjugate V is defined by V .y/ WD sup .U.x/ x2R

xy/;

y 2 R:

(2.4)

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In the standard language of convex analysis, V is the convex conjugate (also called the Fenchel-Legendre transform) of the convex function ˚.x/ D U. x/. By (A1), V is also differentiable on .0; 1/ with V 0 .y/ D .U 0 / 1 .y/, and V 0 .0/ WD lim V 0 .y/ D y#0

1 V 0 .C1/ WD lim V 0 .y/ D C1; y!C1

by the Inada condition, hence V is bounded from below. Also, V .0/ D U.1/. The assumption (A2) of reasonable asymptotic elasticity is equivalent to any of the following two conditions (Frittelli and Rosazza Gianin [14, Proposition 1]): 1. For any compact interval Œa; b .0; 1/, there exist C1 ; C2 > 0 such that V .y/ C1 V .y/ C C2 .y C 1/;

8 2 Œa; b; y > 0I

(2.5)

2. There exist C10 ; C20 > 0 such that yjV 0 .y/j C10 V .y/ C C20 .y C 1/;

8y > 0:

(2.6)

2.1.2 Measures and V -divergences In this paper, we always identify any set Q of positive finite measures absolutely continuous w.r.t. P with a subset of L1 D L1 .˝; F; P / by identifying 2 Q with d=d P 2 L1 . Also, when we speak of weak topology (e.g. weakly closed, weakly compact), we mean .L1 ; L1 /-topology on L1 . For any pair of positive finite measures .; /, we denote by d=d the RadonNikodym density of w.r.t. in the sense of Lebesgue decomposition: d d=d P WD 1fd=d P >0g C 11fd=d P D0;d=d P >0g : d d=d P To the conjugate V defined by (2.4), we associate another map on R2C by

0

V .x; y/ WD

limz!C1 V .z/ z

x yV .x=y/

if x D y D 0; if x > 0; y D 0; if y > 0:

(2.7)

The map V .; / is lower semicontinuous on R2C . Then the V -divergence is defined for any pair of positive measures .; / by V .j/ WD EŒV .d=d P ; d=d P /:

(2.8)

.; / 7! V .j/ is convex and weakly lower semicontinuous on L1C L1C (see [12, Lemma 2.7]). Let P be a set of probability measures absolutely continuous w.r.t. P , expressing the model uncertainty. For this set, we assume: (A3) P is convex and weakly compact as a subset of L1 .

Robust Utility Maximization with Endowment

5

Given such a P, we define the robust V -divergence functional by 7! V .jP/ WD infP 2P V .jP /. The functional 7! V .jP/ is also convex by the convexity of P. Here we recall a characterization of weak compactness in L1 for convex sets, which is a combination of the Dunford-Pettis theorem and the fact that the .L1 ; L1 /topology shares the same closed convex sets with the strong (norm) topology. Lemma 2.1 A convex set A L1 is weakly compact if and only if it is norm closed and uniformly integrable. Proof By definition, A is weakly compact if and only if it is closed and relatively compact for the .L1 ; L1 /-topology. The .L1 ; L1 /-relative compactness coincides with the uniform integrability by the Dunford-Pettis theorem [11, Theorem II.25], while A is weakly closed if and only if strongly closed since it is convex (see e.g. [1, Theorem 5.98]). t u A probability measure Q is called a P -absolutely continuous (resp. P -equivalent) local martingale measure for S if Q P (resp. Q P ) and S is a Q-local martingale. The set of P -absolutely continuous (resp. equivalent) local martingale measures is denoted by Mloc .P / (resp. Meloc .P /), and we write Mloc D Mloc .P / (resp. Meloc D Meloc .P /). Define also: MV WD fQ 2 Mloc W V .QjP/ < 1g;

(2.9)

and set MeV WD fQ 2 MV W Q P g. Note that Mloc is convex and closed as a subset of L1 .P / since S is locally bounded (see e.g., Delbaen and Schachermayer [8, Theorem 5.4]). Also, Meloc , MV and MeV are convex since V .jP/ is convex. We assume a kind of no-arbitrage condition: (A4) MeV ¤ ;. Remark 2.2 By the condition (2.5) (, (A2)), we have, for any pair .Q; P / of probability measures, that V .QjP / < 1 if and only if V .QjP / < 1 for all > 0. Therefore, Q 2 Mloc is in MV if and only if V .QjP/ < 1 for all > 0. (A4) implies the existence of a pair .Q0 ; P0 / 2 MV P such that Q0 P0 P , and V .Q0 jP0 / < 1. In particular, P P in the sense that for all A 2 F, P .A/ D 0; 8P 2 P

,

P .A/ D 0

This yields a kind of no-arbitrage condition of the following type: for any 2 bb , if P . ST 0/ D 1 for all P 2 P, then P . ST D 0/ D 1 for all P 2 P. Indeed, P . ST 0/ D 1 for all P 2 P , P . ST 0/ D 1 , Q0 . ST 0/ D 1 , Q0 . ST D 0/ D 1 (since S is a Q0 -local martingale and ST is bounded from below) , P . ST D 0/ D 0 for all P 2 P. However, arbitrage in the usual sense is possible under each model P 2 P.

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Keita Owari

2.1.3 Random Endowment For the random endowment B, we assume: (A5) B is FT -measurable and there exists " > 0 such that inf E P ŒU. "B C / >

P 2P

1;

(2.10)

fU. .1 C "/B /.dP =d P /gP 2P is uniformly integrable.

(2.11)

For a positive finite measure and a random variable X, we denote .X / WD Xd, whenever the integral makes sense. In the dual problem, we will work with ˝ the functional 7! V .jP/ C .B/. As B is allowed to be unbounded, we have to check that this functional is well-defined. By the definition of V , we have YD V .Y / U. D/ for any positive random variables Y and D. In particular, taking Y D d=dP , R

.D/ V .jP /

E P ŒU. D/ V .jP /

0

inf E P ŒU. D/: 0

P 2P

(2.12)

Taking D D "B C and D D .1 C "/B , assumption (A5) implies that B 2 L1 ./ whenever V .jP/ < 1. Noting that cone.MV / WD fQ W 0; Q 2 MV g is written as cone.MV / D f0g [ f 2 coneMloc W V .jP/ < 1g;

(2.13)

by Remark 2.2, the functional 7! V .jP/ C .B/ is well-defined on cone.MV / by (A2) and (A5) (the case D 0 is trivial since then V .jP/ C .B/ D V .0/). Remark 2.3 Assumptions (A3) and (A5) seem to depend on the choice of reference measure P , but actually not: these assumptions remain true even if P is replaced by another probability P 0 P . For example, the family P is uniformly integrable if and only if for any " > 0, there exists a ı > 0 such that P .A/ < ı implies supP 2P P .A/ ". Since fd P =d P 0 g is clearly P 0 -uniformly integrable, fdP =d P 0 W P 2 Pg is again P 0 -uniformly integrable. Important Note on Assumptions. In the sequel, (A1) – (A5) are always in force (excepting Section 2.3) as the standing assumptions, and we do not cite them in the statements. On the other hand, we explicitly cite (A10 ) when it is used.

2.2 Main Theorems Recall that the primal problem of this paper is: maximize

inf E P ŒU. ST C B/;

P 2P

over all 2 bb :

(2.14)

The dual of the problem (2.14) is now stated as: minimize V .QjP / C E Q ŒB;

over > 0; .Q; P / 2 MV P:

(2.15)

Robust Utility Maximization with Endowment

7

This problem has several equivalent forms. Note that the assumption (A3) of weak compactness together with the lower semicontinuity of the V -divergence imply that the infimum V .jP/ D infP 2P V .jP / is always attained by some P 2 P. Therefore, the problem (2.15) is equivalent to the minimization of .; Q/ 7! V .QjP/ C E Q ŒB over .0; 1/ MV , or equivalently, minimize V .jP/ C .B/;

over 2 cone.MV / n f0g:

(2.16)

The formulation (2.15) may be familiar and conformable with other existing research, while the second formulation (2.16) is convenient for discussing abstract results. Thus we will make proper use of two equivalent formulations depending on the situation. Here we state all the main results of the paper, which will be proved in subsequent sections. Recall that (A1) – (A5) are always assumed without mentioning. The first one concerns the relation between the utility maximization and the dual problem: Theorem 2.4 (Duality) Assume (A10 ). Then the following duality equality holds: sup inf E P ŒU. ST C B/ D inf 2bb

P 2P

inf

>0 .Q;P /2MV P

D

inf

2cone.MV /nf0g

.V .QjP / C E Q ŒB/

.V .jP/ C .B//: (2.17)

Using this duality, we can compute the maximal admissible utility via the dual problem. This will ease the computation of the robust utility indifference prices as in the subjective case (see Section 2.3). Also, (2.17) justifies the term dual problem. Next we consider the dual problem (2.16). Theorem 2.5 (Existence for the Dual Problem) The problem (2.16) admits a solution O 2 cone.MV / n f0g, i.e., V .jP/ O C .B/ O D

inf

.V .jP/ C .B//:

2cone.MV /nf0g

(2.18)

Moreover, there exists a solution having the maximal support among all solutions. y already gives a minimizer As noted above, the existence of a solution O D O Q O y Py / of .; P / 7! V .jP / C .B/ on .cone.MV / n f0g/ P. Then .; O Py / D .Q; the pair .; O Py / is also called a solution to the dual problem. y is unique, and When P is a singleton (say fP g), we have further that O D O Q is equivalent to P . In the robust case, however, we can hope neither the uniqueness nor the equivalence any more, and the best we can do is to construct a solution with the maximal support, which has a kind of uniqueness in a weaker sense: the density d =d O Py is a.s. unique. See Proposition 4.7 below. Aside from the generality, it sometimes happens in special cases that we can take a y Py / 2 MV P which minimizes .Q; P / 7! V .QjP / C E Q ŒB common pair .Q; for all > 0. In such a case, the function v./ D inf.Q;P /2MV P .V .QjP / C y is uniquely deterE Q ŒB/ is strictly convex, hence the scaling factor O of O D O Q mined.

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Example 2.6 (Exponential Utility) Let U.x/ D e x . This is called the exponential utility with the risk aversion parameter 1. Then V .y/ D y log y y, hence V .QjP / C E Q ŒB D .H.QjP / C E Q ŒB/ C log

;

where H.QjP / is the relative entropy of Q w.r.t. P . Therefore, the dual problem amounts to minimize H.QjP / C E Q ŒB in .Q; P /, and the unique minimizer O is given by O D exp. inf.Q;P /2MV P .H.QjP /CE Q ŒB//. See Owari [28] for detail. Once the existence of a solution is obtained, our next interest is in the characterization of solutions. y Py / be a Theorem 2.7 (Characterization for Dual Optimizers) Let .; O Py / D .O Q; solution to the dual problem. Then: y Py . (a) Q y (b) There exists an .S; Q/-integrable predictable process O with 0 D 0 such that O y S is a Q-martingale, and ! y 0 O dQ y D O ST C B; Q-a.s. (2.19) V d Py We now go on to the discussion of optimal strategy. Unfortunately, the class bb is too small to admit an optimizer, even if P D fP g and B 0 (see e.g., [33]). Thus, we need to enlarge the admissible class. We proceed as follows. We first show that the duality (2.17) is stable enough under the change of admissible class . This implies in particular that the maximal admissible utility is unchanged under certain changes of . Then the following special class will turn out to admit an optimal strategy, under an additional assumption: V;Py WD f 2 L.S / W 0 D 0; S is a Q-supermartingale, 8Q 2 MV .Py /g; (2.20) where MV .Py / D fQ 2 MV W V .QjPy / < 1g. Of course, bb V;Py by Ansel and Stricker [2, Corollary 3.5]. Note also that if S is well-defined under P , then it is also well-defined under any P P . Theorem 2.8 (Optimal Strategy) Assume (A10 ). (a) Let L.S / be such that bb V;Py . Then sup inf E P ŒU. ST C B/ D sup inf E P ŒU. ST C B/

2 P 2P

2bb P 2P

D inf

inf

>0 .Q;P /2MV P

.V .QjP / C E Q ŒB/: (2.21)

In particular, the maximal utility is unchanged if bb is replaced by V;Py . (b) O S is a supermartingale under all Q 2 MV .Py /, where O is the integrand appearing in (2.19).

Robust Utility Maximization with Endowment

9

y P , then O 2 y , and the pair .O ; Py / 2 (c) If we assume in addition that Q V;P V;Py P is a saddle point of the map V;Py P 3 .; P / 7! E P ŒU. ST CB/, i.e., y

y

min E P ŒU.O ST C B/ D E P ŒU.O ST C B/ D max E P ŒU. ST C B/:

P 2P

2V;Py

(2.22) In particular, inf E P ŒU.O ST C B/ D max

P 2P

inf E P ŒU. ST C B/:

2V;Py P 2P

(2.23)

O In the assertion (c) on the saddle point argument, the “-part” of the saddle point Q y z is unique, while P is not, in the sense that if . ; P / 2 V;Pz P is another saddle point with Pz P , then O S D Q S up to P -indistinguishability. This is a consequence of the uniqueness of the density of maximal solution in the sense mentioned above. Remark 2.9 The reader may be curious why we choose bb for the primal domain even though it can not admit an optimizer. This is a matter of universality. In the y P ) depends on robust case, the class V;Py admitting an optimal strategy (if Q Py , implying that it is not available until we solve the dual problem. Furthermore, the dependence on Py implies the dependence on B, which is quite undesirable when we consider the application to indifference valuation. On the other hand, bb is a priori well-defined, and depends neither on P nor on B. From this point of view, Theorem 2.8 (c) is understood that an optimal strategy is obtained in a “completion” of bb in a suitable sense.

2.3 Indifference Valuation We now apply our duality result to a valuation problem. Consider an investor thinking whether to buy a contingent claim B. Suppose that his/her preference is represented by the robust utility functional X 7! infP 2P E P ŒU.X /. If he/she buys the claim B at the price p, the maximal admissible utility is: sup inf E P ŒU. ST

2bb P 2P

p C B/;

(2.24)

while if he/she does not buy, sup inf E P ŒU. ST /:

2bb P 2P

(2.25)

From economic point of view, the following decision seems reasonable: if (2.24) > (resp. <) (2.25), buy (resp. not buy) the claim. If both quantities are equal, buying the claim does not change the maximal utility, hence indifference. A similar reasoning apply also to the seller’s decision. Noting that p 7! sup 2bb infP 2P E P ŒU. ST p C B/ is decreasing, we now arrive at the following definition.

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Definition 2.10 (Indifference Prices) Buyer’s robust utility indifference price of B is defined with the convention sup ; D 1 by pb .B/ WD supfp W sup inf E P ŒU. ST 2bb

P 2P

p C B/ sup inf E P ŒU. ST /g: 2bb P 2P

Also, seller’s indifference price of B is defined with the convention inf ; D C1 by ps .B/ WD inffp W sup inf E P ŒU. ST C p

B/ sup inf E P ŒU. ST /g:

2bb P 2P

2bb P 2P

In this subsection, the claim B acts as a “variable” rather than a part of “setting”. Thus we drop (A5) from the “standing assumptions”, and instead,we introduce: B WD fB 2 L0 .˝; FT ; P / W B satisfies (A5)g: This is the set of admissible claims for the buyer. Similarly, the set of admissible claims for the seller is B D fB W B 2 Bg. By (A1) – (A4), which are still in force, B is a convex set satisfying: 1. L1 B \ . B/, hence R B \ . B/ in particular. 2. B 2 B ) x C B 2 B for all x 2 R. To see this, observe that U

" 1 1 .x C B/C U. "x C / C U. "B C /: 2 2 2

Thus, if B satisfies (2.10) with " > 0, so does x C B with "=2 > 0. Similarly, for every > 1, p

1

x 1 p U. U C p U. B /:

.x C B/ / p p

1

Hence if B satisfies (2.11) with " > 0, so does x C B with "0 D

p 1C"

1.

Theorem 2.4 gives us the following risk measure representations of pb ; ps . Proposition 2.11 Assume (A10 ), and let 1 .Q/ WD inf V .QjP/ >0

V . Q jP/ : 0

(2.26)

.Q// 8B 2 B;

(2.27)

inf

inf

0 >0 Q0 2MV

0

Then pb .B/ D

inf .E Q ŒB C

Q2MV

ps .B/ D sup .E Q ŒB

.Q// 8B 2

B:

(2.28)

Q2MV

In particular, pb (resp. ps ) is finite on B (resp. ps .B/ D pb . B/.

B), and if B 2 B \ . B/, then

Robust Utility Maximization with Endowment

11

Proof By the argument above, we can apply Theorem 2.4 for all x C B. We prove only the case of pb , but the same argument works also for ps . By Theorem 2.4, sup inf E P ŒU. ST 2bb

P 2P

p C B/ D inf

inf .V .QjP/

>0 Q2MV

sup inf E P ŒU. ST / D inf

p C E Q ŒB/

inf V .QjP/ DW v0 :

>0 Q2MV

2bb P 2P

Hence, p C B/ sup inf E P ŒU. ST /

sup inf E P ŒU. ST

2bb P 2P

2bb P 2P

, inf

inf .V .QjP/

>0 Q2MV

Q

p C E ŒB/ v0

, inf .V .QjP/ C E Q ŒB/ p v0 ; 8 > 0 Q2MV 1 Q .V .QjP/ v0 / C E ŒB p; 8 > 0 , inf Q2MV ,p

inf .E Q ŒB C

Q2MV

.Q//:

Therefore, we have (2.27). Also, pb .B/ < 1 since .Q/ V .QjP/ v0 < 1 and B 2 L1 .Q/ for all Q 2 MV . Finally, the next lemma shows that the set “f g” appearing in the definition of pb is non-empty, hence pb .B/ > 1. t u Lemma 2.12 For every B 2 B, sup sup inf E P ŒU. ST C x C B/ > sup inf E P ŒU. ST /

x0 2bb P 2P

2bb P 2P

(2.29)

Proof By the concavity of U and the fact that bb is a cone, we have sup inf E P ŒU. ST C x C B/

2bb P 2P

" 1 sup inf E P ŒU . ST / C inf E P ŒU..1 C "/.x C B//; 1 C " 2bb P 2P 1 C " P 2P

where " is taken so that (2.10) and (2.11) are satisfied. Thus, it suffices to see that sup inf E P ŒU..1 C "/.x C B// > sup inf E P ŒU. ST /:

x0 P 2P

2bb P 2P

(2.30)

First, it is easy to deduce from (2.11) that fU..1 C "/.x C B// dP =d P gP 2P is uniformly integrable for every x 0, hence P 7! E P ŒU..1 C "/.x C B// is weakly lower semicontinuous on P (see the proof of Lemma 3.1). Also, noting that B 2 L1 .P / for all P 2 P by (2.12) applied to D P , E P ŒU..1 C "/.x C B//

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Keita Owari

V .1/ C .1 C "/E P Œx C B < 1 for all P 2 P and x 0. Therefore, we can apply a minimax theorem as in the proof of Proposition 3.2 to obtain: sup inf E P ŒU..1 C "/.x C B// D inf sup E P ŒU..1 C "/.x C B//

x0 P 2P

P 2P x0

D inf lim E P ŒU..1 C "/.x C B// P 2P x%1 P D inf E lim U..1 C "/.x C B// P 2P

x%1

D U.1/ D V .0/: Here the third equality follows from the monotone convergence theorem. Finally, the RHS of (2.30) is written as inf>0 infQ2MV V .QjP/ by Theorem 2.4, which is strictly less than V .0/ (see the proof of Theorem 2.5 in Section 4.2). t u y 0 ; Py0 / be a solution to the dual problem (2.15) with B 0, whose Let .O 0 Q existence is guaranteed by Theorem 2.5. Corollary 2.13 Let B; B 0 2 B \ . B/. Then (a) (b) (c) (d) (e)

pb .0/ D ps .0/ D 0. ps .B C x/ D ps .B/ C x, pb .B C x/ D pb .B/ C x, for all x 2 R. B B 0 ) ps .B/ ps .B 0 / and pb .B/ pb .B 0 /. ps (resp. pb ) is a proper convex (resp. concave) function on B (resp. B); y infQ2MeV E Q ŒB pb .B/ E Q0 ŒB ps .B/ supQ2Me E Q ŒB. V

y0/ D Proof Noting that .Q/ 0 for all Q 2 MV , and minQ2MV .Q/ D .Q 0, (a) – (d) are immediate from the risk measure expressions (2.27) and (2.28). y 0 / D 0, (e) Since .Q inf E Q ŒB

Q2MV

inf .E Q ŒB C

Q2MV

.Q//

y

y 0 / D E Q0 ŒB .Q

y

y0/ .Q

E Q0 ŒB C D E Q0 ŒB

y

sup .E Q ŒB Q2MV

.Q// sup E Q ŒB: Q2MV

It remains only that infQ2MV E Q ŒB D infQ2MeV E Q ŒB, and supQ2MV E Q ŒB D x of Me , which is non-empty, we set Q˛ WD supQ2Me E Q ŒB. Taking an element Q V V x ˛ QC.1 ˛/Q for each Q 2 MV . Then Q˛ 2 MeV for ˛ 2 .0; 1, and ˛ 7! E Q˛ ŒB is continuous on Œ0; 1, since B 2 L1 .Q/ for all Q 2 MV . This concludes the proof. t u Remark 2.14 The concept of utility indifference valuation is quite popular, which goes back to Hodges and Neuberger [18]. Especially, the case of exponential utility with risk measure representation has been extensively studied by many authors including Rouge and El Karoui [31], the “six-author paper” [10], Mania and Schweizer

Robust Utility Maximization with Endowment

13

[24], among others. More recently, Owen and Žitkovi´c [29], Biagini et al. [6] consider the case of utility functions other than the exponential utility. Our Proposition 2.11 extends these existing results to the framework of robust utility. This extension is mathematically minor, and the only apparent difference emerges as the robustification of the penalty term. It proposes, however, a method of pricing contingent claims in the presence of model uncertainty. In view of (a) – (d), pb (resp. B 7! ps . B/) indeed defines a concave monetary utility function (resp. convex risk measure) on the convex set B in the sense of Biagini and Frittelli [5]. Although B is not a linear space, the concepts of convex risk measure and monetary utility function still make sense in view of the properties 1 and 2 of B listed just before Proposition 2.11. Finally, ps .B/ and ps .B/ can be viewed as arbitrage-free prices in view of (e).

3 Duality This section aims at proving the duality equality (Theorem 2.4). Our strategy is as follows. When P is a singleton, duality equalities are available in various settings. Thus, we first show that the order of “infP 2P ” and “sup2bb ” in the LHS of (2.17) are interchangeable. This reduces the robust problem to a family of subjective problems. Given this minimax equality, the duality will follow if we can apply the duality result of [27] to all P 2 P. Although this is not applicable to all P 2 P, we can take, in a rather trivial way, a “dense” subset of P on which the duality holds true. Then some approximation arguments conclude the proof. We begin with a simple lemma. Recall that we identify P with D WD fdP =d P W P 2 Pg L1 by the injection P 7! dP =d P , and (A1) – (A5) are always assumed without particular mention. Lemma 3.1 The map P 7! E P ŒU. ST C B/ is weakly lower semicontinuous on P for each 2 bb . Proof Fix 2 bb . By the definition of lower semicontinuity, it suffices to show that for any ˛ 2 R, the level set A˛ WD fD 2 D W EŒDU. ST C B/ ˛g is weakly closed in L1 , and since A˛ is clearly convex, we have only to show A˛ is strongly closed. Let .Dn /n A˛ be a convergent sequence, and D D limn Dn in L1 . Taking a subsequence, we may assume Dn ! D a.s. Since U is concave, and ST is bounded from below by a (say), 1C" 1 " Dn U a C Dn U. .1 C "/B / Dn U. ST C B/ 1C" " 1C" DW Gn" : The sequence .Gn" /n is a.s. convergent, and is uniformly integrable for some " > 0 by (A5). Therefore, we can apply Fatou’s lemma to conclude that EŒDU. ST C B/ lim inf EŒDn U. ST C B/ ˛; n!1

hence D 2 A˛ .

t u

14

Keita Owari

Proposition 3.2 We have sup inf E P ŒU. ST C B/ D inf sup E P ŒU. ST C B/:

2bb P 2P

P 2P 2 bb

(3.1)

Proof The map P 7! E P ŒU. ST C B/ is linear (hence convex), and weakly lower semicontinuous for every 2 bb by Lemma 3.1. Also, the set P is assumed to be weakly compact. On the other hand, 7! E P ŒU. ST C B/ is concave since U is concave, and bb is convex. Therefore, we can apply Fan’s minimax theorem (Simons [36, Theorem 3.2]) to get (3.1). t u We proceed to the next step. By (A4) and Remark 2.2, we can take a pair .Q0 ; P0 / 2 MV P such that Q0 P0 P and V .Q0 jP0 / < 1. Fixing such a pair, we set P˛ WD ˛P C .1

8P 2 P:

˛/P0 ;

Then define P0 WD fP˛ W P 2 P; ˛ 2 Œ0; 1/g:

(3.2) 0

An important consequence of the assumption U.1/ < 1 in (A1 ) is the next lemma, which is pointed out in Föllmer and Gundel [12, Remark 3.10] for a special choice of .Q0 ; P0 /, and the same proof applies to our case. Lemma 3.3 Assume (A10 ). Then for every P 2 P0 , V .Q0 jP / < 1. Note that (A5) implies in particular that E P ŒU. "B C / >

1 and

E P ŒU. .1 C "/B / >

1;

8P 2 P:

(3.3)

Also, by Lemma 3.3, we have MeV .P / ¤ ;;

8P 2 P0 :

(3.4)

Therefore, we can apply Owari [Theorem 2.1 of 27] to get: Lemma 3.4 For every P 2 P0 , we have sup E P ŒU. ST C B/ D inf 2bb

inf

>0 Q2MV .P /

.V .QjP / C E Q ŒB/:

(3.5)

In particular, inf

sup E P ŒU. ST CB/ D inf inf

P 2P0 2 bb

inf

>0 P 2P0 Q2MV .P /

.V .QjP /CE Q ŒB/: (3.6)

Proof of Theorem 2.4. We first show: inf sup E P ŒU. ST C B/ D inf

P 2P 2 bb

sup E P ŒU. ST C B/:

P 2P0 2 bb

(3.7)

The inequality “” is trivial since P0 P. To see the converse inequality, it suffices to check that the map ˛ 7! sup2bb E P˛ ŒU. ST C B/ is upper semicontinuous on Œ0; 1. This map is convex as a pointwise supremum of affine functions, and finite valued since U is bounded from above by (A10 ), hence upper semicontinuous.

Robust Utility Maximization with Endowment

15

Next we show that for all > 0, inf

inf

P 2P0 Q2MV .P /

.V .QjP / C E Q ŒB/ D

inf

.Q;P /2MV P

.V .QjP / C E Q ŒB/:

(3.8) Since B 2 L1 .Q/ for all Q 2 MV , Q 2 MV n MV .P / implies V .QjP / C E Q ŒB/ D C1. Thus, inf

inf

.V .QjP / C E Q ŒB/ D inf

P 2P0 Q2MV .P /

inf .V .QjP / C E Q ŒB/:

P 2P0 Q2MV

Now (3.8) follows by showing that for each P 2 P and > 0, inf .V .QjP / C E Q ŒB/ inf

Q2MV

inf .V .QjP˛ / C E Q ŒB/:

˛2Œ0;1/ Q2MV

(3.9)

If the LHS is C1, there is nothing to prove, hence we assume that the LHS is finite. Note that P 7! infQ2MV .V .QjP / C E Q ŒB/ is convex, since the .Q; P / 7! V .QjP / and the set MV are convex. Therefore, ˛ 7! infQ2MV .V .QjP˛ / C E Q ŒB/ is convex, and finite on Œ0; 1 by assumption, hence upper semicontinuous. Thus (3.9) follows, and we get (3.8). Finally, combining Proposition 3.2, Lemma 3.4, (3.7) and (3.8), we have sup inf E P ŒU. ST C B/ D inf 2bb

P 2P

sup E P ŒU. ST C B/

P 2P0 2 bb

D inf inf

inf

.V .QjP / C E Q ŒB/

>0 P 2P0 Q2MV .P /

D inf

inf

>0 .Q;P /2MV P

.V .QjP / C E Q ŒB/; t u

and the proof is complete.

Remark 3.5 Some related results are found. When the utility function is defined on RC and B 0, Schied and Wu [35] and Schied [34] proved similar duality results, on which our idea of proof is based. Wittmüss [37] dealt with the case of bounded endowment with utility function on RC . Our Theorem 2.4 is an extension of Theorem 2.4 of Owari [28] which investigated the case of exponential utility with unbounded endowment.

4 Solution to the Dual Problem This section studies the dual problem (2.16). We begin with a brief review of a related result due to Föllmer and Gundel [12], explaining our idea and what our contribution is. Suppose for a moment that B 0, and consider the problem: minimize V .QjP/;

over Q 2 MV :

(4.1)

This is nothing other than the robust f -projection problem of [12] which shows the existence of a solution. The heart of the proof of [12] consists of getting the weak

16

Keita Owari

lower semicontinuity of the divergence .Q; P / 7! V .QjP /, and a uniform integrability criterion in the spirit of the de la Vallée-Poussin criterion. These arguments can be modified, in a rather obvious way, to deal with the case where the V -divergence is penalized by E Q ŒB with a bounded B. When we pass from (4.1) to (2.16), two apparent differences arise. First, the domain is no longer a set of probability measures, but of positive finite measures, and the term .B/ with unbounded B appears as a penalty function. The first point is not very problematic, and the crucial is the second. Actually, the penalty term 7! .B/ is no longer lower semicontinuous unless B is bounded from below. Thus, it is not a priori trivial whether .; P / 7! V .jP / C .B/ has an appropriate lower semicontinuity. Also, we need an estimate between V .jP/ and V .jP/ C .B/ to apply the uniform integrability criterion of [12]. In the case where U is the exponential utility, hence V .j/ is the relative entropy, Owari [28] considered the same problem, where (uniform) integrability conditions on B, corresponding to (A5) of our setup, clear everything. The proofs employed there rely on some explicit computation appealing to the nature of exponential functions, which is no longer possible in our general setup. However, we can fill the gap by some knowledge from elementary convex analysis. We proceed as follows. After establishing some elementary estimates concerning the penalty term, we show that the penalized robust V -divergence functional 7! V .jP/ C .B/ is in fact weakly lower semicontinuous. Then the existence of a solution will follow from a similar argument with [12]. Again, recall that the assumptions (A1) – (A5) are in force throughout.

4.1 Preliminary Estimates Lemma 4.1 There are real numbers c; C , as well as positive numbers c 0 ; C 0 depending only on P, B and V such that c C c 0 V .jP/ V .jP/ C .B/ C C C 0 V .jP/;

8 2 cone.MV /:

(4.2)

In particular, inf

2cone.MV /

.V .jP/ C .B// c C c 0 V .1/ >

1:

(4.3)

Proof Let 2 cone.MV /, and take any P 2 P such that V .jP / < 1. Then P , and .B/ is finite. Fixing " > 0 as in (A5), we have d 1 d 1 d " d B D .1 C "/B .1 C "/B C B dP dP 1 C " dP 1 C " dP 1C" 1 d " V C U. .1 C "/B / 1 C " dP 1C" 1 d " V C V .1/ U. .1 C "/B /: 1C" dP 1C"

Robust Utility Maximization with Endowment

17

Taking P -expectation, 1 " V .jP / C V .1/ 1C" 1C" 1 " V .jP / C V .1/ 1C" 1C"

.B/

E P ŒU. .1 C "/B / 0

inf E P ŒU. .1 C "/B /:

P 0 2P

The last term in the second line is finite by (A5), and this holds for every P 2 P, since the last side is C1 if V .jP / D 1. Therefore, taking the infimum over P, we get V .jP/ C .B/

" V .jP/ 1C"

" V .1/ C inf E P ŒU. .1 C "/B /: P 2P 1C"

This shows the first inequality in (4.2) with cD

" V .1/ C inf E P ŒU. .1 C "/B /; P 2P 1C"

c0 D

" : 1C"

Similarly, for any P 2 P, ".B/ ".B C / V .jP /

E P ŒU. "B C / V .jP /

0

inf E P ŒU. "B C /:

P 0 2P

This implies V .jP/ C .B/

1C" V .jP/ "

1 inf E P ŒU. "B C /: " P 2P

We thus obtain the second inequality in (4.2) with C D and C 0 D .1 C "/=".

infP 2P E P ŒU. "B C /=" t u

Let M denote the space of positive finite measures on .˝; F/ absolutely continuous w.r.t. P , which we identify with L1C by the injection 7! d=d P . It is known that the map .; P / 7! V .jP / is weakly lower semicontinuous on M2 (see [12, Lemma 2.7]). Further, since P is weakly compact, we have even that 7! V .jP/ is weakly lower semicontinuous on M. The next lemma extends this to the penalized robust V -divergence functional 7! V .jP/ C .B/. Note that cone.Mloc / D fQ W 0; Q 2 Mloc g is closed, hence the set f 2 cone.Mloc / W V .jP/ ˛g is a closed convex set for each ˛ 2 R. Lemma 4.2 The functional 7! V .jP/ C .B/ is weakly lower semicontinuous on cone.MV /, that is, for every ˛ 2 R, the level set A˛ WD f 2 cone.MV / W V .jP/ C .B/ ˛g is weakly closed in L1 . Note that the convex set cone.MV / is not closed in general. But this will not cause any problem. The lemma states that if we extend the functional V .jP/ C .B/ to the whole M by setting C1 outside cone.MV /, then it is lower semicontinuous in the usual sense.

18

Keita Owari

Proof Since 7! V .jP/C.B/ is convex, it suffices to show that the level set A˛ is strongly closed for each ˛. Let .n /n be a Cauchy sequence in A˛ with the limit 2 M. We first note that 2 cone.MV /. Indeed, by Lemma 4.1, V .n jP/ C n .B/ ˛ implies n 2 f 0 2 cone.Mloc / W V . 0 jP/ .˛ c/=c 0 g which is a closed subset of cone.MV /. Thus .B/ is finite and V .jP/ C .B/ is well-defined. Also, by taking subsequence, we may assume that dn =d P ! d=d P , a.s. Since P is assumed to be weakly compact, there exists for each n a Pn 2 P such that V .n jP/ D V .n jPn /. Then the Komlós theorem (Delbaen and Schachermayer [9, Theorem 1.3]) shows the existence of another sequence f.Q n ; Pzn /gn such that .Q n ; Pzn / 2 convf.n ; Pn /; .nC1 ; PnC1 /; :::g; and d Pzn =d P ! D, a.s. for some positive random variable D, and again since P is weakly compact (, closed and uniformly integrable by Lemma 2.1), the convergence takes place in L1 , hence D D dP =d P with P 2 P. Noting that Q n Pzn since V .Q n jPzn / .˛ c/=c 0 < 1, we have for each n, d Q n d Q n d Q n B B V U. B /; d Pzn d Pzn d Pzn hence multiplying both sides by d Pzn =d P and rearranging the terms, ! d Q n d Q n d Pzn d Pzn C V ; B U. B /; dP dP dP dP for each n. Since V .; / is lower semicontinuous, ! ! d Q n d d Q n d Pzn d dP C ; C B lim inf V ; B ; a.s. V n!1 dP dP dP dP dP dP Also, f.d Pzn =d P /U. B /gn is uniformly integrable by (A5) and the monotonicity of U , and converges a.s. to .dP =d P /U. B / by construction. Therefore, we can apply Fatou’s lemma to conclude: V .jP/ C .B/ V .jP / C .B/ lim inf.V .n jPn / C n .B// ˛: n!1

Thus, we have 2 A˛ , and the assertion is proved.

t u

Recall that limx!1 V .x/=x D C1 by (A1). Then the next one is a restatement of Föllmer and Gundel [12], Lemma 2.12. Although the original version in [12] is stated with Q consisting of probability measures since their primal interest is in the problem (4.1), the exactly same proof applies to the case where Q is a set of positive finite measures. Lemma 4.3 Let Q be a set of positive finite measures. If sup V .jP/ < 1;

(4.4)

2Q

then Q is uniformly integrable as a subset of L1 , hence weakly relatively compact.

Robust Utility Maximization with Endowment

19

Proof We build a bridge between [12, Lemma 2.12] and the current assertion. Since limx!1 V .x/=x D C1 and P is weakly compact by (A3), [12, Lemma 2.12] together with the comment above shows the existence of a function l W Œ0; 1/ ! Œ0; 1/ such that limx!1 l.x/=x D C1 and 8c > 0; 9c0 > 0 s.t. V .jP/ c ) EŒl.d=d P / c0 : Taking c D sup2Q V .jP/, we have sup2Q EŒl.d=d P / < 1. The assertion then follows from the de la Vallée-Poussin criterion [11, Theorem II.22]. t u 4.2 Existence of a Solution Proof of Theorem 2.5 (existence). We first prove that the problem minimize V .jP/ C .B/;

over 2 cone.MV /

(4.5)

admits a minimizer, and then we verify that the minimizer must not be zero. Let ˛ 2 R be such that the level set A˛ WD f 2 cone.MV / W V .jP/ C .B/ ˛g is non-empty (such ˛ exists by (A4)). Then Lemma 4.1 shows that A˛ is contained in the set: f 2 cone.MV / W V .jP/ ˛ 0 g; for some ˛ 0 , which is uniformly integrable by Lemma 4.3. Therefore, A˛ is weakly compact, and 7! V .jP/ C .B/ is weakly lower semicontinuous on A˛ , by Lemma 4.2. We thus obtain the existence of a minimizer O 2 cone.MV /. It remains to verify that O ¤ 0. We may assume V .0/ < 1 since otherwise the assertion is trivial. To see this, it suffices to find an element .; N Px / of cone.MV / P x Px / 2 MV P be with N Px P and V .j N Px / C .B/ N < V .0/. Let .Q; x Px and V .Qj x Px / < 1, whose existence is guaranteed by (A4) such that Q x Consider the random map 7! G./ WD and Remark 2.2, and set D Q. x Px / C .d Q=d x Px /B, which is a.s. convex, hence .G./ G.0//= is inV .d Q=d x x Px / C E Qx ŒB. Then the monotone convercreasing in , and E P ŒG./ D V .Qj gence theorem shows: G.0/ x G./ x lim E P D V 0 .0/ C E Q ŒB D 1: &0 This implies that V . jPx / C .B/ < V .0/ for small enough > 0.

t u

Proof of Theorem 2.5 (construction of maximal solution). Let S be the set of all solutions to (2.16). Letting ˛ D inf2cone.MV / .V .jP/ C .B//, this set is written as: S WD f 2 cone.MV / W V .jP/ C .B/ ˛g:

(4.6)

In particular, S is a weakly compact convex set. Therefore, we P see that S is even n countably convex, i.e., if . / S, and f˛ g R with n n C n ˛n D 1, then P n ˛ 2 S. n n

20

Keita Owari

Next, define U WD ffd=d P > 0g W 2 Sg : By the countable convexity of S, the set U is -additive. Indeed, if An 2 U and n is a correspondingPelement of S for each n, the countable convexity of S implies that the measure D n 2 n n is in S, thus [ n

An D

[ fd n =d P > 0g D fd=d P > 0g 2 U: n

Now let fAn g be a sequence in U such that P .An / % supfP .A/ W A 2 Ug: Replacing An by A1 [ [ An , we may assume that An AnC1 for each n. Define Ay WD [n An , which belongs to U by countable additivity. Then y D lim P .An / D supfP .A/ W A 2 Ug: P .A/ n!1

Finally, an element O 2 S corresponding to Ay is a desired maximal solution. Indeed, if this is not maximal, then there exists some 2 S with 6 , O or equivalently y > 0 where A D fd=d P > 0g. Then P .A n A/ y D P .A/ y C P .A n A/ y > P .A/: y P .A [ A/ y hence O must be maximal. This contradicts to the definition of A,

t u

4.3 Variational Characterizations y Py / 2 .cone.MV /nf0g/P to the dual problem. Let us fix a solution .; O Py / D .O Q; We characterize this solution via variational inequalities. For any Q 2 MV and ˛ 2 Œ0; 1, we set Q˛ WD ˛Q C .1

y ˛/Q:

Similar notations apply to 2 cone.MV / and P 2 P in an obvious way. y Py / be as above. Theorem 4.4 Let .; O Py / D .O Q; O Q=d y Py /CB 2 (a) if Q 2 MV satisfies V .Q˛0 jPy / < 1 for some ˛0 2 .0; 1, V 0 .d 1 L .Q/, and ! # " ! # " y y d Q d Q y C B E Q V 0 O CB : (4.7) 0 D E Q V 0 O d Py d Py

Robust Utility Maximization with Endowment

21

O Q=d y ˛0 / < 1 for some ˛0 2 .0; 1, we have V .d y Py / (b) if P 2 P satisfies V .QjP 0 1 O Q=d O Q=d y Py /V .d y Py / 2 L .P /, and (with the convention 0 1 D 0), .d E

Py

"

!# y y d Q d Q 0 O V O V d Py d Py ! !# " y y y d Q d Q d Q O V 0 O E P V O d Py d Py d Py ! d Py C V .0/P D0 : dP y dQ O d Py

!

(4.8)

Here the second term of the RHS does not appear if either V .0/ D C1 or P Py . y Py . (c) Q We first derive an auxiliary variational inequality from which Theorem 4.4 follows easily. Proposition 4.5 Let .; P / 2 .cone.MV / n f0g/ P with V .˛0 jP˛0 / < 1 for some ˛0 2 .0; 1/. (a) Let

d O d d O .; P / W D 1f d Py >0g V CB dP dP dP d Py d O d O 0 d O dP V C 1f d Py >0g V y y y d P dP dP dP dP d dP d ; B : C 1f d Py D0; dP >0g V C dP dP dP dP dP 0

d Py dP

! (4.9)

Then .; P / 2 L1 .P / and EŒ .; P / 0:

(4.10)

(b) We have P

! d Py d O d > 0; D 0; > 0 D 0: dP dP dP

(4.11)

Proof Fixing .; P / as above, define G.˛; ; P / WD V .d˛ =d P ; dP˛ =d P / C .d˛ =d P /B;

˛ 2 Œ0; 1/:

Since .; P / 7! V .d=d P ; dP =d P / is convex, the map ˛ 7! G.˛; ; P / is convex, a.s., hence ˛ 7! .˛I ; P / WD .G.˛; ; P / G.0; ; P //=˛ decreases a.s. to

22

Keita Owari

a limit .; P / as ˛ tends to 0. Noting that .; P / .˛0 ; ; P / 2 L1 by assumption, thus .; P /C 2 L1 , the dominated convergence theorem applied to the non-negative increasing sequence f .˛0 ; ; P / .˛; ; P /g˛ shows: EŒ .; P / D lim EŒ .˛I ; P / 0: ˛&0

Here the last inequality follows from the fact:

EŒ .˛; ; P / D

V .˛ jP˛ / C ˛ .B/

V .j O Py / C .B/ O

˛

0:

Thus we have .; P / 2 L1 , and the latter assertion in part (a) follows. It remains to compute .; P / explicitly. To facilitate the notations, we denote y WD d =d y WD d Py =d P , and Z WD d=d P , Z O P , Z˛ WD d˛ =d P , D WD dP =d P , D y D D D 0g, since then D˛ WD dP˛ =d P . Note first that .; P / D 0 on the set fD y D 0, hence G.˛/ 0. ZDZ y D 0; D > 0g, we have Z y D 0, thus D˛ D ˛D and Z˛ D ˛Z. Therefore, On fD G.˛/ D V .˛Z; ˛D/ C ˛ZB D ˛DV .Z=D/ C ˛ZB, hence .; P / D V .Z; D/ C ZB D DV .Z=D/ C ZB; y > 0; Z y > 0g, while on fD n o y D// y CB Z Z y .; P / D V 0 .Z= n o y D/ y y D/V y 0 .Z= y D/ y C V .Z= .Z= D

y : D

y > 0; Z y D 0g needs more care. Note that this set has P -probability The case of fD zero if V .0/ D C1, hence we assume V .0/ < 1. Then y V .˛Z; D˛ / V .0; D/ C ZB ˛ ˛&0

.; P / D lim

y V .˛Z=D˛ / V .0/ D˛ D C V .0/ lim C ZB ˛ ˛ ˛&0 ˛&0 V .˛Z=D˛ / V .0/ y C ZB D lim Z C V .0/.D D/ ˛Z=D˛ ˛&0 y C ZB: D ZV 0 .0/ C V .0/.D D/ D lim D˛

Summing up the terms, we have (a). Also, since V 0 .0/ D 1, ( ) d Py d O d .; P / D 1 on > 0; D 0; >0 : dP dP dP If this set has positive probability, then we have a contradiction to the fact that .; P / 2 L1 .P /, hence (b). t u

Robust Utility Maximization with Endowment

23

x Px / 2 Proof of Theorem 4.4. (c) By (A4) and Remark 2.2, there exists a pair .Q; x x x x MV P such that Q P P and V .QjP / < 1, which satisfies the assumption x P , Proposition 4.5 (b) yields of Proposition 4.5. Since Q ! y dQ d Py P > 0; D 0 D 0: dP dP y hence Py Q. y This implies Py Q, (a) Let P D Py , then P˛0 D Py . Note first that .d =d O Py /jV 0 .d =d O Py /j 0 0 1 y 0 O y y y Py /C C1 V .d =d O P /CC2 .d =d O P C1/ 2 L .P / by (2.6) (, (A2)), hence V .d Q=d 0 1 y y B D V .d =d O P / C B 2 L .Q/. For any D Q 2 cone.MV / n f0g with O Q=d y Py /C V .˛ jPy / < 1 for some ˛ 2 .0; 1/, Proposition 4.5 then shows that V 0 .d B 2 L1 ./ D L1 .Q/ and ! ! ! ! y y d Q d Q C B V 0 O CB : (4.12) O V 0 O d Py d Py O with V .Q˛ jPy / < 1 for some ˛ 2 .0; 1/, we have the second Taking D Q y with > 0 satisfies V .˛ jPy / < inequality in (4.7). On the other hand, every D Q 1, hence " ! # " ! # y y y y Q 0 O dQ Q 0 O dQ E V C B E V C B ; 8 > 0: d Py d Py This is possible only if the LHS is zero, thus we obtain the first equality in (4.7). y Then noting that V .0; y/ D yV .0/ by (2.7), (b) Similarly, we set D O D O Q. y and D O P , (4.9) is written (with the convention 0 1 D 0) as ( ! !) ! y y y dQ dQ dP d Py 0 O dQ O O .; O P/ D V V dP dP d Py d Py d Py C

dP V .0/1f d Py D0; dP >0g : dP dP dP

y Here the set f dd P D 0; ddPP > 0g is a null set if P Py , and note that when V .0/ D P y ˛ / < 1 implies Q y P˛ , hence P P˛ Q y Py . C1, the assumption V .QjP The assertion now follows by taking the expectation of .; O P /. t u

Proof of Theorem 2.7. The assertion (a) is nothing other than Theorem 4.4 (c). (b) Given the variational inequality (4.7), the representation (2.19) follows from a standard argument using a version of Hahn-Banach theorem and Yor’s closedness theorem as Goll and Rüschendorf [15, Theorems 3.2 and 7.1]. However, we give a proof for the convenience of the reader. Let L be the vector space of all variables of the form f D ST where y runs through all .S; Q/-integrable processes such that 0 D 0 and each S is O Q=d y y Py / C a Q-martingale (not only local). It suffices to show that ' WD V 0 .d

24

Keita Owari

O Q=d y Py /B 2 R ˚ L where “˚” denotes the direct sum, since this already implies .d ' 2 L in view of the first equality in (4.7). Now Yor’s theorem ([38, Corollaire 2.5.2], [9, Theorem 1.6]) shows that L is y hence so is R ˚ L. Thus, if ' 62 R ˚ L, the Hahn-Banach theorem closed in L1 .Q/, y shows the existence of an h 2 L1 with khk1 D 1=2 such that E Q Œh' > 0, y y E Q Œh D 0, and E Q Œhf D 0 for all f 2 L. z by setting d Q=d z Q y WD 1 h. Noting that d Q=d z Py D Define a new probability Q y y z .1 h/d Q=d P and .1 h/ 2 Œ1=2; 3=2 by construction, we have V .QjPy / < 1 z y y by (2.5). Also, E Q Œf D E Q Œf E Q Œhf D 0 for all f 2 L, which implies that z is a local martingale measure, hence Q z 2 MV . Finally, Q y

z

E Q Œ' D E Q Œ'

y

y

E Q Œh' < E Q Œ':

This contradicts to (4.7), hence we must have ' 2 R ˚ L.

t u

Remark 4.6 When B 0, parts (a) and (b) of Theorem 4.4 are contained in [12, y is a Lemma 3.12], although an additional assumption is required for part (c). Since Q O Py / C E O Q ŒB, the variational inequality (4.7) follows esminimizer of Q 7! V .Qj sentially from[15, Proposition 7.2]. A similar remark applies also to (4.8). However, our auxiliary characterization (Proposition 4.5) allows us to prove all at once without y Py . any additional assumption for the equivalence Q We conclude this section by proving the fact noted in the comment after Theorem 2.5. Proposition 4.7 Let .0 ; P0 / and .1 ; P1 / be two solutions to the dual problem, which have the maximal support among all solutions. Then we have d0 d1 D ; dP0 dP1

a.s.

(4.13)

Proof Since .0 ; P0 / and .1 ; P1 / are solutions, .˛ ; P˛ / WD .˛1 C.1 ˛/0 ; ˛P1 C .1 ˛/P0 / is also a solution for every ˛ 2 .0; 1/. Therefore, ˛.V .1 jP1 / C 1 .B// C .1

˛/.V .0 jP0 / C 0 .B// D V .˛ jP˛ / C ˛ .B/: (4.14)

Since 0 1 P0 P1 P˛ by Theorem 4.4 and the maximality of solutions, the LHS is written as dP1 d1 dP0 d0 P˛ E ˛ V C .1 ˛/ V C ˛ .B/; dP˛ dP1 dP˛ dP0 while the RHS is written as: d˛ E P˛ V C ˛ .B/: dP˛

Robust Utility Maximization with Endowment

25

Noting that ˛.dP1 =dP˛ / C .1 ˛/.dP0 =dP˛ / D 1 and ˛.dP1 =dP˛ /.d1 =dP1 / C .1 ˛/.dP0 =dP˛ /.d0 =dP0 / D d˛ =dP˛ , the strict convexity of V shows that (4.14) holds only if d0 dP1 dP0 d1 D ; a.s. on A WD > 0 and >0 : dP1 dP0 dP˛ dP˛ Since d0 =dP0 D d1 =dP1 D 0 outside A by 0 1 P0 P1 and the definition of the Radon-Nikodym densities, we have (4.13). u t

5 Optimal Strategy 5.1 Proof of Theorem 2.8 (a) Let 2 V;Py , i.e., S is a supermartingale under 8Q 2 MV .Py /. Then y

inf E P ŒU. ST C B/ E P ŒU. ST C B/ ! # " y y d Q d Q y E P V O C O . ST C B/ d Py d Py " ! # y y dQ dQ Py O O E V C B d Py d Py

P 2P

O Qy ŒB y Py / C E D V .O Qj D inf

inf

>0 .Q;P /2MV P

.V .QjP / C E Q ŒB/:

Here the second inequality follows from Young’s inequality, while the third from the y fact that S is a Q-supermartingale. On the other hand, since bb V;Py , sup

inf E P ŒU. ST C B/ sup inf E P ŒU. ST C B/

2V;Py P 2P

2bb P 2P

D inf

inf

>0 .Q;P /2MV P

.V .QjP / C E Q ŒB/;

by Theorem 2.4. Thus we have (2.21).

5.2 Proof of Theorem 2.8 (b), (c) The supermartingale property of O S under all Q 2 MV .Py / will be verified by y using the m-stability considering a dynamic version of variational inequality for Q, (multiplicative stability, see Delbaen [7, Definition 1]) of the set of local martingale measures. This is more or less a standard route, and similar arguments are found in literature with slight differences in assumptions. We follow the line of Föllmer and

26

Keita Owari

Gundel [12], Lemma 3.12, with modifications involving the presence of unbounded endowment B. y Py / be a solution to the dual problem, and O be the inteLet .; O Py / D .O Q; y is also a minimizer of Q 7! grand appearing the representation (2.19). Note that Q Q O O y y y y V .QjP / C E ŒB. Let Z t WD .d Q=d P /jF t , and for each Q, we set Z tQ WD .dQ=d Py /jF t . Lemma 5.1 For all stopping times T and Q 2 MeV .Py /, " h ˇ i y y T C O Z y T B ˇ F E P V E V O Z Py

y O Z

ZTQ ZQ

! y C O Z

ZTQ

# ˇ ˇ B ˇ F : Q

Z

(5.1) Proof We first show that the RHS of (5.1) is well-defined and finite, Py -a.s. For each n, set y =ZQ 2 .1=n; n/g 2 F : An WD fZ

(5.2)

y Q Py , Z y =ZQ is finite and strictly positive, Py -a.s., hence limn Py .An / D Since Q; 1. By (2.5), there exist constants Kn ; Kn0 > 0, for each n, such that

V

y Z O Q ZTQ Z

! Kn V .ZTQ / C Kn0 .ZTQ C 1/ 2 L1 .Py /; on An :

(5.3)

y

y .Z Q =ZQ //jF < 1 a.s. on each An , hence it is finite a.s. On the Thus E P ŒV .O Z T y y E Q ŒjBjjF < 1 a.s., since B 2 y .Z Q =ZQ /jBjjF D O Z other hand, E P ŒO Z T 1 L .Q/. Set (

" h ˇ i Py O O ˇ y y C WD E V ZT C ZT B F > E V Py

Q

Z y T O Z ZQ

!

ZQ ˇ y T B ˇˇ F C O Z ZQ

#)

Suppose that Py .C / > 0. Then taking a large n, Py .C \An / > 0. Setting A WD An \C , x with density process Z x by: we define a new probability measure Q

x T WD 1Ac Z y T C 1A Z y Z

ZTQ ZQ

:

Robust Utility Maximization with Endowment

27

x 2 Mloc .Py /. Note first that Mloc .Py / is m-stable by [7, Proposition 5], hence Q x y Also, using (5.3), we have Q 2 MV .P /. Finally, O Qx ŒB x Py / C E V .O Qj h i y y y T / C O Z y T BjF D E P 1Ac E P ŒV .O Z " " ! ## Q Q ˇ Z Z ˇ Py Py T T y y CE 1A E V O Z C O Z B ˇ F ZQ ZQ h i h i y y y T / C O Z y T BjF C E Py 1A E Py ŒV .O Z y T / C O Z y T BjF < E P 1Ac E P ŒV .O Z O Qy ŒB: y Py / C E D V .O Qj y hence Py .C / D 0. This contradicts to the minimality of Q,

t u

Lemma 5.2 For any Q 2 MV .Py /, and T , y y T / C BjF E Q ŒV 0 .O Z y T / C BjF ; E Q ŒV 0 .O Z

Q-a.s.

(5.4)

Proof Since both sides of (5.4) are F -measurable, it suffices to show that the desired inequality holds Py -a.s. on the set fZQ > 0g, or equivalently: n o y T / C BjF E Qy ŒV 0 .O Z y T / C BjF 0; Py -a.s. ZQ E Q ŒV 0 .O Z (5.5) y for each ˛ 2 Œ0; 1/, and Z ˛ the Let Q 2 MV .Py /, Q˛ WD ˛Q C .1 ˛/Q y y Py . Therefore, density process of Q˛ w.r.t. P . Note that Q˛ 2 MeV .Py /, since Q Lemma 5.1 shows: for each ˛ 2 Œ0; 1/, ZT˛ ZT˛ ˇˇ Py Py O O O O y y y y E ŒV .ZT / C ZT BjF E V Z ˛ C Z ˛ B F : (5.6) Z Z We first compute ZT˛ =Z˛ . ˛ZTQ C .1 ZT˛ D Z˛ ˛ZQ C .1 D D

yT ˛/Z y ˛/Z ZTQ

˛ZQ ˛ZQ

C .1

ZQ .˛/ TQ Z

y ZQ ˛/Z

C .1

.˛//

C

.1 ˛ZQ

y ˛/Z

C .1

yT Z y y Z ˛/Z

yT Z ; y Z

y /. .˛/ is F -measurable for each ˛, while where .˛/ WD ˛ZQ =.˛ZQ C .1 ˛/Z ˛ 7! .˛/ is increasing, differentiable, and .0/ D 0;

0 .0/ D

ZQ : y Z

28

Keita Owari

Set G.˛/ WD V

Z˛ Z˛ y T B: y T C O Z O Z Z˛ Z˛

We see that on fZQ > 0g, n o ZQ G.˛/ G.0/ T y V 0 .O Z yT / C B & O Z .˛/ ZQ

yT Z y Z

! :

Therefore, the conditional monotone convergence theorem together with (5.6) shows: G.0/ ˇˇ y G.˛/ 0 EP ˇ F ˛ .˛/ Py G.˛/ G.0/ ˇˇ D E ˇ F ˛ .˛/ " ! # Q n o ZQ Z y T ˇˇ Z Py O y 0 O y T ! E Z V .ZT / C B ˇ F y y Z Z ZQ " ! # n o ZQ Z y T ˇˇ y Q P 0 T O yT / C B D Z V .O Z ˇ F ; E y Z ZQ t u

and we have (5.5) by the Bayes’ formula.

Remark 5.3 The inequality (5.4) can not be extended beyond the support of Q. In y T / C O Z y T B on fZQ D 0g, hence (5.6) tells us nothing outside fact, G.˛/ V .O Z the support of Q. y Proof of Theorem 2.8, (b). We know from Theorem 2.7 (b) that O S is a Q-martingale and O Q=d y Py / D V 0 .d

.O ST C B/

O Q=d y Py / C B D V 0 .d

,

O ST ;

Py -a.s.

O Q=d y Py / C BjF ; E Q ŒV 0 .d

Q-a.s.,

Thus, for each Q 2 MV .Py / and T , Lemma 5.2 shows: O S D

y O Q=d y Py / C BjF E Q ŒV 0 .d

i.e., the stochastic integral O S is bounded from below by a Q-martingale. We have therefore O S is a Q-supermartingale as a local martingale (by [2]) bounded from below by a martingale. t u y Py P , and part (b), Proof of Theorem 2.8, (c). By the additional assumption Q we have O 2 V;Py , and the representation (2.19) holds P -a.s., hence P -a.s. for all P 2 P. Also, by (A10 ) and Lemma 3.3, the variational inequality (4.8) applies to all P 2 P without the bizarre term V .0/P .d Py =d P D 0/. Now using the relation U. V 0 .y// D V .y/ yV 0 .y/ for all y > 0, which follows from the definition of V and (A10 ), the inequality (4.8) is rewritten as: y E P ŒU.O ST C B/ D min E P ŒU.O ST C B/ P 2P

(5.7)

Robust Utility Maximization with Endowment

29

Finally, we have Py

E ŒU.O ST C B/ D E DE

Py

Py

"

!

V

y dQ O d Py

y y dQ dQ O V 0 O d Py d Py

!

V

y dQ O d Py

y dQ C O .O ST C B/ d Py

"

!#

#

O Qy ŒB y Py / C E D V .O Qj D

y

sup E P ŒU. ST C B/: 2V;Py

Here the last equality follows from the computation in the beginning of the proof of Theorem 2.8 (a). Therefore, .O ; Py / is a saddle point. t u

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