Robust expected utility maximization with medial limits

arXiv:1712.07699v1 [q-fin.PM] 20 Dec 2017

Robust expected utility maximization with medial limits Daniel Bartl∗

Patrick Cheridito†

Michael Kupper∗

December 2017 Abstract We study a robust expected utility maximization problem with random endowment in discrete time. We give conditions under which an optimal strategy exists and derive a dual representation of the optimal utility. Our approach is based on medial limits, a functional version of Choquet’s capacitability theorem and a general representation result for monotone convex functionals. The novelty is that it works in cases where robustness is described by a general family of probability measures that do not have to be dominated or time-consistent. Keywords: Robust expected utility maximization, random endowment, nonlinear expectation, medial limit, Choquet’s capacitability theorem, convex duality. MSC 2010 Subject Classification: 91B16, 90C47, 93E20

1

Introduction

We consider a robust expected utility maximization problem of the form ! T X ϑt ∆St , U (X) = sup inf EP u X + ϑ∈Θ P∈P

(1.1)

t=1

where P is a family of probability measures, u is a random utility function, S0 , S1 , . . . , ST are the discounted prices of a tradable asset, Θ is a set of dynamic trading strategies and X a discounted random endowment. Expected utility has a long history in decision theory, economics and finance; see e.g. [3, 20, 25, 17, 12, 16, 8] and the references therein. Robust expected utility preferences were axiomatized by [13]. In the existing literature on robust expected utility maximization it is common to either assume that the set P is dominated1 (see e.g. [24, 14, 26, 23, 1]) or timeconsistent2 (see e.g. [22, 5, 19, 2]). If P is dominated, one can, like in the classical case, apply Komlós’ theorem to construct an optimal strategy from a sequence of approximately optimal strategies. The existence of optimal strategies can then be used to deduce a dual representation ∗

Department Mathematics and Statistics, University of Konstanz, 78464 Konstanz, Germany. Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland and Swiss Finance Institute. 1 i.e., all P ∈ P are absolutely continuous with respect to a common probability measure P∗ 2 i.e., P is stable under concatenation of transition probabilities †

1

for U . If P is time-consistent, the existence of optimal strategies and dual representations can be derived step by step backwards in time with dynamic programming arguments. In this paper we do not assume that P is dominated or time-consistent. As a consequence, we cannot use Komlós’ theorem or dynamic programming arguments. Instead, we suppose that a medial limit exists, which for our purposes, is a positive linear functional lim med : l∞ → R satisfying lim inf ≤ lim med ≤ lim sup with the following property: for any uniformly bounded sequence of universally measurable3 functions Xn : M → R on a measurable space (M, F), X = lim med Xn is universally measurable and EP X = lim med EP Xn for every probability measure P on the universal completion of F. Mokobozki proved that medial limits exist under the usual axioms of ZFC together with the continuum hypothesis; see [18]. Later, Normann [21] showed that it is enough to assume ZFC and Martin’s axiom. We show that the existence of a medial limit implies that problem (1.1) admits an optimal strategy. From there we derive a dual representation of U using convex analysis arguments together with a functional version of Choquet’s capacitability theorem [7]. We work with a sample space of the form Ω = Ω0 × · · · × ΩT for non-empty Borel subsets Ωt ⊆ [a, b] × R, where 0 < a ≤ b are fixed numbers. We suppose there is a money market account evolving according to Bt (ω) = ωt1 and a risky asset whose price in units of B is given by St (ω) = ωt2 . As usual, ∆St denotes the increment St − St−1 . P is assumed to be a non-empty set of Borel probability measures on Ω. The set Θ consists of all strategies (ϑt )Tt=1 such that for each t, ϑt : Ω0 × · · · × Ωt−1 → R is a universally measurable function4 , and u : Ω × R → R is a random utility function satisfying the following three conditions: (U1) u(ω, x) is increasing and concave in x, (U2) u : Ω × [−n, +∞) → R is uniformly continuous and bounded for all n ∈ N, (U3) limx→−∞ supω∈Ω u(ω, x)/|x| = −∞. Note P that if u does not depend on ω, (1.1) measures the utility of the discounted terminal wealth X + Tt=1 ϑt ∆St . On the other hand, if u is of the form u(ω, x) = u ˜(ωT1 x) for a deterministic P function u ˜, then (1.1) evaluates the undiscounted terminal wealth BT X + BT Tt=1 ϑt ∆St . By v we denote the convex conjugate of u, given by v(ω, y) := supx∈R (u(ω, x)−xy) for y ∈ R+ . By (U2), one has v(ω, y) = supx∈Q (u(ω, x) − xy), and as a consequence, v : Ω × R+ → R is Borelmeasurable. For q ∈ R+ and Borel probability measures Q and P on Ω, we define the v-divergence between qQ and P by P E v qdQ/dP if qQ ≪ P Dv (qQ, P) := +∞ otherwise, Recall that the universal completion F ∗ of a σ-algebra F is defined as the intersection of σ(F ∪ N P ), where P ranges over all probability measures on F and N P denotes the collection of all P-null sets. By saying that X : M → R is universally measurable, we mean that it is measurable with respect to the universal completion F ∗ of F and the Borel σ-algebra on R. This is equivalent to saying that X is measurable with respect to F ∗ and the universal completion of the Borel σ-algebra on R. 4 We call a function X : Ω → R on a subset Ω ⊆ Rd universally measurable if it is measurable with respect to the universal completion of the Borel σ-algebra on Ω and the Borel σ-algebra on R (or equivalently, the universal completion of the Borel σ-algebra on R). 3

2

and the v-divergence between qQ and P by Dv (qQ, P) := inf Dv (qQ, P). P∈P

We sayP a Borel probability measureP P on Ω admits arbitrage if there exists a strategy ϑ ∈ Θ such that P[ Tt=1 ϑt ∆St > 0] > 0 and P[ Tt=1 ϑt ∆St ≥ 0] = 1. P We suppose there exists a continuous function Z : Ω → R+ such that Z ≥ 1 ∨ Tt=0 |St | and all sublevel sets {ω ∈ Ω : Z(ω) ≤ z}, z ∈ R+ , are compact. Let BZ be the space of all Borelmeasurable functions X : Ω → R such that X/Z is bounded. By MZ we denote the set of all Borel probability measures P on Ω satisfying EP Z < +∞ and by QZ the subset of measures P ∈ MZ under which S is a martingale. Then EP X is well-defined for all P ∈ MZ and X ∈ BZ . The following is our main result: Theorem 1.1. Assume a medial limit exists and u satisfies (U1)–(U3). If (P1) every P ∈ P is dominated by a P∗ ∈ P that does not admit arbitrage, then the supremum in (1.1) is attained for every Borel-measurable function X : Ω → R such that U (X) ∈ R. If in addition, (P2) P is a σ(MZ , CZ )-closed convex subset of MZ , and (P3) there exists an increasing function w : R+ → R+ such that limx→+∞ w(x)/x = +∞ and inf P∈P EP u(−w(Z)) > −∞, then U (X) ∈ R

and

U (X) =

inf

(q,Q)∈R+ ×QZ

qEQ X + Dv (qQ, P)

for all X ∈ BZ .

(1.2)

In the special case, where u is of the form u(x) = − exp(−λx) for a risk-aversion parameter λ > 0, the dual expression in (1.2) simplifies if instead of (1.1), one considers the equivalent problem ! T X 1 ϑt ∆St . W (X) = sup inf − log EP exp −λX − λ λ ϑ∈Θ P∈P t=1

Corollary 1.2. Assume a medial limit exists and (P1)–(P3) hold for u(x) = − exp(−λx). Then 1 Q W (X) ∈ R and W (X) = inf E X + H(Q, P) for all X ∈ BZ , Q∈QZ λ where H(Q, P) := inf P∈P H(Q, P) is the robust version of the relative entropy Q E log(dQ/dP) if Q ≪ P H(Q, P) := +∞ otherwise. In the following we give a typical example of a family of probability measures that is not dominated or time-consistent but still satisfies our assumptions. 3

2(T +1)

be of the form Ω = Ω0 × · · · × ΩT , where Ω0 = {(b0 , s0 )} for Example 1.3. Let Ω ⊆ R+ b0 , s0 ∈ (0, +∞) and Ωt = [at , bt ] × (0, +∞) for 0 < at ≤ bt and t = 1, . . . , T . Let M ⊆ R be a finite set containing at least one strictly positive and one strictly negative element. For every m ∈ M and t = 1, . . . , T , fix Etm ∈ R+ , and consider the set P := P : EP [(Bt St )m ] ≤ Etm for all t = 1, . . . , T and m ∈ M .

Then, it is easy to see that P satisfies condition (P2) for every continuous function Z : Ω → [1, +∞). Moreover, if u : Ω × R → R is bounded from above, increasing as well as concave in the second argument and satisfies u(x) ≥ −c|x|p for all x ≤ −1, where c and p are positive constants such ∗ that 0 0, φ∗CV (µ) ≥ hyXn , µi − φ(yXn ), and therefore, hXn , µi ≤

φ(yXn ) φ∗CV (µ) + . y y

It follows that hXn , µi ↓ 0. Hence, by the Daniell–Stone theorem, µ is in ca+ V . This shows that ˜ . In particular, Λ is equal to Λ and therefore, σ(ca+ φ∗CV (µ) = +∞ for all µ ∈ CV∗,+ \ ca+ a a V , CV )V compact. Now we show that if φ satisfies (R2), the dual representation extends to UV . To do that, we use that on a perfectly normal space, every upper semicontinuous function is the pointwise limit of a decreasing sequence of continuous functions; see [27]. This implies that for a given X ∈ UV , there exists a sequence (Xn ) in CV such that Xn ↓ X. It follows from (R2) and the definition of φ∗CV that φ(X) = lim φ(Xn ) ≥ lim hXn , µi − φ∗CV (µ) ≥ hX, µi − φ∗CV (µ) for all µ ∈ ca+ V. n

n

On the other hand, by (2.2), one has φ(X) ≤ φ(Xn ) = max (hXn , µi − φ∗CV (µ)) for every n. µ∈ca+ V

7

(2.7)

Since hXn , µi − φ∗CV (µ) ≤ hX1 , µi − φ∗CV (µ) ≤ kX1 kV kµkCV∗ − φ∗CV (µ) ≤ kX1 kV ϕ−1 (φ∗CV (µ)) − φ∗CV (µ), this implies that there exists an a ∈ R such that φ(Xn ) = max (hXn , µi − φ∗CV (µ)) for all n, µ∈Λa

Note that hXn , µi − φ∗CV (µ) is decreasing in n as well as upper semicontinuous and concave in µ. So it follows from the minimax result, Theorem 2 of [10], and the monotone convergence theorem that φ(X) = inf φ(Xn ) = inf max (hXn , µi − φ∗CV (µ)) n

= max inf (hXn , µi − µ∈Λa n

n µ∈Λa ∗ φCV (µ)) =

max (hX, µi − φ∗CV (µ)),

µ∈Λa

which together with (2.7), proves (2.3). The last part of Theorem 2.2 follows from Proposition 2.1. Indeed, if (R2)–(R3) hold, fix a constant r > 0 and let G be the set of X ∈ BV satisfying |X| ≤ r|V |. Then, φ, G and H = CV ∩ G satisfy the assumptions of Proposition 2.1, and one has Hδ = UV ∩ G. Hence, Proposition 2.1 and (2.3) yield φ(X) =

sup

φ(Y ) =

Y ≤X, Y ∈UV ∩G

sup

max (hY, µi − φ∗CV (µ))

Y ≤X, Y ∈UV ∩G µ∈ca+ V

for all X ∈ G ∩ S(H).

Since for fixed µ ∈ ca+ V , the mapping X 7→ hX, µi together with G and H also satisfies the assumptions of Proposition 2.1, one has φ(X) = sup µ∈ca+ V

sup Y ≤X, Y ∈UV ∩G

(hY, µi − φ∗CV (µ)) = sup (hX, µi − φ∗CV (µ)) for all X ∈ G ∩ S(H). µ∈ca+ V

So, if we can show that G ⊆ S(H), the representation (2.4) holds for all X ∈ BV since r was arbitrary. To prove G ⊆ S(H), we note that a function X ∈ G can be written as X = sup(qV 1{X≥qV } − rV 1{X 0, then since lim medn ϑt ∆St = +∞ on At ∩ {∆St < 0}, one must have Q(∆St < + 0|A+ t ) = 0. But by the martingale property, this implies Q(∆St = 0|At ) = 1, and therefore, − n ϑnt ∆St = 0 Q-almost surely on A+ t . The same argument applied to At gives ϑt ∆St = 0 Q-almost − n ∆St | < +∞ Q-almost surely, which implies (3.2). As surely on At . It follows that n |ϑt P PTlim med n a result, one has lim medn t=1 ϑt ∆St = Tt=1 ϑ∗t ∆St P-almost surely for all P ∈ P. Since u is increasing, concave and bounded from above, an application of (iii) and (v) of Lemma 3.1 to −u

12

gives P

U (X) ≥ inf E u X + P∈P

≥

inf lim med EP u Xn +

P∈P

n

T X

ϑ∗t ∆St

t=1 T X

!

ϑnt ∆St

t=1

!

≥ inf E P∈P

P

"

lim med u Xn + n

≥ lim med inf EP u Xn + n

P∈P

T X

ϑnt ∆St

t=1

T X

ϑnt ∆St

t=1

!#

!

= inf U (Xn ). n

P This shows that U (Xn ) ↓ U (X) and U (X) = inf P∈P EP u X + Tt=1 ϑ∗t ∆St .

Lemma 3.4. Assume u fulfills (U2)–(U3) and P has the property (P1). Then the functional D(X) := inf qEQ X + Dv (qQ, P) , X ∈ BZ , (q,Q)∈R+ ×QZ

satisfies U (X) ≤ D(X) < +∞ for all X ∈ BZ . Proof. By (P1), there exists a P ∈ P that does not admit arbitrage. It follows as in footnote 6 that there exists a Q ∈ QZ with bounded density dQ/dP. By Lemma 3.2, one has EQ X+EPv(dQ/dP) < +∞ for all X ∈ BZ . This shows that D(X) < +∞ forall X ∈ BZ . Nowchoose X ∈ BZ , P ∈ P, P ϑ ∈ Θ, Q ∈ QZ and q ∈ R+ such that qQ ≪ P. If EP u X + Tt=1 ϑt ∆St > −∞, it follows as in P P the proof of Lemma 3.3 that ts=1 ϑs ∆Ss is a Q-martingale, and therefore, EQ Tt=1 ϑt ∆St = 0. Moreover, it is immediate from the definition of v that u(ω, x) ≤ xy + v(ω, y) for all ω, x and y. P So, setting x = X + Tt=1 ϑt ∆St , y = qdQ/dP and taking expectation under P gives P

E u X+

T X t=1

ϑt ∆St

!

≤ qE

Q

"

X+

T X

#

dQ dQ = qEQ X + EP v q . ϑt ∆St + EP v q dP dP t=1

Finally, first taking the infimum over all P ∈ P, Q ∈ QZ , q ∈ R+ such that qQ ≪ P and then the supremum over all ϑ ∈ Θ gives U (X) ≤ D(X). In the next step we aim to show that under suitable assumptions, the inequality U (X) ≤ D(X) established in Lemma 3.4 is in fact, an equality. Note that under (U1)–(U3) and (P1)–(P3), U is an increasing concave mapping from BZ to R. Therefore, φ(X) = −U (−X) is an increasing convex function from BZ to R which satisfies any of the conditions (R1)–(R3) of Theorem 2.2 if and only if U has the corresponding of the following properties: (R1’) U (Xn ) ↑ U (0) for every sequence (Xn ) in CZ such that Xn ↑ 0, (R2’) U (Xn ) ↑ U (X) for every sequence (Xn ) in CZ such that Xn ↑ X ∈ LZ := −UZ (R3’) U (Xn ) ↓ U (X) for every sequence (Xn ) in BZ such that Xn ↓ X ∈ BZ .

13

Let us define UC∗ Z (qQ) := φ∗CZ (qQ) = sup (U (X) − qEQ X), X∈CZ

for q ∈ R+ and Q ∈ MZ .

˜ be the set of all strategies ϑ ∈ Θ such that ϑt is continuous and bounded for all Moreover, let Θ t = 1, . . . , T , and denote ! T X ˜ (X) := sup inf EP u X + X ∈ BZ . ϑt ∆St , U ˜ P∈P ϑ∈Θ

t=1

Then the following holds:

Lemma 3.5. Assume a medial limit exists, u fulfills (U1)–(U3) and P satisfies (P1)–(P3). Then ˜ is an increasing concave mapping from BZ to R with the property (R2’), and for all q ∈ R+ U and Q ∈ MZ , one has Dv (qQ, P) if q = 0 or Q ∈ QZ ∗ ˜ UCZ (qQ) = (3.4) +∞ otherwise. ˜ it follows from (P3) that U ˜ is an increasing concave Proof. Since ϑ ≡ 0 is a possible strategy in Θ, mapping from BZ to R . ˜ satisfies (R1’). To do that we first prove Next, we show that U sup inf EP u(Xn ) = inf EP u(X) n P∈P

(3.5)

P∈P

for every sequence (Xn ) in CZ such that Xn ↑ X ∈ CZ . But since inf P∈P EP u(.) is a realvalued concave functional on CZ , it is enough to show (3.5) for X = 0; see Proposition 1.1 in [6]. So assume X = 0 and fix an ε > 0. Note that by (U2), u is uniformly continuous on Ω × [−1, +∞). Therefore, there exists a δ > 0 such that u(ω, −δ) ≥ u(ω, 0) − ε for all ω. Moreover, there is an m ∈ R+ such that X1 ≥ −mZ, and by (P3), there exists an increasing function w : R+ → R+ such that limx→+∞ w(x)/x = +∞ and inf P∈P EP u(−w(Z)) > −∞. Since by (U3), supω u(ω, x)/|x| → −∞ for x → −∞ and by (U2), u(ω, −δ) is bounded in ω, it follows that there exists a z ∈ R+ such that inf EP u(−mZ)1{Z>z} ≥ −ε and inf EP u(−δ)1{Z≤z} ≥ inf EP [u(−δ)] − ε. P∈P

P∈P

P∈P

By Dini’s lemma, there is an n such that Xn ≥ −δ on {Z ≤ z}. So inf EP u(Xn ) ≥ inf EP u(−δ)1{Z≤z} + inf EP u(−mZ)1{Z>z} P∈P

P∈P

≥

P∈P

P

inf E [u(−δ)] − 2ε ≥ inf EP [u(0)] − 3ε,

P∈P

P∈P

˜ and let (Xn ) be a sequence in CZ such which shows (3.5) for X = Now fix a strategy ϑ ∈ Θ P0. T that Xn ↑ 0. Then, since t=1 ϑt ∆St belongs to CZ , it follows from (3.5) that ! ! T T X X P P ϑt ∆St , ϑt ∆St = inf E u sup inf E u Xn + n P∈P

P∈P

t=1

14

t=1

P T ˜ (0) = sup ˜ inf P∈P EP u ˜ which, since U ϑ ∆S t , implies that U satisfies (R1’). t=1 t ϑ∈Θ To show (3.4), we choose q ∈ R+ , Q ∈ MZ and note that ! ! T X ˜C∗ (qQ) = sup sup inf EP u X + ϑt ∆St − qEQ X U Z

˜ P∈P X∈CZ ϑ∈Θ

= sup sup inf

˜ P∈P X∈CZ ϑ∈Θ

t=1

P

Q

E u(X) − qE X + qE

Q

T X t=1

!

ϑt ∆St .

P ˜ if and only if S is a Q-martingale, one has U ˜ ∗ (qQ) = +∞ Since EQ Tt=1 ϑt ∆St = 0 for all ϑ ∈ Θ CZ for q > 0 and Q ∈ MZ \ QZ . On the other hand, if q = 0 or Q ∈ QZ , then ˜C∗ (qQ) = sup inf EP u(X) − qEQ X . U Z X∈CZ P∈P

By (P2), P is a σ(MZ , CZ )-closed convex subset of MZ , and by (P3), there exists an increasing function w : R+ → R+ such that limx→+∞ w(x)/x = +∞ and inf P∈P EP u(−w(Z)) > −∞. So, since by (U3), supω u(ω, x)/|x| → −∞ for x → −∞, one has (3.6) lim inf EP u(−mZ)1{Z>z} = 0 for all m ∈ R+ . z→+∞ P∈P

This shows that {ZdP : P ∈ P} is a tight family of measures, which, by Prokhorov’s theorem, implies that P is σ(MZ , CZ )-compact. In addition, it can be seen from (3.6) that EP u(X) is σ(MZ , CZ )-continuous in P ∈ P for all X ∈ CZ . Therefore, it follows from the minimax result, Theorem 2 of [10], that ˜C∗ (qQ) = inf sup EP u(X) − qEQ X . U Z P∈P X∈CZ

If qQ is not absolutely continuous with respect to P, then supX∈CZ EP u(X) − qEQ X = +∞. Otherwise, since u satisfies (U2), one obtains from the monotone convergence theorem, " # dQ dQ P P sup u(x) − xq = sup E = sup EP u(X) − qEQ X , E v q dP dP n X∈Cb x∈[−n,n] where Cb is the set of all bounded continuous functions on Ω, and for the last equality we used a standard selection argument8 and the fact that every universally measurable function can be approximated in measure by continuous functions. Hence ˜C∗ (qQ) ≥ inf Dv (qQ, P) = Dv (qQ, P) U Z P∈P

8

The function u(ω, x) − xqdQ/dP(ω) is Borel measurable in (ω, x). By Proposition 7.50 of [4], there exists a universally measurable mapping X : Ω → [−n, n] such that u(ω, X(ω)) − X(ω)qdQ/dP(ω) = maxx∈[−n,n] u(ω, x) − xqdQ/dP(ω) for all ω.

15

if q = 0 or Q ∈ QZ . Moreover, it follows from the definition of v that EP v(qdQ/dP) ≥ EP u(X) − qEQ X for all q ∈ R+ , Q ∈ QZ and X ∈ LZ . Hence, ! T X ˜L∗ (qQ) := sup U (X) − qEQ X = sup sup inf u(X) − qEQ X + qEQ ϑt ∆St U Z

˜ P∈P X∈LZ ϑ∈Θ

X∈LZ

≤

t=1

inf sup EP u(X) − qEQ X ≤ Dv (qQ, P).

P∈P X∈LZ

˜ ∗ (qQ) = Dv (qQ, P). This shows ˜ ∗ (qQ) = U ˜ ∗ (qQ), one obtains U ˜ ∗ (qQ) ≤ U Since, clearly, U LZ CZ LZ CZ ∗ ∗ ˜ ˜ ˜ ˜ = −U(−X). So Proposition (3.4) as well as φCZ = φUZ for the increasing convex functional φ(X) ˜ satisfies (R2’). 2.3 yields that U We now are ready for the proof of Theorem 1.1. Proof of Theorem 1.1. Assume a medial limit exists, u satisfies (U1)–(U3) and P has the property (P1). Then an application of Lemma 3.3 with Xn = X yields that the supremum in (1.1) is attained for every Borel measurable function X : Ω → R satisfying U (X) ∈ R. ˜ is an increasing concave If in addition, P fulfills (P2)–(P3), we know from Lemma 3.5 that U mapping from BZ to R satisfying (R2’) as well as Dv (qQ, P) if q = 0 or Q ∈ QZ ∗ ˜ UCZ (qQ) = +∞ otherwise. So one obtains from Theorem 2.2 that ˜ (X) = ˜ ∗ (qQ) = U inf qEQ X + U CZ (q,Q)∈R+ ×MZ

inf

(q,Q)∈R+ ×QZ

qEQ X + Dv (qQ, P) = D(X)

˜ ≤ U ≤ D on BZ , U is an increasing concave mapping for all X ∈ LZ . Since, by Lemma 3.4, U ˜ ∗ . Moreover, we know from Lemma 3.3 from BZ to R satisfying (R1’)–(R2’) as well as UC∗ Z = U CZ that U fulfills (R3’). Hence, it follows from Theorem 2.2 that U (X) = inf qEQ X + UC∗ Z (qQ) = inf qEQ X + Dv (qQ, P) (q,Q)∈R+ ×MZ

(q,Q)∈R+ ×QZ

for all X ∈ BZ .

3.3

Proof of Corollary 1.2

Note that

1 W (X) = − log(−U (X)), λ

where P

U (X) = sup inf E u X + ϑ∈Θ P∈P

T X

ϑt ∆St

t=1

16

!

for u(x) = − exp(−λx).

By Theorem 1.1 , the supremum is attained for all X ∈ BZ . Therefore, U (X) ∈ (−∞, 0), and hence, W (X) ∈ R for all X ∈ BZ . Moreover, a direct computation yields v(ω, y) = λy log λy − 1 . Therefore, q q q log − 1 , Dv (qQ, P) = H(Q, P) + λ λ λ and so by Theorem 1.1, 1 1 W (X) = − log(−U (X)) = − log − inf qEQ X + Dv (qQ, P) λ λ (q,Q)∈R+ ×QZ q q 1 1 log − 1 . = − log − inf q inf (EQ X + H(Q, P)) + q∈R+ Q∈QZ λ λ λ λ Solving for the minimizing q gives W (X) = inf Q∈QZ (EQ X + λ1 H(Q, P)).

A

Example 1.3 satisfies the no-arbitrage condition (P1)

For notational simplicity, assume that T = 2 and at = bt = 1 for t = 1, 2. Then Ω can be identified with (0, +∞) × (0, +∞). The general case follows from the same arguments by induction. m So assume sm 0 < Et for t = 1, 2 and all m ∈ M . A given P ∈ P can be disintegrated as P = P1 ⊗ K, where P1 is the first marginal distribution (corresponding to the distribution of S1 ) and K is a transition probability kernel (corresponding to the conditional distribution of S2 given S1 ). For every ε ∈ (0, s0 ), denote by Pε1 and K ε the measure and kernel given by Pε1 :=

1 1 (δs0 −ε + δs0 +ε ) and K ε (x) := (δx/(1+ε) + δx(1+ε) ). 2 2

Note that Pε := (εP1 + (1 − ε)Pε1 ) ⊗ (εK + (1 − ε)K ε ) dominates P and does not admit arbitrage. It remains to show that Pε ∈ P for some ε > 0. Since EP S1m < +∞, one has (s0 − ε)m + (s0 + ε)m → sm for ε → 0. 0 2 m If sm 0 < E1 , this shows that the moment conditions for S1 are satisfied for all ε > 0 small enough. Moreover, by definition of Pε , one has ε

EP S1m = εEP S1m + (1 − ε)

ε

ε

ε

ε

ε

EP S2m = ε2 EP S2m + ε(1 − ε)EP1 ⊗K S2m + ε(1 − ε)EP1 ⊗K S2m + (1 − ε)2 EP1 ⊗K S2m . ε

ε

m The term EP1 ⊗K S2m clearly converges to sm 0 < E2 for ε → 0. Therefore it just remains to show that all other expectation terms are bounded in ε. The first expectation EP S2m is independent of ε and finite since P belongs to P. The second expectation satisfies

EP1 S1m /(1 + ε)m + EP1 S1m (1 + ε)m → EP1 S1m ≤ Etm . 2 Finally, note that one can change K on a P1 -zero set and still have P = P1 ⊗ K. Therefore, one εn can assume that K(s0 ± εn ) = δs0 for a sequence εn converging to 0. Then EP1 ⊗K S2m = sm 0 . So S2 also satisfies the moment conditions and the proof is complete. ε

EP1⊗K S2m =

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