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On Admissible Strategies in Robust Utility Maximization

The existence of optimal strategy in robust utility maximization is addressed when the utility function is finite on the entire real line. A delicate problem in this case is to find a"good definition"of admissible strategies, so that an optimizer is obtained. Under suitable assumptions, especially a time-consistency property of the set of probabilities which describes the model uncertainty, we show that an optimal strategy is obtained in the class of strategies whose wealths are supermartingales under all local martingale measures having a finite generalized entropy with at least one of candidate models (probabilities).


INTRODUCTION
This paper analyzes a qualitative aspect of the problem of robust utility maximization. Given a utility function U and a set P of probabilities which describes the model uncertainty, the basic problem of this paper is to maximize the robust utility functional exponential utility. [21] shows that a "good definition" which yields us an optimal strategy is that θ · S is a supermartingale under all local martingale measures Q which has a "finite entropy" with the physical probability P. We denote the class of such θ by Θ V (P) (see Section 2 for precise definitions including the meaning of "finite entropy"). Note that this class contains the usual admissible class, and the supermartingale property is consistent to the "No-Arbitrage philosophy". Thus Θ V (P) is acceptably natural choice when a single physical probability is specified.
In the general robust case with P containing (infinitely) many elements, [8] (see also [17] for a slight generalization) provides a partial analogue of the above result which states that, under certain stronger assumptions, an optimal strategy is obtained in the class of θ with θ · S being a supermartingale under all local martingale measures Q having a finite entropy w.r.t. a certain elementP ∈ P called a least favorable measure, i.e., in the class Θ V (P). Here a dissatisfaction comes of course from the dependence of admissibility on P. In philosophy, P is the set of candidates of real world models, and we do not know which one is true. Thus an "admissible strategy" should be universally admissible for all candidates P ∈ P. Also, the least favorable probabilityP is a part of solution to the dual problem of robust utility maximization, hence the class Θ V (P) is not a priori available.
In this view, a seemingly natural admissible class is P∈P Θ V (P) which is universal and contains all θ whose stochastic integrals are bounded below. Thus our central question in this paper is: Question 1. Does the class P∈P Θ V (P) admit an optimal strategy?
The main result (Theorem 3.2) states that this is indeed the case if (in addition to standard assumptions) the set P of candidate models has a time-consistency property. We proceed as follows. The first step is to construct a so-called "optimal claim" for the abstract version of robust utility maximization, from which a candidate of optimal strategyθ is derived through a predictable representation argument. This part is mostly standard excepting some technicality, but we give a slightly better description of optimal claim. Note that the additional time-consistency assumption is not required at this stage. The crucial step is to verify the supermartingale property ofθ · S under all local martingale measures Q which has a finite entropy with some P ∈ P but its entropy withP is infinite. We shall do this by a (slight surprisingly) simple trick.

FORMULATION
We fix a complete probability space (Ω, F, P) as well as a filtration F = (F t ) t∈[0,T ] satisfying the usual conditions, where T ∈ (0, ∞) is a fixed time horizon. Though many probabilities on (Ω, F) will appear in the sequel, the probability P plays the role of reference probability, i.e., every probabilistic notion is defined under P unless other probability is explicitly specified as E P [·], L 1 (P) etc. In particular, the underlying asset prices S is a d-dimensional P-càdlàg semimartingale, and we assume: S is P-locally bounded.
Let P be a set of probabilities P P, which we can (and do) embed into L 1 via the mapping P → dP/dP. In this sense, we assume: P is convex and σ (L 1 , L ∞ )-compact.
We work with a utility function U : R → R which we assume (A3) U is differentiable, strictly concave on R, and U (−∞) = ∞, U (∞) = 0, and satisfies the condition of reasonable asymptotic elasticity: The conjugate of utility function U is denoted by V , i.e., The assumptions (A3) and (A4) guarantee that V is a "nice" convex function (see [10], [19] for details). Using this function, we introduce a generalized entropy: +∞ otherwise for any positive finite measure ν P and P ∈ P. When U(x) = 1 − e −x (exponential utility) and Q is a probability with Q P, we have V (Q|P) = E Q [log(dQ/dP)], i.e., the relative entropy. Abusing the terminology, we still call the map V (·|·) the generalized entropy associated to V . We define also the robust generalized entropy by Let M loc be the set of all local martingale measures for S, i.e., probabilities Q P under which S is a local martingale. We then set Generically, for any set Q of probabilities Q P, we denote by Q e the set of Q ∈ Q with Q ∼ P. We assume the existence of equivalent local martingale measure with finite entropy in the following sense: In particular, this implies the existence of (Q, P) ∈ M e V × P such that Q ∼ P ∼ P and V (Q|P) < ∞. See [17] for detail and other consequences of these assumptions.
Let L(S) be the totality of all (S, P)-integrable d-dimensional predictable processes, L 0 (S) := {θ ∈ L(S) : θ 0 = 0}, and we denote by θ · S the stochastic integral of θ ∈ L(S) w.r.t. S. See e.g., [12] or [13] for more information. When the utility function is finite on the entire real line, a conceptually natural choice of Θ is Then the value function of the robust utility maximization problem is given by When we seek an optimal strategy, however, the class Θ bb is typically too small to admit an optimal strategy. We thus have to enlarge the admissible class. Our choice is the following. When a single P ∈ P is fixed as the physical probability, the class Θ V (P) is shown to be an appropriate domain of utility maximization in [21]. Recalling (2.3), our choices M V and Θ V are rewritten respectively as Thus our definition (2.7) is consistent to what we wrote in introduction.
Under the assumptions (A1) -(A5), a duality result (Theorem 2.3 of [18]) is applicable, which states in our case that for any Θ with Θ bb ⊂ Θ ⊂ Θ V , we have In particular, the value function is unchanged if we replace Θ bb by the larger class Θ V . Under the same assumptions, the right hand side, the dual problem of the (2.6), admits a solution (λ ,Q) ∈ (0, ∞) × M V , and the infimum V (λQ|P) = inf P∈P V (λQ|P) is attained by aP ∈ P since P is weakly compact, and V (·|·) is lower semicontinuous. Thus the right hand side of (2.10) is also written as V (λQ|P), and we call the triplet (λ ,Q,P) a dual optimizer.
A way of proving (2.10) and the existence of a solution (λ ,Q) is to closely analyze the robust utility functional X → inf P∈P E P [U(X)] on L ∞ characterizing V (·|P) as its conjugate. Then the duality and the existence of (λ ,Q) follow simultaneously from Fenchel's duality theorem. See [18] for detail. Alternatively, one can separate the dual problem into the minimization of λ → inf Q∈M V V (λ Q|P) + λ x and of Q → V (λ Q|P) for each λ . For the latter problem, called the robust f -projection, [8] proves the existence by establishing a uniform integrability criterion in terms of V (·|P) in the spirit of the de la Vallée-Poussin theorem.
In contrast to the standard utility maximization, neither the uniqueness of (λ ,Q) (hence of the triplet (λ ,Q,P)) nor the equivalenceQ ∼ P hold in the robust case, as the following trivial example illustrates: Suppose M e loc = ∅, and that M loc contains an element Q 0 which is not equivalent to P. Then we take P so that Q 0 ∈ P ⊂ M loc . In this case,λ is uniquely determined as the minimizer of λ → V (λ ) + λ x. Then a triplet (λ , Q, P) is a dual optimizer if (and only if) P = Q ∈ P ⊂ M loc . Indeed, by Jensen's inequality and the strict convexity of V , V (λ Q|P) = E P [V (λ dQ/dP)] ≥ V (λ ) whenever Q P, and the "equality" holds if and only if Q = P. Hence (λ ,Q,P) is not unique, and (λ , Q 0 , Q 0 ) is a solution with Q 0 ∼ P.
As for the equivalence, we still haveQ ∼P whenever (λ ,Q,P) is a dual optimizer (see [17], Theorem 2.7). Also, by an exhaustion argument, there exists a maximal solution (λ ,Q,P) in the sense that if (λ , Q, P) is another dual optimizer, then P P (hence Q Q ) and λ dQ/dP =λ dQ/dP, P-a.s., where the density dQ/dP is defined P-a.s. in the sense of Lebesgue decomposition. In particular, if (λ ,Q,P) and (λ ,Q,P) are two maximal solution, then (2.11)λ dQ/dP =λ dQ/dP, P-a.s., See [17, Theorem 2.5 and Proposition 4.7]. This uniqueness is still useful in our purpose. Note finally that even such a maximalQ may fail to be equivalent to the reference probability P. See [23, Example 2.5] for a counter example. In the sequel, we fix such a maximal dual optimizer, and callP a least favorable measure.
The duality (2.10) completely characterizes the quantitative nature of the problem (2.6). But our aim in this paper is to discuss the qualitative nature, especially the existence of optimal strategy in Θ V . To do this, assumptions (A1) -(A5) are not enough, and we assume additionally Remark 2.3. Several remarks on assumption (A6) are in order.
1. This assumption is automatically satisfied if U(∞) := sup x U(x) < ∞ as exponential utility, and in this case, U(X) + ∈ P∈P L 1 (P) for any random variable X. Therefore, the robust utility functional This is further equivalent to saying that v P (y) < ∞ for all y > 0 and P ∈ P e , where v P is the dual value function 3. We could state (A6) with the whole P rather than P e . But for our purpose, (A6) is enough. Recall that (A5) implies in particular P e = ∅. IfP ∈ P e , we have (P +P)/2 ∈ P e for all P ∈ P, and for all P ∈ P e , then the same is true for all P ∈ P.
In particular, (A6) (hence (2.12)) guarantees even in the case U(∞) = ∞ that In fact, if V (Q|P) < ∞ and X ∈ L 1 (Q), Young's inequality implies U(X) ≤ V (dQ/dP)+ (dQ/dP)X ∈ L 1 (P), and we can take such a Q ∈ M V by (2.12) for all P ∈ P e . Remark 2.4 (Continuation of Remark 2.1). We give a brief comparison of admissible classes considered in literature. In [17], the class Θ V (P) is used to discuss the existence of optimal strategy, while [8] considered (implicitly) a slightly smaller class: .
Thus Θ V (P) and Θ 0 V (Q,P) are essentially equivalent for the existence of optimal strategy (see Theorem 3.2). We just emphasize here that our class Θ V depends neither on particular P ∈ P nor Q ∈ M V , while Θ V (P) and Θ 0 V (Q,P) do.
We conclude this section by recalling a stability property of a set of probability measures, called m-stability, which will be used in Theorem 3.2 below. Definition 2.5 ([5], Definition 1). A set Q of probability measures is said to be mstable (multiplicatively stable) if for any Q ∈ Q, Q ∈ Q e with the density processes Z t = (dQ/dP)| Ft and Z t = (dQ /dP)| Ft , as well as any stopping time τ ≤ T , a new prob-abilityQ defined by dQ/dP : This property is equivalent to the time-consistency of the corresponding dynamic coherent monetary utility function φ τ (X) . This is further equivalent (under (A3)) to the time-consistency of the dynamic robust utility functional U τ (X) [5,Theorem 12] for details and precise formulation. Note that the set M loc of all local martingale measures is m-stable.

MAIN RESULTS
We first state a result on a "weak solution" to the problem (2.6), which yields a candidate of optimal strategy. Let Theorem 3.1. Suppose (A1) -(A6), and let x ∈ R and (λ ,Q,P) be a maximal dual optimizer. Then there exists anX ∈ X such that U(x +X) ∈ P∈P L 1 (P) and where the infimum is attained byP. Moreover, there exists an (S,Q)-integrable predictable processθ withθ 0 = 0 such thatθ · S is aQ-martingale (not only local) and x +X = −V (λ dQ/dP) = x +θ · S T ,Q-a.s.
In particular,X isQ-a.s. unique, andθ is unique in the sense thatθ · S is unique up tô Q-indistinguishability.
The proof is given in Section 4. The first equality in (3.2) states that the robust utility maximization over terminal wealths x + θ · S T is (quantitatively) equivalent to the indirect utility maximization : while the random variableX is the so-called optimal contingent claim. Such arguments are quite standard in (non-robust) utility maximization, and also in the robust case, [8,Theorem 3.11] shows a similar result: under (A1) -(A5), the assertions of Theorem 3.1 hold true except that the sets M V (in the definition (3.1)) and P are replaced by M 0 V (Q,P) and P 0 (Q,P) defined respectively by Remark 2.4 and Note that our finite utility assumption (A6) is automatic if P is replaced by P 0 (Q,P). Also, when U(∞) < ∞, the set P 0 (Q,P) actually coincides with the whole set P ([8], Remark 3.10). However, M 0 V (Q,P) still depends on (Q,P) which is the solution to the dual problem, hence not a priori available. On the other hand, our formulation is universal, which is a slight, but qualitatively crucial contribution. Theorem 3.1 suggests that the "strategy"θ is a candidate of optimal strategy. However, we still have to prove that this strategy is indeed admissible.
Theorem 3.2. In addition to (A1) -(A6), we assume thatQ ∼ P and P is m-stable. Then θ is (S, P)-integrable (hence (S, P)-integrable for all P ∈ P), andθ · S is a supermartingale under all Q ∈ M V . In particular,θ belongs to Θ V and is an optimal strategy.
The proof is given in Section 5. When P = {P}, the question of uniform supermartingale property of this type goes back to the "six-author paper" [7] which shows that the optimal wealth in exponential utility maximization is a martingale under all local martingale measures having a finite relative entropy with P, under an additional assumption on reverse Hölder inequality which is later removed by [14]. Although this uniform martingale property is no longer true for other utility functions, [21] shows that the optimal wealth is a supermartingale under all Q ∈ M V (P), for any utility functions on R with reasonable asymptotic elasticity. There are also some extensions to the case where the semimartingale S is not locally bounded. See e.g. [3] and [4].
In the robust case, the Q-supermartingale property for all Q ∈ M V (P) (hence all Q ∈ M 0 V (Q,P) sinceθ · S is aQ-martingale) is shown by [8] (see also [17] for a slight extension). We emphasize that the difference between M V (P) and M V is essential here. Note thatX is also optimal for the utility maximization problem under the fixed measureP, and the same is true for (λ ,Q) in the dual side. Thus the result of [21] cited in the previous paragraph is still applicable (under the assumptionQ ∼ P) for Q with V (Q|P) < ∞, while we have to consider the case where V (Q|P) < ∞ for some P ∈ P but possibly V (Q|P) = ∞.
To grasp the situation, we try to describe the heuristics behind the argument in [21] (from our point of view), and our idea of extending it. In what follows in this section, we suppose all the assumptions of Theorem 3.2, especiallyQ ∼ P.
For a moment, we suppose thatθ · S is a Q-supermartingale for some Q ∈ M V . Then theQ-martingale property and the representation (3.3) imply: for any stopping time τ ≤ T , On the other hand, Ansel-Stricker's lemma [1] shows thatθ · S is a Q-supermartingale if and only if there exists a Q-martingale lower bound, i.e., a Q-martingale M Q such that θ · S ≥ M Q , Q-a.s. In particular, if (3.4) holds true for any stopping time τ ≤ T , the process defined by M Q τ = −E Q [V (λ dQ/dP)|F τ ] provides a desired lower bound, hence (3.4) is a necessary and sufficient condition forθ · S to be a Q-supermartingale.
When V (Q|P) < ∞, the inequality (3.4) is obtained as the variational inequality which characterizesQ as a minimizer of the functional Q → V (λ Q|P) when τ = 0, and a "Bellmantype" principle using the m-stability of the set of local martingale measures shows the case of general τ ≤ T .
If inf P∈P V (Q|P) < ∞ but V (Q|P) = ∞, this argument is no longer applicable at least directly. Mathematically speaking, we loose some important estimates to guarantee the necessary convergences, or more intuitively, any element Q with V (Q|P) = ∞ is in no way optimal at very early stage, and we can not draw further information from the optimality of Q in the minimization of Q → V (λ Q|P). However, we have used only a part of information ofQ so far, and it is natural to expect that a better information may improve the result. More specifically, Step 1 the optimality of (Q,P) in the minimization of (Q, P) → V (λ Q|P) should yield a variational inequality similar to (3.4) but with an additional term involving P: Step 2 Though we may not take P =P in general, it seems natural to expect that we may take P "arbitrarily close toP" keeping V (Q|P) < ∞ with fixed Q.
Step 3 If this is the case, we may expect (3.4) by an approximation argument: The formal inequality in Step 1 will be realized as Proposition 5.4 below, where the mstability of P will play an important role. On the other hand, Steps 2 and 3 will be justified in a certain sense by a simple trick which is a consequence of reasonable asymptotic elasticity (Lemma 5.5).

Remark 3.3 (What happens whenQ ∼ P?).
The equivalenceQ ∼ P is automatic if all elements of P are equivalent to P. When the filtration F is continuous (i.e., every (F, P)martingale is continuous, especially if it is generated by a Brownian motion), the latter condition is already implied by the m-stability of P and (A2) (see [5,Theorem 8]), thus it is not a further restriction in that case.
In general, however, the equivalenceQ ∼ P may fail (see [23, Example 2.5] for a counter example), thus it is worth asking what happens in that case. When U is finite only on the positive half-line, the optimal claimX (which does not require the assumptionQ ∼ P) is super-hedged by some (S, P)-integrable processθ withθ · S =θ · S,Q-a.s. By the monotonicity of robust utility functional, we see thatθ is an optimal strategy without the additional assumptionQ ∼ P (see [23] and [22]). However, this argument essentially relies on the fact thatX is bounded below by −x (since U(x) = −∞ for x < 0), and no longer works when the utility function is finite on the entire real line. Thus we can not drop the assumptionQ ∼ P (at now).

Remark 3.4 (Random Endowment
). The results of this paper may also be stated with a random endowment B as long as it is an F T -measurable random variable satisfying ∀P ∈ P, ∃ε P > 0 such that U(−ε P B + ) ∈ L 1 (P), Then the robust utility maximization problem (2.6) reads as With the same assumptions, the dual problem admits a maximal solution with the unique density in the sense of (2.11). Then Theorems 3.1 and 3.2 remain true with similar proofs, and with obvious modifications, e.g., (3.3) is replaced by x +X + B = −V (λ dQ/dP) = x +θ · S T + B,Q-a.s. We omit the details. See [18] for the treatment of random endowment and other implications of (3.5).

OPTIMAL CLAIM
We first note that we have only to consider the case x = 0. Indeed, assumptions (A3) and (A4) on the utility function are invariant under the translation of utility function from U to U x (ξ ) := U(x + ξ ), and all the results for x = 0 follow from those for x = 0 applied to the new utility function U x . Thus we assume x = 0 in what follows. The next technical lemma is a collection of several arguments in [4]. (c) If in addition k n converges a.s. to some k ∈ L 0 , we have k ∈ L 1 (Q), U(k) ∈ L 1 (P) and that Proof. We just fill the gap from [4]. As we are assuming the reasonable asymptotic elasticity (A4), assertions (a) and (b) are contained in Proposition 6.3 of [4]. The assertion (c) also appears (implicitly) in the proof of their Theorem 4.10, which we briefly recall here. Assume k n → k, P-a.s. Since (k n ) (resp. (U(k n )) is bounded in L 1 (Q) (resp. L 1 (P)), Fatou's lemma applied to the sequence (|k n |) n (resp. (|U(k n )|) n ) shows that k ∈ L 1 (Q) (resp. U(k) ∈ L 1 (P)). By Young's inequality, we have U(k n ) − λ (dQ/dP)k n ≤ V (λ dQ/dP) ∈ L 1 (P) for all n ∈ N and λ > 0, where the P-integrability of the right hand side for all λ follows from the reasonable asymptotic elasticity. By this integrable upper bound as well as the assumption E Q [k n ] ≤ 0, (reverse) Fatou's lemma shows that Letting λ ↓ 0, we have (4.1), while E Q [k] ≤ 0 follows by letting λ ↑ ∞.
By construction, eachk n is again the terminal value of a stochastic integralθ n · S T wherẽ θ n is the convex combination of (θ n , θ n+1 , · · · ) with the same convex weights ask n , hencẽ θ n ∈ Θ bb and E Q [k n ] ≤ 0 for each n and Q, in particular.
Since the robust utility functional X → inf P∈P E P [U(X)] is concave as a point-wise infimum of concave functionals, we have inf P∈P E P [U(k n )] ≥ inf P∈P E P [U(θ n · S T )] for each n. Hence we still have lim n inf P∈P E P [U(k n )] = u(0), and the sequence (E P [U(k n )]) n is bounded from below for all P ∈ P.
If Q ∈ M V , there is a P ∈ P with V (Q|P) < ∞ by the definition of M V , hence another application of Lemma 4.1 to the sequence (k n ) with the pair (Q, P) shows thatX ∈ L 1 (Q) and E Q [X] ≤ 0. HenceX ∈ X .
We next show that U(X) ∈ P∈P L 1 (P) and This is immediate from Fatou's lemma if U is bounded from above. When U(∞) = ∞ and P ∈ P e , we can take a Q ∈ M V with V (Q|P) < ∞ by (2.12), hence Lemma 4.1 shows (4.3) and that (U(k n )) n is bounded in L 1 (P). Then Remark 2.3 shows that U(X) ∈ L 1 (P) and (U(k n )) n is still bounded in L 1 (P) for arbitrary P ∈ P which need not be equivalent to P. To prove (4.3) in the case P ∼ P, we take (Q,P) as above, and set P α := αP + (1 − α)P for α ∈ (0, 1). Since P α ∼ P, the claim is true for P α for all α ∈ (0, 1), while we see Hence (4.3) holds for all P ∈ P. We now prove (3.2). Note first that for all λ > 0, X ∈ X , Q ∈ M V and P ∈ P, In particular, On the other hand, (4.3) shows This concludes the proof of (3.2). We proceed to (3.3). Notice that U(X) = V (λ dQ/dP) +λ (dQ/dP)X,P-a.s. (4.4) Indeed, "≤" is just a Young's inequality, while "≥" follows from Here (i) follows from the "≤" part, and (ii) fromX ∈ X . In particular,P attains the infimum in (3.2) and we obtain (4.4). But an elementary knowledge from convex analysis shows that this is possible only ifX = −V (λ dQ/dP),P-a.s.

UNIFORM SUPERMARTINGALE PROPERTY OF OPTIMAL WEALTH
We now proceed to the uniform supermartingale property of the optimal wealth, that is, we shall show thatθ · S is a supermartingale under all local martingale measures Q with finite entropy w.r.t. some P ∈ P. As outlined in Section 3, this will follow if we can prove the dynamic variational inequality (3.4) for every Q ∈ M V . Therefore, the key of this section is the next proposition which should be compared with [8,Lemma 3.12]. Recall that we have only to consider the case x = 0. In what follows, all the assumptions of Theorem 3.2 are in force, and we do not cite them in each statement.
Proposition 5.1. We have 1. for all Q ∈ M V , and for all stopping time τ ≤ T , 2. for all P ∈ P, and for all stopping time τ ≤ T , We introduce some notations. If L is a strictly positive martingale, we denote L τ,T := L T /L τ , for any stopping time τ ≤ T . Recall that any probability Q P is identified with a (uniformly integrable) martingale, namely the density process Z Q · = (dQ/dP)| F· . In what follows, we denote byẐ (resp.D) the density process ofQ (resp.P). Also, when a pair (Q, P) ∈ M loc × P is fixed, the density process of Q (resp. P) is denoted by Z (resp. D), and set: Z α τ,T := αZ τ,T + (1 − α)Ẑ τ,T , D α τ,T := αD τ,T + (1 − α)D τ,T , α ∈ [0, 1]. We make a couple of simple reductions. The first one is just a notational reduction. In our purpose, we can assume without loss of generality thatλ = 1 since we already knoŵ λ . Indeed, (λQ,P) minimizes (ν, P) → V (ν|P) if and only if (Q,P) minimizes (ν, P) → Vλ (ν|P) := 1 λ V (λ ν|P). Next, we have only to prove (5.1) and (5.2) for all Q ∈ M e V and P ∈ P e , respectively. Indeed, if we could show (5.1) for all Q ∈ M e V for instance, we havē Q := (Q +Q)/2 ∈ M e V for any Q ∈ M V on the one hand, and on the other hand, Bayes' formula implies where Φ = V (dQ/dP), hence (5.1). A similar argument applies also to (5.2). The first step is to show a "Bellman-type" principle for a time-consistent optimization. Note that the set M loc of all local martingale measures is m-stable, while M V is not. The next simple lemma allows us to avoid this difficulty.
is F τ -locally integrable i.e., there exists an increasing sequence A n ∈ F τ such that (see the proof of Lemma 5.5 below), and the second term is integrable, it suffices to prove the case α = 1.
Since V is bounded from below by U(0), we can choose the constant C a so that the right hand side is always positive. For the sequence A n , we take Noting that ϕ : Thus 1 An ϕ ∈ L 1 for each n. Finally, P(A n ) 1 sinceQ ∼P ∼ Q ∼ P ∼ P by assumption.
Proof. Note first that the conditional expectation of the right hand side is well-defined and a.s. finite by Lemma 5.2. Let C be the set on which the inequality (5.6) fails, which is F τ -measurable. Then we suppose by way of contradiction that P(C ) > 0.
Take a sequence (A n ) ⊂ F τ as in Lemma 5.2 and a large n so that P(C ∩A n ) > 0. Setting C := C ∩ A n , we define a new pair (Q,P) (Z,D) bȳ First, (Q,P) ∈ M loc × P by the m-stability of M loc and P. Also, sincē we have V (Q|P) < ∞ by the construction of C and Lemma 5.2, henceQ ∈ M V . Finally, This contradict to the optimality of (Q,P).

Now the formal inequality in
Step 1 at the end of Section 3 is realized as follows.
We proceed to Step 2. Fixing Q ∈ M V , we want to take P "arbitrarily close" toP. The next simple lemma gives a precise form of this argument.