Log-optimal portfolio without NFLVR: existence, complete characterization, and duality
aa r X i v : . [ q -f i n . M F ] J u l log-optimal portfolio without NFLVR: existence, completecharacterization, and duality ∗ Tahir Choulli and Sina Yansori
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada
July 18, 2018
Abstract
This paper addresses the log-optimal portfolio for a general semimartingale model. The mostadvanced literature on the topic elaborates existence and characterization of this portfolio underno-free-lunch-with-vanishing-risk assumption (NFLVR). There are many financial models violatingNFLVR, while admitting the log-optimal portfolio on the one hand. On the other hand, for fi-nancial markets under progressively enlargement of filtration, NFLVR remains completely an openissue, and hence the literature can be applied to these models. Herein, we provide a complete char-acterization of log-optimal portfolio and its associated optimal deflator, necessary and sufficientconditions for their existence, and we elaborate their duality as well without NFLVR.
Since the seminal papers of Merton [26, 27]), the theory of utility maximization and optimal portfoliohas been developed successfully in many directions and in different frameworks. These achievementscan be found in [20, 21, 9, 19], and the references therein to cite few. Besides the Markowitz’ portfolio,thanks to the nice properties of the logarithm utility, the log-optimal portfolio draw tremendous atten-tion since a while. This resulted in a large literature on the topic for different level of generalities. Themost advanced of this literature, under the assumption of no-free-lunch-with-vanishing-risk assump-tion (NFLVR herefater), provides explicit characterization for this optimal portfolio for the generalsemimartingale market models, see [8, 13, 14] and the references therein to cite few. However, thereare many financial models that violates NFLVR, while they might admit the log-optimal portfolio, see[6, 22, 24], and hence these assert somehow that NFLVR might be too strong for a financial marketmodel to be “acceptable and worthy”. For market models under progressive enlargement of filtration,which incorporate the two important settings of credit risk and life insurance, NFLVR remains anopen issue, and hence the existing literature is not applicable to these models on the one hand. Onthe other hand, for these latter market models, the no-unbounded-profit-with-bounded-risk receivedfull attention as it is the minimal no-arbitrage condition for a market model to be financial and quan-titatively “acceptable” and viable, see [1, 2] and the references therein. Furthermore, recently therehas been an interest to extend the existing results on the optimal portfolio and utility maximizationwithout NFLVR, see [5]. ∗ This research is supported by the Natural Sciences and Engineering Research Council of Canada, through GrantRES0020459
Throughout the paper, we consider a filtered probability space (Ω , F , H := ( H t ) t ≥ , P ) satisfying theusual conditions of right continuity and completeness. On this stochastic basis, we suppose givena d -dimensional semimartingale, X , that represents the discounted price process of d risky assets.Throughout the paper, the set M ( Q ) denotes the set of all martingales under Q , while A ( Q ) (re-spectively A + ( Q )) denotes the set of all optional processes with integrable variation (respectivelynondecreasing and integrable) under Q . When Q = P , we simply omit the probability for the sake ofsimple notations. For a semimartingale Y , by L ( Y ) we denote the set of predictable processes thatare X -integrable in the semimartingale sense. For ϕ ∈ L ( Y ), the resulting integral of ϕ with respectto X is denoted by ϕ · Y . For any local martingale M , we denote by L loc ( M ) the set H -predictableprocesses ϕ that are Y -integrable and the resulting integral ϕ · M is a local martingale. If C is the setof processes, then C loc is the set of processes, Y , for which there exists a sequence of stopping times,( T n ) n ≥ , that increases to infinity and Y T n belongs to C , for each n ≥
1. For any semimartinagle, L , wedenote by E ( L ) the Doleans-Dade (stochastic) exponential, it is the unique solution to the stochasticdifferential equation dY = Y − dL, X = 1 , and is given by E t ( L ) = exp( L t − h L c i t ) Y I { ∆ X u =0 } δ ( u, ∆ X u ) ( dt, dx ) . For a product-measurable functional W ≥ × R + × R d , we denote W ⋆ µ (or sometimes, withabuse of notation, W ( x ) ⋆ µ ) the process( W ⋆ µ ) t := Z t Z R d \{ } W ( u, x ) µ ( du, dx ) = X
The model ( X, H ) is called σ -special if there exists a real-valued and H -predictableprocess ϕ such that < ϕ ≤ and X ϕ | ∆ X | I {| ∆ X | > } ∈ A + loc ( H ) . (2.2)It is clear that (2.2) is equivalent to R ( | x | > | x | F ( dx ) < + ∞ P ⊗ A -a.e., or to ϕ · X being a specialsemimartingale (i.e. sup
Let Z be a process. Z is called a deflator for ( X, H ) if Z > and Z E ( ϕ · X ) is an H -supermartingale, for any ϕ ∈ L ( X, H ) such that ϕ ∆ X ≥ − .Throughout the paper, the set of all deflator for ( X, H ) will be denoted by D ( X, H ) . Theorem 3.2.
Suppose ( X, H ) is σ -special and quasi-left-continuous with predictable characteristics ( b, c, F, A ) . Then the following assertions are equivalent. (a) The set D log ( X, H ) , given by D log ( X, H ) := (cid:8) Z ∈ D ( X, H ) (cid:12)(cid:12) E [ − ln( Z T )] < + ∞ (cid:9) , (3.2)3 s not empty (i.e. D log ( X, H ) = ∅ ). (b) There exists an H -predictable process e ϕ ∈ L ( X, H ) such that, for any ϕ belonging to L ( X, H ) , thefollowing hold E (cid:20) e V T + 12 ( e ϕ tr c e ϕ · A ) T + ( Z ( − e ϕ tr x e ϕ tr x + ln(1 + e ϕ tr x )) F ( dx ) · A ) T (cid:21) < + ∞ , (3.3) e V := (cid:12)(cid:12)(cid:12) e ϕ tr ( b − c e ϕ ) + Z (cid:20) e ϕ tr x e ϕ tr x − e ϕ tr h ( x ) (cid:21) F ( dx ) (cid:12)(cid:12)(cid:12) · A, (3.4)( ϕ − e ϕ ) tr ( b − c e ϕ ) + Z (cid:18) ( ϕ − e ϕ ) tr x e ϕ tr x − ( ϕ − e ϕ ) tr h ( x ) (cid:19) F ( dx ) ≤ . (3.5)(c) There exists a unique e Z ∈ D ( X, H ) such that inf Z ∈D ( X, H ) E [ − ln( Z T )] = E [ − ln( e Z T )] < + ∞ . (3.6)(d) There exists a unique e θ ∈ Θ( X, H ) such that sup θ ∈ Θ( X, H ) E [ln(1 + ( θ · X ) T )] = E [ln(1 + ( e θ · X ) T )] < + ∞ . (3.7) Furthermore, when these assertions hold, the following hold. e ϕ ∈ L ( X c , H ) ∩ L ( X, H ) , p ((1 + e ϕ tr x ) − − ⋆ µ ∈ A + loc ( H ) , (3.8)1 e Z = E ( e ϕ · X ) , e Z := E ( K − e V ) , K := e ϕ · X c + − e ϕ tr x e ϕ tr x ⋆ ( µ − ν ) . (3.9) e ϕ = e θ (1 + ( e θ · X ) − ) − and e θ = e ϕ E − ( e ϕ · X) P ⊗ A − a . e .. (3.10)It is important to notice that, for any Z ∈ D ( X, H ), we always have E ln + ( Z T ) ≤ ln(2). Furthermore,one can easily prove that the following two assertions are equivalent:(a) Z ∈ D log ( X, H ) (i.e. − ln( Z T ) is integrable or equivalently (ln( Z T )) − is integrable),(b) {− ln( Z t ) , ≤ t ≤ T } , or equivalently { (ln( Z t )) − , ≤ t ≤ T } , is uniformly integrable submartin-gale.Besides this, for a positive local martingale Z , the condition E [ − ln( Z T )] < + ∞ does not guar-antee that this Z is a martingale, while it implies that K := Z − − · Z is a martingale satisfying E [sup ≤ t ≤ T | K t | ] < + ∞ instead, see Lemma A.1 for this latter fact. As a result of this discussion,we conclude that Theorem 3.2 extends deeply the existing literature on the log-optimal portfolio bydropping the no-free-lunch-with-vanishing-risk condition on the model. This assumption is really avital assumption for the analysis of [13]. This achievement is due to our approach that differs funda-mentally from that of [13], while it is inspired from the approach of [7] with a major difference. Thisdifference lies in dropping all assumptions on the model ( X, H ) considered in [7], which guarantee thatthe minimizer of a functional belongs to the interior of its effective domain. We recall our aforemen-tioned claim that the “ σ -special assumption” for ( X, H ) is purely technical and is not related at all tothe minimizer of this functional. In conclusion, our theorem establishes the duality, under basically noassumption, besides describing the optimal dual solution when it exists as explicit as possible. This,furthermore, proves that in general, this optimal deflator might not be a local martingale deflator. Remark 3.3.
It is clear that the process V is well defined. This is due to Z [ e ϕ tr x e ϕ tr x − e ϕ tr h ( x )] F ( dx ) = − Z ( | x |≤ ( e ϕ tr x ) e ϕ tr x F ( dx ) − Z ( | x | >
11 + e ϕ tr x F ( dx ) + F ( | x | > , hich is a well defined integral with values in [ −∞ , + ∞ ) .Similarly for the LHS term of (3.5), the integral term is well defined for any ϕ ∈ L ( X, H ) . Indeed, dueto Ω × [0 , + ∞ ) = ∪ n ≥ ( | ϕ | ≤ n ) , for any process ϕ ∈ L ( X, H ) , it is enough to prove that the integralterm is well defined for bounded ϕ ∈ L ( X, H ) . To this end, on the one hand, we write Z (cid:18) ϕ tr x e ϕ tr x − ϕ tr h ( x ) (cid:19) F ( dx ) = − Z ( | x |≤ ( ϕ tr x )( e ϕ tr x )1 + e ϕ tr x F ( dx ) − F ( | x | > Z ( | x | > ϕ tr x + 11 + e ϕ tr x F ( dx ) + Z ( | x | > e ϕ tr x e ϕ tr x F ( dx ) . On the other hand, since ϕ is X -integrable (as it is bounded), both processes I {| ∆ X |≤ } · [ K, ϕ · X ] and [ P I {| ∆ X | > } , K ] have locally integrable variations and their compensators are − Z ( | x |≤ ( ϕ tr x )( e ϕ tr x )1 + e ϕ tr x F ( dx ) · A and − Z ( | x | > e ϕ tr x e ϕ tr x F ( dx ) · A respectively. This proves that the integral is well defined with values in ( −∞ , + ∞ ] . Proof. of Theorem 3.2.
It is clear that (c)= ⇒ (a) is obvious, and hence the proof of the theorem reducesto proving (a)= ⇒ (b)= ⇒ (c), (b)= ⇒ (d), (d)= ⇒ (a), and as long as assertion (b) holds the properties in(3.8)-(3.9) hold also. Thus, the rest of this proof is divided into three steps. The first step proves thatassertion (b) implies both assertions (c) and (d) and (3.8)-(3.9). The second step deals with (d)= ⇒ (a), while the third step addresses (a)= ⇒ (b). Step 1.
Here, we assume that assertion (b) holds, and focus on proving assertions (c) and (d), and(3.8)-(3.9). Then due to (3.3) and(1 + y ) ln(1 + y ) − y ≥ − δ y y I {| y |≤ δ } + δ δ ) | y | I {| y | >δ } for any δ ∈ (0 ,
1) and any y ≥ −
1, we deduce that for δ ∈ (0 ,
1) the following e ϕ tr c e ϕ · A, Z R d \{ } (cid:18) e ϕ tr x e ϕ tr x (cid:19) I {| e ϕ tr x |≤ δ } F ( dx ) · A, and Z R d \{ } | e ϕ tr x | e ϕ tr x I {| e ϕ tr x | >δ } F ( dx ) · A are integrable processes, and hence e ϕ ∈ L ( X c , H ) and p ((1 + e ϕ tr x ) − − ⋆ µ ∈ A + loc due to LemmaA.2 (see Appendix A). Hence K , defined in (3.9), is a well defined local martingale satisfying ∆ K +1 =(1 + e ϕ tr ∆ X ) − >
0. Furthermore, thanks to Yor’s formula and the continuity of A , we conclude thatfor any bounded ϕ ∈ L ( X, H ), E ( ϕ · X ) e Z = E (cid:16) ϕ · X + [ ϕ · X, K ] + K − e V (cid:17) . It is easy to check that (3.4) and (3.5) imply that ϕ · X + [ ϕ · X, K ] is a special semimartingale andits compensator ( ϕ · X + [ ϕ · X, K ]) p, H is dominated by e V . This proves that the process ϕ · X + [ ϕ · , K ] + K − e V is a local supermartingale. As a consequence, E ( ϕ · X ) e Z is a positive supermatingale,and hence e Z ∈ D ( X, H ) on the one hand. On the other hand, due to Itˆo, we derive − ln( e Z ) = local martingale + e V + 12 e ϕ tr c e ϕ · A + (cid:20) − e ϕ tr x e ϕ tr x + ln(1 + e ϕ tr x ) (cid:21) ⋆ µ. By combining this with (3.3), we deduce that e Z ∈ D log ( X, H ). This proves that assertion (a) holds.Furthermore, since e Z is a positive supermartingale, e Z − is a positive semimartingale, and e Z − I {| e ϕ |≤ n } · ( e Z ) − = e ϕI {| e ϕ |≤ n } · X. Since the LHS term, of the above equality converges (in probability at any time t ∈ (0 , T ]), we deducethat e ϕ ∈ L ( X, H ) (i.e. it is X -integrable in the semimartingale sense), and ( e Z ) − = E ( e ϕ · X ) . Therefore,on the one hand, this ends the proof for the properties (3.8)-(3.9). On the other hand, we notice thatln( E ( e ϕ · X ) T ) = − ln( e Z T ) is an integrable random variable, and for any ϕ ∈ L ( X, H ) ∩ L ( X, H )satisfying the condition E ln − ( E ( ϕ · X ) T ) < + ∞ , we get E [ln( E ( ϕ · X ) T / E ( e ϕ · X ) T )] = E [ln( E ( ϕ · X ) T ) − E ln( E ( e ϕ · X ) T )] ≤ . Thus, assertion (d) and (3.10) follow, and the rest of this step proves assertion (c).Let Z ∈ D log ( X, H ), and by applying Theorem B.2, we deduce the existence of ( β, f, V ) such that ϕ tr xf ( x ) ≥ − [ f ( x ) − − ln( f ( x ))] + ln(1 + ϕ tr x ) , for any ϕ ∈ L ( X, H ) ,V (cid:23) (cid:18) ϕ tr b + ϕ tr cβ + Z (cid:0) ϕ tr xf ( x ) − ϕ tr h ( x ) (cid:1) F ( dx ) (cid:19) · A,E [ − ln( Z T )] ≥ E (cid:20) V T + 12 β tr cβ · A T + Z [ f ( x ) − − ln( f ( x ))] F ( dx ) · A T (cid:21) . Then by combining these properties (take ϕ = e ϕ ) with (3.3)-(3.4)-(3.5) and the fact that under (3.5)we have e ϕ tr ( b − c e ϕ ) + Z [ e ϕ tr x e ϕ tr x − e ϕ tr h ( x )] F ( dx ) ≥
0, we derive E [ − ln( e Z T )] = E (cid:20) e V T + 12 ( e ϕ tr c e ϕ · A ) T + ( − e ϕ tr x e ϕ tr x + ln(1 + ( e ϕ tr x )) ⋆ µ ) T (cid:21) = E (cid:20) e V T + (cid:18) e ϕ tr c e ϕ + Z ( − e ϕ tr x e ϕ tr x + ln(1 + ( e ϕ tr x )) F ( dx ) (cid:19) · A T (cid:21) = E (cid:20) ( e ϕ tr b − e ϕ tr c e ϕ ) · A T + Z (ln(1 + e ϕ tr x ) − e ϕ tr h ( x )) F ( dx ) · A T (cid:21) ≤ E (cid:20) ( e ϕ tr b − e ϕ tr c e ϕ ) · A T + Z ( e ϕ tr xf ( x ) − e ϕ tr h ( x )) F ( dx ) · A T (cid:21) ++ E (cid:20)(cid:18)Z [ f ( x ) − − ln( f ( x ))] F ( dx ) (cid:19) · A T (cid:21) ≤ E (cid:20)(cid:18) − e ϕ tr cβ − e ϕ tr c e ϕ + Z [ f ( x ) − − ln( f ( x ))] F ( dx ) (cid:19) · A T + V T (cid:21) ≤ E (cid:20) V T + 12 β tr cβ · A T + Z [ f ( x ) − − ln( f ( x ))] F ( dx ) · A T (cid:21) ≤ E [ − ln( Z T )] . This proves assertion (c), and the first step is complete.
Step 2.
This step proves (d)= ⇒ (a). Thus, we suppose that assertion (d) holds. Then there exists a6ortfolio e θ ∈ Θ( X, H ) such that (3.7) holds. Thanks to [6, Theorem 2.8] (see also [8] and [14, Theorem2.3]), we deduce that D ( X, H ) = ∅ . By combining this with 1 + ( e θ · X ) T >
0, we conclude the positivityof both processes 1 + e θ · X and 1 + ( e θ · X ) − , and hence the existence of e ϕ ∈ L ( X, H ) ∩ L ( X, H ) such that1 + e θ · X = E ( e ϕ · X ) on the one hand. On the other hand, the condition D ( X, H ) = ∅ is equivalent tothe existence of the num´eraire portfolio, that we denote by b ϕ (see [6, 17, 18] and the references thereinto cite few). This means that there exists b ϕ ∈ L ( X, H ) such that E ( b ϕ · X ) > E ( ϕ · X ) / E ( b ϕ · X )is a supermartingale for any ϕ ∈ L ( X, H ) with 1 + ϕ ∆ X ≥
0. In particular, the process M := E ( e ϕ · X ) E ( b ϕ · X ) − , is a suprermartingale. Due to ln( x ) ≤ x −
1, we get − ln( E ( b ϕ · X )) ≤ − ln( E ( e ϕ · X )) + E ( e ϕ · X ) E ( b ϕ · X ) − , (4.1)and deduce that ln − ( E T ( b ϕ · X )) is integrable. As a result, b θ := b ϕ E ( b ϕ · X ) − ∈ Θ( X, H ), and thefollowing hold E [ln( E T ( b ϕ · X ))] = E [ln(1 + ( b θ · X ) T )] ≤ E [ln(1 + ( e θ · X ) T )] = E [ln( E T ( e ϕ · X ))] . (4.2)This, in particular, implies that ln( E T ( b ϕ · X )) is an integrable random variable, or equivalently ln( E T ( e ϕ · X ) / E T ( b ϕ · X ) = ln( E T ( e ϕ · X )) − ln( E T ( b ϕ · X ) is integrable. Then using Jensen’s inequality, we deducethat E [ln( E T ( e ϕ · X ) / E T ( b ϕ · X )] ≤ ln( E [ E T ( e ϕ · X ) / E T ( b ϕ · X )]) ≤ . This combined with (4.2) implies that E [ln( E T ( b ϕ · X ))] = E [ln( E T ( e ϕ · X ))] . (4.3)A combination of this with (4.1) leads to E [ E T ( e ϕ · X ) / E T ( b ϕ · X )] = 1, and hence the process M + 1 isin fact a martingale (a positive supermartingale with constant expectation is a martingale). It is clearthat f ( x ) := x − ln(1 + x ) , x > −
1, is a nonnegative and strictly convex function that vanishes at x = 0 only. Since E [ f ( M T )] < + ∞ , we conclude that f ( M ) is a nonnegative submartingale satisfying0 = E [ f ( M )] ≤ E [ f ( M t )] ≤ E [ f ( M T )] = 0 , where the last equality follows from combining (4.3) with the fact that M is martingale. Thus, weconclude that M ≡ E ( b ϕ · X ) ≡ E ( e ϕ · X ). As a consequence the process Z := 1 / E ( e ϕ · X )belongs to D ( X, H ). Therefore, assertion (a) follows immediately from this and E [ − ln( Z T )] = E [ln( E T ( e ϕ · X ))] = E [1 + ( e θ · X ) T ] < + ∞ , and the proof of (d)= ⇒ (a) is complete. Step 3.
This step proves the implication (a) = ⇒ (b). Hence, we assume that assertion (a) holdsfor the rest of this proof. In virtue of Theorem B.2, which guarantees the existence of ( β, f, V ) suchthat β ∈ L ( X c , H ), f is e P ( H )-measurable, positive and p ( f − ⋆ µ ∈ A + loc , V is a predictable and7ondecreasing process, and the following hold for any bounded θ ∈ L ( X, H ). E (cid:20) V T + 12 ( β tr cβ · A ) T + (cid:18)Z ( f ( x ) − − ln( f ( x ))) F ( dx ) (cid:19) · A T (cid:21) ≤ E [ − ln( Z T )] < + ∞ , (4.4) (cid:18)Z | f ( x ) θ tr x − θ tr h ( x ) | F ( dx ) (cid:19) · A T < + ∞ P -a.s. , and (4.5) (cid:18) θ tr b + θ tr cβ + Z [ f ( x ) θ tr x − θ tr h ( x )] F ( dx ) (cid:19) · A (cid:22) V, (4.6)The rest of this proof is divided into two sub-steps, and uses these properties. The first sub-step provesthat a functional L , that we will define below, attains its minimal value, while the second sub-stepproves that this minimum fulfills (3.3)-(3.4)-(3.5). Step 3.a.
Throughout the rest of the proof, we denote by L ( ω,t ) – P ⊗ A -almost all ( ω, t ) ∈ Ω × [0 , + ∞ )–the function given by L ( ω,t ) ( λ ) := − λ tr b ( ω, t ) + 12 λ tr c ( ω, t ) λ + Z (cid:0) λ tr h ( x ) − ln((1 + λ tr x ) + ) (cid:1) F ( ω,t ) ( dx ) , (4.7)for any λ ∈ R d with the convention ln(0 + ) = −∞ . This sub-step proves the existence of a predictableprocess e ϕ such that P ⊗ A -almost all ( ω, t ) ∈ Ω × [0 , + ∞ ) e ϕ ( ω, t ) ∈ L ( ω,t ) ( X, H ) and L ( ω, t) ( e ϕ ( ω, t)) = min λ ∈L ( ω, t) (X , H ) L ( ω, t) ( λ ) . (4.8)To this end, we start by noticing that in virtue of a combination of Remark 3.3 (which implies thatthis functional takes values in ( −∞ , + ∞ ]), Lemma C.1, and [12, Proposition 1] (which guarantees theexistence of a predictable selection for the minimizer when it exists), this proof boils down to provethat L ( ω,t ) attains in fact its minimum for all ( ω, t ) ∈ Ω × [0 , + ∞ ). This is the aim of the rest of thissub-step. For the sake of simplicity, we denote L ( ω,t ) ( · ) by L throughout the rest of this proof. In orderto prove that L attains it minimum value, we start by proving that this function L is convex, properand closed. Let first recall some definitions from convex analysis. Consider a convex function f . Theeffective domain of f , denoted by dom( f ), is the set of all x ∈ R d such that f ( x ) < + ∞ . The function f is said to be proper if, for any x ∈ R d , f ( x ) > −∞ and if its effective domain dom( f ) is not empty.For all undefined or unexplained concepts from convex analysis, we refer the reader to Rockafellar [28].Let θ be a bounded element of L ( X, H ), and due to ln(1+( θ tr x ) + ) ≤ ( θ tr x ) + ≤ | θ || x | and R ( | x | > | x | F ( dx ) < + ∞ (since X is σ -special), we obtain P ⊗ A -a.e. Z ( | x | > ln(1 + ( θ tr x ) + ) F ( dx ) ≤ Z ( | x | > ( θ tr x ) + F ( dx ) ≤ | θ | Z ( | x | > | x | F ( dx ) < + ∞ . (4.9)Then by combining this with Z (cid:0) λ tr h ( x ) − ln(1 + λ tr x ) (cid:1) F ( dx ) ≥ − Z ( | x | > ln(1 + ( λ tr x ) + ) F ( dx ) > −∞ , and L (0) = 0 < + ∞ (i.e. 0 ∈ dom( L ) ⊂ L ( X, H )), we deduce that L is a convex and proper function.Now we prove that L is closed or equivalently L is lower semi-continuous. Let θ n be a sequence in R d that converges to θ such that L ( θ n ) converges. Then it is clear that θ trn b + θ trn cβ converges to θ tr b + θ tr cβ and R ( | x | > ln(1 + ( θ trn x ) + ) F ( dx ) converges to R ( | x | > ln(1 + ( θ tr x ) + ) F ( dx ). This latter is8ue to a combination of ln(1 + ( θ trn x ) + ) ≤ ( θ trn x ) + ≤ (sup n | θ n | ) | x | , (4.9), and dominated convergencetheorem. Now consider the assumption θ ∈ L ( X, H ) and there exists n such that for all n ≥ n θ n ∈ L ( X, H ) . (4.10)Under (4.10), by combining Fatou’s lemma and the above remarks, we get L ( θ ) = − θ tr b + 12 θ tr cθ + Z (cid:0) θ tr h ( x ) − ln(1 + θ tr x ) (cid:1) F ( dx )= − θ tr b + 12 θ tr cθ − Z | x | > ln(1 + ( θ tr x ) + ) F ( dx ) − Z | x | > ln(1 − ( θ tr x ) − ) F ( dx ) + Z | x |≤ ( θ tr x − ln(1 + θ tr x )) F ( dx ) ≤ lim n −→ + ∞ L ( θ n ) . This proves that L is closed under (4.10) on the one hand. On the other hand, it is clear that, when(4.10) is violated, there exists a subsequence ( θ k ( n ) ) n such that θ k ( n )
6∈ L ( X, H ) for all n ≥
1. As aresult, since L ( θ n ) converges, we conclude that L ( θ ) ≤ lim n −→ + ∞ L ( θ n ) = lim n −→ + ∞ L ( θ k ( n ) ) = + ∞ .This proves that L is closed, convex and proper. Thus, we can apply [28, Theorem 27.1(b)] whichstates that, for L to attain its minimal value, it is sufficient to prove that the set of recession for L is contained in the set of directions in which L is constant. To check this last condition, we calculatethe recession function for L . For λ ∈ dom( L ) and y ∈ R d , the recession function for L is by definition L + ( y ) := lim α −→ + ∞ L ( λ + αy ) − L ( λ ) α . Consider the following setsΓ + ( λ ) := { x ∈ R d (cid:12)(cid:12) λ tr x > } , Γ − ( λ ) := { x ∈ R d (cid:12)(cid:12) λ tr x < } , and remark that we have L ( λ + αy ) − L ( λ ) α = − y tr b + α y tr cy + y tr cλ + Z (cid:18) y tr h ( x ) − α ln(1 + αy tr x λ tr x ) (cid:19) F ( dx )= − y tr b + α y tr cy + y tr cλ + Z Γ + ( y ) (cid:18) y tr h ( x ) − α ln(1 + αy tr x λ tr x ) (cid:19) F ( dx )+ Z Γ − ( y ) (cid:18) y tr h ( x ) − α ln(1 + αy tr x λ tr x ) (cid:19) F ( dx ) . Then, on the one hand, we calculate the recession function L + ( y ) as follows. L + ( y ) = (cid:26) + ∞ if either F (Γ − ( y )) > y tr cy > , − y tr b + R Γ + ( y ) y tr h ( x ) F ( dx ) otherwiseOn the other hand, we have α { y ∈ R d : cy = 0 and F (Γ − ( y )) = 0 } ⊂ L ( X, H ) for any α ∈ (0 , + ∞ ) , − α { y ∈ R d : cy = 0 and F (Γ + ( y )) = 0 } ⊂ L ( X, H ) for any α ∈ (0 , + ∞ ) . − y tr b + Z Γ + ( y ) y tr h ( x ) F ( dx ) > F (Γ − ( y )) = 0 < F (Γ + ( y )) , cy = 0 ,y tr b − Z Γ − ( y ) y tr h ( x ) F ( dx ) > F (Γ + ( y )) = 0 < F (Γ − ( y )) , cy = 0 . Thus, thanks to these remarks, the recession cone for L and the set of directions in which L is constant,that we denote RC and CD respectively, are defined and calculated as follows RC := { y ∈ R d (cid:12)(cid:12) L + ( y ) ≤ } = { y ∈ R d (cid:12)(cid:12) cy = y tr b = F (Γ − ( y )) = F (Γ + ( y )) = 0 } CD := { y ∈ R d (cid:12)(cid:12) L + ( y ) ≤ , L + ( − y ) ≤ } = { y ∈ R d (cid:12)(cid:12) y tr b = cy = F (Γ − ( y )) = F (Γ + ( y )) = 0 } . This proves that both sets (RC and CD) are equal. Hence, thanks to [28, Theorem 27.1(b)], weconclude that L ( ω,t ) attains its minimal value at ϕ ( ω, t ) which satisfies 1 + x tr ϕ ( ω, t ) > F ( ω,t ) ( dx )-a.e. since L ( ω,t ) ( ϕ ( ω, t )) ≤ L ( ω,t ) (0) = 0 < + ∞ . This ends the first part of the third step. Step 3.b.
This sub-step proves that e ϕ , a minimizer for L proved in the previous sub-step, fulfills infact the conditions of assertion (b) (i.e. the properties (3.3)-(3.4)-(3.5)).Since L ( e ϕ ) ≤ L ( ϕ ) for any ϕ ∈ L ( X, H ). Let ϕ ∈ L ( X, H ) and α ∈ (0 , L ( e ϕ ) − L ( e ϕ + α ( ϕ − e ϕ )) α = ( ϕ − e ϕ ) tr b − α ϕ − e ϕ ) tr c ( ϕ − e ϕ ) − ( ϕ − e ϕ ) tr c e ϕ ++ Z (cid:18) α ln(1 + α ( ϕ − e ϕ ) tr x e ϕ tr x ) − ( ϕ − e ϕ ) tr h ( x ) (cid:19) F ( dx ) . It is clear that, as a function of α , α − ln(1 + α ( ϕ − e ϕ ) tr x e ϕ tr x ) is decreasing and henceln(1 + ϕ tr x ) − ln(1 + e ϕ tr x ) ≤ α ln(1 + α ( ϕ − e ϕ ) tr x e ϕ tr x ) ≤ ( ϕ − e ϕ ) tr x e ϕ tr x . As a result of this, combined with the convergence monotone theorem, we deduce that
Z (cid:18) α ln(1 + α ( ϕ − e ϕ ) tr x e ϕ tr x ) − ( ϕ − e ϕ ) tr h ( x ) (cid:19) F ( dx )converges to R [ ( ϕ − e ϕ ) tr x e ϕ tr x − ( ϕ − e ϕ ) tr h ( x )] F ( dx ) , when α goes to zero and hence (3.5) is proved. Byusing (3.5) for ϕ = 0, and L ( e ϕ ) ≤ L (0) = 0, we get0 ≤ e ϕ tr b − e ϕ tr c e ϕ + Z (cid:18) − e ϕ tr h ( x ) + e ϕ tr x e ϕ tr x (cid:19) F ( dx ) (4.11)0 ≤ e ϕ tr b − e ϕ tr c e ϕ + Z (cid:0) ln(1 + e ϕ tr x ) − e ϕ tr h ( x ) (cid:1) F ( dx ) . (4.12)Therefore, by combining these two inequalities with − e ϕ tr c e ϕ − β tr cβ ≤ e ϕ tr cβ , and f ( x ) − − f ( x )) ≥ ln(1 + e ϕ tr x ) − f ( x ) e ϕ tr x (Young’s inequality), we derive e ϕ tr b − e ϕ tr c e ϕ − β tr cβ + Z [ln(1 + e ϕ tr x ) − e ϕ tr h ( x )] F ( dx ) + − Z [ f ( x ) − − ln( f ( x ))] F ( dx ) ≤ e ϕ tr b − e ϕ tr c e ϕ − β tr cβ + Z [ f ( x ) e ϕ tr x − e ϕ tr h ( x )] F ( dx ) ≤ e ϕ tr b + e ϕ tr cβ + Z [ f ( x ) e ϕ tr x − e ϕ tr h ( x )] F ( dx ) . Therefore, thanks to this latter inequality and (4.6), we deduce that0 (cid:22) (cid:18) e ϕ tr b − e ϕ tr c e ϕ + Z [ln(1 + e ϕ tr x ) − e ϕ tr h ( x )] F ( dx ) (cid:19) · A (cid:22) (cid:18)Z [ f ( x ) − − ln( f ( x ))] F ( dx ) + 12 β tr cβ (cid:19) · A + V. By combining this, (4.4), the fact that e ϕ tr b − e ϕ tr c e ϕ + Z [ln(1 + e ϕ tr x ) − e ϕ tr h ( x )] F ( dx )= (cid:18) e ϕ tr c e ϕ + Z (cid:18) ln(1 + e ϕ tr x ) − e ϕ tr x e ϕ tr x (cid:19) F ( dx ) (cid:19) + (cid:18) e ϕ tr b − e ϕ tr c e ϕ + Z (cid:18) − e ϕ tr h ( x ) + e ϕ tr x e ϕ tr x (cid:19) F ( dx ) (cid:19) , where both terms of the RHS are nonnegative, and the second term of this RHS coincides with d e VdA ,we conclude that E (cid:20) e V T + (cid:18) e ϕ tr c e ϕ + Z (cid:18) ln(1 + e ϕ tr x ) − e ϕ tr x e ϕ tr x (cid:19) F ( dx ) (cid:19) · A T (cid:21) < + ∞ . This proves (3.3), and assertion (b) follows. This ends the proof of the theorem.
A Some useful integrability properties
The results of this section are new and are general, not technical at all, and very useful, especially thefirst lemma and the proposition.
Lemma A.1.
Consider K ∈ M ,loc ( H ) with K > . If E [ h K c i T + X δ ∈ (0 , K − ln(1 + ∆ K ) ≥ δ | ∆ K | max(2(1 − δ ) , δ ) I {| ∆ K | >δ } + (∆ K ) δ I {| ∆ K |≤ δ } .
11y using this inequality and (A.1), on the one hand, we deduce that E h K c i T + X Lemma A.2. Let λ ∈ L ( X, H ) , and δ ∈ (0 , such that | λ tr x | λ tr x I {| λ tr x | >δ } ⋆ µ + (cid:18) λ tr x λ tr x (cid:19) I {| λ tr x |≤ δ } ⋆ µ ∈ A + loc ( H ) . (A.2) Then p ((1 + λ tr x ) − − ⋆ µ ∈ A + loc ( H ) .Proof. By using qP i x i ≤ P i | x i | , we derive p ((1 + λ tr x ) − − ⋆ µ = sX (cid:18) λ tr ∆ X λ tr ∆ X (cid:19) ≤ sX ( λ tr ∆ X ) (1 + λ tr ∆ X ) I {| λ tr ∆ X |≤ δ } + X | λ tr ∆ X | λ tr ∆ X I {| λ tr ∆ X | >δ } . Thus, the lemma follows immediately from the latter inequality. B Martingales and deflators via predictable characteristics For the following representation theorem, we refer to [15, Theorem 3.75] and to [16, Lemma 4.24]. Theorem B.1. Suppose that X is quasi-left-continuous, and let N ∈ M ,loc ( H ) . Then, there exist φ ∈ L ( X c , H ) , N ′ ∈ M ,loc ( H ) with [ N ′ , X ] = 0 and functionals f ∈ e P ( H ) and g ∈ e O ( F ) such that thefollowing hold. (a) (cid:16) t X s =0 ( f ( s, ∆ S s ) − I { ∆ S s =0 } (cid:17) / and (cid:16) t X s =0 g ( s, ∆ S s ) I { ∆ S s =0 } (cid:17) / belong to A + loc . (b) M Pµ ( g | e P ) = 0 , P ⊗ µ -.a.e., and the process N is given by N = φ · X c + ( f − ⋆ ( µ − ν ) + g ⋆ µ + N ′ . (B.1)The following theorem describes how general deflators can be characterized using the predictablecharacteristics. A version of this theorem can be found in [25].12 heorem B.2. Suppose X is quasi-left-continuous. Z ∈ D log ( X, H ) if and only if there exists atriplet ( β, f, V ) such that β ∈ L ( X c , H ) , f is e P ( H ) -measurable, positive and p ( f − ⋆ µ belongsto A + loc ( H ) , V is an H -predictable and nondecreasing process, and the following hold for any boundedprocess θ ∈ L ( X, H ) . Z = E (cid:16) β · X c + ( f − ⋆ ( µ − ν ) (cid:17) exp( − V ) , (B.2) E (cid:20) V T + (cid:18) β tr cβ + Z ( f ( x ) − − ln( f ( x ))) F ( dx ) (cid:19) · A T (cid:21) ≤ E [ − ln( Z T )] , (B.3) (cid:18)Z | f ( x ) θ tr x − θ tr h ( x ) | F ( dx ) (cid:19) · A T < + ∞ P -a.s. (B.4) (cid:18) θ tr b + θ tr cβ + Z [ f ( x ) θ tr x − θ tr h ( x )] F ( dx ) (cid:19) · A (cid:22) V, (B.5) Proof. Let Z ∈ D log ( X, H ), then Z − − · Z a local supermartingale (which follows from Z ∈ D ( X, H )only). Hence, there exists a local martingale N and a nondecreasing and predictable process V suchthat Z = E ( N ) exp( − V ). Then we derive − ln( Z ) = − N + V + 12 h N c i + X (∆ N − ln(1 + ∆ N )) . Thus Z ∈ D log ( X, H ) if and only if V + h N c i + P (∆ N − ln(1 + ∆ N )) is integrable. Then thereexists a positive and e P ( H )-measurable functional f such that p ( f − ⋆ µ is locally integrable, and β ∈ L ( X c , H ) such that N can be chosen to be N := β · X c + ( f − ⋆ ( µ − ν ) and V = v · A .Then Z ∈ D log ( X, H ) if and only if V + β tr cβ · A + ( f − − ln( f )) ⋆ ν ∈ A + ( H ) and Z E ( θ · X ) is asupermartingale, for any locally bounded H -predictable process θ such that 1 + θ tr x > P ⊗ A -a.e..Here ( b, c, ν := F ⊗ A ) is the predictable characteristics of ( X, H ).On the one hand, we have Z E ( θ · X ) = E ( N − v · A + θ · X + [ θ · X, N ]) is a positive supermartingale andhence N − v · A + θ · X + [ θ · X, N ] is a local supermartingale. This is equivalent, (after simplificationand transformation), to the conditions (B.4)-(B.5). This ends the proof of theorem. C A measurability result Lemma C.1. Consider the triplet (Ω × [0 , + ∞ ) , P ( H ) , P ⊗ A ) , and L ( ω, t, λ ) := L ( ω,t ) ( λ ) , definedin (4.7) for any λ ∈ R d and any ( ω, t ) ∈ Ω × [0 , + ∞ ) . Then the functional L , as map ( ω, t, λ ) −→ L ( ω, t, λ ) , is P ( H ) × B ( R d ) -measurable.Proof. The proof of the lemma will be achieved in two steps. The first step defines a family offunctionals { L δ ( ω, t, · ) , δ ∈ (0 , } for ( ω, t ) ∈ Ω × [0 , + ∞ ), and proves that these functionals areindeed P ( H ) × B ( R d )-measurable (i.e. jointly measurable in ( ω, t ) and λ ). Then the second stepproves that L δ ( ω, t, λ ) converges to L ( ω, t, λ ) when δ goes to one for any ( ω, t, λ ) ∈ Ω × [0 , + ∞ ) × R d . Step 1: Let ( ω, t ) ∈ Ω × [0 , + ∞ ) and δ ∈ (0 , λ ∈ R d , put L δ ( ω, t, λ ) := − λ tr b ( ω, t ) + 12 λ tr c ( ω, t ) λ + Z R d f δ ( λ, x ) F ( ω,t ) ( dx ) ,f δ ( λ, x ) := δλ tr h ( x ) − ln(1 − δ + δ (1 + λ tr x ) + ) . It is clear that for any λ ∈ R d , L δ ( ω, t, λ ) is predictable. Thus, in order to prove that L δ is jointlymeasurable (i.e. P ( H ) × B ( R d )-measurable), it is enough to prove that this functional is continuous13n λ (in this case our functional L δ falls into the class of Carath´eodory functions), and hence one canconclude immediately that it is jointly measurable due to [3, Lemma 4.51]. Thus, the rest of this stepfocuses on proving that L δ is continuous in λ . To this end, we first remark that − λ tr b ( ω, t )+ λ tr c ( ω, t ) λ is continuous, and we derive − δ | λ | | x | ≤ f δ ( λ, x ) ≤ max( 12(1 − δ ) , − δ − ln(1 − δ )) | λ | | x | on {| x | ≤ }− δ | λ || x | ≤ f δ ( λ, x ) ≤ − ln(1 − δ ) on {| x | > } . Therefore, thanks to the dominated convergence theorem and these inequalities, we deduce that infact L δ is continuous in λ , and the first step is complete. Step 2: Herein, we prove that for any ( ω, t ) ∈ Ω × [0 , + ∞ ) and any λ ∈ R d , L δ ( ω, t, λ ) converges to L ( ω, t, λ ) when δ goes to one. To this end, we first write L δ ( ω, t, λ ) = δ Z { λ tr x ≤− } λ tr h ( x ) F ( dx ) − δ Z { λ tr x ≤− } λ tr xI {| x | > } F ( dx ) − ln(1 − δ ) F (cid:0) { λ tr x ≤ − } (cid:1) + Z I { λ tr x> − } (cid:0) δλ tr x − ln(1 + δλ tr x ) (cid:1) F ( dx ) . Remark that R { λ tr x ≤− } λ tr h ( x ) F ( dx ) and R { λ tr x ≤− } λ tr xI {| x | > } F ( dx ) are well defined and take finitevalues, while I { λ tr x> − } (cid:0) δλ tr x − ln(1 + δλ tr x ) (cid:1) is nonnegative and increasing in δ . By distinguishingthe cases whether F (cid:0) { λ tr x ≤ − } (cid:1) is null or not, thanks to the convergence monotone theorem, weconclude that L δ ( ω, t, λ ) converges to L ( ω, t, λ ). This ends the second step and the proof of thelemma. 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