Martingale Inequalities for the Maximum via Pathwise Arguments

Transcription

1 Martingale Inequalities for the Maximum via Pathwise Arguments Jan Ob lój : Peter Soida ; Nizar Touzi March 24, 2015 Abstract We study a class of martingale inequalities involving the running maximum rocess. They are derived from athwise inequalities introduced by Henry-Labordère et al. [15] and rovide an uer bound on the exectation of a function of the running maximum in terms of marginal distributions at n intermediate time oints. The class of inequalities is rich and we show that in general no inequality is uniformly shar for any two inequalities we secify martingales such that one or the other inequality is sharer. We then use our inequalities to recover Doob s L inequalities. For 1 we obtain new, or refined, inequalities. 1 Introduction In this article we study certain martingale inequalities for the terminal maximum of a stochastic rocess. We thus contribute to a research area with a long and rich history. In seminal contributions, Blackwell and Dubins [7], Dubins and Gilat [14] and Azéma and Yor [4; 3] showed that the distribution of the maximum X T : su tt X t of a martingale X t q is bounded above, in stochastic order, by the so called Hardy-Littlewood transform of the distribution of X T, and the bound is attained. This led to series of studies on the ossible distributions of X T, X T q, see Carraro, El Karoui and Ob lój [10] for a discussion and further references. More recently, such roblems aeared very naturally within the field of mathematical finance. The original result was extended to the case of a non trivial starting law in Hobson [16] and to the case of a fixed intermediate law in Brown, Hobson and Rogers [9]. Jan Ob lój thankfully acknowledges suort from the ERC Starting Grant RobustFin- Math , the Oxford-Man Institute of Quantitative Finance and St John s College in Oxford. Peter Soida gratefully acknowledges scholarshis from the Oxford-Man Institute of Quantitative Finance and the DAAD. Nizar Touzi gratefully acknowledges financial suort from the ERC Advanced Grant ROFIRM, the Chair Financial Risks of the Risk Foundation sonsored by Société Générale, and the Chair Finance and Sustainable Develoment sonsored by EDF and CA-CIB. : University of Oxford, Mathematical Institute, the Oxford-Man Institute of Quantitative Finance and St John s College, Jan.Obloj@maths.ox.ac.uk ; University of Oxford, Mathematical Institute and the Oxford-Man Institute of Quantitative Finance, Peter.Soida@maths.ox.ac.uk Ecole Polytechnique Paris, Centre de Mathématiques Aliquées, nizar.touzi@olytechnique.edu 1

2 The novelty of our study here, as comared with the works mentioned above, is that we look at inequalities which use the information about the rocess at n intermediate time oints. One of our goals is to understand how the bound induced by these more elaborate inequalities comares to simler inequalities which do not use information about the rocess at intermediate time oints. We show that in our context these bounds can be both, better or worse. We also note that knowledge of intermediate moments does not induce a necessarily tighter bound in Doob s L -inequalities. Our main result is slit into two Theorems. First, in Theorem 2.1, we resent our class of inequalities, indexed with an n-tule of functions ζ, and show that they are shar: for a given ζ we find a martingale which attains equality. Second, in Theorem 3.1, we show that no inequality is universally better than another: for ζ ζ we find two rocesses X and X which show that either of the inequalities can be strictly better than the other. Throughout, we emhasise the simlicity of our arguments, which are all elementary. This is illustrated in Sections where we obtain amongst others the shar versions of Doob s L -inequalities for all ą 0. While the case ě 1 is already known in the literature, our Doob s L -inequality in the case P 0, 1q aears new. The idea of deriving martingale inequalities from athwise inequalities is already resent in work on robust ricing and hedging by Hobson [16]. Other authors have used athwise arguments to derive martingale inequalities, e.g. Doob s inequalities are considered by Acciaio et al. [1] and Ob lój and Yor [19]. The Burkholder-Davis-Gundy inequality is rediscovered with athwise arguments by Beiglböck and Sioraes [6]. In this context we also refer to Cox and Wang [13] and Cox and Peskir [12] whose athwise inequalities relate a rocess and time. In a similar sirit, bounds for local time are obtained by Cox et al. [11]. Beiglböck and Nutz [5] look at general martingale inequalities and exlain how they can be obtained from deterministic inequalities. This aroach builds on the so-called Burkholder s method, a classical tool in robability used to construct shar martingale inequalities, see Osȩkowski [20, Ch. 2] for a detailed discussion. In a discrete time and quasi-sure setu, the results of Bouchard and Nutz [8] can be seen as general theoretical underinning of many ideas we resent here in the secial case of martingale inequalities involving the running maximum. Organization of the article We first recall a remarkable athwise inequality obtain by Henry-Labordère et al. [15] and some related results. The body of the aer is then slit into two sections. In Section 2 we derive our class of submartingale inequalities and demonstrate how they can be used to derive, amongst others, Doob s inequalities. Then, in Section 3, we study if a given inequality can be universally better than another one for all submartingales. 1.1 Preliminaries We assume that a filtered robability sace Ω, F, F t q, Pq is fixed which suorts a standard real-valued Brownian motion B with some initial value P R. We will tyically use X X t q to denote a (sub/suer) martingale and, unless otherwise secified, we always mean this with resect to X s natural filtration. Throughout, we fix arbitrary times 0 t 0 t 1 t 2... t n : T. 2

3 Before we roceed to the main result, we recall a remarkable athwise inequality from Henry-Labordère et al. [15]. The version we give below aears in the roof of Proosition 3.1 in [15] and is best suited to our resent context. Proosition 1.1 (Proosition 3.1 of Henry-Labordère et al. [15]). Let ω be a càdlàg ath and denote ω t : su 0st ω s. Then, for m ą ω 0 and ζ 1 ζ n ă m: 1 t ωtn ěmu Υ n ω, m, ζq : nÿ ˆωti ζ i q` i 1 n1 ÿ i 1 m ζ i m ω ti ` 1 t ωti1 ăm ω ti u m ζ i ˆωti ζ i`1 q` ` 1 m ζ tm ωti,ζ i`1ω ti i`1 u (1.1) ω ti`1 ω ti m ζ i`1 Next, we recall a rocess with some secial structure in view of (1.1). This rocess has been analysed in more detail by Ob lój and Soida [18]. Definition 1.2 (Iterated Azéma-Yor Tye Embedding). Let ξ 1,..., ξ n be nondecreasing functions on, 8q and denote B t : su ut B u. Set τ 0 0 and for i 1,..., n define τ i : inf t ě τ i1 : B t ξ i B t q (. (1.2) A continuous martingale X is called an iterated Azéma-Yor tye embedding based on ξ ξ 1,..., ξ n q if X ti, X ti q B τi, B τi q a.s. for i 0,..., n. (1.3) Note from the non-decrease of the ξ i s that τ 0 inftt ě H 1 : B t ξ 1 1qu for H 1 inftt ě 0 : B t ě 1u and then τ i inftt ě τ i1 : B t ξ i B τi1 qu, i 2,..., n. It follows that τ i ă 8 a.s. for all i 1,..., n. Further, X being a martingale imlies that B τi are integrable and all have mean. In articular, τ n ă 8 a.s. More imortantly, it follows from the characterisation of uniform integrable martingales in Azéma et al. [2] that B t^τn, t ě 0q is uniformly integrable. Indeed, we have, with H x inftt ě 0 : B t xu, lim xp xñ8 j su B t^τn ą x tě0 lim xp xñ8 lim xñ8 H x ă H ξ1 maxi i xq ˆxmaxi ξ1 i xq q xq ` x max i ξ1 i ı ` xp Bt^τn ą x ` xp Xtn ą x 0, since X t : t t n q is uniformly integrable and max ξ1 i i xq Œ 0. Conversely, if B t^τn : t ě 0q is uniformly integrable then an examle of an iterated Azéma-Yor tye embedding is obtained by taking X t : B τi^ τ i1_ tt i1 t it, for t i1 ă t t i, i 1,..., n. (1.4) Finally, we recall a version of Lemma 4.1 from Henry-Labordère et al. [15].. 3

4 Proosition 1.3 (Pathwise Equality). Let ξ ξ 1,..., ξ n q be non-decreasing right-continuous functions and let X be an iterated Azéma-Yor embedding based on ξ. Then, for any m ą with ξ n mq ă m, X achieves equality in (1.1), i.e. 1 t Xtn ěmu Υ n `X, m, ζmq a.s., (1.5) where ζ i mq min jěi ξ jmq, i 1,..., n. (1.6) We note that if we work on the canonical sace of continuous functions then (1.5) holds athwise and not only a.s. We also note that the assumtion that X is an iterated Azéma-Yor tye embedding, or that B τn^tq is a uniformly integrable martingale, may be relaxed as long as X satisfies (1.3). 2 (Sub)martinagle inequality and its alications We resent now an inequality on the exected value of a function of the running maximum of a submartingale which is obtained by taking exectations in the athwise inequality of Proosition 1.1. We then demonstrate how this inequality can be used to derive and imrove Doob s inequalities. Related work on athwise interretations of Doob s inequalities can be found in Acciaio et al. [1] and Ob lój and Yor [19]. Peskir [21, Section 4] derives Doob s inequalities and shows that the constants he obtains are otimal. We give below an alternative roof of these statements and rovide new shar inequalities for the case ă Submartingale inequality We first deduce a general martingale inequality for E φ X T q, similarly as in Proosition 3.2 in [15], and rove that it is attained under some conditions. Define Z :! ζ ζ 1,..., ζ n q : ζ i :, 8q Ñ R is right-continuous, ) (2.1) ζ 1 mq ζ n mq ă m, n P N. In order to ensure that the exectations we consider are finite we will occasionally need the technical condition that ζ 8 1 : lim inf mñ8 ζ 1 mq m ą 0 and lim su mñ8 i 1 φmq m γ 1 0 for some γ ă 1 ζ1 8. (2.2) Theorem 2.1. Let ζ ζ 1,..., ζ n q P Z. Then, (i) for any càdlàg submartingale X: for any m ą we have P XT ě m «ff nÿ n1 X ti ζ i mqq` ÿ X ti ζ i`1 mqq` E m ζ i mq m ζ i`1 mq i 1 (2.3) and, more generally, for a right-continuous non-decreasing function φ, E φ X T q ż nÿ ı UB X, φ, ζq : φ q ` E X ti q dφmq (2.4),8q i 1 λ ζ,m i 4

5 where λ ζ,m i xq : x ζ imqq` x ζ i`1mqq` m ζ i mq m ζ i`1 mq 1 tiănu, (2.5) (ii) if ζ 1 is non-decreasing and satisfies, together with φ, the condition (2.2), there exists a continuous martingale which achieves equality in (2.4). Remark 2.2 (Otimization over ζ). If X and t 1,..., t n are fixed we can otimize (2.4) over ζ P Z to obtain a minimizer ζ. Clearly, more intermediate oints t i in (2.4) can only imrove the bound for this articular rocess X. However, only for very secial rocesses (e.g. the iterated Azéma-Yor tye embedding) there is hoe that (2.4) will hold with equality. This is, loosely seaking, because a finite number of intermediate marginal law constraints does not, in general, determine uniquely the law of the maximum at terminal time t n. Proof of Theorem 2.1. Equation (2.3) follows from (1.1) by taking exectations. Then, (2.4) follows from (2.3) by integration and Fubini s theorem: E φ X T q «ż ff E φ q ` 1 t XT ěmudφmq.,8q ı Note that for a fixed m, E λ ζ,m i X ti q ă 8 for i 1,..., n, since E X`t i ă 8 by the submartingale roerty. If ζ 1 is non-decreasing and ζ 1 mq ě αm for m large, α ą 0, we define X by # B t t X t ^τ ζ 1t 1 if t ă t 1, B τζ1 if t ě t 1. where B is a Brownian motion, B 0, and τ ζ1 : inf u ą 0 : B u ζ 1 B u q (. Excursion theoretical considerations, cf. e.g. Rogers [22], combined with asymtotic bounds on ζ 1 in (2.2), allow us to comute P Xtn ě y ż ex,ys const y 1 1α 1 z ζ 1 zq dz ż const ex 1,ys 1 z αz dz for large y. We may take α such that γ ă 1{1 αq in (2.2) which then ensures that E φ X tn q ă 8. Further, note that for large y, inf tě0 X t y imlies X 8 X tn ě y{α and hence it follows that j lim yp su X t ě y const lim y 1 1 1α 0 yñ8 tě0 yñ8 which in turn imlies that X t : t ě 0q is a uniformly integrable martingale, see Azéma et al. [2]. Finally, one readily verifies together with Proosition 1.3 that Υ n X, m, ζq Υ 1 X, m, ζq 1 t Xt1 ěmu 1 t X tn ěmu. 5

6 and then the claim follows from E φ X tn q ż φ q ` ż φ q ` UB X, φ, ζq where we alied Fubini s theorem.,8q,8q 2.2 Doob s L -Inequalities, ą 1 ı E 1 t Xtn ěmu dφmq UB `X, 1 rm,8q, ζ dφmq Using a secial case of Theorem 2.1 we obtain an imrovement to Doob s inequalities. Denote ow mq m, ζ α mq : αm. be a non- Proosition 2.3 (Doob s L -Inequalities, ą 1). Let X t q tt negative càdlàg submartingale. (i) Then, E X T UB X, ow, ζ 1 ˆ 1 E rx T s 1. (2.6a) (2.6b) (ii) For every ɛ ą 0, there exists a martingale X such that ˆ 0 E rx T 1 s 1 E X T ă ɛ. (2.7) (iii) The inequality in (2.6b) is strict if and only if either holds: E X T ă 8 and XT ă 1 with ositive robability. (2.8a) E X T ă 8 and X is a strict submartingale. (2.8b) Proof. Let us first rove (2.6a) and (2.6b). If E rx T s 8 there is nothing to show. In the other case, equation (2.6a) follows from Theorem 2.1 alied with n 1, φyq ow yq y and ζ 1 ζ 1. To justify this choice of ζ 1 and to simlify further the uer bound we start with a more general ζ 1 ζ α, α ă 1 and comute E X T X «ż X T α _ E y 1 X T αy y αy dy ż 8 0 UB X, ow, ζ α q X 0 E y 1 X T αyq` y αy ff «ż X T ff α E y 1 X T αy y αy dy + ff 1 1 «#ˆXT 1 1 α E X 1 0 X T α j ˆXT α 1 α E X 0 α j dy αqα 1 E α 1q rx T s 1q1 αq, (2.9) 6

7 where we used Fubini in the first equality and the submartingale roerty of 1 X in the last inequality. We note that the function α ÞÑ 1αqα attains its 1 minimum at α 1. Plugging α α into the above yields (2.6b). We turn to the roof that Doob s L -inequality is attained asymtotically in the sense of (2.7), a fact which was also roven by Peskir [21, Section 4]. Let ą 0, otherwise the claim is trivial. Set α 1 and take α ă α : `ɛ1 `ɛ ă 1. Let X T B τα where B is a Brownian motion started at and τ α : inftu ą 0 : B u α B u u. Then by using excursion theoretical results, cf. e.g. Rogers [22], P XT ě y ˆ ż y ˆ 1 1 y ex z αz dz 1α and then direct comutation shows By Doob s L -inequality, E ` ɛ X T X 0 ɛ. E ˆ X T E rx T 1 s α 1 E X α T 1 and one verifies "ˆ j ` ɛ 1 1* 1 ` ɛ ` ɛ X 0 ɛ ÝÝÝÝÝÑ ɛó0 1. This establishes the claim in (2.7). Finally, we note that in the calculations (2.9) which led to (2.6b) there are three inequalities: the first one comes from Theorem 2.1 and does not concern the claim regarding (2.8a) (2.8b). The second one is clearly strict if and only if (2.8a) holds. The third one is clearly strict if and only if (2.8b) holds. Remark 2.4 (Asymtotic Attainability). For the martingales in (ii) of Proosition 2.3 we have ˆ UB X, ow, ζ 1 E rx T 1 s 1 and E rx T s Ñ 8 as ɛ Ñ Doob s L 1 -Inequality Using a secial case of Theorem 2.1 we focus on Doob s L log L tye inequalities. We recover here the classical constant e{e 1q, see (2.11b), with a refined structure on the inequality. A further imrovement to the constant will be obtained in subsequent section in Corollary 2.7. Denote idmq m, and # 8 if m ă 1, ζ α mq : (2.10) αm if m ě 1. 7

8 Proosition 2.5 (Doob s L 1 -Inequality). Let X t q tt be a non-negative càdlàg submartingale. Then: (i) with 0 log0q : 0 and V xq : x x logxq, E XT UB X, id, ζ 1 (2.11a) e e E rx T log X T qs ` V 1 _ q. (2.11b) e 1 (ii) in the case ě 1 there exists a martingale which achieves equality in both, (2.11a) and (2.11b) and in the case ă 1 there exists a submartingale which achieves equality in both, (2.11a) and (2.11b). (iii) the inequality in (2.11b) is strict if and only if either holds: E XT ă 8 and XT ě 1, X T ă 1 e with ositive robability, (2.12a) E XT ă 8 and XT ě 1, E rx T s ą _ 1. (2.12b) E XT ă 8 and XT ă 1 with ositive robability. (2.12c) Proof. Let us first rove (2.11a) and (2.11b). If E XT 8 there is nothing to show. In the other case, equation (2.11a) follows from Theorem 2.1 alied with n 1, φyq idyq y and ζ 1 ζ 1. e In the case ě 1 we further comute using ζ 1 ζ α, α ă 1, E XT X0 UB X, id, ζ α «ż X T ff «α _ ż X T ff X T αy E y αy dy α X T αy E 1 αqy dy " *j α 1 α E XT ˆXT log log q α j α α 1 α E XT α Choosing α e1 gives a convenient cancellation. Together with the submartingale roerty of X, this rovides E XT X0 e e 1 E rx T s log q ` 1 e 1 e e 1 E rx T log X T qs e e 1 E rx T q log X T qs e log q ` e 1 This is (2.11b) in the case ě 1. For the case 0 ă ă 1 we obtain from Proosition 1.1 for n 1, for α ă 1 and therefore P XT ě y E rx T ζq`s inf E rx T αyq`s ζăy y ζ y αy E XT X0 ż 8 P XT ě y dy 1 q ` ż 8 1 P XT ě y dy 1 q ` e e 1 E rx T q log X T qs ` 1 e 1 8 X0 e 1. (2.13) (2.14)

9 by (2.13). This is (2.11b) in the case ă 1. Now we rove that Doob s L 1 -inequality is attained. This was also roven by Peskir [21, Section 4]. Firstly, let ě 1. Then the martingale X ˆ B t T t ^τ 1 e tt, where τ 1 e inftt : eb t B t u, (2.15) and B is a Brownian motion with B 0, achieves equality in both (2.11a) and (2.11b). Secondly, let ă 1. Then the submartingale X defined by $ & if t ă T {2, (2.16) % B tt {2 if t ě T {2, T {2tT {2q ^τ 1 e where B is a Brownian motion, B 0 1, achieves equality in both, (2.11a) and (2.11b). Finally, we note that in the calculations (2.13) which led to (2.6b) there are three inequalities: the first one comes from Theorem 2.1 and does not concern the claim regarding (2.12a) (2.12c). The second one is clearly strict if and only if (2.12a) holds. The third one is clearly strict if and only if (2.12b) holds. In addition, in the case ă 1 there is an additional error coming from (2.14). Note that, in the case when E XT ă 8, E X T ζq`ı y ζ ˇ ˇζ 8 : E X T ζq`ı lim ζñ8 y ζ 1. Hence, the first inequality in (2.14) is strict if and only if (2.12c) holds. The second inequality in (2.14) is strict if and only if (2.12a) or (2.12b) holds. 2.4 Doob Tye Inequalities, 0 ă ă 1 It is well known that if X is a ositive continuous local martingale converging a.s. to zero, then X 8 U (2.17) where U is a uniform random variable on r0, 1s. More generally, for any nonnegative suermartingale X, with a deterministic, we have P X8 ě x {x, for all x ě. Hence, for any non-negative suermartingale X and ą 1 E j ˆX0 X T E U ż 1 0 ˆX0 du u 1 (2.18) and (2.18) is attained. We now generalize (2.18) to a non-negative submartingale. Proosition 2.6 (Doob Tye Inequalities, 0 ă ă 1). Let X be a non-negative càdlàg submartingale, ą 0, and P 0, 1q. Denote m r : Xr 0 E rxr T s for r 1. Then: 9

10 (i) there is a unique ˆα P 0, 1s which solves m ˆα 1 ` m 1 1 ` ˆα (2.19) and for which we have E X T X 0 m ˆα 1 ` X1 0 1 ă X 0 1 ` ˆα ` X1 0 E rx T s 1 ` ˆα (2.20a) E rx T s. (2.20b) (ii) there exists a martingale which attains equality in (2.20a). Further, for every ɛ ą 0 there exists a martingale such that 0 1 ` X1 0 E rx T s E X T ă ɛ. (2.21) 1 Proof. Following the calculations in (2.9), we see that E 1 X T 1 α ` 1 ı 1 αq1 q E α 1 X T ` X1 0 X T X 0 fαq, where, with the notation m r introduced in the statement of the Proosition, fαq : 1 α1 1 α ` m ` m 1, α P r0, 1s. 1 αq1 q Next we rove the existence of a unique ˆα P 0, 1s such that fˆαq min αpr0,1s fαq. To do this, we first comute that f 1 αq hαq 1 q1 αq 2, where hαq : 1 ` m 1 1 ` αqm α. By direct calculation, we see that h is continuous and strictly increasing on 0, 1s, with h0`q 8 and h1q 1 ` m 1 m. Moreover, it follows from the Jensen inequality and the submartingale roerty of X that m m 1 and m 1 ě 1. This imlies that h1q ě 0 since 1 ` x x ě 0 for x ě 1. In consequence, there exists ˆα P 0, 1s such that h 0 on 0, ˆαs and h ě 0 on rˆα, 1s. This imlies that f is decreasing on r0, ˆαs and increasing on rˆα, 1s, roving that ˆα is the unique minimizer of f. Now the first inequality (2.20a) follows by lugging the equation hˆαq 0 into the exression for f. The bound in (2.20b) is then obtained by adding strictly ositive terms. It also corresonds to taking α 0 in the exression for f. This comletes the roof of the claim in (i). As for (ii), the claim regarding a martingale attaining equality in (2.20a) follows recisely as in the roof of Proosition 2.3. Let α P 0, 1q and recall that τ α inftt : B t α B t u for a standard Brownian motion B with B 0 ą 0. Then, similarly to the roof of Proosition 2.3, we comute directly P B τα ě yq PB τα ě αyq 1 1α ˆX0, y ě X0. (2.22) y 10

11 Comuting and simlifying we obtain E B τα 1 1`α, and hence E B τ α α 1`α, while E rb τ α s. It follows that ˆα α solves (2.19) and equality holds in (2.20a). Taking α arbitrarily small shows (2.21) holds true. We close this section with a new tye of Doob s L ln L tye of L 1 inequality obtained taking Õ 1 in Proosition 2.6. Since ˆαq defined in (2.19) belongs to r0, 1s there is a converging subsequence. So without loss of generality, we may assume ˆαq ÝÑ ˆα1q for some ˆα1q P r0, 1s. In order to comute ˆα1q, we re-write (2.19) into gq g1q 1 m where gq : m ˆαq 1 ` m 1 qˆαq.(2.23) We see by a direct differentiation, invoking imlicit functions theorem, that ˆ g 1 XT 1q ˆα1q 1 ` E ln X j j T XT ˆα1q ln ˆα1qE. Then, sending Ñ 1 in (2.23), we get the following equation for ˆα1q: ˆ XT ˆα1q 1 ` E ln X j j T XT E 1 ` ˆα1q ln ˆα1qq. (2.24) We note that this equation does not solve exlicitly for ˆα1q. Sending Ñ 1 in the inequality of Proosition 3.4 we obtain the following imrovement to the classical Doob s L log L inequality resented in Proosition 2.5 above. Corollary 2.7 (Imroved Doob s L 1 Inequality). Let X be a non-negative càdlàg submartingale, ą 0. Then: E XT ErX T s ˆα where ˆα P 0, 1q is uniquely defined by (2.25). E rx T ln X T s ` E rx T s ln 1 ` ˆα ln ˆα (2.25) Note that the equality in (2.25) is a rewriting of (2.24). To the best of our knowledge the above inequality in (2.25) is new. It bounds E XT in terms of a function of E rx T s and E rx T ln X T s, similarly to the classical inequality in (2.11b). However here the function deends on ˆα which is only given imlicitly and not exlicitly. In exchange, the bound refines and imroves the classical inequality in (2.11b). This follows from the fact that 1 ` α ln α ě e 1, α P 0, 1q. e We note also that for X t : B t T t ^τα, α P 0, 1q, we have ˆα α and equality is attained in (2.25). This follows from the roof above or is verified directly using (2.22). The corresonding classical uer bound in (2.11b) is strictly greater exect for α 1{e when the two bounds coincide. 11

12 3 Universally best submartingale inequalities As mentioned in the introduction, the novelty of our martingale inequality from Theorem 2.1 is that it uses information about the rocess at intermediate times. In the revious section we saw that careful choice of functions ζ in Theorem 2.1 allowed us to recover and imrove the classical Doob s inequalities. In this section we study the finer structure of our class of inequalities and the question whether the information from the intermediate marginals gives us more accurate bounds than e.g. in the case when no information about the rocess at intermediate times is used. In short, the answer is negative, i.e. we demonstrate that for a large class of ζ s there is no universally better choice of ζ in the sense that it yields a tighter bound in the class of inequalities for E φ X T q from Theorem No inequality is universally better than other To avoid elaborate technicalities, we imose additional conditions on ζ P Z and φ below. Many of these conditions could be relaxed to obtain a slightly stronger, albeit more involved, statement in Theorem 3.1. We define! ) Z cts : ζ P Z : ζ are continuous (3.1) and Z :! ζ P Z cts : ζ are strictly increasing, lim inf mñ8 ζ 1 mq m ą 0, ) and ζ 1 ζ n on, ` ɛs, for some ɛ ą 0. (3.2) Before we roceed, we want to argue that the set Z arises quite naturally. In the setting of Remark 2.2, if X is a martingale such that its marginal laws µ 1 : L X t1 q,..., µ n : L X tn q ş satisfy Assumtion f of Ob lój and Soida [18], x ζq`µ i dxq ă ş x ζq`µ i`1 dxq for all ζ in the interior of the suort of µ i`1 and their barycenter functions satisfy the mean residual value roerty of Madan and Yor [17] close to and have no atoms at the left end of suort, then the otimization over ζ as described in Remark 2.2 yields a unique ζ P Z. Hence, the set of these Z seems to be a good candidate set for ζ s to be used in Theorem 2.1. The statement of the Theorem 3.1 concerns the negative orthant of Z cts, Z cts φ, ζq! : ζ P Z cts : UB X, φ, ζq UB X, φ, ζ for all càdlàg ) submartingales X and ă for at least one X, (3.3) and hence it comlements Theorem 2.1. Part (ii) in Theorem 2.1 studied sharness of (2.4) for a fixed ζ with varying X while Theorem 3.1 studies (2.4) for a fixed X with varying ζ. Theorem 3.1. Let φ be a right-continuous, strictly increasing function. Then, for ζ P Z such that (2.2) holds we have Z cts φ, ζq H. (3.4) 12

13 The above result essentially says that no martingale inequality in (2.4) is universally better than another one. For any choice ζ P Z, the corresonding martingale inequality (2.4) can not be strictly imroved by some other choice of ζ P Z cts, i.e. no other ζ would lead to a better uer bound for all submartingales and strictly better for some submartingale. The key ingredient to rove this statement is isolated in the following Proosition. Proosition 3.2 (Positive Error). Let ζ P Z and ζ P Z cts satisfy ζ ζ. Then there exists a non-emty interval m 1, m 2 q Ď, 8q such that UB X, 1 rm,8q, ζ ă UB `X, 1 rm,8q, ζ for all m P m 1, m 2 q, where X is an iterated Azéma-Yor tye embedding based on some ξ. Proof. To each ζ P Z we can associate non-decreasing and continuous stoing boundaries ξ ξ 1,..., ξ n q which satisfies ζ i mq min jěi ξ j ą. (3.5) Further, since ζ P Z imlies that ζ i are all equal on some, ` ɛs we may take ξ such that ξ n mq ă ă ξ 1 mq ă P, ` ɛq, (3.6a) ξmq ě ` ɛ, (3.6b) for some ɛ ą 0. A ossible choice is given by ξ i mq ζ i mq ` m ζ i mqq n i n ` ɛ mq`, m ą, i 1,..., n, ɛ but we may take any ξ satisfying (3.5) (3.6b). Let X be an iterated Azéma- Yor tye embedding based on this ξ, e.g. we may take X given by (1.4) since B t^τn : t ě 0q is uniformly integrable by the same argument as in the roof of Theorem 2.1. Let j ě 1. Using the notation of Definition 1.2, it follows by monotonicity of ξ, (3.6b) and (3.5) that on the set tb τj ξ j B τj q, Bτj ě `ɛu we have B τj ξ j B τj q ξ j`1 B τj q. Therefore, the condition of (1.2) in the definition of the iterated Azéma-Yor tye embedding is not satisfied and hence τ j`1 τ j. Consequently, X tj X tj`1 X tn and Xtj X tj`1 X tn! on the set X tj ξ j X ) (3.7) tj q, Xtj ě ` ɛ for all j ě 1. Take 1 j n. Denote χ : maxtk n : Dt H X0`ɛ s.t. B t ξ k B t qu_0, where H x : inftu ą 0 : B u xu and H : tχ j 1, H X0`ɛ ă 8u. By (3.6a) we have P rhs ą 0. Further, by using ζ 1 mq ζ n mq ă m we conclude by the roerties of Brownian motion that P H X t B τj P Ou ą 0 for O Ď ` ɛ, 8q an oen set. Relabelling and using (3.6b) yields P X tj ζ j X tj q, X tj P O, X ı tj1 ă ` ɛ ą 0 for all oen O Ď ` ɛ, 8q. (3.8) 13

14 By ζ ζ either Case A or Case B below holds (ossibly by changing ɛ above). In our arguments we refer to the roof of the athwise inequality of Proosition 1.1 given by Henry-Labordère et al. [15] and argue that certain inequalities in this roof become strict. Case A: There exist m 2 ą m 1 ą ` ɛ and j n s.t. ζj! m 1 q ą ζ j m 2 q. Set O : m 1, m 2 q, and take m ą m 2. Then, on X tj ζ j X ) tj q, Xtj P O, it follows from (3.7) and Proosition 1.3 that Υ n X, m, ζq Υ j X, m, ζq ą 0 1 tm Xtj u 1 tm X tn u Υ nx, m, ζq, a.s. where the strict inequality holds by noting that X tj ζ j mqq` ą 0 for all m P m 1, m 2 q on the above set and then directly verifying that the second inequality of equation (4.3) of Henry-Labordère et al. [15] alied with ζ and X is strict. Case B: There exist m 2 ą m 1 ą ` ɛ and j n s.t. ζj! m 2 q ă ζ j m 1 q. Take m P O. Then, on X tj ζ j X ) tj q, Xtj P O X m, 8q, ă X Xtj1 0 ` ɛ, it follows again from (3.7) and Proosition 1.3 that Υ n X, m, ζq Υ j X, m, ζq ą 1 1 tm Xtj u 1 tm X tn u Υ nx, m, ζq, a.s. where the strict inequality holds by observing that the last inequality in equation (4.3) of Henry-Labordère et al. [15] alied with ζ and X is strict because X j ζ j mqq` 0 ą X j ζ j mq for all m P O on the above set. Combining, in both cases A and B the claim (3.5) follows from (3.8). Proof of Theorem 3.1. Take ζ P Z cts such that strict inequality holds for one submartingale in the definition of Z cts, see (3.3). We must have ζ ζ. As in the roof of Proosition 3.2 we choose a ξ such that (3.6a) (3.6b), (3.5) hold and let X be an iterated Azéma-Yor tye embedding based on this ξ. Proositions 1.1 and 1.3 yield E 1 rm,8q X tn q UB X, 1 rm,8q, ζ UB `X, 1 rm,8q, ą and by Proosition 3.2 UB X, 1 rm,8q, ζ ă UB `X, 1 rm,8q, ζ for all m P O where O Ď, 8q is some oen set. Now the claim follows as in the roof of Theorem 2.1. Remark 3.3. In the setting of Theorem 3.1 let ζ 1, ζ 2 P Z, ζ 1 ζ 2, and assume that (2.2) holds for φ, ζ 1 q and φ, ζ 2 q. Then there exist martingales X 1 and X 2 such that 1 2 UB X 1, φ, ζ ă UB X 1, φ, ζ, 1 2 UB X 2, φ, ζ ą UB X 2, φ, ζ. This follows by essentially reversing the roles of ζ 1 and ζ 2 in the roof of Theorem

15 3.2 No Further Imrovements with Intermediate Moments We now use the results of the revious section to show that beyond the imrovement stated in Proosition 2.3 no sharer Doob s L bounds can be obtained from the inequalities of Theorem 2.1. Proosition 3.4 (No Imrovement of Doob s L -Inequality from Theorem 2.1). Let ą 1 and ζ P Z be such that ζ j mq ζ 1 mq 1 m for some m ą and some j. Then, there exists a martingale X such that ˆ E rx T 1 s 1 ă UB X, ow, ζ. (3.9) Proof. Let α ą 1 : α and take X α satisfying 0 X α t 1 X α t j1, B τα X α t j X α t n where B is a Brownian motion started at and τ α inftu ą 0 : B u ζ α B u qu. It follows easily that for this rocess X α, UB X α, ow, j ζ UB X α, ow, ζ and hence it is enough to rove the claim for n 1 and ζ ζ j. For all α P α, α ` ɛq, ɛ ą 0, Proosition 3.2 yields existence of a nonemty, oen interval I α such that UB `X α, 1 rm,8q, ζ α ă UB X α, 1 rm,8q, j ζ for all m P I α. In fact, taking ɛ ą 0 small enough, I α can be chosen such that č I α Ě m 1, m 2 q, ă m 1 ă m 2. (3.10) αpα,α `ɛq We can further (recalling the arguments in Case A and Case B in the roof of Proosition 3.2) assume that for all α P α, α ` ɛq: UB X α, 1 rm,8q, j ζ UB `X α, 1 rm,8q, ζ α ě δ ą 0 for all m P m1, m 2 q. The claim follows by letting α Ó α and using the asymtotic otimality of X α q α, see (2.7). In addition to the result of Proosition 3.4 we rove that there is no intermediate moment refinement of Doob s L -inequalities in the sense formalized in the next Proosition. Intuitively, this could be exlained by the fact that the th moment of a continuous martingale is continuously non-decreasing and hence does not add relevant information about the th moment of the maximum. Only the final th moment matters in this context. Proosition 3.5 (No Intermediate Moment Refinement of Doob s L -Inequality). Let ą 1, 0 t 0 t 1... t n T and a 0,..., a n P R. 15

16 (i) If E ř n X T i 0 a ie X t i for every continuous non-negative submartingale X, then also ˆ 1 E rx T s 1 nÿ a i E X t i. (i) If `E 1{ ř n X T i 1 a i `E Xti X ti1 1{ for every continuous nonnegative submartingale X with 0, then also ˆ 1 E rx T sq1{ i 0 nÿ a i `E Xti X ti1 1{. i 1 Proof. From Peskir [21, Examle 4.1] or our Proosition 2.3 we know that Doob s L -inequality given in (2.6b) is enforced by a sequence of continuous martingales Y ɛ q in the sense of (2.7), i.e. ˆ 1 E r Y ɛ T s E j max Y t ɛ ` tt 1 Y ɛ 0 ` ɛ. Recall that 0 t 0 t 1... t n. We first rove (i). We take Y0 ɛ. By scalability of the asymtotically otimal martingales Y ɛ q we can assume E X t n E Y ɛ tn. In addition we can find times u 1 u n1 such that E X t i E Y ɛ ui, 1 i n 1. Furthermore, by a simle time-change argument, we may take u i t i. Therefore, writing u n t n T and using asymtotic otimality of Y ɛ q, ˆ E ˆ X t 1 n E Yt ɛ 1 n j E max Y t ɛ ` tt 1 Y 0 ɛ ` ɛ nÿ i 0 ã i E Y ɛ t i ` ɛ nÿ ã i E X t i ` ɛ where ã 0 a 0 ` { 1q, ã i a i for i 1,..., n. We obtain the required inequality by sending ɛ Œ 0 in the above. We next rove (ii). Taking a martingale which is constant until time t i1 and constant after time t i and using the fact that Doob s L inequality is shar yields ˆ 1 i a i for all i 1,..., n. The required inequality follows using triangular inequality for the L norm. Remark 3.6. Note that it follows instantly from the revious roof that we may also formulate Proosition 3.5 in terms of L norms instead of the exectations of the th moment. Also, analogous statements as in Proosition 3.5 hold for Doob s L 1 inequality. This can be argued in the same way by using that Doob s L 1 inequality is attained (cf. e.g. Peskir [21, Examle 4.2] or our Proosition 2.5) and observing that the function x ÞÑ x logxq is convex. 16

17 References [1] Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W., and Temme, J. (2013). A trajectorial interretation of Doob s martingale inequalities. Ann. Al. Probab., 23(4): [2] Azéma, J., Gundy, R. F., and Yor, M. (1980). Sur l intégrabilité uniforme des martingales continues. In Seminar on Probability, XIV (Paris, 1978/1979) (French), volume 784 of Lecture Notes in Math., ages Sringer, Berlin. [3] Azéma, J. and Yor, M. (1979a). Le roblème de Skorokhod: comléments à Une solution simle au roblème de Skorokhod. In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), volume 721 of Lecture Notes in Math., ages Sringer, Berlin. [4] Azéma, J. and Yor, M. (1979b). Une solution simle au roblème de Skorokhod. In Dellacherie, C., Meyer, P.-A., and Weil, M., editors, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), volume 721 of Lecture Notes in Math., ages Sringer, Berlin. [5] Beiglböck, M. and Nutz, M. (2014). Martingale inequalities and deterministic counterarts. Electron. J. Probab., 19(95):1 15. [6] Beiglböck, M. and Sioraes, P. (2013). Pathwise versions of the Burkholder- Davis-Gundy inequality. arxiv.org, v1. [7] Blackwell, D. and Dubins, L. E. (1963). A converse to the dominated convergence theorem. Illinois J. Math., 7: [8] Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Ann. Al. Prob, 25(2): [9] Brown, H., Hobson, D., and Rogers, L. C. G. (2001). The maximum maximum of a martingale constrained by an intermediate law. Probab. Theory Rel. Fields, 119(4): [10] Carraro, L., El Karoui, N., and Ob lój, J. (2012). On Azéma-Yor rocesses, their otimal roerties and the Bachelier-Drawdown equation. Ann. Probab., 40(1): [11] Cox, A. M. G., Hobson, D., and Ob lój, J. (2008). Pathwise inequalities for local time: Alications to Skorokhod embeddings and otimal stoing. Ann. Al. Prob, 18(5): [12] Cox, A. M. G. and Peskir, G. (2015). Embedding Laws in Diffusions by Functions of Time. Ann. Probab., to aear. Available at arxiv: v3. [13] Cox, A. M. G. and Wang, J. (2013). Root s barrier: Construction, otimality and alications to variance otions. Ann. Al. Prob, 23(3): [14] Dubins, L. E. and Gilat, D. (1978). On the distribution of maxima of martingales. Proc. Amer. Math. Soc., 68(3): [15] Henry-Labordère, P., Ob lój, J., Soida, P., and Touzi, N. (2015). The maximum maximum of a martingale with given n marginals. Ann. Al. Probab. To aear. Available at arxiv: v3. 17

18 [16] Hobson, D. G. (1998). Robust hedging of the lookback otion. Finance Stoch., 2(4): [17] Madan, D. B. and Yor, M. (2002). Making Markov Martingales Meet Marginals: With Exlicit Constructions. Bernoulli, 8(4): [18] Ob lój, J. and Soida, P. (2013). An Iterated Azéma-Yor Tye Embedding for Finitely Many Marginals. arxiv.org, v2. [19] Ob lój, J. and Yor, M. (2006). On local martingale and its suremum: harmonic functions and beyond. In From Stochastic Calculus to Mathematical Finance, ages Sringer-Verlag. [20] Osȩkowski, A. (2012). Shar martingale and semimartingale inequalities. Birkhäuser. [21] Peskir, G. (1998). Otimal Stoing of the Maximum Process: The Maximality Princile. Ann. Probab., 26(4): [22] Rogers, L. C. G. (1989). A guided tour through excursions. Bull. London Math. Soc., 21(4):