A TRAJECTORIAL INTERPRETATION OF DOOB S MARTINGALE INEQUALITIES

Transcription

1 A RAJECORIAL INERPREAION OF DOOB S MARINGALE INEQUALIIES B. ACCIAIO, M. BEIGLBÖCK, F. PENKNER, W. SCHACHERMAYER, AND J. EMME Abstract. We resent a unified aroach to Doob s L maximal inequalities for 1 <. he novelty of our method is that these martingale inequalities are obtained as consequences of elementary deterministic counterarts. he latter have a natural interretation in terms of robust hedging. Moreover our deterministic inequalities lead to new versions of Doob s maximal inequalities. hese are best ossible in the sense that equality is attained by roerly chosen martingales. Keywords: Doob maximal inequalities, martingale inequalities, athwise hedging. Mathematics Subject Classification (21): Primary 6G42, 6G44; Secondary 91G2. 1. Introduction In this aer we derive estimates for the running maximum of a martingale or nonnegative submartingale in terms of its terminal value. Given a function f we write f (t) = su u t f (u). Among other results, we establish the following martingale inequalities: heorem 1.1. Let (S n ) n= be a non-negative submartingale. hen E [ ] ( S ) E[S 1 ], 1 < <, (Doob-L ) E[ S ] e [ E[S log(s )] + E [ ( S 1 log(s ) )] ]. (Doob-L 1 ) e 1 Here (Doob-L ) is the classical Doob L -inequality, (1, ), [8, heorem 3.4]. he second result (Doob-L 1 ) reresents the Doob L 1 -inequality in the shar form derived by Gilat [1] from the L log L Hardy-Littlewood inequality. rajectorial inequalities. he novelty of this note is that the above martingale inequalities are established as consequences of deterministic counterarts. We ostone the general statements (Proosition 2.1) and illustrate the sirit of our aroach by a simle result that may be seen as the trajectorial version of Doob s L 2 -inequality: Let s,..., s be real numbers. hen s 2 + 4[ 1 n= s n(s n+1 s n ) ] 4s 2. (Path-L2 ) Inequality (Path-L 2 ) is comletely elementary and the roof is straightforward: it suffices to rearrange terms and to comlete squares. he significance of (Path-L 2 ) rather lies in the fact that it imlies (Doob-L 2 ). Indeed, if S = (S n ) n=1 is a non-negative submartingale, we may aly (Path-L 2 ) to each trajectory of S. he decisive observation is that, by the submartingale roerty, E [ 1 n= S n (S n+1 S n ) ], (1.1) hence (Doob-L 2 ) follows from (Path-L 2 ) by taking exectations. All authors situated at the University of Vienna, Faculty of Mathematics, Nordbergstraße 15, A-19 Wien, Corresonding author: B. Acciaio, beatrice.acciaio@univie.ac.at, Phone/Fax: /5727. he authors thank Jan Obłoj for insightful comments and remarks. 1

2 2 B. ACCIAIO, M. BEIGLBÖCK, F. PENKNER, W. SCHACHERMAYER, AND J. EMME Inequalities in continuous time sharness. Passing to the continuous time setting, it is clear that (Doob-L ) and (Doob-L 1 ) carry over verbatim to the case where S = (S t ) t [,] is a non-negative càdlàg submartingale, by the usual limiting argument. It is not surrising that also in continuous time one has trajectorial counterarts of those inequalities, the sum in (Path-L 2 ) being relaced by a carefully defined integral. Moreover, in the case = 1 the inequality can be attained by a martingale in continuous time (cf. [1] and [11]). Notably, this does not hold for 1 < <. We discuss this for the case = 2 in the L 2 -norm formulation: Given a non-negative càdlàg submartingale S = (S t ) t [,] we have S 2 2 S 2. (Doob-L 2 ) Dubins and Gilat [9] showed that the constant 2 in (Doob-L 2 ) is otimal, i.e. can not be relaced by a strictly smaller constant. It is also natural to ask whether equality can be attained in (Doob-L 2 ). It turns out that this haens only in the trivial case S ; otherwise the inequality is strict. Keeing in mind that equality in (Doob-L 1 ) is attained, one may try to imrove also on (Doob-L 2 ) by incororating the starting value of the martingale. Indeed, we obtain the following result: heorem 1.2. For every non-negative càdlàg submartingale S = (S t ) t [,] S 2 S 2 + S S 2. (1.2) Inequality (1.2) is shar. More recisely, given x, x 1 R, < x x 1, there exists a ositive, continuous martingale S = (S t ) t [,] such that S 2 = x, S 2 = x 1 and equality holds in (1.2). In heorem 3.1 we formulate the result of heorem 1.2 for 1 < <, thus establishing an otimal a riori estimate on S. We emhasize that the idea that (Doob-L ) can be imroved by incororating the starting value S into the inequality is not new. Cox [7], Burkholder [5] and Peskir [18] show that E[ S 2 ] 4E[S 2 ] 2E[S 2 ]. (1.3) Here the constants 4 res. 2 are shar (cf. [18]) with equality in (1.3) holding iff S. 1 Financial interretation. We want to stress that (Path-L 2 ) has a natural interretation in terms of mathematical finance. Financial intuition suggests to consider the ositive martingale S = (S n ) n= as the rocess describing the rice evolution of an asset under the so-called risk-neutral measure, so that Φ(S,..., S ) = ( S ) 2, res. ϕ(s ) = S 2, have the natural interretation of a socalled exotic otion, res. a Euroean otion, written on S. In finance, a Euroean otion ϕ, res. exotic otion Φ, is a function that deends on the final value S of S, res. on its whole ath S,..., S. he seller of the otion Φ ays the buyer the random amount Φ(S,..., S ) after its exiration at time. Following [2] we may interret E[Φ] as the rice that the buyer ays for this otion at time (Cf. [19, Ch. 5] for an introductory survey on risk-neutral ricing). Here we take the oint of view of an economic agent who sells the otion Φ and wants to rotect herself in all ossible scenarios ω Ω, i.e., against all ossible values Φ(S (ω),..., S (ω)), which she has to ay to the buyer of Φ. his means that she will trade in the market in order to arrive at time with a ortfolio value which is at least as 1 hat (1.2) imlies (1.3) follows from a simle calculation. Alternatively the sharness of (1.3) is a consequence of the fact that equality in (1.2) can be attained for all ossible values of S 2, S 2.

3 A RAJECORIAL INERPREAION OF DOOB S MARINGALE INEQUALIIES 3 high as the value of Φ. By buying a Euroean otion ϕ(s ) = S 2, she can clearly rotect herself in case the asset reaches its maximal value at maturity. However, she still faces the risk of S having its highest value at some time n before. o rotect against that ossibility, one way for her is to go short in the underlying (i.e., to hold negative ositions in S ). By scaling, her rotecting strategy should be roortional to the running maximum S n. At this oint our educated guess is to follow the strategy H n = 4 S n, meaning that from time n to time n + 1 we kee an amount H n of units of the asset S in our ortfolio. he ortfolio strategy roduces the following value at time : 1 1 H n (S n+1 S n ) = 4 S n (S n+1 S n ). (1.4) n= he reason why we have chosen the secial form H n = 4 S n now becomes aarent when considering (Path-L 2 ) and (1.1). In our financial mind exeriment this may be interreted as follows: by buying 4 Euroean otions S 2 and following the self-financing trading strategy H = (H n ) 1 n=, the seller of the otion Φ = ( S ) 2 covers her osition at maturity, whatever the outcome (S (ω),..., S (ω)) of the rice evolution is. hus an uer bound for the rice of the exotic otion Φ in terms of the Euroean otion ϕ is given by E[( S ) 2 ] 4E[S 2 ]. n= We note that Henry-Labordère [12] derived (Doob-L ) in a related fashion. he idea of robust ricing and athwise hedging of exotic otions seemingly goes back to Hobson [13], see also [4, 6, 15]. We refer the reader to [14] for a thorough introduction to the toic. Organisation of the aer. In Section 2 we rove Doob s inequalities (Doob-L ) and (Doob-L 1 ) after establishing the trajectorial counterarts (Path-L ) and (Path-L 1 ). We rove heorem 1.2 and its L version in Section Proof of heorem 1.1 he aim of this section is to rove Doob s maximal inequalities in heorem 1.1 by means of deterministic inequalities, which are established in Proosition 2.1 below. he roof of heorem 1.1 is given at the end of this section. As regards (Doob-L ), we rove the stronger result E [ S which was obtained in [7, 18]. ] ( ) E[S 1 ] Proosition 2.1. Let s,..., s be non-negative numbers. (I) For 1 < < and h(x) := 2 1 x 1, we have 1 s h( s i ) ( ) s i+1 s i (II) For h(x) := log (x), we have s e e 1 ( 1 1 E[S ], 1 < <, (2.1) 1 s + ( ) s 1. (Path-L ) h( s i ) ( ) s i+1 s i + s log (s ) + s (1 log(s )) ). (Path-L 1 )

4 4 B. ACCIAIO, M. BEIGLBÖCK, F. PENKNER, W. SCHACHERMAYER, AND J. EMME We note that for = 2 inequality (Path-L ) is valid also in the case where s,..., s are real (ossibly negative) numbers. A continuous time counterart of (Path-L ) is given in Remark 3.5 below. In the roof of Proosition 2.1, we need the following identity: Lemma 2.2. Let s,..., s be real numbers and h : R R any function. hen 1 h( s i ) ( ) 1 s i+1 s i = h( s i ) ( s ) i+1 s i + h( s )(s s ). (2.2) Proof. his follows by roerly rearranging the summands. Indeed, observe that for a term on the right-hand side there are two ossibilities: if s i+1 = s i res. s = s, it simly vanishes. Otherwise it equals a sum h( s k )(s k+1 s k ) h( s m )(s m+1 s m ) where s k =... = s m. In total, every summand on the left-hand side of (2.2) is accounted exactly once on the right. We note that Lemma 2.2 is a secial case of [17, Lemma 3.1]. Proof of Proosition 2.1. (I) By convexity, x + x 1 (y x) y, x, y. Lemma 2.2 yields 1 h( s i ) ( ) 2 s i+1 s i = s 1 i+1 s i s 1 i ( s i+1 s i ) 2 1 s 1 (s s ) = s 2 1 s 1 s + 1 s. Hence 2 1 s 1 (s s ) (2.3) We therefore have 1 h( s i ) ( ( ) ) s i+1 s i + s 1 ( ) 1 s s ( 1) s 2 1 s 1 s + s 1. (2.4) o establish (Path-L ) it is thus sufficient to show that the right-hand side of (2.4) is nonnegative. Defining c such that S n = c S n, this amounts to showing that g(c) = ( 1) 2 1 c + ( 1 ) c. (2.5) Using standard calculus we obtain that g reaches its minimum at ĉ = 1 where g(ĉ) =. (II) By Lemma 2.2 we have 1 h( s i ) ( ) 1 s i+1 s i = log( s i )( s i+1 s i ) log( s )(s s ) 1 ( si+1 s i s i+1 log ( s i+1 ) + s i log ( s i ) ) log ( s ) (s s ) = s s + s log(s ) s log( s ),

5 A RAJECORIAL INERPREAION OF DOOB S MARINGALE INEQUALIIES 5 where the inequality follows from the convexity of x x + x log (x), x >. If s = then the above inequality shows that (Path-L 1 ) holds true. Otherwise, we have 1 s h( s i ) ( ( ) ) s s i+1 s i + s s log(s ) + s log(s ) + s log. Note that the function x x log (y/x) on (, ), for any fixed y >, has a maximum in ˆx = y/e, where it takes the value y/e. his means that s log ( s /s ) s /e which concludes the roof. We are now in the osition to rove heorem 1.1. Proof of heorem 1.1. By Proosition 2.1 (I), for h(x) := 2 1 x 1 we have 1 S h( S i ) ( ( ) ) S i+1 S i 1 S + S 1. (2.6) Since S is a submartingale and h is negative, E[ 1 h( S i ) ( ) S i+1 S i ] and thus (2.1) and consequently (Doob-L ) follow from (2.6) by taking exectations. Inequality (Doob-L 1 ) follows from Proosition 2.1 (II) in the same fashion. Remark 2.3. Given the terminal law µ of a martingale S, Hobson [14, Section 3.7] also rovides athwise hedging strategies for lookback otions on S. As oosed to the strategies given in Proosition 2.1, we emhasize that the strategies in [14] deend on µ. 3. Qualitative Doob L Inequality Proof of heorem 1.2 In this section we rove heorem 1.2 as well as the following result which ertains to (1, ). heorem 3.1. Let (S t ) t [,] be a non-negative submartingale, S, and 1 < <. hen S 1 S 1 S. (3.1) 1 S 1 Given the values S and S, inequality (3.1) is best ossible. More recisely, given x, x 1 R, < x x 1, there exists a ositive, continuous martingale S = (S t ) t [,] such that S = x, S = x 1 and equality holds in (3.1). Moreover, equality in (3.1) holds if and only if S is a non-negative martingale such that S is continuous and S = αs, where α [1, 1 ). Remark 3.2. We rove heorem 3.1 by introducing a athwise integral in continuous time. Note that inequality (3.1) can also be obtained without defining such an integral. However, the definition of the athwise integral will allow us to characterize all submartingales for which equality in (3.1) holds. Connection between heorem 1.2 and heorem 3.1. We now discuss under which conditions heorem 1.2 and heorem 3.1 are equivalent for = 2. Recall that heorem 1.2 asserts that and heorem 3.1 reads in the case of = 2 as S 2 S 2 + S S 2 (3.2) S 2 2 S 2 S 2 2 S 2. (3.3) s

6 6 B. ACCIAIO, M. BEIGLBÖCK, F. PENKNER, W. SCHACHERMAYER, AND J. EMME If S is a martingale, then (3.2) and (3.3) are equivalent. Indeed, rearranging (3.3) yields ψ( S 2 ) := 1 2 S 2 + S S 2 S 2, (3.4) and by inverting the strictly monotone function ψ on [ S 2, ) we obtain S 2 ψ 1 ( S 2 ) = S 2 + S 2 2 S 2 2. Since S is a martingale, S 2 2 S 2 2 = S S 2, which gives (3.2). If S is a true submartingale, then the estimate in (3.2) is in fact stronger than (3.3). his follows from the above reasoning and the fact that for a true submartingale we have S 2 2 S 2 2 > S S 2. Clearly, it would be desirable to obtain also for general an inequality of the tye (3.2), which is in the case of a martingale S equivalent to (3.1), and where S only aears on the left-hand side. By similar reasoning as for = 2, finding such an inequality is tantamount to inverting the function ψ(x) = 1 x + S x, 1 which is strictly monotone on [ S, ). Since finding ψ 1 amounts to solving an algebraic equation, there is in general no closed form reresentation of ψ 1 unless {2, 3, 4}. Definition of the continuous-time integral. For a general account on the theory of athwise stochastic integration we refer to Bichteler [3] and Karandikar [16]. Here we are interested in the articular case where the integrand is of the form h( S ) and h is monotone and continuous. In this setu a rather naive and ad hoc aroach is sufficient (see Lemma 3.3 below). Fix càdlàg functions f, g : [, ] [, ) and assume that g is monotone. We set ( ) g t- d f t := lim g ti - fti+1 f ti n t i π n (3.5) if the limit exists for every sequence of finite artitions π n with mesh converging to. he standard argument of mixing sequences then imlies uniqueness. We stress that (3.5) exists if and only if the non redictable version g ( ) t d f t = lim n t i π n g ti fti+1 f ti exists; in this case the two values coincide. By rearranging terms one obtains the identity t i π g ti ( f ti+1 f ti ) = t i π { }} { f ti (g ti+1 g ti ) + g f g f (g ti+1 g ti )( f ti+1 f ti ). (3.6) If it is ossible to ass to a limit on either of the two sides, one can do so on the other. Hence, g t d f t is defined whenever f t dg t is defined and vice versa, since the monotonicity of g imlies that ( ) converges. In this case we obtain the integration by arts formula g t d f t = f t dg t + g f g f (g t g t- )( f t f t- ). (3.7) t i π t ( )

7 A RAJECORIAL INERPREAION OF DOOB S MARINGALE INEQUALIIES 7 Below we will need that the integrals h( f t ) d f t and whenever h is continuous, monotone and f is càdlàg. In the case of seen by slitting f in its continuous and its jum art. Existence of f t dh( f t ) are well defined f t dh( f t ) this can be h( f t ) d f t is then a consequence of (3.7). he following lemma establishes the connection of the just defined athwise integral with the standard Ito-intgral. Lemma 3.3. Let S be a martingale on (Ω, F, (F t ) t, P) and h be a monotone and continuous function. hen (h( S ) S ) (ω) = h( S t- (ω)) ds t (ω) P-a.s., (3.8) where the left hand side refers the Ito-integral while the right hand side aeals to the athwise integral defined in (3.5). Proof. Karandikar ([16, heorem 2]) roves that (h( S ) S ) (ω) = lim h( S ti n -(ω)) ( S ti+1 (ω) S ti (ω) ) t i π n for a suitably chosen sequence of random artions π n, n 1. According to the above discussion, h( S t- (ω)) ds t (ω) = lim n t i π n h( S ti -(ω)) ( S ti+1 (ω) S ti (ω) ) for any choice of artions π n (ω), n 1 with mesh converging to. We are now able to establish a continuous-time version of Proosition 2.1 Proosition 3.4. Let f : [, ] [, ) be càdlàg. hen for h(x) := 2 1 x 1 we have f 1 h( f t ) d f t + f 1 1 f 1 1 f. (3.9) Equality in (3.9) holds true if and only if f is continuous. Similarly, a continuous-time version of (Path-L 1 ) also holds true. Proof. Inequality (3.9) follows from (2.3) by assing to limits. We now show that equality in (3.9) holds iff f is continuous. o simlify notation, we consider the case = 2. (3.7) imlies h( f t ) d f t = 4 f t d f t 4 f f + 4 f ( f t f t- )( f t f t- ), t 2 f 2 4 f f + 2 f 2, (3.1) where equality in (3.1) holds iff f is continuous. Hence, equality in (3.9) holds true iff f is continuous. If we choose f to be the ath of a continuous martingale, the integral in (3.9) is a athwise version of an Azéma-Yor rocess, cf. [17, heorem 3]. Remark 3.5. Passing to limits in (Path-L ) in Section 2 we obtain that for every càdlàg function f : [, ] [, ) ( t d f t + f 2 ) f 1 1 f 1 Alternatively this can be seen as a consequence of (3.9). 1 f, 1 < <.

8 8 B. ACCIAIO, M. BEIGLBÖCK, F. PENKNER, W. SCHACHERMAYER, AND J. EMME Lemma 3.6. Let (S t ) t [,] be a non-negative submartingale and 1 < <. Set S = M + A, where M is a martingale and A is an increasing, redictable rocess with A =. hen E[ S ] 1 E[S 1 A ] + 1 E[ S 1 S ] 1 1 E[S ]. (3.11) Equality holds in (3.11) if and only if S is a martingale such that S is a.s. continuous. Proof. By Proosition 3.4 we find for h(x) = 2 1 x 1 S 1 h( S t ) ds t + 1 S S S, (3.12) where equality holds iff S is continuous. Since [ ] E 1 h( S t ) da t 1 E[S 1 A ], (3.13) (3.11) follows by taking exectations in (3.12). As the estimate in (3.13) is an equality iff A =, we conclude that equality in (3.11) holds iff S is a martingale such that S is continuous. We note that in the case of = 2 also [1, Corollary ] imlies that equality in (3.11) holds for every continuous martingale S. Proof of heorem 3.1 and heorem 1.2. By Lemma 3.6 and Hölder s inequality we have S 1 1 E[S 1 A ] + 1 S 1 S S, (3.14) E[S 1 A ] + 1 S 1 S 1 1 S, (3.15) where equality in (3.14) holds for every martingale S such that S is continuous, and equality in (3.15) holds whenever S is a constant multile of S. Since E[S 1 A ] we obtain (3.1) after dividing by S 1. In order to establish (1.2) in heorem 1.2 for = 2, we rearrange terms in (3.15) to obtain ψ( S 2 ) := 1 2 S 2 + 2E[S A ] + S 2 2 S 2. 2 S 2 Similarly as in the discussion after Remark 3.2 above, inverting ψ on [ S 2, ) imlies S 2 S 2 + S 2 2 2E[S A ] S 2 2. Since for every submartingale S we have S 2 2 2E[S A ] S 2 2 = S S 2, this roves (1.2). In order to rove that (3.1), res. (1.2), is attained, we have to ensure the existence of a -integrable martingale S such that S is continuous and S is a constant multile of S. o this end we may clearly assume that x = 1. Fix α (1, 1 ) and let B = (B t) t be a Brownian Motion starting at B = 1. Consider the rocess B τ α = (B t τα ) t obtained by stoing B at the stoing time τ α := inf { t > : B t B t /α }.

9 A RAJECORIAL INERPREAION OF DOOB S MARINGALE INEQUALIIES 9 his stoing rule corresonds to the Azéma-Yor solution of the Skorokhod embedding roblem (B, µ), cf. [1], where the robability measure µ is given by dµ dx = α 1 α 1 2α 1 x α 1 1[α (α 1) 1, )(x). Clearly B τ α is a uniformly integrable martingale. herefore the rocess (S t ) t [,] defined as S t := B t t τ is a non-negative martingale satisfying S α = S /α. S is -integrable for α (1, 1 ) and S runs through the interval (1, ) while α runs in (1, 1 ). his concludes the roof. Note that the roof in fact shows that equality in (3.1) holds if and only if S is a nonnegative martingale such that S is continuous and S = αs, where α [1, References 1 ). [1] J. Azéma and M. Yor. 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, [2] L. Bachelier. héorie de la séculation. Ann. Sci. École Norm. Su. (3), 17:21 86, 19. [3] K. Bichteler. Stochastic integration and L -theory of semimartingales. Ann. Probab., 9(1):49 89, [4] H. Brown, D. Hobson, and L.C.G. Rogers. Robust hedging of barrier otions. Math. Finance, 11(3): , 21. [5] D.L. Burkholder. Exlorations in martingale theory and its alications. In École d Été de Probabilités de Saint-Flour XIX 1989, volume 1464 of Lecture Notes in Math., ages Sringer, Berlin, [6] A.M.G. Cox and J. Obłój. Robust ricing and hedging of double no-touch otions. ArXiv e-rints, January 29. [7] D.C. Cox. Some shar martingale inequalities related to Doob s inequality. In Inequalities in statistics and robability (Lincoln, Neb., 1982), volume 5 of IMS Lecture Notes Monogr. Ser., ages Inst. Math. Statist., Hayward, CA, [8] J.L. Doob. Stochastic rocesses. Wiley Classics Library. John Wiley & Sons Inc., New York, 199. Rerint of the 1953 original, A Wiley-Interscience Publication. [9] L.E. Dubins and D. Gilat. On the distribution of maxima of martingales. Proc. Amer. Math. Soc., 68(3): , [1] D. Gilat. he best bound in the L log L inequality of Hardy and Littlewood and its martingale counterart. Proc. Amer. Math. Soc., 97(3): , [11] S.E. Graversen and G. Peškir. Otimal stoing in the L log L-inequality of Hardy and Littlewood. Bull. London Math. Soc., 3(2): , [12] P. Henry-Labordère. Personal communication. [13] D. Hobson. he maximum maximum of a martingale. In Séminaire de Probabilités, XXXII, volume 1686 of Lecture Notes in Math., ages Sringer, Berlin, [14] D. Hobson. he Skorokhod embedding roblem and model-indeendent bounds for otion rices. In Paris- Princeton Lectures on Mathematical Finance 21, volume 23 of Lecture Notes in Math., ages Sringer, Berlin, 211. [15] D. Hobson and M. Klimmek. Model indeendent hedging strategies for variance swas. rerint, 211. [16] R.L. Karandikar. On athwise stochastic integration. Stochastic Process. Al., 57(1):11 18, [17] J. Obłój and M. Yor. On local martingale and its suremum: harmonic functions and beyond. In From stochastic calculus to mathematical finance, ages Sringer, Berlin, 26. [18] G. Peškir. he best Doob-tye bounds for the maximum of Brownian aths. In High dimensional robability (Oberwolfach, 1996), volume 43 of Progr. Probab., ages Birkhäuser, Basel, [19] S. Shreve. Stochastic calculus for finance. II. Sringer Finance. Sringer-Verlag, New York, 24. Continuous-time models.