Convex Stochastic Duality and the Biting Lemma

Transcription

1 Jurnal f Cnvex Analysis Vlume 9 (2002), N. 1, Cnvex Schasic Dualiy and he Biing Lemma Igr V. Evsigneev Schl f Ecnmic Sudies, Universiy f Mancheser, Oxfrd Rad, Mancheser, M13 9PL, UK igr.evsigneev@man.ac.uk Sjur D. Flåm Deparmen f Ecnmics, Universiy f Bergen, Fsswinckels gae 6, 5007 Bergen, Nrway sjur.flaam@ecn.uib.n Received April 10, 2000 Revised manuscrip received April 26, 2001 A sandard apprach dualiy in schasic pimizain prblems wih cnsrains in L relies upn he Ysida - Hewi herem. We develp an alernaive echnique which emplys nly elemenary means. The echnique is based n an ɛ-regularizain f he riginal prblem and n passing he limi as ɛ 0 wih he help f a simple measure-hereic fac he biing lemma. Keywrds: schasic pimizain, cnvex dualiy, cnsrains in L, schasic Lagrange mulipliers, bunded ses in L 1, biing lemma, Gale s ecnmic mdel 1991 Mahemaics Subjec Classificain: 90C15, 52A41, 90C19, 90A16 1. Inrducin The paper analyzes dynamic prblems f schasic pimizain in discree ime. The prblems under cnsiderain are cncerned wih maximizing cncave funcinals n cnvex ses f feasible sraegies (prgrams). Feasibiliy is defined in erms f linear inequaliy cnsrains in L hlding alms surely. The fcus f he wrk is he exisence f dual variables schasic Lagrange mulipliers in L 1 relaxing he cnsrains. Such Lagrange mulipliers are impran in varius applicains. In paricular, hey play key rles in he analysis f sabiliy and sensiiviy f sluins schasic pimizain prblems, as well as in he design f algrihms fr cmpuing hese sluins [14]. Als, such mulipliers fen have clear inerpreains, especially in mdels relaed ecnmics, which sheds addiinal ligh n he issues under sudy [1]. The ms cmmn apprach Lagrangian relaxain fr he class f pimizain prblems in quesin is as fllws. Firs, by using an infinie-dimensinal versin f he Kuhn - Tucker herem, ne cnsrucs mulipliers belnging he dual, (L ), f he space L. Linear funcinals in (L ) are finiely addiive measures. Accrding he Ysida - Hewi herem [23], any funcinal π (L ) can be decmpsed in he sum π = π a +π s, where π a is absluely cninuus and π s is singular. Abslue cninuiy means ha π a is represenable by a funcin in L 1. The funcins in L 1 represening he absluely cninuus cmpnens f he funcinals cnsruced urn u be he desired Lagrange mulipliers. ISSN / $ 2.50 c Heldermann Verlag

2 238 I. V. Evsigneev, S. D. Flåm / Cnvex Schasic Dualiy and he Biing Lemma The idea f he mehd ulined ges back Dubviskii and Milyuin [6] (wh deal riginally wih deerminisic cninuus-ime prblems). Varians f he mehd applicable schasic mdels were develped by Evsigneev [9], [10], Radner [18], Taksar [22], Rckafellar and Wes [19], [20], Flåm [12], Dempser [5], and hers. Analgus echniques fr equilibrium (raher han pimizain) prblems arising in ecnmics were prpsed by Bewley [2]. In he presen paper, we sugges a differen prcedure fr cnsrucing he Lagrange mulipliers. We firs cnsider an apprpriae regularizain f he given prblem, depending n a small parameer, ɛ > 0. We cnsruc (L ) -mulipliers fr his versin f he prblem. Is regulariy permis shw ha he mulipliers bained are absluely cninuus, i.e., represenable by funcins in L 1. Having esablished his, we le ɛ 0 and prve ha he crrespnding vecr funcins in L 1 are nrm-bunded. Then we use an elemenary measure-hereic fac he biing lemma. By virue f his lemma, any L 1 -bunded sequence f funcins cnains a subsequence which can be mdified n a family f ses wih measures ending zer s ha he resuling sequence will be unifrmly inegrable. Any unifrmly inegrable se f funcins in L 1 has a weak limi pin. Such a limi pin prvides he sugh-fr vecr f Lagrange mulipliers. The abve prcedure can be applied a whle range f pimizain prblems wih cnsrains in L. In his paper, we have chsen as a cnvenien vehicle fr presening he mehd a schasic analgue f Gale s [13] ecnmic mdel. Schasic versins f Gale s mdel represen a class f dynamic schasic pimizain prblems which is invesigaed in much deail see, e.g., Dynkin [7], Radner [18], Arkin and Evsigneev [1]. Our assumpins are milder han hse in he lieraure cied (in paricular, hey d n imply ha he pimal values are aained). The paper is rganized as fllws. In Secin 2, we describe he pimizain prblem and sae he resuls. In Secin 3, we presen he prf f he main herem. 2. The prblem and he resul Le (Ω, F, P ) be a prbabiliy space and F 0 F 1... F T a sequence f sub-σ-algebras f F. Le d be a naural number. We dene by L 1 (resp. L ) he space f inegrable (resp. essenially bunded) F-measurable vecr funcins wih values in he Euclidean space R d. The analgus spaces f F -measurable vecr funcins are dened by L 1 () and L (). We wrie X fr he sandard nnnegaive cne in L (). In he mdel under cnsiderain we are given: nn-empy cnvex ses Z X ( 1) X (), {1,..., T }, real-valued cncave funcinals F (v), v Z, and a vecr y 0 X 0. The pimizain prblem is as fllws. (P) Maximize ver he se f sequences saisfying F (z) = F (x 1, y ) z = {(x 1, y )} T (1) (x 1, y ) Z, {1, 2,..., T }, (2)

3 I. V. Evsigneev, S. D. Flåm / Cnvex Schasic Dualiy and he Biing Lemma 239 and y x, {0,..., T 1}. (3) All inequaliies beween randm vecrs are suppsed hld crdinaewise and alms surely (a.s.). If a sequence z = {(x 1, y )} T saisfies (2), we call z a prgram and wrie z Z. If z saisfies (3) as well, hen z is ermed a feasible prgram. The pimal value f prblem (P) (which is n necessarily aained) is dened by sup(p). We assume ha sup(p)<. Sluins prblem (P) are called pimal prgrams. In he ecnmic applicains f he abve mdel, elemens (x, y) in Z are inerpreed as echnlgical prcesses (feasible inpu-upu pairs). The ses Z are called echnlgy ses. Inequaliies (3) express resurce cnsrains. The bjecive funcinals F (x, y) may represen, fr example, expeced uiliies Eu (ω, x(ω), y(ω)). Fr a finie-dimensinal vecr a = (a i ), we wrie a = a i. The L 1 -nrm f a randm vecr v(ω) is defined as v 1 = E v(ω), where E sands fr he expecain wih respec he given prbabiliy P. We pu e = (1,..., 1) R d. If Γ is a se in Ω, hen χ Γ sands fr he indicar funcin f Γ. We impse he fllwing assumpins. (A.1) We have (0, 0) Z. (A.2) If (x, y) Z, (x, y ) Z, and Γ F 1, hen χ Γ (x, y)+ (1 χ Γ )(x, y ) Z. (A.3) There exiss a prgram z= {( x 1, y )} T such ha, fr each = 0,..., T 1, we have y x +δe, where δ > 0 is sme nn-randm cnsan (fr = 0, we define y 0 = y 0 ). (A.4) Fr all v, v Z, as P (Γ) 0 (Γ F 1 ). lim[f (v) F (χ Γ v + (1 χ Γ )v)] = 0 Requiremens (A.1), (A.2) and (A.4) are suppsed be fulfilled fr all = 1,..., T. Cndiin (A.1) says ha he sequence f zer inpus and upus frms a feasible prgram (inaciviy is feasible). Hyphesis (A.2) expresses a pssibiliy f chice beween w inpu-upu pairs (x, y) Z and (x, y ) Z depending n an even Γ F 1. Assumpin (A.3) guaranees he exisence f a prgram which saisfies he resurce cnsrains wih excess. Cndiin (A.4) a cninuiy prpery f F ( ) hlds, fr example, if F (z) = Eu (ω, z(ω)), where u (ω, b) is a measurable funcin f ω Ω and b R 2d meeing he fllwing requiremen: here exiss a real-valued randm variable h (ω) 0 such ha Eh (ω) < and u (ω, v(ω)) h (ω) fr any v Z and ω Ω. Dene by P he sandard nnnegaive cne in L 1 (). The main resul, hlding under assumpins (A.1) - (A.4), is as fllws. Therem 2.1. There exis p 0 P 0,..., p T 1 P T 1 such ha fr all prgrams z = {(x 1, y )} T. F (z) + Ep (y x ) sup(p) (4) The randm vecrs p 0,..., p T described in Therem 1 represen schasic Lagrange mulipliers relaxing cnsrains (3). As is well-knwn and easy prve (see, e.g., [15], Secin

4 240 I. V. Evsigneev, S. D. Flåm / Cnvex Schasic Dualiy and he Biing Lemma 8.6), he exisence f such vecrs is equivalen he ruh f he fllwing dualiy herem: sup(p) = min(p ), where he dual prblem (P ) is defined as fllws: (P ) Minimize ver all p = (p 0,..., p T ) P 0... P T. F (p) = sup{f (z) + Ep (y x )} z Z Frm (4), i fllws immediaely ha a feasible prgram z = {( x 1, ȳ )} T is pimal if and nly if F (z) + Ep (y x ) F ( z) (5) fr all z Z. Simple argumens shw ha (5) hlds if and nly if F (x, y) + Ep y Ep 1 x F ( x 1, ȳ ) + Ep ȳ Ep 1 x 1 [(x, y) Z, = 1,..., T ], p (ȳ x ) = 0 [ = 0,..., T 1], where ȳ 0 = y 0 and p T = 0. In ecnmic erms, his means ha p 0,..., p T are suppring prices assciaed wih he resurce cnsrains (see [1], Chaper 4). In he prf f he abve herem, we will use he fllwing fac. Lemma 2.2 ( biing lemma ). Le {q m (ω)} (ω Ω, m = 1, 2,...) be a sequence f randm d-dimensinal vecrs such ha {E q m } is bunded. Then here exis measurable ses and naural numbers m 1 < m 2 <... such ha P ( k ) 1 and he sequence q mk χ k is unifrmly inegrable. Varius versins f his lemma have been esablished by many auhrs. A number f impran resuls relaed i are cnained in he papers by Casaing [3] and Saadune and Valadier [21], where ne can find furher references. A prf is given, e.g., in [21], p Remark 2.3. Accrding he Dunfrd Peis herem (see, e.g., [16], Therem II.23), every unifrmly inegrable sequence cnains a subsequence cnverging in he weak plgy σ(l 1, L ). Thus, here exis naural numbers n 1 < n 2 <..., measurable ses D 1 D 2..., and a randm vecr q L 1 such ha P (D k ) 1 and q nk χ Dk q [σ(l 1, L )]. (6) 3. Prf f he main herem 1s sep. Fix a sequence f numbers 1 > ɛ 1 > ɛ 2 >... > 0 cnverging zer. Fr each m {1, 2,...}, cnsider an auxiliary prblem (P m ) which is frmulaed like (P) wih he nly difference ha Z is replaced by Z m = {(x, y) Z : y ɛ m y }.

5 I. V. Evsigneev, S. D. Flåm / Cnvex Schasic Dualiy and he Biing Lemma 241 The definiins f prgrams and feasible prgrams fr prblem (P m ) are analgus hse fr (P) (wih Z m in place f Z ). We wrie z Z m if z is a prgram fr (P m ). Clearly sup(p m ) sup(p m+1 ) sup(p). By virue f (A.1) and (A.3), he sequence {ɛ m ( x 1, y )} belngs Z m, and, fr any {(x 1, y )} Z m, we have y ɛ m y, = 0,..., T. (7) (Fr = 0, he las inequaliy hlds because y 0 = y 0 and ɛ m < 1.) Furhermre, ɛ m y ɛ m x ɛ m δe ( = 1, 2,..., T ), and y 0 ɛ m x0 ɛ m δe. (8) Viewing inequaliies (3) as linear inequaliy cnsrains in he spaces L (0), L (1),..., L (T 1), we can use an infinie-dimensinal versin f he Kuhn-Tucker herem (see, e.g., Therem in [15]) and cnsruc nnnegaive linear funcinals π m (L ()) such ha F (x 1, y ) + < π m, y x > sup(p m ) (9) fr any prgram z = {(x 1, y )} T 1 Z m. The Slaer cnsrain qualificain, which allws apply he abve resul, hlds by virue f (8). 2nd sep. Le us shw ha he funcinals π m invlved in (9) are in fac absluely cninuus, i.e., represenable in he frm < π m, x >= Ep m x, p m P. Fix any {0, 1,..., T 1} and any naural number m. Furhermre, fix any real number κ > 0 and cnsider sme feasible prgram z = {(x 1, y )} f prblem (P m ) fr which Fr any Γ F, define a sequence z Γ = {(x Γ j 1, y Γ j )} by (x Γ j 1, y Γ j ) = F (z) sup(p m ) κ. (10) { (xj 1, y j ), j ; (1 χ Γ )(x j 1, y j ) + χ Γ ɛ m ( x j 1, y j ), j >. Observe ha z Γ is a feasible prgram fr (P m ). Indeed, if j >, hen Γ F j 1, and s (x Γ j 1, y Γ j ) = (1 χ Γ )(x j 1, y j ) + χ Γ ɛ m ( x j 1, y j ) Z j by virue f (A.2). Furher, we have y 0 x Γ 0 because x Γ 0 = x 0 if > 0 and x Γ 0 = (1 χ Γ )x 0 + χ Γ ɛ m x0 (1 χ Γ )y 0 + χ Γ ɛ m y0 y 0 if = 0. If T > j > 0 and j, hen he inequaliy y Γ j x Γ j hlds because bh {(x j 1, y j )} and {( x j 1, y j )} are feasible prgrams f prblem (P m ). If j =, hen, by virue f (7) and (8), we have y Γ j x Γ j y [(1 χ Γ )x + χ Γ ɛ m x ] (1 χ Γ )(y x ) + χ Γ (y ɛ m x ) χ Γ ɛ m ( y x ) χ Γ ɛ m δe 0. (11)

6 242 I. V. Evsigneev, S. D. Flåm / Cnvex Schasic Dualiy and he Biing Lemma By applying (9) and (11) z Γ, we find ɛ m δ < π m, χ Γ e > < π m, y Γ x Γ > sup(p m ) F (x Γ 1, y Γ ) F (x 1, y ) F (x Γ 1, y Γ ) + κ, where y Γ 0 = y 0 = y 0. Since κ > 0 is arbirary, we bain ɛ m δ < π m, χ Γ e > F (x 1, y ) F (x Γ 1, y Γ ). By virue f (A.4), he las expressin ends zer as P (Γ) 0. Thus < π m, χ Γ e > 0 as P (Γ) 0. This implies ha he funcinal π m ( 0) is absluely cninuus, i.e., represenable in he frm < π m, x >= Ep m x, x L (), where p m P (see, e.g., [17], Crllary Prpsiin IV.2.1). By subsiuing {( x 1, y )} in (9), we ge δe p m = Ep m δe Ep m ( y x ) sup(p) This shws ha he randm vecrs p m F j ( x j 1, y j ). j=1 are unifrmly bunded in he L 1 nrm. 3rd sep. By using Lemma 2.2, we find ses m F such ha P ( m ) 1, , and he sequence {χ m p m } is unifrmly inegrable. Wihu lss f generaliy, we may assume in addiin ha m 0 m 1... m T 1 (he se m can be replaced by m 0 m 1... m ). Since he sequence {χ m p m } is unifrmly inegrable, i cnains a subsequence which cnverges in he plgy σ(l 1 (), L ()). This is rue fr each = 0,..., T 1. Thus, by passing a subsequence, we may suppse, again wihu lss f generaliy, ha, fr each, χ m p m p [σ(l 1 (), L ())], where p is a nnnegaive randm vecr in L 1 (). Le z = {(x 1, y )} be any prgram f prblem (P) saisfying y ɛ y, where ɛ is sme number in (0, 1). Then z is a prgram f prblem (P m ) fr all m fr which ɛ m < ɛ (and s fr all m large enugh). Fix any such m and pu = m. Define ( x 1, ỹ ) = (1 χ 1 )ɛ( x 1, y ) + χ 1 (x 1, y ), = 1,..., T. By virue f (A.2), ( x 1, ỹ ) Z. Furhermre, ỹ ɛ y, where ɛ > ɛ m, and s z {( x 1, ỹ )} is a prgram f (P m ). Cnsequenly, where ỹ 0 = y 0. F ( x 1, ỹ ) + Ep m (ỹ x ) sup(p m ) sup(p), (12) Observe nex ha (ỹ x ) χ (y x ). Indeed, (ỹ x ) χ (y x ) (1 χ 1 )ɛ y +χ 1 \ y (1 χ )ɛ x

7 I. V. Evsigneev, S. D. Flåm / Cnvex Schasic Dualiy and he Biing Lemma 243 (1 χ 1 )ɛ y +χ 1 \ ɛ y (1 χ )ɛ x = (1 χ ) ɛ(y x ) 0, because 1 and y ɛ y. Cnsequenly, frm his and (12), we find F ( x 1, ỹ ) + Ep m χ m (y x ) sup(p). By passing he limi (using (A.4)), we cnclude F (x 1, y ) + Ep (y x ) sup(p). (13) The abve inequaliy was bained under he assumpin ha {(x 1, y )} is a prgram f (P) saisfying y ɛ y fr sme ɛ > 0. Nw cnsider any prgram {(x 1, y )} f (P). Fr each ɛ [0, 1], se (x 1, (ɛ) y (ɛ) ) = ɛ( x 1, y )+ (1 ɛ)(x 1, y ). By using he resul jus bained, we find F (x 1, (ɛ) y (ɛ) ) + Ep (y (ɛ) x (ɛ) ) sup(p) (14) fr all ɛ (0, 1]. The lef-hand side f (14) is a cncave funcin f ɛ [0, 1]. Hence, if inequaliy (14) hlds fr each ɛ (0, 1], i hlds fr ɛ = 0 as well. This prves (13) fr any prgram f (P). The prf is cmplee. Remark 3.1. We cnjecure ha he mehd described can be exended schasic mdels f ecnmic equilibrium [11] generalizing he pimizain mdel cnsidered in his paper. In ha, mre general, seing, he cunerpar f he prblem f cnsrucing schasic Lagrange mulipliers is he prblem f cnsrucing equilibrium saes (sluins cerain variainal inequaliies). The cnveninal apprach emplying he Ysida Hewi herem seems be inapplicable in ha cnex. I wuld be f ineres find u wheher he echnique based n he biing lemma culd replace he cnveninal ne in he framewrk f equilibrium mdels. Acknwledgemens. The auhrs hank an annymus referee fr helpful cmmens, which were aken in accun in he final versin f he paper. Financial suppr frm he Universiy f Bergen, NFR prjec Quanec, and frm he grans INTAS and RFBR is graefully acknwledged. References [1] V. I. Arkin, I. V. Evsigneev: Schasic Mdels f Cnrl and Ecnmic Dynamics, Academic Press, Lndn (1987). [2] T. Bewley: Exisence f equilibria in ecnmies wih infiniely many cmmdiies, Jurnal f Ecnmic Thery 4 (1972)

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