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1 SIAM J. CONTROL OPTIM. Vol. 52, No. 5, pp c 204 Socey for Indural and Appled Mahemac Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// ON THE ROBUST OPTIMAL STOPPING PROBLEM ERHAN BAYRAKTAR AND SONG YAO Abrac. We udy a robu opmal oppng problem wh repec o a e P of muually ngular probable. Th can be nerpreed a a zero-um conroller-opper game n whch he opper ryng o maxmze payoff whle an advere player wan o mnmze h payoff by choong an evaluaon crera from P. We how ha he upper Snell envelope Z of he reward proce Y a upermarngale wh repec o an appropraely defned nonlnear expecaon E, and Z furher an E -marngale up o he fr me τ when Z mee Y. Conequenly, τ he opmal oppng me for he robu opmal oppng problem and he correpondng zero-um game ha a value. Alhough he reul eem mlar o he one obaned n he clacal opmal oppng heory, he muual ngulary of probable and he game apec of he problem gve re o maor echncal hurdle, whch we crcumven ung ome new mehod. Key word. robu opmal oppng, zero-um game of conrol and oppng, volaly uncerany, dynamc programmng prncple, Snell envelope, nonlnear expecaon, wea ably under pang, pah-dependen ochac dfferenal equaon wh conrol AMS ubec clafcaon. Prmary, 60G40, 93E20; Secondary, 49L20, 9A5, 60G44, 9G80 DOI. 0.37/ Inroducon. We olve a connuou-me robu opmal oppng problem wh repec o a nondomnaed e P of muually ngular probable on he canoncal pace Ω of connuou pah. Th opmal oppng problem can alo be nerpreed a a zero-um conroller-opper game n whch he opper ryng o maxmze payoff whle an advere player wan o mnmze h payoff by choong an evaluaon crera from P. In our man reul, Theorem 5., we conruc an opmal oppng me and how ha he correpondng game ha a value. More precely, we oban ha. up τ T nf P P E P Yτ =nf E P Yτ =nf P P P P up τ T E P Yτ. Here T denoe he e of all oppng me wh repec o he naural flraon F of he canoncal proce B, Y an F-adaped rgh-connuou-wh-leflm RCLL càdlàg proce afyng an one-ded unform connuy condon ee 3., and τ he fr me Y mee upper Snell envelope Z ω = nf P P,ω up τ T E P Yτ,ω,, ω 0,T Ω. Pleae refer o econ 2 5 for he noaon. The proof of h reul urn ou o be que echncal for hree reaon. Fr, nce he probably e P doe no adm a domnang probably, here no domnaed convergence heorem for he nonlnear expecaon E ω =nf P P,ω E P, Receved by he edor December 23, 203; acceped for publcaon n reved form July 2, 204; publhed elecroncally Ocober 9, 204. hp:// Deparmen of Mahemac, Unvery of Mchgan, Ann Arbor, MI 4809 erhan@umch.edu. Th auhor reearch wa uppored n par by he Naonal Scence Foundaon, a career gran DMS , and an Appled Mahemac Reearch gran DMS-8673, and n par by he Suan M. Smh Profeorhp. Any opnon, fndng, and concluon or recommendaon expreed n h maeral are hoe of he auhor and do no necearly reflec he vew of he Naonal Scence Foundaon. Deparmen of Mahemac, Unvery of Pburgh, Pburgh, PA 5260 ongyao@p. edu. 335

2 336 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// ω 0,T Ω. So we canno follow echnque mlar o he one ued n he clacal heory of opmal oppng due o El Karou 5 o oban he marngale propery of he upper Snell envelope Z. Second, we do no have a meaurable elecon heorem for oppng raege, whch complcae he proof of he dynamc programmng prncple. Moreover, he local approach ha ued comparon prncple of vcoy oluon o how he exence of game value ee, e.g., 6 and doe no wor for our pah-dependen e-up. In Theorem 5., we demonrae ha Z an E -upermarngale, and an E - marngale up o τ, he fr me Z mee Y, from whch. mmedaely follow. To prove h heorem, we ue a more global approach raher han he local approach. We ar wh a dynamc programmng prncple, ee Propoon 4., whoe uper-oluon par echncally dffcul due o he lac of meaurable elecon for oppng me. We overcome h ue by ung a counable dene ube of T o conruc a uable approxmaon. Th dynamc programmng reul ued o how he connuy of he upper Snell envelope, whch play an mporan role n he man heorem a our reul heavly rely on he conrucon of approxmang oppng me for τ. However, he dynamc programmng prncple drecly ener he proof of Theorem 5. o how he upermarngale propery of Z only afer we upgrade he dynamc programmng prncple for random horzon n Propoon 4.3. We would le o emphaze ha he ubmarngale propery of he upper Snell envelope Z unl τ doe no drecly follow from he dynamc programmng prncple. Inead, we buld a delcae approxmaon cheme ha nvolve carefully pang probable and leveragng he marngale propery of he ngle-probably Snell envelope unl hey mee Y. Le u ay a few word abou our aumpon. I hould no come u a urpre ha a a funcon of, ω, he probably e P, ω need o be adaped. The mo mporan aumpon on he probably cla {P, ω},ω 0,T Ω he wea ably under pang; ee P2 n econ 3. I hard o envon ha a dynamc programmng reul could hold whou a ably under pang aumpon. Th aumpon along wh he aforemenoned connuy aumpon 3. on Y he regulary aumpon on he reward are common and can be verfed for example of payoff of all fnancal dervave allow u o conruc approxmae raege for he conroller by appropraely choong condonal drbuon. Our ably aumpon weaer han counerpar n Eren, Touz, and Zhang 2; ee, for example, our Remar 3.4 for a furher dcuon. We how n econ 6 ha h aumpon along wh oher aumpon we mae on he probably cla are afed for ome pah-dependen SDE wh conrol, whch repreen a large cla of model on mulaneou drf and volaly uncerany. A ronger ably aumpon a n 2 lead o reul whch applcable only for volaly uncerany. We ee econ 6, whch we dedcae one hrd of our paper o, a one of he man conrbuon of our paper. Anoher aumpon we mae on he probably cla ha he augmenaon of he flraon generaed by he canoncal proce wh repec o each probably n he cla rgh-connuou. Th becaue, a menoned above, we explo he reul from he clac opmal oppng heory on he marngale propery of he Snell envelope for a gven probably. Agan, he example n econ 6 hown o afy h aumpon. Relevan leraure. Snce he emnal wor 34, he marngale approach wa exenvely ued n opmal oppng heory ee, e.g., 27, 5, and Appendx D of

3 ON THE ROBUST OPTIMAL STOPPING PROBLEM 337 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// 2 and ha been appled o varou problem emmng from mahemacal fnance, he mo mporan example of whch he compuaon of he uper hedgng prce of he Amercan conngen clam 7, 8, 9, 23. Opmal oppng under Knghan uncerany/nonlnear expecaon/r meaure or he cloely relaed conrolleropper-game have araced a lo of aenon n he recen year 24, 25, 7, 9, 0, 32, 2, 3, 4, 5, 8, 26. In h leraure, he e of probable aumed o be domnaed by a ngle probably or he conroller only allowed o nfluence he drf. When he e of probable conan muually ngular probable or he conroller can nfluence no only he drf bu alo he volaly, reul are avalable only n ome parcular cae. Karaza and Sudderh 22 condered he conrolleropper-game n whch he conroller allowed o conrol he volaly a well a he drf and reolved he addle pon problem for cae of one-dmenonal ae varable ung he characerzaon of he value funcon n erm of he cale funcon of he ae varable. In he muldmenonal cae, howed he exence of he value of a game ung a comparon prncple for vcoy oluon. Our echncal e-up follow cloely ha of 2, whch analyzed a conrol problem wh dcreonary oppng.e., up τ T up P P E P Y τ n a non-marovan framewor wh muually ngular probably pror. The oluon of h problem wa an mporan echncal ep n exendng he noon of vcoy oluon o he fully nonlnear pah-dependen PDE n 3 and 4. Nuz and Zhang 30 ndependenly and around he ame me addreed he problem we are conderng by ung a dfferen and an elegan approach: They exploed he ower propery of he nonlnear expecaon E developedn29odervehee-marngale propery of he dcree me veron of he lower Snell envelope Z ω =up τ T nf P P,ω E P Yτ,ω,, ω 0,T Ω. In conra, we ae an approach we conder o be very naural: We wor wh he upper Snell envelope and buld our approxmaon drecly n connuou me leveragng he nown reul from he clacal opmal oppng heory. In he nroducon, 30 ae ha hey canno wor on upper Snell envelope due o he meaurably elecon ue; ee paragraph 3 on page 3 of ha paper. Our paper overcome h ue. A maor benef of our approach ha we do no have o aume ha he reward proce bounded nce we do no have o rely on he approxmaon from dcree o connuou me. Anoher benef he weaer connuy aumpon we mpoe on he value funcon n he pah; compare Aumpon 4. n our paper and Aumpon 3.2 n 30. The laer requre he value of any oppng raegy o be connuou wh he ame modulu of connuy, whch an aumpon ha no ealy verfable. One rong u of 30 he addle pon analy, whch wor under he wea formulaon of he problem. The re of he paper organzed a follow: In econ 2 we wll nroduce noaon and ome prelmnary reul uch a he regular condonal probably drbuon. In econ 3, we e up he age for our man reul by mpong ome aumpon on he reward proce and he clae of muually ngular probable. Then econ 4 ude propere of he upper Snell envelope of he reward proce uch a pah regulary and dynamc programmng prncple. They are he eence o reolve our man reul on he robu opmal oppng problem aed n econ 5. In econ 6, we gve an example of pah-dependen SDE wh conrol ha afe all our aumpon. The proof of our reul are deferred o econ 7, and he appendx conan ome echncal lemmaa needed for he proof of he man reul. 2. Noaon and prelmnare. Le M,ϱ M be a generc merc pace and le BM beheborelσ-feld of M. For any x M and δ > 0, O δ x = {x

4 338 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// M : ϱ M x, x <δ} and O δ x = {x M : ϱ M x, x δ}, repecvely, denoe he open and cloed ball cenered a x wh radu δ. Fx d N. Le S d >0 and for all R d d -valued povely defne marce. We denoe by BS d >0 heborelσ-feld of under he relave Eucldean opology. S >0 d Gven 0 T<, leω,t = {ω C, T ; R d :ω =0} be he canoncal pace over he perod, T, whoe null pah ω 0 wll be denoed by 0,T. For any S T, we nroduce a emnorm,s on Ω,T : ω,s =upr,s ωr for all ω Ω,T.Inparcular,,T a norm on Ω,T, called unform norm, under whch Ω,T a eparable complee merc pace. Alo, he runcaon mappng Π,T,S from Ω,T o Ω,S defned by Π,T,S ω r = ωr ω ω Ω,T, r, S. The canoncal proce B,T on Ω,T a d-dmenonal Brownan moon under he Wener meaure P,T 0 on Ω,T, BΩ,T. Le F,T = {F,T = σbr,t ; r, },T be he naural flraon of B,T and le C,T collec all cylnder e n F,T T : C,T = { m E : m N, < < < m T, {E } m = BRd }. Iwellnownha B,T = { B BΩ,T =σc,t,t E =σ r :r, T, E BR } d = F,T T. Le P,T denoe he F,T -progrevely meaurable σ-feld of, T Ω,T and le T,T collec all F,T -oppng me. We e T,T = {τ T,T : τ } for each, T and wll ue he convenon nf =. From now on, we hall fx a me horzon T 0, and drop from he above Ω, 0,, B, P 0, noaon,.e., Ω,T, 0,T,,T, B,T, P,T 0, F,T, P,T, T,T F, P, T. When S = T,Π,T,T wll be mply denoed by Π. For any 0 T, ω Ω,andδ>0, defne Oδ ω = {ω Ω : ω ω, <δ} In parcular, Oδ T ω= O δ ω ={ω Ω : ω ω,t <δ}. Snce Ω he e of R d -valued connuou funcon on, T arng from 0, Oδ ω = { ω Ω : ω ω, δ δ/n } n N 2. = n N r, Q = n N r, Q { ω Ω : ω r ωr δ δ/n } { ω Ω : B r ω O δ δ/n ωr } F. We fx a counable dene ube { ω } N of Ω under,andeθ = {Oδ ω : δ Q +, N} F. Gven 0,T and a probably P on Ω, BΩ = Ω, FT, le u e N P = {N Ω : N Afor ome A FT wh PA =0}. The P-augmenaon FP of F con of F P = σf N P,, T. In parcular, we wll wre N for N P 0 and F = {F },T for F P P 0 = {F 0 },T. We denoe by T P he collecon of all F P -oppng me and e T P = {τ T P : τ } for each, T. The compleon of Ω, FT, P he probably pace Ω, FT P, P wh P FT = P; we ll wre P for P for convenence. In parcular, he expecaon on Ω, F T, P 0

5 ON THE ROBUST OPTIMAL STOPPING PROBLEM 339 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// wll be mply denoed by E. A probably pace Ω, F, P calledanexenon of Ω, FT, P ff T F and P F T = P. For any merc pace M and any M-valued proce X = {X },T, we e F X = {F X = σx r ; r, },T a he naural flraon of X and le F X,P = {F X,P = σf X N P },T. If X F P -adaped, hold for any, T ha F X F P X,P and hu F F P. The followng pace abou P wll be frequenly ued n wha follow: For any ub-σ-feld G of FT P, le L G, P be he pace of all real-valued, G-meaurable random varable ξ wh ξ L G,P = E P ξ <. 2 Le D F, P rep.,s F, P be he pace of all real valued, F -adaped procee {X },T whoe pah are all rgh-connuou rep., connuou and afy E P X <, wherex =up,t X. Alo, by eng φx= x ln + x, x 0,, we defne DF, P = {X D F, P: E P φx < }. For any x, y 0,, f z = x y<2, φx + y φ2z <φ4; oherwe, f z 2, φx + y φ2z=2z ln2z < 2z ln z 2 =4z ln z =4φz 4φx+φy. So 2.2 φx + y 4φx+4φy+φ4. If he upercrp =0, we wll drop hem from he above noaon. For example, 0 = 0 0,T and T = T 0,T. 2.. Concaenaon of ample pah. In he re of h econ, le u fx 0 T. We concaenae an ω Ω and an ω Ω a me by ω ω r = ωr {r,} + ω+ ωr {r,t } r, T, whch ll of Ω. For any nonempy à Ω,weeω = and ω à = {ω ω : ω Ã}. The nex reul how ha A F con of elemen ω Ω wh ω A. Lemma 2.. Le A F.Ifω A, henω Ω A. Oherwe,fω/ A, hen ω Ω A c. For any F -meaurable random varable η, nce{ω Ω : ηω =ηω} F, Lemma 2. how ha 2.3 ω Ω {ω Ω : ηω =ηω},.e., ηω ω=ηω ω Ω. On he oher hand, for any A Ω we e A,ω = { ω Ω : ω ω A} a he proecon of A on Ω along ω. Inparcular,,ω =. For any r, T, he operaon,ω proec an Fr-meaurable e o an Fr - meaurable e whle he operaon ω ae an Fr -meaurable e a npu and reurn an Fr -meaurable e. Lemma 2.2. Gven ω Ω and r, T, we have A,ω F r for any A F r,and ω à F r for any à F r. Corollary 2.. Gven τ T and ω Ω,fτω Ω r, T for ome r, T, hen τ,ω T r. For any D, T Ω, we accordngly e D,ω ={r, ω, T Ω :r, ω ω D}.

6 340 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// Regular condonal probably drbuon. Le P be a probably on Ω, BΩ. In vrue of Theorem.3.4 and.3.5 of 37, here ex a famly {P ω } ω Ω of probable on Ω, BΩ, called he regular condonal probably drbuon of P wh repec o F, uch ha for any A F T, he mappng ω Pω A F -meaurable; for any ξ L F T, P, E P ω ξ =E Pξ F ω forp-a.. ω Ω ; for any 2.4 ω Ω, P ω ω Ω =. Gven ω Ω, by Lemma 2.2, ω à FT for any à F T. So we can deduce from 2.4 ha P,ω à = P ω 2.5 ω à à F T } r,t F -adaped rep., F -progrevely meaurable. Propoon 2.2. If ξ L FT, P for ome probably P on Ω, BΩ, hen hold for P-a.. ω Ω ha he hfed random varable ξ,ω L FT, P,ω and 2.6 defne a probably on Ω, FT. The Wener meaure, however, are nvaran under pah hf. Lemma 2.3. Le 0 T. I hold for P 0-a.. ω Ω ha P 0,ω = P 0. Than o he exence of regular condonal probably drbuon we can defne condonal drbuon ung 2.5. Then by nroducng pah regulary for he reward proce Y, one can rea pah-dependen problem n way mlar o ae-dependen problem. Th can be een a he general dea behnd a dynamc programmng n he pah-dependen eng and he pah-dependen PDE nroduced n Shfed random varable and hfed procee. Gven a random varable ξ and a proce X = {X r } r,t on Ω, for any ω Ω we defne he hfed random varable ξ,ω by ξ,ω ω = ξω ω, ω Ω andhehfedprocex,ω by Xr,ω ω =Xr, ω ω, r, ω, T Ω. In lgh of Lemma 2.2 and he regular condonal probably drbuon, hfed random varable/procee nher meaurably and negrably a follow. Propoon 2.. Le M be a generc merc pace and le ω Ω. If an M-valued random varable ξ on Ω Fr-meaurable for ome r, T, hen ξ,ω Fr -meaurable. 2 If an M-valued proce {X r } r,t F -adaped rep., F -progrevely meaurable, hen he hfed proce {X,ω r E P,ω ξ,ω = E P ξ F ω R. A a conequence of 2.6, a hfed P 0 null e or dr dp 0-null e alo ha zero meaure. Lemma 2.4. For any N N, hold for P 0 -a.. ω Ω ha N,ω N ;for any D B, T FT wh dr dp 0D, T Ω = 0, hold for P 0-a.. ω Ω ha dr dp 0D,ω =0. The proof of reul n h econ can be found n 36, 35, ee alo 6. In he nex hree econ, we wll gradually provde he echncal e-up and preparaon for our man reul Theorem 5. on he robu opmal oppng problem.

7 ON THE ROBUST OPTIMAL STOPPING PROBLEM 34 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// 3. Wea ably under pang. In he proof of Theorem 5., we wll ue an approxmaon cheme whch explo reul from he clac opmal oppng heory for a gven probably. For h purpoe, we conder he followng probably e. Defnon 3.. For any 0,T, lep collec all probable P on Ω, BΩ uch ha F P rgh-connuou. We wll alo need ome regulary aumpon on he reward proce. Sandng aumpon on reward proce Y. Y Y an F-adaped proce ha afe an one-ded connuy condon n, ω wh repec o ome modulu of connuy funcon ρ 0 n he followng ene: Y ω Y 2 ω 2 ρ 0 d,ω, 2,ω T, ω,ω 2 Ω, where d,ω, 2,ω 2 = 2 + ω ω 2 2 0,T. Remar 3.. A poned ou n Remar 3.2 of 2, 3. mple ha each pah of Y RCLL wh pove ump. 2 Alo, one can deduce from 3. ha he proce Y lef upper emconnuou,.e., for any, ω 0,T Ω, Y ω lm Y ω. I follow ha he hfed proce Y,ω alo lef upper emconnuou. Then we can apply he clacal opmal oppng heory o Y,ω under each P P. Acually, he proof of Theorem 5. rele on he comparon of Z,ω wh he Snell envelope of Y,ω under each P P. The nex reul how ha L ln L-negrably of hfed reward proce ndependen of he gven pah hory. Lemma 3.. Aume Y. For any 0,T and any probably P on Ω, BΩ, f Y,ω DF, P for ome ω Ω, heny,ω DF, P for all ω Ω. We hall focu on he followng ube of P ha mae he hfed reward proce L ln L-negrable. = {P P : Y,0 DF, P} no Aumpon 3.. For any 0,T, he e P Y empy. Remar 3.2. If Y DF, P 0, hen Lemma 2.3, 2.6, and Lemma 3. mply ha P 0 PY for any 0,T. 2 A we wll ee n Lemma 6., when he modulu of connuy ρ 0 ha polynomal growh, he law of oluon o he conrolled SDE 6. over perod, T belongop Y. Under Y and Aumpon 3., we ee from Lemma 3. ha for any 0,Tand P P Y, 3.2 Y,ω D F, P ω Ω. Nex, we need he probably clae o be adaped and wealy able under pang n he followng ene. Sandng aumpon on probably cla. P0 For any 0,T, le u conder a famly {P, ω =P Y, ω} ω Ω of ube of P Y whch adaped n he ene ha P, ω =P, ω 2 fω 0, =ω 2 0,. So P =P0, 0=P0,ω for all ω Ω.

8 342 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// We furher aume ha he probably cla {P, ω},ω 0,T Ω afy he followng wo condon for ome modulu of connuy funcon ρ 0 : for any 0 < T, ω ΩandP P, ω. P There ex an exenon Ω, F, P ofω, F T, P andω F wh P Ω = uch ha for any ω Ω, P, ω P, ω ω. P2 For any δ Q + and N, le{a } =0 be a F -paron of Ω uch ha for =,...,, A O δ ω forome ω Ω. Then for any P P, ω ω, =,...,, here ex a P P, ω uch ha PA A 0 =PA A 0, A F T ; for any =,..., and A F, PA A =PA A and 3.3 up τ T E P A A Yτ,ω E P { ω A A} up E P Y,ω ω + ρ0 δ. T From now on, when wrng Yτ,ω,wemeanY,ω τ no Y τ,ω. Remar 3.3. A we wll how n econ 7, boh de of 3.3 are fne. In parcular, he expecaon on rgh-hand de well-defned nce he mappng ω up T E P Y,ω ω connuou. 2 The condon P2 can be vewed a a wea ably under pang nce mpled by he ably under fne pang ee, e.g., 4.8 of 35: for any 0 < T, ω Ω, P P, ω, δ Q +,and N, le{a } =0 be a F-paron of Ω uch ha for =,...,, A Oδ ω forome ω Ω. Then for any P P, ω ω, =,...,, here ex a P P, ω uch ha 3.4 PA=PA A 0 + E P { ω A}P A, ω A FT. = Remar 3.4. The reaon we aume P2 raher han he ably of fne pang 3.4 le n he fac ha he laer doe no hold for our example of pah-dependen SDE wh conrol econ 6 a poned ou n Remar 3.6 of 28, whle he former uffcen for our approxmaon mehod n provng he man reul. 4. The dynamc programmng prncple. The ey o olvng problem. he followng upper Snell envelope of he reward procee: 4. Z ω = nf up E P Y,ω τ P P,ω τ T, ω 0,T Ω. In h econ, we derve ome bac propere of Z and he dynamc programmng prncple afe. Thee reul wll provde an mporan echncal ep for he proof of Theorem 5.. Le Y, P0, P, and P2 hold hroughou he econ. Gven, ω 0,T Ω, nce Y F -meaurable, 2.3 mple ha Y,ω =Y ω. I hen follow from 4. ha 4.2 Z ω nf E P Y,ω = Y ω, ω 0,T Ω. P P,ω

9 ON THE ROBUST OPTIMAL STOPPING PROBLEM 343 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// We need wo addonal aumpon on Z before dcung pah regulary propere and dynamc programmng prncple. Aumpon 4.. There ex a modulu of connuy funcon ρ ρ 0 uch ha for any 0,T Z ω Z ω ρ ω ω 2 0, ω,ω 2 Ω. Remar 4.. If P, ω doe no depend on ω for all 0,T, hen 3. mple Aumpon 4.. Remar 4.2. Aumpon 4. on Z mple ha Z F-adaped. Aumpon 4.2. For any α>0, here ex a modulu of connuy funcon ρ α uch ha for any 0,T 4.4 up ω O α 0 up E P ρ δ +2 up Br ρ α δ δ 0,T. P P,ω r,+δ T Smlar o 3.2, one ha he followng negrably reul of hfed procee of Z. Lemma 4.. Gven, ω 0,T Ω, hold for any P P, ω and, T ha E P Z,ω <. A o he dynamc programmng prncple, we preen fr a bac veron n whch he ran horzon deermnc. Propoon 4.. For any 0 T and ω Ω, 4.5 Z ω = nf up E P {τ<} Yτ,ω + {τ } Z,ω. P P,ω τ T Conequenly, all pah of Z are connuou. Propoon 4.2. For any, ω 0,T Ω and P P, ω, Z,ω S F, P. The connuy of Z allow u o derve a general veron of dynamc programmng prncple wh random horzon. Propoon 4.3. For any, ω 0,T Ω and ν T, 4.6 Z ω nf up E P {τ<ν} Yτ,ω P P,ω τ T + {τ ν} Z,ω ν The revere nequaly hold under an addonal condon; ee 6 for deal. Bu h no needed for our man reul. 5. Robu opmal oppng. In h econ, we ae our man reul on robu opmal oppng problem. Le Y, P0, P, P2, and Aumpon hold hroughou he econ. For any 0,T, we e L = {random varable ξ on Ω : ξ,ω L FT, P for all ω Ω, P P, ω} and defne on L a nonlnear expecaon: E ξω = nf P P,ω E P ξ,ω for all ω Ω, ξ L. Remar 5.. Gven τ T, Y τ, Z τ L for any 0,T, han o 3.2 and Propoon 4.2. Smlar o he clac opmal oppng heory, we wll how ha he fr me Z mee Y. τ =nf{ 0,T:Z = Y }

10 344 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// an opmal oppng me for., and he upper Snell envelope Z ha a marngale characerzaon wh repec o he nonlnear expecaon E = {E } 0,T. Theorem 5.. Le Y, P0, P, P2 and Aumpon hold. If up,ω 0,T Ω Y ω =, we furher aume ha for ome L>0 5. Y 2 ω Y ω L + φ up Y r ω + ρ r 0, Then Z an E -upermarngale and {Z ene ha 0 2 T, ω Ω. up r, 2 ωr ω = Z τ } 0,T an E -marngale n he 5.2 Z ω E Zτ ω and Z ω =E Z τ ω, ω 0,T Ω, τ T. In parcular, he F-oppng me τ afe.. Remar 5.2. Smlar o 30, we can apply. o ubhedgng of Amercan opon n a fnancal mare wh volaly uncerany. 2 A o a wor-cae r meaure Rξ =up P P E P ξ defned for any bounded fnancal poon ξ, applyng. o a gvenbounded rewardproce Y yeld ha nf τ T RY τ = up τ T nf P P E P Y τ = nf P P E P Y τ =RY τ. So τ alo an opmal oppng me for he opmal oppng problem of R. 3 From he perpecve of a zero-um conroller-opper game n whch he opper chooe he ermnaon me whle he conroller elec he drbuon law from P,. how ha uch a game ha a value E 0 Y τ = nf P P E P Y τ a lower value up τ S nf P P E P Y τ concde wh he upper one nf P P up τ S E P Y τ. 6. Example: Pah-dependen conrolled SDE. In h econ we wll preen an example of he probably cla {P, ω},ω 0,T Ω n he cae of pahdependen SDE wh conrol. Le κ>0 and le b: 0,T Ω R d d R d be a P BR d d /BR d -meaurable funcon uch ha b, ω, u b, ω,u κ ω ω 0, and b, 0,u κ+ u ω, ω Ω,, u 0,T R d d. Le, ω 0,T Ω. b,ω r, ω, u =br, ω ω, u, r, ω, u, T Ω R d d clearly a P BR d d /BR d - meaurable funcon ha afe b,ω r, ω, u b,ω r, ω,u κ ω ω,r and b,ω r, 0,u κ + ω 0, + u, ω, ω Ω, r, u, T R d d. For any 0,T, le U collec all S d >0-valued, F -progrevely meaurable procee {μ },T uch ha μ κ, d dp 0 -a.. Gven μ U, mlar o he

11 ON THE ROBUST OPTIMAL STOPPING PROBLEM 345 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// clacal SDE heory, an applcaon of fxed-pon eraon how ha he followng SDE on he probably pace Ω, FT, P 0: 6. X = b,ω r, X, μ r dr + μ r dbr,, T, adm a unque oluon X,ω,μ,whchanF -adaped connuou proce. Noe ha he SDE 6. depend on ω 0, va he generaor b,ω. Whou lo of generaly, we aume ha all pah of X,ω,μ are connuou and arng from 0. Oherwe, by eng N = {ω Ω : X,ω,μ ω 0 or he pah X,ω,μ X,ω,μ ω no connuou} N,onecanae = N cx,ω,μ,, T. I an F -adaped proce ha afe 6. and whoe pah are all connuou and arng from 0. Applyng he Burholder Dav Gundy nequaly and Gronwall nequaly and ung he Lpchz connuy of b n ω, one can ealy derve he followng emae for X,ω,μ : for any p E ϕ p ω 0, p/2 and E X,ω,μ p 6.2 up Xr,ω,μ p r, C p ω ω p 0, p ω Ω, up r, r X,ω,μ r where ϕ p a modulu of connuy funcon dependng on p, κ, T and C p denoe a conan dependng on p, κ, T. Smlar o Lemma 3.3 of 30, he followng reul how ha he hf of X,ω,μ exacly he oluon of SDE 6. wh hfed drf coeffcen and hfed conrol. See 6 for proof. Propoon 6.. Gven 0 T, ω Ω, andμ U,leX = X,ω,μ.I hold for P 0 -a.. ω Ω ha μ, ω U and ha X, ω = X,ω X ω,μ, ω + X ω. A a mappng from Ω o Ω, X,ω,μ F /F -meaurable for any, T : To ee h, le u pc up an arbrary E BR d. The F -adapne of X,ω,μ how ha for any r, X,ω,μ B r E 6.3 Thu Br E G X,ω,μ ha F GX,ω,μ,.e., 6.4 = { ω Ω : X,ω,μ ω Br E } = { ω Ω : X,ω,μ r ω E } F. = {A Ω :X,ω,μ A F },aσ-feld of Ω. I follow X,ω,μ A F A F, provng he meaurably of he mappng X,ω,μ. We defne he law of X,ω,μ under P 0 by p,ω,μ A = P 0 X,ω,μ A A G X,ω,μ T, and denoe by P,ω,μ he rercon of p,ω,μ on Ω, FT. The flraon FP,ω,μ are all rgh-connuou. Propoon 6.2. For any, ω 0,T Ω and μ U, P,ω,μ belong o P. Remar 6.. ThereaonweconderhelawofX,ω,μ under P 0 over GX,ω,μ T he large σ-feld o nduce P 0 under he mappng X,ω,μ raher han FT a

12 346 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// follow. Our proof for Propoon 6.2 and 6.3 rely heavly on he nvere mappng W,ω,μ of X,ω,μ,whchanF -progevely meaurable procee havng only p,ω,μ - a.. connuou pah. Conequenly, a we wll how n he proof of he followng Propoon 6.3, hold for p,ω,μ -a.. ω Ω ha he hfed probably P,ω,μ, ω he law of he oluon o he hfed SDE and hu belong o P, ω ω. Th explan why our aumpon P need an exenon Ω, F, P of he probably pace Ω, F T, P. Gven ϖ, le ρ 0 be a modulu of connuy funcon uch ha 6.5 ρ 0 δ κ+δ ϖ δ>0, and le Y afy Y wh ρ 0.WeeP, ω ={P,ω,μ : μ U }. Lemma 6.. Aume Y and 6.5. For any, ω 0,T Ω, we have P, ω P Y. For any ω,ω 2 Ω wh ω 0, = ω 2 0,, nce 6. depend only on ω 0,,we ee ha X,ω,μ = X,ω2,μ and hu P,ω,μ = P,ω2,μ for any μ U. I follow ha P, ω =P, ω 2. So aumpon P0 afed. Propoon 6.3. Aume Y and 6.5. Then he probably cla {P, ω},ω 0,T Ω afe P, P2, and Aumpon 4. and Proof. 7.. Proof of he reul n econ 3. ProofofLemma3.. Le 0,TandP be a probably on Ω, BΩ. Suppoe ha Y,ω DF, P foromeω Ω and fx ω Ω. The F-adapne of Y and Propoon 2. 2 how ha Y,ω F -adaped. Gven ω Ω, 3. mple ha for any, T 7. Y,ω I follow ha Y,ω Y,ω ω Y,ω ω = Y ω ω Y ω ω ρ 0 ω ω ω ω 0, =ρ0 ω ω 0,. ω =up,t Y,ω ω up,t Y,ω ω +ρ 0 ω ω 0, = 4E P φy,ω + ω +ρ 0 ω ω 0,. Then 2.2 mple ha E P φy,ω 4φρ 0 ω ω 0, +φ4<. SoY,ω DF, P. Proof of Remar 3.3. Le ω, ω 2 Ω. For any T, mlar o 7., we can deduce ha Y,ω ω I follow ha 7.2,ω ω2 ω Y ω ρ 0 ω ω ω ω ω 2 ω 0, ω = ρ 0 ω ω 2, ω Ω. E P Y,ω ω EP Y,ω ω 2 + ρ0 ω ω 2,. Tang upremum over T yeld ha up T E P Y,ω ω up T E P Y,ω ω2 +ρ 0 ω ω 2,T. Exchangng he role of ω and ω 2 how,ω ω ha he mappng ω up T E P Y connuou and hu FT - meaurable. Then he expecaon on he rgh-hand de of 3.3 welldefned.

13 ON THE ROBUST OPTIMAL STOPPING PROBLEM 347 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// ha Nex, le u how ha boh de of 3.3 are fne: For any τ T, 3.2 how ha E P A A Yτ,ω,ω,ω E P Yτ E PY <, whch lead o he,ω fac ha < E PY up E P τ T A A Yτ,ω,ω E PY <. On he oher hand, gven ω A A and T, applyng 7.2 wh ω, ω 2 = ω, ω and ω, ω 2 = ω, ω, repecvely, yeld ha E P Y,ω ω EP,ω Y ω EP + Y,ω ω Y,ω ω,ω E P Y ω,ω + ρ0 ω ω, EP Y ω + ρ0 δ. I hen follow from 3.2 ha E P { ω A A} up E P Y,ω ω + ρ0 δ T,ω E P Y ω E P { ω A A} E P Y,ω ω + ρ0 δ+ ρ 0 δ PA A <, and up E P Y,ω ω + ρ0 δ T ρ0 δ+ ρ 0 δ PA A >. 2 Gven A FT, for any =,..., and ω A,nceA F, Lemma 2. how ha A, ω =Ω or A, ω, whch mple ha A A 0, ω =. So eay o calculae ha PA A 0 =PA A 0. Nex, le =,..., and A F. We ee from Lemma 2. agan ha 7.3 f ω A A rep., / A A, and hen A A, ω =Ω rep., =. Then PA A = = E P { ω A }P A A, ω = = E P { ω A A} { ω A }P Ω = PA A. Gven τ T,nceτ, ω T by Corollary 2., we can deduce from 7.3 agan E P A A Y,ω τ = A A E P { ω A }E P Yτ,ω, ω = where we ued he fac ha Y,ω τ = E P { ω A A}E P = E P { ω A A}E P E P { ω A A} up E P T Yτ,ω, ω,ω ω Yτ, ω Y,ω ω,, ω ω =Yτ,ω ω ω =Y τ ω ω,ω ω ω = Y τ, ω ω, ω ω ω =Y,ω ω τ, ω ω, ω =Y,ω ω ω ω Ω. τ, ω 7.2. Proof of he reul n econ 4. ProofofRemar4.. Le 0,Tandω,ω 2 Ω. For any P P, τ T, and ω Ω,7.howha Y,ω ω Y,ω2 ω ρ 0 ω ω 2 0,,, T. In parcular, Y,ω τ ω, ω Y,ω2 τ ω, ω ρ 0 ω ω 2 0,. I hen follow ha 7.4 E P Y,ω τ EP Y,ω 2 τ + ρ0 ω ω 2 0,.

14 348 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// Tang upremum over τ T and hen ang nfmum over P P yeld ha Z ω Z ω 2 +ρ 0 ω ω 2 0,. Exchangng he role of ω and ω 2,weoban 4.3 wh ρ = ρ 0. Proof of Lemma 4.. Le 0 T, ω Ω, and P P, ω. If =, az F - meaurable by Remar 4.2, 2.3 how ha E P Z,ω =E P Z ω = Z ω <. So le u aume <. For any ω Ω, one can deduce ha 7.5,ω ω Y ω = up r,t = Y,ω Y r, ω ω ω ω ω = Y,ω up r,t Y r,ω, ω ω ω Ω. ω ω By P, here ex an exenon Ω, F, P ofω, FT, P andω F wh P Ω = uch ha for any ω Ω, P, ω P, ω ω. Snce Y,ω DF, P D F, P by 3.2, we ee from 2.6 ha for all ω Ω excep on ome N N P, E P, ωy,ω, ω =E P Y,ω F ω. Le A be he F T -meaurable e conanng N wh PA =0. Forany ω Ω A c F, 4.2 and 7.5 mply ha Y ω ω Z ω ω up τ T E P, ω E P, ω Y,ω Y,ω ω τ EP, ω Y,ω ω, ω = E P Y,ω F ω, o Ω A c à = {Y,ω how ha à F, and hen follow ha Pà =P à P Ω A c =. Tow, 7.6 Z,ω Y,ω Z,ω E P Y,ω F }. Remar 4.2 and Propoon 2.2 E P Y,ω F, P-a.., whch lead o ha E P Z,ω E P Y,ω + E P Y,ω F 2E P Y,ω <. Proof of Propoon 4.. Fx 0 T and ω Ω. If =, Remar 4.2 and 2.3 mply ha Z,ω = Z ω. Then 4.5 clearly hold. So we u aume < and defne 7.7 To how 7.8 Y r = Y,ω r and Z r = Z,ω r r, T. Z ω nf up E P {τ<} Y τ + {τ } Z, P P,ω τ T we hall pae he local approxmang mnmzer P ω of Z,ω ω accordng o P2 and hen mae ome emaon. Fx ε>0 and le δ>0 uch ha ρ 0 δ ρ 0 δ ρ δ <ε/4. Gven ω Ω,we can fnd a P ω P, ω ω uch ha Z ω ω up E P ω Y,ω 7.9 ω τ ε/4. τ T Clearly, Oδ ω anopeneofω. For any ω Oδ ω, an analogy o 7.4 how ha E P ω Y,ω ω τ EP ω Y,ω ω τ + ρ0 ω ω ω ω 0, + ρ0 ω ω, = E P ω Y,ω ω τ τ T.

15 ON THE ROBUST OPTIMAL STOPPING PROBLEM 349 Tang upremum over τ T, we can deduce from 4.3 and 7.9 ha Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// 7.0 up E P ω Y,ω ω τ Z ω ω + τ T 2 ε Z ω ω +ρ ω ω, + 2 ε Z ω ε ω Oδ ω. Nex, fx P P, ω and N. For =,...,,weea =O δ ω \ < Oδ ω F by 2. and e P = P ω where ω defned rgh afer 2.. Le P be he probably of P, ω n P2 ha correpond o he paron {A } =0 and he probable {P } =,wherea 0 = = A c F.So 7. E P ξ=e P ξ, ξ L F, P L F, P and E P A0 ξ=e P A0 ξ ξ L FT, P L FT, P. Gven τ T, one can deduce from 3.2, 3.3, 7., and 7.0 ha E P Yτ = EP {τ<} Y τ + {τ } A0 Y τ + = E P {τ } A Yτ,ω E P {τ<} Y τ + {τ } A0 Y τ + E P {τ ω } { ω A} up E P Y,ω ω + ρ0 δ = T E P {τ<} Y τ + {τ } A0 Y τ + {τ } A c 0 Z + ε E P {τ<} Y τ + {τ } Z + E P A0 Y + Z + ε. Tang upremum over τ T yeld ha 7.2 Z ω up E P Yτ up E P {τ<} Y τ + {τ } Z τ T τ T + E P c Y + Z + ε. A = Snce N A = N Oδ ω =Ω and nce E P Y + Z < by 3.2 and Lemma 4., leng n 7.2, we can deduce from he domnaed convergence heorem ha Z ω up τ T E P {τ<} Y τ + {τ } Z +ε. Evenually, ang nfmum over P P, ω on he rgh-hand de and hen leng ε 0, we oban A o he revere of 7.8, uffce o how for a gven P P, ω ha 7.3 up E P {τ<} Y τ + {τ } Z up E P Yτ. τ T τ T Le u ar wh he man dea of provng 7.3: Conrary o 7.9, we need upper bound for Z,ω h me. Fr noe ha Z,ω ω up T E P, ω

16 350 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// Y,ω ω ω Ω. Gven T, 2.6 mple ha,ω ω 7.4 E P, ω Y = E P YΠ F ω EP Y τ F ω hold for any ω Ω excep on a P-null e N,where τ an opmal oppng me. Snce T an uncounable e, we canno ae upremum over T for P-a.. ω Ω n 7.4 o oban 7.5 Z E P Y τ F P, P-a.. To overcome h dffculy, we hall conder a dene counable ube Γ of T n ene of a Conrucon of Γ: For any n N, weed n =, T {2 n } N {T } and D = n N D n. Gven q D, we mply denoe he counable ube Θ q of Fq by {O q } N and defne Υ q = {q I O q + T I O q : c I {,...,}} T N. For any n, N, we e Γ n, = { q Dn τ q : τ q Υ q } T.ThenΓ= n, N Γ n, clearly a counable ube of T. Snce he flraon F P rgh-connuou, and nce he proce Y rgh-connuou and lef upper em-connuou by Remar 3.2, he clac opmal oppng heory how ha eup τ T P E P Y τ F P adm an opmal oppng me τ T P, whch he fr me afer he proce Y mee he RCLL modfcaon of Snell envelope {eup τ T P r E P Y τ Fr P} r,t. Fx ε>0. We clam ha here ex a τ T uch ha 7.6 E P Y τ Y τ <ε/4. To ee h, le n be an neger 2. Gven =,...,n,wee n + n T and An = { n < τ n } FP wh n n 0 =. By, e.g., Problem of 20, here ex an A n F uch ha n A n ΔA n N P. Defne A n =A n \ <A n F and A n n = n = A n = n = A n F T.Thenτ n = n = A n n a T P-oppng me whle τ n = n = A n n + A n ct defne an T -oppng me. Clearly, τ n concde wh τ n over n = An A n, whoe complemen n = An \A n nfacofn P becaue for each {,...,n} A A n \A n = An n c < A n = A n \A n A n < An 7.7 A n ΔA n A n < An c A n ΔA n N P. To w, τ n = τ n, P-a.. Snce lm n τ n = τ and nce E P Y < by 3.2, we can deduce from he rgh-connuy of he hfed proce Y =

17 ON THE ROBUST OPTIMAL STOPPING PROBLEM 35 and he domnaed convergence heorem ha Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// 7.8 lm n E P Y τ n Y τ = lmn E P Y τn Y τ =0. So here ex a N N uch ha E P Y τ N Y τ <ε/4,.e., 7.6 hold for τ = τ N. 2b In he nex wo ep, we wll gradually demonrae 7.5. Snce E P Y < and nce Π T T P for any T by Lemma A., applyng Lemma A.2 wh X = B how ha excep on an N N P E P YΠ F = EP YΠ F P eup E P Yτ F P 7.9 τ T P = E P Y τ F P = EP Y τ F Γ. Alo n lgh of 2.6, here ex anoher Ñ N P uch ha for any ω Ñ c, YΠ E P YΠ F, ω ω =EP, ω 7.20 = E P, ω Γ, Y,ω ω whereweuedhefachaforany ω Ω YΠ, ω ω =YΠ ω ω =Y Π ω ω,ω ω ω = Y ω, ω ω ω = Y,ω ω ω. By P, here ex an exenon Ω, F, P ofω, FT, P andω F wh P Ω = uch ha for any ω Ω, P, ω P, ω ω. Le  be he FT -meaurable e conanng N Ñ and wh P =0. Now, fx ω Ω Âc and e Tr, ω ω P, = Tr, r, T. Analogou o τ, he fr me ω T, ω ω P, = T when he proce Y,ω ω mee he RCLL modfcaon of Snell envelope {eup T, ω E r P, ωy,ω ω ω P, Fr } r,t an opmal oppng me for up T, ω E P, ωy,ω ω. Smlar o 7.6, here ex a ω T uch ha 7.2 E P, ω Y,ω ω ω Y,ω ω ω <ε/4. 2c Nex, we wll approxmae ω by a equence {n } n N n Γ: A P, ω P, ω ω, 3.2 how ha E P, ωy,ω ω <. So here ex a δ>0 uch ha,ω ω E P, ω A Y <ε/4 for any A F 7.22 T wh P, ω A <δ. à n Θ q n Gven n N and { 2 n,..., 2 n T }, leq n = { n = {O qn à n + ω < 2 } F n q n } N uch ha = + 2 T D n n and } l N of. We can fnd a ubequence {On, l O n, l N l and P, ω Ãn > P, ω O n, δ l N l 2 n T 2.

18 352 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// See Lemma A.7 of 6 for deal. Moreover, here ex an l n uch ha P, ω O n > P, ω O n, δ 7.24 l N l 2 n T 2 wh O n = ln l= On, l Fq. Clearly, n n ome n N. Se Ôn how ha Ãn \Ôn = O n \ = 2 n On F q n N = q n O n + T O n c Υqn for n. An analogy o 7.7 = Ãn On c = 2 n On l NO n, l \O n = 2 n On Ãn c. I hen follow from 7.23 and 7.24 ha P, ω Ãn \Ôn P, ω 7.25,ω ω Y n + = 2 n l N On, l \O n P, ω l N On, l Ãn < δ 2 n T 2 δ 2 n T. Se Ôn = 2n T = 2 n Ôn = 2n T = 2 n On and n =max{ n : = 2 n,..., 2 n T }, weeeha n = 2n T = 2 n n = 2 n T = 2 n qn Ô + n Ôc T a n oppng me of Γ n,n, whch equal o n = 2 n T = 2 n qn à T over n 2 A n = n T = 2 n Ãn Ôn F T. A T 2n = 2 n Ãn =Ω, 7.25 mple ha P, ω A c n=p, ω 2n T = 2 n Ãn \Ôn = 2 n T = 2 n P, ω Ãn \Ôn <δ.,ω ω,ω ω I hen follow from 7.22 ha E P, ω Y Y n =E n P ω, A c n Y,ω ω 2E n P, ω A c n Y,ω ω <ε/2, whch ogeher wh 7.9 and 7.20 how ha E P, ωy < E n P, ωy +ε/2 n E P Y τ F ω+ε/2. Snce lm n n = ω and nce E,ω ω P, ωy <, leng n, we can deduce from 7.2, he rgh-connuy of he hfed proce Y,ω ω and he domnaed convergence heorem ha Z ω =Z ω ω up up T, ω E P, ω E P, ω Y,ω ω ω T E P, ω Y,ω ω,ω ω Y,ω ω = EP, ω Y,ω ω ω,ω ω + ε/4 = lmn E P, ω Y,ω ω n + ε/4 E P Y τ F 3 ω+ 4 ε ω Ω Âc. Snce Z F by Remar 4.2 and Propoon 2.2, an analogy o 7.6 yeld ha 7.26 Z E P Y τ F 3 + ε, P-a.. 4 If endng ε o 0 and applyng Lemma A.2 wh X = B now, we wll mmedaely oban 7.5.

19 ON THE ROBUST OPTIMAL STOPPING PROBLEM 353 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// 2d Gven τ T,leτ = {τ<} τ + {τ } τ T. We can deduce from 7.26 and 7.6 ha E P {τ<} Y τ + {τ } Z E P {τ<} Y τ + {τ } E P Y τ F ε = E P E P {τ<} Y τ + {τ } Y τ F ε =E P {τ<} Y τ + {τ } Y τ ε E P = E P Yτ + ε up τ T E P Yτ + ε. {τ<} Y τ + {τ } Y τ + ε Tang upremum over τ T on he lef-hand de and hen leng ε 0 yeld 7.3. So we proved he propoon. Proof of Propoon 4.2. Fx ω Ω. Leng 0 < T uch ha up r, ωr ω T.We hall how ha 7.27 Z ω Z ω 2ρα δ,, where α =+ ω 0,T and δ, = upr, ωr ω T. Gven ε>0, here ex a P=P, ω, ε P, ω uch ha 7.28 Z ω up E P Y,ω τ ε τ T up E P {τ<} Yτ,ω τ T E P ε, Z,ω + {τ } Z,ω ε where we ued 7.3 n he econd nequaly and oo τ = n he la nequaly. In lgh of 4.3 Z ω Z,ω ω = Z ω Z, ω ω ρ ω ω ω 0, 7.29 = ρ ρ ρ up r, up r, ωr+ω ωr ωr + up up r,+δ, T r, ωr ω B r ω + δ, ω Ω. Snce ω 0, ω 0,T <α, 7.28 and 4.4 mply ha Z ω Z ω E P Z ω Z,ω + ε E P ρ δ, + up B r + ε ρ α δ, +ε. r,+δ, T

20 354 ERHAN BAYRAKTAR AND SONG YAO Leng ε 0 yeld ha Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// Z ω Z ω ρ α δ,. On he oher hand, le P be an arbrary probably n P, ω. Applyng Propoon 4. yeld ha 7.3 Z ω Z ω up E P {τ<} Yτ,ω + {τ } Z,ω Z ω. τ T For any τ T and ω {τ<}, 3.howha Yτ,ω ω Y,ω ω =Y τ ω,ω ω Y, ω ω ρ 0 d τ ω,ω ω,, ω ω Z ω Z ω up τ T E P ρ 0 + up ω r τ ω ω r r,t ρ +2 up B r ω. r, Pluggng h no 7.3, we can deduce from 4.4, 4.2, and 7.29 ha {τ<} ρ B r + {τ<} Y,ω +2 up r, + {τ } Z,ω Z ω ρ α +E P Z,ω Z ω 2ρ α δ,, whch ogeher wh 7.30 prove A lm δ, = lm δ, =0, he connuy of Z ealy follow. 2 Le, ω 0,T ΩandP P, ω. A E P Y,ω < by 3.2, ung 7.6 and applyng Lemma A.2 wh X = B how ha for any, T, Z,ω E P Y,ω F=E P Y,ω F P, P-a.. Then by he connuy of proce Z and he rgh connuy of proce {E P Y,ω F P },T,holdP-a.. ha Z,ω E P Y,ω F P for any, T. I follow ha Z,ω up,t E P Y,ω F P, P-a.. Applyng Doob marngale nequaly and Jenen nequaly and ung he convexy of φ yeld ha E P Z,ω e e e e = e e + up,t + up,t E P φ EP Y,ω F P E P EP φy,ω F P +E P φy,ω <. ProofofPropoon4.3. When = T, 4.6 rvally hold a an equaly. So le u fx, ω 0,T Ω and ll defne Y, Z a n 7.7. For 4.6, uffce o how for a gven P P, ω ha 7.32 up E P {τ<ν} Y τ + {τ ν} Z ν up E P Yτ. τ T τ T

21 ON THE ROBUST OPTIMAL STOPPING PROBLEM 355 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// Fx ε>0, ν, τ T,andn N. We defne τ n = {τ n } n + n =2 { n <τ n } n T.Lebean neger 2. For =,...,, applyng 7.5 wh = yeld ha 7.33 where τ T P Z 7.6, we can fnd a τ T 7.34 E P Yτ F P, P a.., = + T he opmal oppng me for eup τ T P E P Y τ F P. Smlar o uch ha E P Y τ Y τ <ε/. Defne ν = {ν } + =2 { <ν } T and τ n = = A {τ n< } τ n + } τ {τn T,whereA = {ν = } F. We can deduce from 7.33 and 7.34 ha E P {τn<ν }Y τn + {τn ν }Z ν 7.35 = = < = = = E P A {τn< } Y τn + {τn } E P Yτ F P E P E P A {τn< } Y τn + {τn } Y τ F P E P A {τn< } Y τn + {τn } Y τ E P A {τn< } Y τn + {τn } Y τ + ε = = E P Yτ n + ε up T E P Y + ε. Snce E P Y + Z < by 3.2 and Propoon 4.2, leng n 7.35, we can deduce from he connuy of Z and he domnaed convergence heorem ha E P {τn ν}y τn + {τn>ν}z ν = lm E P {τn<ν }Y τn + {τn ν }Z ν up T E P Y + ε. A n, he rgh connuy of Y and he domnaed convergence heorem mply ha E P {τ<ν} Y τ + {τ ν} Z ν = lm n E P {τn ν}y τn + {τn>ν}z ν up E P Y +ε. T Tang upremum over τ T on he lef-hand de and hen leng ε 0 yeld 7.32.

22 356 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// Proof of he reul n econ 5. Proof of Remar 5.. Le, ω 0,T Ω. A Y τ F T -meaurable, Lemma 2. how ha Y τ,ω n urn FT -meaurable. Snce Y τ F, we can deduce from 2.3 ha Yτ,ω ω = Y τω ω,ω ω {τ ω ω<} Yτ ω ω + {τ ω ω }Y,ω ω = {τ ω ω<} Yτ ω + {τ ω ω }Y,ω ω ω Ω. For any P P, ω, hen follow from 3.2 ha E P Y τ,ω Y τ ω + E P Y,ω <. Thu, Y τ L. Smlarly, one can deduce from Remar 4.2 and Propoon 4.2 ha Z τ L. Proof of Theorem 5.. When = T, 5.2 clearly hold. So le u fx, ω 0,T Ωandν T. We ll defne Y and Z a n 7.7. By Corollary 2., ν,ω T. Tang τ = ν = ν,ω n 4.6 yeld ha Z ω nf up E P {τ<ν,ω }Y τ + {τ ν,ω }Z ν,ω P P,ω τ T 7.36 nf E P Zν,ω = E Zν ω, P P,ω whch how ha Z an E -upermarngale. Nex, le u how he E -marngaly of Z :If = τ ω,.e., ω {τ = } F F, Lemma 2. mple ha ω Ω {τ = }. Then for any, ω, T Ω, we have νω ω = τ ω ω. Applyng 2.3 o Z F F yeld ha Z ν,ω ω =Z ν τ ω ω =Zνω ω τ ω ω,ω ω =Z, ω ω =Z, ω. I follow ha Z E Z ν ω = nf E,ω P ν = nf E P Z, ω 7.37 P P,ω P P,ω = Z, ω = Z τ ω,ω = Z ω We now uppoe τ ω >,.e., ω {τ >} F. Lemma 2. agan how ha ω Ω {τ >}. By Corollary 2., τ,ω T. Smlar o 7.36, ang τ = ν,ω τ,ω =ν τ,ω n 4.6 yeld ha Z ω =Z τ ω,ω 7.39 = Z, ω E Zν τ ω =E Z ν ω The demonraon of Z ω E Z ν ω nhecaeofτ ω > relavely lenghy. We pl no everal ep. The man dea he followng: We approxmae τ by he hng me τ n = nf{ 0,T: Z Y +/n} and hen approxmae he correpondng hfed oppng me n =ν τ n,ω by oppng me n ha ae fne value = + T, =,...,. We wll pae n accordance wh P2 he local approxmang mnmzer P ĩ ω of Z ω over he e {n = } bacwardly o ge a probably P P, ω ha

23 ON THE ROBUST OPTIMAL STOPPING PROBLEM 357 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// afe E P Y τ F P Z n n + ε for all oppng me τ. Tang eenal upremum over τ how ha 7.4 Z P n Z n + ε, where Z P denoe he Snell envelope of Y under he ngle probably P. By he marngale propery of Z P, 7.42 Z ω Z P E P Z P, n τ P where τ P he opmal oppng me for Z P. A he fr me Z P mee Y, τ P τ,ω. Snce τ = lm n τ n and lm n = n,forn, large enough we have τ P n excep for a ny probably. Then combnng 7.42 wh 7.4 and applyng a ere of emaon yeld ha Z ω E P Z n +ε E P Z n +ε. Fnally, leng, n, ε 0 and ang nfmum over P P, ω lead o a In he fr ep, we pae he local approxmang mnmzer P ĩ ω of Z ω over he e { n = } bacwardly. Fx P P, ω, ε 0, and α, n,, N wh 2. We le {ω α} N be a ubequence of { ω } N n O α 0 and defne an F-oppng me τ n =nf{ 0,T:Z Y +/n}. By Corollary 2. and 7.38, boh n =ν τ n,ω and =τ,ω are T -oppng me. We e = = + T for =,..., and defne n = { n } + =2 { < n } T. There ex a δ>0uch ha ρ 0 δ ρ 0 δ ρ δ <ε/4. Gven, {,...,} {,...,}, weea = { n = } O δ ωα \ <O δ ωα F by 2.. There ex a P P,ω ω α uch ha Z ω ω α up τ T E P Y,ω ωα τ ε/4. For any ω A wh A, one can deduce from 3. and 4.3 ha up E P τ T Yτ, ω,ω ω = up E P τ T Yτ up E P τ T Y,ω ωα τ + ρ 0 ω ω α, 7.43 Z ω ω α + ε 4 + ρ 0 ω ω α, < Z ω ω+ρ ω ω α, + 2 ε<z ω+ 3 4 ε, where we ued n he fr nequaly he fac ha for any τ T and ω Ω,ω ω Yτ ω Y,ω ωα τ ω = Y τ ω, ω ω ω Y τ ω, ω ω α ω ρ 0 ω ω ω ω ω α ω 0,τ ω =ρ 0 ω ω α,. Seng P = P, we recurvely pc up P, =,...,fromp, ω uch ha P2 hold for, P, P, {A } =0, {P } = =, P, P +, {A } =0,

24 358 ERHAN BAYRAKTAR AND SONG YAO Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// {P } = wh A 0 = = A c. In parcular up E P A A τ T Y,ω τ 7.44 E P + { ω A A } up T =,...,, A F. E P Y,ω ω + ρ0 δ Smlar o 7., we have E P ξ =E P + ξ ξ L F, P L F, P and E P A 0 ξ=e P + A 0 ξ ξ L FT, P L FT, P +. b Now, le u conder he Snell envelope Z P of Y under P,.e., Z P eup P E τ T P Y τ F P,, T. Snce he flraon F P rgh-connuou, and nce he proce Y rghconnuou and lef upper emconnuou by Remar 3. 2, he clac opmal oppng heory how ha Z P P adm an RCLL modfcaon {Z uch ha for any, T, τ P opmal oppng me for eup τ T P =nf{r, T :Z P r = Y r } T P = },T E P Y τ F P. Smply denong τ P an τ, we alo now ha Z P P rep., {Z τ },T a upermarngale rep., marngale wh repec o F P, P. I follow from oponal amplng heorem ha Z ω = nf up E P Yτ up E P Yτ P P,ω τ T τ T 7.47 up τ T P E P Yτ = Z P = Z P = E P Z P n τ Moreover, for any, T, applyng 7.5 wh P = P yeld ha Z E P Y τ F P P = eup P E τ T P Y τ F P = Z P = Z P, P -a.. By he connuy of Z and he rgh connuy of Z P,holdforP -a.. ω Ω ha Z ω Z P ω for any, T. Snce τ ω ω >by 7.38, one can deduce ha ω =τ ω ω = nf{ 0,T:Z ω ω =Y ω ω} =nf{, T :Z ω ω =Y ω ω} 7.48 =nf{, T :Z ω =Y ω} nf{, T :Z P ω =Y ω} = τ ω. Nex, le u ue o how ha 7.49 Z P n = A 0 c = A 0 c Z n + ε, P a... by

25 ON THE ROBUST OPTIMAL STOPPING PROBLEM 359 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// To ee h, we le, {,..., } {,...,}, τ T and A F. Snce A A 0 for {,..., }\{}, we can deduce from 7.46, 3.2, 7.44, 7.43, 7.45, and Propoon 4.2 ha E P A A Y τ = = E P A A Y τ E P + { ω A A } up T E P Y,ω ω + ρ0 δ E P + A A Z + ε = E P A A Z + ε = = E P A A Z + ε, where we ued he fac ha Z F by Remar 4.2 and Propoon 2.. Leng A vary over F and applyng Lemma A.2 wh P,X=P,B yeld ha A Z + ε E P A Y τ F F P = E P A Y τ, 7.50 P a.. For any τ T P, mlar o 7.8, one can fnd a equence {τl } l N of T uch ha lm l E P Y τ l Y τ = 0. Then {τl } l N n urn ha a ubequence we ll denoe by {τl } l N uch ha lm l Y τ l = Y τ, P - a.. A E P Y < by 3.2, a condonal-expecaon veron of he domnaed convergence heorem and 7.50 mply ha E P A Y τ F P = lm l E P A Y τ l F P A Z + ε, P a.. Snce A F, follow ha A Z P n = A Z P = eup τ T P = eup E P τ T P F E P A P Y τ A Y τ F P A Z n + ε, P a.. Summng hem up over {,...,} and hen over {,..., } yeld c In h ep, we wll ue 7.47 and 7.49 o how 7.5 Z ω E P A Z n + A c Y τ + ε, where A = { n } = A 0 c ={ n } = = A. We fr clam ha A F n FP. To ee h clam, we e an n τ auxlary e  = { n τ } = A 0 c. Gven, T, f <,hen A { n } = A { n } = and  { n τ } =  { n } =. Oherwe, le be he large neger from {,..., } uch ha

26 360 ERHAN BAYRAKTAR AND SONG YAO. Snce A 0 c = = A {n = } for =,...,, Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// A { n } = A { n } = { n } = A 0 c { n } and  { n τ } =  { n } = { n τ } = A 0 c { n }. Clearly, = A 0 c F F F P. A { n } F n F and n { n τ } F P n τ F P, we alo have { n n } { n } F and { n τ } {n } FP. I follow ha A { n } F and  { n τ } FP. Hence A F and n  F P n τ. By 7.48, N = { >τ } N P. Snce A N c { n τ } and nce { n τ } F P n τ F P n τ one can deduce ha, A N c = A { n τ } Nc = { n τ } = A 0 c N c = { n τ }  N c F P n τ. A A N N P,weeehaA F P. n τ Snce {Z P τ },T a marngale wh repec o F P, P, follow from oponal amplng heorem ha A c Z P n τ = A c E P Z P τ F P = n τ E P A c Z P τ F P, P n τ -a.. Tang expecaon E P 7.52 E P A c Z P n τ yeld ha = E P A c Z P τ = E P A c Y τ. Snce n τ hold P -a.. on A by 7.48, we can deduce from 7.47, 7.52, and 7.49 ha Z ω E P Z P n τ = E P A Z P + n A c Y τ E P A Z n + A c Y τ + ε. d In he nex ep, we replace E P A Z n + A c Y τ on he rgh-hand de of 7.5 by an expecaon under P. For =,...,, a A F n F, one ha A n = A { n = } = { n } A 0 c F. By 7.46, 7.45, Remar 4.2, and Propoon 4.2, E P A Z = = E P A Z =E P + A Z = = E P A Z = E P A Z. Ther um over {,..., } 7.53 E P A Z n = E P A Z n.

27 ON THE ROBUST OPTIMAL STOPPING PROBLEM 36 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// Snce Z T ω = nf P PT,ω E P Y T,ω T = nf P PT,ω E P Y T,ω = Y T,ω for all ω Ω, 7.48 mple ha E P {T = n }Y τ = E P {T = n }Y T 7.54 = E P {T = n }Z T = E P {T = n }Z n. A {T = n } { n = T } = A 0, one can deduce from 7.46 and Propoon 4.2 agan ha E P {T = n }Z n = E P 2 {T = n }Z n 7.55 and mlarly ha 7.56 E P Y = A 0 {T = n τ } = = E P {T = n }Z n = E P {T = n }Z n = E P = A 0 {T = n } Y τ E P = A 0 {T Y = n } Smlar o 7.8, one can fnd a equence {τ l} l N of T uch ha lm l E P Y τ l Y τ =0. Lel N and, {,..., } {,...,}. Snce { < n} F F n,wehave{ < n n } A = { < n} {n = } A F. We can deduce from 3.2 and ha E P { < n Y } A τ l = = E P { < Y n} A τ l = E P { < n} A {τ l } Y τ l + { < n} A {τ l>} Y τ l 7.57 E P + { < n } A {τ l } Y τ l + E P + { ω< n ω} { ω A } {τ l ω> } up E P T Y,ω ω + ρ 0 δ. If M =up,ω 0,T Ω Y ω <, follow ha 7.58 E P { < n Y } A τ E l P + { < n + M +. } A Suppoe oherwe ha M =. The rgh connuy of proce Y and Propoon 2. mply ha ξ =upr, Y r =up r Q, Y r Y F -meaurable. For any T, ω Ω,and ω Ω,nce = ω and nce Y r ω ω ω = Y r ω for any r 0, by 2.3 agan, 5..

28 362 ERHAN BAYRAKTAR AND SONG YAO mple ha Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// Y,ω ω ω =Y, ω ω ω Y,ω ω ω + L + φ Y r, ω ω ω + ρ up r 0, = Y, ω ω + L + φ up φ up r, Y r, ω ω L + ξ ω ω+φ + φ ξ ω ω + ρ = L + ξ ω+φ up + ρ up r,t B r 0, r 0, up r 0, r ω. up r 0,T Y r, ω + ρ Yr ω r,t Yr ω r, up B r ω + φ ξ ω up r, up B r ω Remar 3. mple ha ee Lemma A.8 of 6 for deal Y ω = Yr ω 7.59 <. ωr Snce ω ω α 0, ω 0, + ω α, ω 0, + ω α,t < ω 0, + α = α,ω ω,4.4howhae P Y L + Y + φy +ρ α T, where L = L + φup r 0, Y r ω <. Pluggng h no 7.57 yeld ha E P { < n Y } A τ E l P + { < n + L + Y } A + φy +ρ α T, whch ogeher wh 7.58, 7.46, and 3.2 how ha E P { < n Y } A τ l E P + { < n + η } A α = = E P { < n + η } A α = E P { < n + η } A α for η α = {M< } M + + {M= } L + Y + φy +ρ α T. Summng hem up over {,...,} andhenover {,..., } gve ha E P { < n} = A c Y τ 0 E P { < n} = A c Y τ l + E Yτ P Y τ l 0 = E P { < n} = A c + η α + E Yτ P Y τ l. 0

29 ON THE ROBUST OPTIMAL STOPPING PROBLEM 363 Downloaded 0/3/6 o Redrbuon ubec o SIAM lcene or copyrgh; ee hp:// A l,weobane P { < n} = A 0 c Y τ E P { < n} = A 0 c + η α. Pung h and bac no 7.5 yeld ha Z ω E P { n } = A c + {T = n } Z n = A 0 {T Y = n + } { < n } = A c + η α 0 e In he la ep, we wll gradually end he parameer,, n, α o o oban Le A α n, = N = A 0 c and O α δ = N O δ ω α. Snce O δω α O δ ωα for, {,..., } N, one can deduce ha A α n, = = N A 0 c = = N A = { n = } = N O δ ωα = {n = } = { n A α n, = = = <T} and { n = } N O δ ωα O α δ = { n <T} O α δ. = {n = } { n = = } O α δ + ε. A E P Z + η α + Y < by 3.2 and Propoon 4.2, leng n 7.60 and applyng he domnaed convergence heorem yeld ha 7.6 { Z ω E P n } A α + n, {T = } n Z n + A α n, c \{T = n }Y + { < n } Aα n, + η α + ε E P { n }Z n + O α δ cz + O α δ c {T = n }Y > + { < n} +η α + ε, whereweuedhefacha { n } A α Z n = n, { n } { n<t } Z n { n } { n<t }\Aα n, Z n { n } { n<t } Z n + { n } { n<t cz } Oα δ { n } { n<t } Z n + O α δ cz. Snce lm n = n τ n,ω < τ,ω = T by 7.38, leng n 7.6, ung he connuy of Z Propoon 4.2, and applyng he domnaed convergence heorem agan yeld ha Z ω E P Z n + O α δ cz + Y + ε = E P Zν τ n,ω + O αcz + Y δ + ε. Snce τ = lm n τ n and α N O α δ =Ω, leng n, leng α, and hen leng ε 0, we can deduce from he connuy of Z, he domnaed convergence heorem, and 7.38 ha Z Z,ω ω =Z ω E P Zν τ,ω = EP Zν τ,ω = EP ν,