Avalable ole at www.scecedrect.com SceceDrect Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 www.elsever.com/locate/spa Optmal stoppg uder olear expectato Ibrahm Ekre a, Nzar Touz b, Jafeg Zhag a, a Uversty of Souther Calfora, Departmet of Mathematcs, Uted States b CMAP, Ecole Polytechque Pars, Frace Receved 9 February 2013; receved revsed form 7 Jauary 2014; accepted 12 Aprl 2014 Avalable ole 10 May 2014 Abstract Let X : [0, T ] Ω R be a bouded càdlàg process wth postve jumps defed o the caocal space of cotuous paths Ω. We cosder the problem of optmal stoppg the process X uder a olear expectato operator E defed as the supremum of expectatos over a weakly compact but odomated famly of probablty measures. We troduce the correspodg olear Sell evelope. Our ma objectve s to exted the Sell evelope characterzato to the preset cotext. Namely, we prove that the olear Sell evelope s a E-supermartgale, ad a E-martgale up to ts frst httg tme of the obstacle X. Ths result s obtaed uder a addtoal uform cotuty property of X. We also exted the result the cotext of a radom horzo optmal stoppg problem. Ths result s crucal for the ewly developed theory of vscosty solutos of path-depedet PDEs as troduced Ekre et al. (2014), the semlear case, ad exteded to the fully olear case the accompayg papers (Ekre et al. [6,7]). c 2014 Elsever B.V. All rghts reserved. MSC: 35D40; 35K10; 60H10; 60H30 Keywords: Nolear expectato; Optmal stoppg; Sell evelope 1. Itroducto O the caocal space of cotuous paths Ω, we cosder a bouded càdlàg process X : [0, T ] Ω R, wth postve jumps, ad satsfyg some uform cotuty codto. Correspodg author. Tel.: +1 213 740 9805. E-mal addresses: ekre@usc.edu (I. Ekre), zar.touz@polytechque.edu (N. Touz), jafez@usc.edu (J. Zhag). http://dx.do.org/10.1016/j.spa.2014.04.006 0304-4149/ c 2014 Elsever B.V. All rghts reserved.
3278 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 Let H 0 be the frst ext tme of the caocal process from some covex doma, ad H := H 0 t 0 for some t 0 > 0. Ths paper focuses o the problem sup E[X τ H ], τ T where E[ ] := sup E P [ ], P P T s the collecto of all stoppg tmes, relatve to the atural fltrato of the caocal process, ad P s a weakly compact o-domated famly of probablty measures. Our ma result s the followg. Smlar to the stadard theory of optmal stoppg, we troduce the correspodg olear Sell evelope Y, ad we show that the classcal Sell evelope characterzato holds true the preset cotext. More precsely, we prove that the Sell evelope Y s a E-supermartgale, ad a E-martgale up to ts frst httg tme τ of the obstacle. Cosequetly, τ s a optmal stoppg tme for our problem of optmal stoppg uder olear expectato. Ths result s proved by adaptg the classcal argumets avalable the cotext of the stadard optmal stoppg problem uder lear expectato. However, such a exteso turs out to be hghly techcal. The frst step s to derve the dyamc programmg prcple the preset cotext, mplyg the E-supermartgale property of the Sell evelope Y. To establsh the E-martgale property o [0, τ ], we eed to use some lmtg argumet for a sequece Y τ, where τ s are stoppg tmes creasg to τ. However, we face oe major dffculty related to the fact that a olear expectato framework the domated covergece theorem fals geeral. It was observed Des, Hu ad Peg [3] that the mootoe covergece theorem holds ths framework f the decreasg sequece of radom varables are quas-cotuous. Therefore, oe ma cotrbuto of ths paper s to costruct coveet quas-cotuous approxmatos of the sequece Y τ. Ths allows us to apply the argumets [3] o Y τ, whch s decreasg uder expectato (but ot potwse!) due to the supermartgale property. The weak compactess of the class P s crucal for the lmtg argumets. We ote that a oe dmesoal Markov model wth uformly o-degeerate dffuso, Krylov [10] studed a smlar optmal stoppg problem the laguage of stochastc cotrol (stead of olear expectato). However, hs approach reles heavly o the smoothess of the (determstc) value fucto, whch we do ot have here. Ideed, oe of the ma techcal dffcultes our stuato s to obta the locally uform regularty of the value process. Our terest ths problem s motvated from the recet oto of vscosty solutos of pathdepedet partal dfferetal equatos, as developed [5] ad the accompayg papers [6,7]. Our defto s the sprt of Cradall, Ish ad Los [2], see also Flemg ad Soer [9], but avods the dffcultes related to the fact that our caocal space fals to be locally compact. The key pot s that the potwse maxmalty codto, the stadard theory of vscosty soluto, s replaced by a problem of optmal stoppg uder olear expectato. Our prevous paper [5] was restrcted to the cotext of semlear path-depedet partal dfferetal equatos. I ths specal case, our defto of vscosty solutos ca be restrcted to the cotext where P cossts of equvalet measures o the caocal space (ad hece P has domatg measures). Cosequetly, the Sell evelope characterzato of the optmal stoppg problem uder olear expectato s avalable the exstg lterature o reflected backward stochastc dfferetal equatos, see e.g. El Karou et al. [8], Bayraktar, Karatzas ad Yao [1]. However, the exteso of our defto to the fully olear case requres to cosder a odomated famly of measures. The paper s orgazed as follows. Secto 2 troduces the probablstc framework. Secto 3 formulates the problem of optmal stoppg uder olear expectato, ad cotas the
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3279 statemet of our ma results. The proof of the Sell evelope characterzato the determstc maturty case s reported Secto 4. The more volved case of a radom maturty s addressed Secto 5. Fally, the Appedx we preset some addtoal results. 2. Nodomated famly of measures o the caocal space 2.1. The caocal spaces Let Ω := ω C([0, T ], R d ) : ω 0 = 0, the set of cotuous paths startg from the org, B the caocal process, F = {F t } 0 t T the atural fltrato geerated by B, P 0 the Weer measure, T the set of F-stoppg tmes, ad Λ := [0, T ] Ω. Moreover, for ay sub-σ -feld G F T, let L 0 (G) deote the set of G-measurable radom varables, ad H 0 (F) the set of F-progressvely measurable processes. Here ad the sequel, for otatoal smplcty, we use 0 to deote vectors or matrces wth approprate dmesos whose compoets are all equal to 0. We defe a semorm o Ω ad a pseudometrc o Λ as follows: for ay (t, ω), (t, ω ) Λ, ω t := sup ω s, d (t, ω), (t, ω ) := t t + ω t ω t T. (2.1) 0 s t The (Ω, T ) s a Baach space ad (Λ, d ) s a complete pseudometrc space. I fact, the subspace {(t, ω t ) : (t, ω) Λ} s a complete metrc space uder d. We ext troduce the shfted spaces. Let 0 s t T. Let Ω t := ω C([t, T ], R d ) : ω t = 0 be the shfted caocal space; B t the shfted caocal process o Ω t ; F t the shfted fltrato geerated by B t, P t 0 the Weer measure o Ω t, T t the set of F t -stoppg tmes, ad Λ t := [t, T ] Ω t. Moreover, for ay G F t T, L0 (G) ad H 0 (F t ) are the correspodg sets of measurable radom varables ad processes, respectvely. For ω Ω s ad ω Ω t, defe the cocateato path ω t ω Ω s by: (ω t ω )(r) := ω r 1 [s,t) (r) + (ω t + ω r )1 [t,t ](r), for all r [s, T ]. Let 0 s < t T ad ω Ω s. For ay ξ L 0 (F s T ) ad X H0 (F s ) o Ω s, defe the shfted ξ t,ω L 0 (F t T ) ad X t,ω H 0 (F t ) o Ω t by: ξ t,ω (ω ) := ξ(ω t ω ), X t,ω (ω ) := X (ω t ω ), for all ω Ω t. 2.2. Capacty ad olear expectato A probablty measure P o Ω s called a semmartgale measure f the caocal process B s a semmartgale uder P. For every costat L > 0, we deote by P L the collecto of all cotuous semmartgale measures P o Ω whose drft ad dffuso characterstcs are bouded by L ad 2L, respectvely. To be precse, let Ω := Ω 3 be a elarged caocal space, B := (B, A, M) be the caocal processes, ad ω = (ω, a, m) Ω be the paths. For ay P P L, there exsts a exteso Q o Ω such that: B = A + M, A s absolutely cotuous, M s a martgale, α P 1 L, 2 tr ((βp ) 2 ) L, where αt P := d A t dt, βt P d M t :=, dt Smlarly, for ay t [0, T ), we may defe P L t o Ω t. Q-a.s. (2.2)
3280 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 Remark 2.1. Let S d + deote the set of d d oegatve defte matrces. () I Q-a.s. sese, clearly β P L 0 (F B ) ad the α P L 0 (F B,M ). () We may also have the followg equvalet characterzato of P L. Cosder the caocal space Ω := Ω 2 wth caocal processes (B, B ). For ay P P L, there exst a probablty measure Q ad α P L 0 (F B,B, R d ), β P L 0 (F B, S d + ) such that α P 1 L, 2 tr ((βp ) 2 ) L, Q F B = P, Q T F B = Weer measure; T (2.3) d B t = αt P (B, B )dt + βt P (B)d B t, Q -a.s. () For ay determstc measurable fuctos α : [0, T ] R d ad β : [0, T ] S d + satsfyg α L, 2 1 tr (β2 ) L, there exsts uque P P L such that α P = α, β P = β, P-a.s., where α P, β P ca be uderstood the sese of ether (2.2) or (2.3). Throughout ths paper, we shall cosder a famly {P t, t [0, T ]} of semmartgale measures o Ω t satsfyg: (P1) there exsts some L 0 such that, for all t, P t s a weakly compact subset of P L 0 t. (P2) For ay 0 t T, τ T t, ad P P t, the regular codtoal probablty dstrbuto P τ,ω P τ(ω) for P-a.e. ω Ω t. (P3) For ay 0 s t T, P P s, {E, 1} Ft s dsjot, ad P P t, the followg ˆP s also P s : ˆP := P t P 1 E + P1 =1 E c =1. (2.4) Here (2.4) meas, for ay evet E FT s ad deotg Et,ω := {ω Ω t : ω t ω E}: ˆP[E] := E P =1 P [E t,b ]1 E (B) + P E ( =1 Ec ). We refer to the semal work of Stroock ad Varadha [16] for the troducto of regular codtoal probablty dstrbuto (r.c.p.d. for short). See also Appedx A.1, partcular (A.2) below for the precse meag of P τ,ω. Remark 2.2. () The weak compactess of (P1) s crucal for the exstece of the optmal stoppg tme. As explaed Itroducto, the major techcal dffculty we face s the falure of the domated covergece theorem our olear expectato framework. To overcome ths, we shall use the regularty of the processes ad the weak compactess of the classes P t. See e.g. Step 2 Secto 4.4. () The regular codtoal probablty dstrbuto s a coveet tool for provg the dyamc programmg prcple, see e.g. Soer, Touz, ad Zhag [14]. I partcular, (P2) s used to prove oe equalty the dyamc programmg prcple, see e.g. Step 1 the proof of Lemma 4.1. () The cocateato Property (P3) s used to prove the opposte equalty the dyamc programmg prcple, see e.g. Step 2 the proof of Lemma 4.1. We remark that ths codto ca be weakeed by usg the more abstract framework Nutz ad va Hadel [11].
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3281 We frst observe that Lemma 2.3. For all L > 0, the famly {P L t, t [0, T ]} satsfes codtos (P1) (P3). The proof s qute straghtforward, by usg the defto of r.c.p.d. We evertheless provde a proof the Appedx. The followg are some other typcal examples of such a famly {P t, t [0, T ]}. Ther Propertes (P1) (P3) ca be checked smlarly. Example 2.4. Let L, L 1, L 2 > 0 be some costats. Weer measure Pt 0 := {P t 0 } = {P : αp = 0, β P = I d }. Fte varato Pt FV (L) := {P : α P L, β P = 0}. Drfted Weer measure Pt 0,ac (L) := {P : α P L, β P = I d }. Relaxed bouds P t (L 1, L 2 ) := {P : α P L 1, 0 β P L 2 I d }. Relaxed bouds, Uformly ellptc Pt UE (L 1, L 2, L) := {P : α P L 1, L I d β P L 2 I d }. Equvalet martgale measures Pt e(l 1, L 2, L) := {P P t (L 1, L 2 ) : γ P L, α P = β P γ P }. We deote by L 1 (F t T, P t) the set of all ξ L 0 (F t T ) wth sup P P t E P [ ξ ] <. The set P t duces the followg capacty ad olear expectato: C t [A] := sup P P t P[A] for A F t T, ad E t[ξ] := sup P P t E P [ξ] for ξ L 1 (F t T, P t). Whe t = 0, we shall omt t ad abbrevate them as P, C, E. Clearly E s a G-expectato, the sese of Peg [12]. We remark that, whe ξ satsfes certa regularty codto, the E t [ξ t,ω ] ca be vewed as the codtoal G-expectato of ξ, ad as a process t s the soluto of a Secod Order BSDE, as troduced by Soer, Touz ad Zhag [13]. We remark that the last three famles of measures Example 2.4 are o-domated, whch are most terestg to us. I partcular, these cases the domated covergece theorem fals uder the correspodg olear expectato as we see the followg smple example. Example 2.5. Cosder the relaxed bouds P t (L 1, L 2 ) Example 2.4 wth d = 1. Let ξ := 1 {0< B T 1 }, where B s the pathwse quadratc varato. The ξ 0 for all ω as, but E 0 [ξ ] = 1 for all 2L 1 2. Gve a famly of probablty measures P o Ω, abusg the termology of Des ad Mart [4], we say that a property holds P-q.s. (quas-surely) f t holds P-a.s. for all P P. Moreover, a radom varable ξ : Ω R s P-quas-cotuous f for ay ε > 0, there exsts a closed set Ω ε Ω such that C(Ω c ε ) < ε ad ξ s cotuous Ω ε, P-uformly tegrable f E[ ξ ]1 { ξ } 0, as. Sce P s weakly compact, by Des, Hu ad Peg [3, Lemma 4 ad Theorems 22, 28], we have: Proposto 2.6. () Let (Ω ) 1 be a sequece of ope sets wth Ω Ω. The C(Ω c ) 0. () Let (ξ ) 1 be a sequece of P-quas-cotuous ad P-uformly tegrable maps from Ω to R. If ξ ξ, P-q.s. the E[ξ ] E[ξ]. (2.5)
3282 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 We fally recall the oto of martgales uder olear expectato. Defto 2.7. Let X H 0 (F) such that X τ L 1 (F τ, P) for all τ T. We say that X s a E-supermartgale (resp. submartgale, martgale) f, for ay (t, ω) Λ ad ay τ T t, E t [X t,ω τ ] (resp., =) X t (ω) for P-q.s. ω Ω. We remark that we requre the E-supermartgale property holds for stoppg tmes. Uder lear expectato E P, ths s equvalet to the P-supermartgale property for determstc tmes, due to the Doob s optoal samplg theorem. However, uder olear expectato, they are geeral ot equvalet. 3. Optmal stoppg uder olear expectatos We ow fx a process X H 0 (F). Assumpto 3.1. X s a bouded càdlàg process wth postve jumps, ad there exsts a modulus of cotuty fucto ρ 0 such that for ay (t, ω), (t, ω ) Λ: X (t, ω) X (t, ω ) ρ 0 d (t, ω), (t, ω ) wheever t t. (3.1) Remark 3.2. There s some redudacy the above assumpto. Ideed, t s show the Appedx that (3.1) mples that X has left-lmts ad X t X t for all t (0, T ]. Moreover, the fact that X has oly postve jumps s mportat to esure that the radom tmes τ (3.2), ˆτ (3.5), ad τ (4.7) ad (5.15) are F-stoppg tmes. We defe the olear Sell evelope ad the correspodg obstacle frst httg tme: Y t (ω) := sup τ T t E t [X t,ω τ ], ad τ := f{t 0 : Y t = X t }. (3.2) Our frst result s the followg olear Sell evelope characterzato of the determstc maturty optmal stoppg problem Y 0. Theorem 3.3 (Determstc Maturty). Uder Assumpto 3.1, the process Y s a E- supermartgale o [0, T ], Y τ = X τ, ad Y τ s a E-martgale. Cosequetly, τ s a optmal stoppg tme for the problem Y 0. To prove the partal comparso prcple for vscosty solutos of path-depedet partal dfferetal equatos our accompayg paper [7], we eed to cosder optmal stoppg problems wth radom maturty tme H T of the form H := f{t 0 : B t O c } t 0, (3.3) for some t 0 (0, T ] ad some ope covex set O R d cotag the org. We shall exted the prevous result to the followg stopped process: X H s := X s1 {s<h} + X H 1 {s H} for s [0, T ]. (3.4) The correspodg Sell evelope ad obstacle frst httg tme are deoted: X H t,ω τ Y H t (ω) := sup τ T t E t, ad τ := f{t 0 : Y t H = X t H }. (3.5)
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3283 Our secod ma result requres the followg addtoal assumpto. Assumpto 3.4. () For some L > 0, Pt FV (L) P t for all t [0, T ], where Pt FV (L) s defed Example 2.4. () For ay 0 t < t + δ T, P t P t+δ the followg sese: for ay P P t we have P P t+δ, where P s the probablty measure o Ω t+δ such that the P-dstrbuto of B t+δ s equal to the P-dstrbuto of {Bs t, t s T δ}. Remark 3.5. The above assumpto s a techcal codto used to prove the dyamc programmg prcple Secto 5.1 below. () All sets Example 2.4 satsfy Assumpto 3.4(), ad the relaxed bouds P t (L 1, L 2 ) satsfes Assumpto 3.4(). We remark that, for the vscosty theory of path-depedet partal dfferetal equatos our accompayg papers [6,7], we shall use P t (L, 2L) whch satsfes both () ad () of Assumpto 3.4. () By a lttle more volved argumets, we may prove the results Secto 5.1 by replacg Assumpto 3.4 () wth: for P UE t defed Example 2.4, for some costats L, L 1, L 2, Pt UE (L 1, L 2, L) P t for all t [0, T ], (3.6) () If P t s uformly odegeerate, amely there exsts c > 0 such that β P ci d for all t ad P P t, (3.7) the we shall use (3.6) stead of Assumpto 3.4 (). I ths case, uder the addtoal codto that X s uformly cotuous (t, ω), Ŷ H s left cotuous at H ad the argumets for our ma result Theorem 3.6 below ca be smplfed sgfcatly, see Lemma A.1 ad Remark 5.10 below. Theorem 3.6 (Radom Maturty). Uder Assumptos 3.1 ad 3.4, the process Y H s a E- supermartgale o [0, H], Y τ H = X τ H, ad Y τ H s a E-martgale. I partcular, τ s a optmal stoppg tme for the problem Y 0 H. Remark 3.7. The ma dea for provg Theorem 3.6 s to show that E[Y τ H ] coverges to E[Y τ H ], where τ s defed by (5.15) below ad creases to τ. However, we face a major dffculty that the domated covergece theorem fals our olear expectato framework. Notce that Y s a E-supermartgale ad thus Y τ are decreasg uder expectato (but ot potwse!). We shall exted the argumets of [3] for the mootoe covergece theorem, Proposto 2.6, to our case. For ths purpose, we eed to costruct certa cotuous approxmatos of the stoppg tmes τ, ad the requremet that the radom maturty H s of the form (3.3) s crucal. We remark that, hs Markov model, Krylov [10] also cosders ths type of httg tmes. We also remark that, a specal case, Sog [15] proved that H s quas-cotuous. 4. Determstc maturty optmal stoppg We ow prove Theorem 3.3. Throughout ths secto, Assumpto 3.1 s always force, ad we cosder the olear Sell evelope Y together wth the frst obstacle httg tme τ, as defed (3.2). Assume X C 0, ad wthout loss of geeralty that ρ 0 2C 0. It s obvous that Y C 0, Y X, ad Y T = X T. (4.1)
3284 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 Throughout ths secto, we shall use the followg modulus of cotuty fucto: ρ 0 (δ) := ρ 0 (δ) ρ 0 (δ 3 1 1 ) + δ 3, (4.2) ad we shall use a geerc costat C whch depeds oly o C 0, T, d, ad the L 0 Property (P1), ad t may vary from le to le. 4.1. Dyamc programmg prcple Smlar to the stadard Sell evelope characterzato uder lear expectato, our frst step s to establsh the dyamc programmg prcple. We start by the case of determstc tmes. Lemma 4.1. For each t, the radom varable Y t s uformly cotuous ω, wth the modulus of cotuty fucto ρ 0, ad satsfes Y t1 (ω) = sup τ T t 1 E t1 X t 1,ω τ 1 {τ<t2 } + Y t 1,ω t 2 1 {τ t2 } for all 0 t 1 t 2 T, ω Ω. (4.3) Proof. () Frst, for ay t, ay ω, ω Ω, ad ay τ T t, by (3.1) we have X t,ω τ Xτ t,ω = X (τ(b t ), ω t B t ) X (τ(b t ), ω t B t ) ρ 0 d (τ(b t ), ω t B t ), (τ(b t ), ω t B t ) = ρ 0 ω ω t. Sce τ s arbtrary, ths proves uform cotuty of Y t ω. () Whe t 2 = T, sce Y T = X T (4.3) cocdes wth the defto of Y. Wthout loss of geeralty we assume (t 1, ω) = (0, 0) ad t := t 2 < T. Recall that we omt the subscrpt 0. Step 1. We frst prove. For ay τ T ad P P: E P [X τ ] = E P X τ 1 {τ<t} + E P t [X τ ]1 {τ t}. By the defto of the regular codtoal probablty dstrbuto, we have E P t [X τ ](ω) = E Pt,ω [X t,ω τ t,ω ] Y t (ω) for P-a.e. ω {τ t}, where the equalty follows from Property (P2) of the famly {P t } that P t,ω P t. The: E P [X τ ] E P X τ 1 {τ<t} + Y t 1 {τ t}. By takg the sup over τ ad P, t follows that: Y 0 = sup E[X τ ] sup E X τ 1 {τ<t} + Y t 1 {τ t}. τ T τ T Step 2. We ext prove. Fx arbtrary τ T ad P P, we shall prove E P X τ 1 {τ<t} + Y t 1 {τ t} Y0. (4.4) Let ε > 0, ad {E } 1 be a F t -measurable partto of the evet {τ t} F t such that ω ω t ε for all ω, ω E. For each, fx a ω E, ad by the defto of Y we have Y t (ω ) E P X t,ω τ + ε for some (τ, P ) T t P t. By (3.1) ad the uform cotuty of Y, proved (), we have Y t (ω) Y t (ω ) ρ 0 (ε), X t,ω τ X t,ω τ ρ 0 (ε), for all ω E.
Thus, for ω E, I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3285 Y t (ω) Y t (ω ) + ρ 0 (ε) E P X t,ω τ + ε + ρ0 (ε) E P X t,ω τ + ε + 2ρ0 (ε). (4.5) Thaks to Property (P3) of the famly {P t }, we may defe the followg par ( τ, P) T P: τ := 1 {τ<t} τ + 1 {τ t} 1 E τ (B t ); P := P t 1 E P + 1 {τ<t} P. 1 1 It s obvous that {τ < t} = { τ < t}. The, by (4.5), E P X τ 1 {τ<t} + Y t 1 {τ t} = E P X τ 1 {τ<t} + Y t 1 E 1 E P = E P X τ 1 {τ<t} + 1 E P [X t, τ ]1 E + ε + 2ρ 0 (ε) X τ 1 { τ<t} + X τ 1 E + ε + 2ρ 0 (ε) 1 = E P X τ + ε + 2ρ0 (ε) Y 0 + ε + 2ρ 0 (ε), whch provdes (4.4) by sedg ε 0. We ow derve the regularty of Y t. Lemma 4.2. For each ω Ω ad 0 t 1 < t 2 T, Y t1 (ω) Y t2 (ω) C ρ 0 d (t1, ω), (t 2, ω). Proof. Deote δ := d (t1, ω), (t 2, ω). If δ 8 1, the clearly Y t 1 (ω) Y t2 (ω) 2C 0 C ρ 0 (δ). So we cotue the proof assumg δ 8 1. Frst, by settg τ = t 2 Lemma 4.1, δy := Y t2 (ω) Y t1 (ω) Y t2 (ω) E t1 Y t 1,ω t 2 E t1 Yt2 (ω) Y t2 (ω t1 B t 1 ) E t1 ρ0 d (t2, ω), (t 2, ω t1 B t 1 ) E t1 ρ0 δ + B t 1 t1 +δ. O the other had, by the equalty X Y, Lemma 4.1, ad (3.1), we have X t δy sup E 1,ω t1 t 2 + ρ 0 d ((τ, ω t1 B t 1 ), (t 2, ω t1 B t 1 )) 1 {τ<t2 } τ T t 1 + Y t 1,ω t 2 1 {τ t2 } Y t2 (ω) E t1 Y t 1,ω t 2 Y t2 (ω) + ρ 0 d ((t 1, ω), (t 2, ω t1 B t 1 ))
3286 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 E t1 ρ 0 d ((t 2, ω), (t 2, ω t1 B t 1 )) + ρ 0 d ((t 1, ω), (t 2, ω t1 B t 1 )) Hece 2E t1 ρ0 δ + B t 1 t1 +δ. δy 2E t1 ρ0 δ + B t 1 t1 +δ Et1 ρ 0 δ + 3 4 δ 1 3 + 2C 0 1 { B t 1 t1 +δ 4 3 δ 3 1. } Sce δ + 3 4 δ 1 3 δ 1 3 for δ 1 8, ths provdes: δy ρ 0 (δ 1 3 ) + Cδ 2 3 Et1 B t 1 2 t 1 +δ ρ 0 (δ 1 3 ) + Cδ 2 3 δ C ρ0 (δ). (4.6) We are ow ready to prove the dyamc programmg prcple for stoppg tmes. Theorem 4.3. For ay (t, ω) Λ ad τ T t, we have Y t (ω) = sup E t X t,ω τ T t τ 1 { τ<τ} + Yτ t,ω 1 { τ τ}. Cosequetly, Y s a E-supermartgale o [0, T ]. Proof. Frst, follow the argumets Lemma 4.1() Step 1 ad ote that Property (P2) of the famly {P t } holds for stoppg tmes, oe ca prove straghtforwardly that Y t (ω) sup E t X t,ω τ T t τ 1 { τ<τ} + Yτ t,ω 1 { τ τ}. O the other had, let τ k τ such that τ k takes oly ftely may values. By Lemma 4.1 oe ca easly show that Theorem 4.3 holds for τ k. The for ay P P t ad τ T t, by deotg τ m := [ τ + m 1 ] T we have E P X t,ω τ m 1 { τm <τ k } + Yτ t,ω k 1 { τm τ k } Y t (ω). Sedg k, by Lemma 4.2 ad the domated covergece theorem (uder P): E P X t,ω τ m 1 { τm τ} + Yτ t,ω 1 { τm >τ} Y t (ω). Sce the process X s rght cotuous t, we obta by sedg m : Y t (ω) E P X t,ω τ 1 { τ<τ} + Yτ t,ω 1 { τ τ}, whch provdes the requred result by the arbtraress of P ad τ. 4.2. Preparato for the E-martgale property If Y 0 = X 0, the τ = 0 ad obvously all the statemets of Theorem 3.3 hold true. Therefore, we focus o the o-trval case Y 0 > X 0. We cotue followg the proof of the Sell evelope characterzato the stadard lear expectato cotext. Let τ := f t 0 : Y t X t 1 T, for > (Y 0 X 0 ) 1. (4.7)
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3287 Lemma 4.4. The process Y s a E-martgale o [0, τ ]. Proof. By the dyamc programmg prcple of Theorem 4.3, Y 0 = sup E τ T X τ 1 {τ<τ } + Y τ 1 {τ τ }. For ay ε > 0, there exst τ ε T ad P ε P such that Y 0 E P ε X τε 1 {τε <τ } + Y τ 1 {τε τ } + ε E P ε Y τε τ 1 1 {τ ε <τ } + ε, (4.8) where we used the fact that Y t X t > 1 for t < τ, by the defto of τ. O the other had, t follows from the E-supermartgale property of Y Theorem 4.3 that E P ε Y τε τ E[Y τε τ ] Y 0, whch mples by (4.8) that P ε [τ ε < τ ] ε. We the get from (4.8) that: Y 0 E P ε (X τε Y τ )1 {τε <τ } + Y τ + ε CP ε [τ ε < τ ] + E P ε [Y τ ] + ε E[Y τ ] + (C + 1)ε. Sce ε s arbtrary, we obta Y 0 E[Y τ ]. Smlarly oe ca prove Y s a E-submartgale o [0, τ ]. By the E-supermartgale property of Y establshed Theorem 4.3, ths mples that Y s a E-martgale o [0, τ ]. By Lemma 4.2 we have Y 0 E[Y τ ] = E[Y τ ] E[Y τ ] CE ρ 0 d (τ, ω), (τ, ω). (4.9) Clearly, τ τ, ad ρ 0 d (τ, ω), (τ, ω) 0. However, geeral the stoppg tmes τ, τ are ot P-quas-cotuous, so we caot apply Proposto 2.6() to coclude Y 0 E[Y τ ]. To overcome ths dffculty, we eed to approxmate τ by cotuous radom varables. 4.3. Cotuous approxmato The followg lemma s crucal for us. Lemma 4.5. Let θ θ θ be radom varables o Ω, wth values a compact terval I R, such that for some Ω 0 Ω ad δ > 0: θ(ω) θ(ω ) θ(ω) for all ω Ω 0 ad ω ω δ. The for ay ε > 0, there exsts a uformly cotuous fucto ˆθ : Ω I ad a ope subset Ω ε Ω such that C Ω c ε ε ad θ ε ˆθ θ + ε Ω ε Ω 0. Proof. If I s a sgle pot set, the θ s a costat ad the result s obvously true. Thus at below we assume the legth I > 0. Let {ω j } j 1 be a dese sequece Ω. Deote O j := {ω Ω : ω ω j < δ 2 } ad Ω := j=1 O j. It s clear that Ω s ope ad Ω Ω as. Let f : [0, ) [0, 1] be defed as follows: f (x) = 1for x [0, δ 2 ], f (x) = 1 2 I
3288 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 for x δ, ad f s lear [ 2 δ, δ]. Defe θ (ω) := φ (ω) θ(ω j )ϕ, j (ω) where ϕ, j (ω) := f ( ω ω j ) j=1 1 ad φ := ϕ, j. j=1 The clearly θ s uformly cotuous ad takes values I. For each ω Ω Ω 0, the set J (ω) := {1 j : ω ω j δ} ad φ (ω) 1. The, by our assumpto, θ (ω) θ(ω) = φ (ω) [θ(ω j ) θ(ω)]ϕ, j (ω) + [θ(ω j ) θ(ω)]ϕ, j (ω) φ (ω) j J (ω) j J (ω) I ϕ, j (ω) φ (ω) j J (ω) j J (ω) 1 2 1. Smlarly oe ca show that θ 1 θ Ω Ω 0. Fally, sce Ω Ω as, t follows from Proposto 2.6() that lm C[Ω c] = 0. 4.4. Proof of Theorem 3.3 We proceed two steps. Step 1. For each, let δ > 0 be such that 3C ρ 0 (δ ) (+1) 1 for the costat C Lemma 4.2. Now for ay ω ad ω such that ω ω T δ, by (3.1), the uform cotuty of Y Lemma 4.1, ad the fact that ρ 0 ρ 0, we have (Y X) τ+1 (ω)(ω ) (Y X) τ+1 (ω)(ω) + 3C ρ 0 (δ ) 1 + 1 + 1 ( + 1) = 1. The τ (ω 1 1 ) τ +1 (ω). Sce 3C ρ 0 (δ ) (+1) ( 1), smlarly we have τ 1(ω) τ (ω ). We may the apply Lemma 4.5 wth θ = τ 1, θ = τ, θ = τ +1, ad Ω 0 = Ω. Thus, there exst a ope set Ω Ω ad a cotuous radom varable τ valued [0, T ] such that C Ω c 2 ad τ 1 2 τ τ +1 + 2 Ω. Step 2. By Lemma 4.4, for each large, there exsts P P such that Y 0 = E[Y τ ] E P [Y τ ] + 2. By Property (P1), P s weakly compact. The, there exsts a subsequece { j } ad P P such that P j coverges weakly to P. Now for ay large ad ay j, ote that τ j τ. Sce Y s a E-supermartgale ad thus a P j -supermartgale, we have Y 0 2 j E P j Yτ j E P j Yτ E P j Y τ + E P j Y τ Y τ. (4.10) By the boudedess of Y (4.1) ad the uform cotuty of Y Lemma 4.2, we have Y τ Y τ C ρ 0 d ( τ, ω), (τ, ω) C ρ 0 d ( τ, ω), (τ, ω) 1 Ω 1 Ω +1 + C1 Ω c 1 Ω c +1.
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3289 Notce that τ 1 2 1 τ τ +1 + 2 1 o Ω 1 Ω +1. The Y τ Y τ C ρ 0 d ( τ, ω), ( τ 1 2 1, ω) 1 Ω 1 Ω +1 + C ρ 0 d ( τ, ω), ( τ +1 + 2 1, ω) 1 Ω 1 Ω +1 + C1 Ω c 1 Ω+1 c C ρ 0 d ( τ, ω), ( τ 1 2 1, ω) + C ρ 0 d ( τ, ω), ( τ +1 + 2 1, ω) + C1 Ω c 1 Ω c +1. The (4.10) together wth the estmate C[Ω c ] 2 lead to Y 0 2 j E P j Y τ + CE P j ρ 0 d ( τ, ω), ( τ 1 2 1, ω) + CE P j ρ 0 d ( τ, ω), ( τ +1 + 2 1, ω) + C2. Notce that Y ad τ 1, τ, τ +1 are cotuous. Sed j, we obta Y 0 E P Y τ + CE P ρ 0 d ( τ, ω), ( τ 1 2 1, ω) + CE P ρ 0 d ( τ, ω), ( τ +1 2 1, ω) + C2. (4.11) Sce P τ τ 2 C τ τ 2 2 < ad τ τ, by the Borel Catell lemma uder P we see that τ τ, P -a.s. Sed (4.11) ad apply the domated covergece theorem uder P, we obta Y 0 E P Yτ E[Yτ ]. Smlarly Y t (ω) E t [Y t,ω τ ] for t < τ (ω). By the E-supermartgale property of Y establshed Theorem 4.3, ths mples that Y s a E-martgale o [0, τ ]. 5. Radom maturty optmal stoppg I ths secto, we prove Theorem 3.6. The ma dea follows that of Theorem 3.3. However, sce X H s ot cotuous ω, the estmates become much more volved. Throughout ths secto, let X, H, O, t 0, X := X H, Y := Y H, ad τ be as Theorem 3.6. Assumptos 3.1 ad 3.4 wll always be force. We shall emphasze whe the addtoal Assumpto 3.4 s eeded, ad we fx the costat L as Assumpto 3.4 (). Assume X C 0, ad wthout loss of geeralty that ρ 0 2C 0 ad L 1. It s clear that Y C 0, X Y, ad Y H = X H = X H. (5.1) By (3.1) ad the fact that X has postve jumps, oe ca check straghtforwardly that, X(t, ω) X(t, ω ) ρ 0 d ((t, ω), (t, ω )) for t t, t H(ω), t H(ω ) (5.2) except the case t = t = H(ω ) < H(ω) t 0. I partcular, X(t, ω) X(t, ω) ρ 0 d ((t, ω), (t, ω)) wheever t t H(ω). (5.3)
3290 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 Moreover, we defe ρ 1 (δ) := ρ 0 (δ) ρ 0 (L 1 δ) 1 1 3 + δ 3, ρ 2 (δ) := [ρ 1 (δ) + δ] [ρ 1 (δ 1 1 3 ) + δ 3 ], (5.4) ad ths secto, the geerc costat C may deped o L as well. 5.1. Dyamc programmg prcple We start wth the regularty ω. Lemma 5.1. For ay t < H(ω) H(ω ) we have: Y t (ω) Y t (ω ) Cρ 1 ω ω t. To motvate our proof, we frst follow the argumets Lemma 4.1() ad see why t does ot work here. Ideed, ote that Y t (ω) Y t (ω ) sup sup E P X t,ω τ T t τ H P P t,ω X t,ω. τ H t,ω t Sce we do ot have H t,ω H t,ω, we caot apply (5.2) to obta the requred estmate. Proof. Let τ T t ad P P t. Deote δ := L 1 ω ω t, t δ := [t +δ] t 0 ad B t δ s := Bs+δ t Bt t δ for s t. Set τ (B t ) := [τ( B t δ) + δ] t 0, the τ T t. Moreover, by Assumpto 3.4 ad Property (P3), we may choose P P t defed as follows: α P := 1 δ (ω t ω t ), βp := 0 o [t, t δ ], ad the P -dstrbuto of B t δ s equal to the P-dstrbuto of B t. We clam that I := E P [X t,ω τ H t,ω ] E P [X t,ω τ H t,ω ] Cρ 1 (Lδ). (5.5) The E P [X t,ω τ H t,ω ] Y t (ω ) E P [X t,ω τ H t,ω ] E P [X t,ω τ H t,ω ] Cρ 1 (Lδ), ad t follows from the arbtraress of P P t ad τ T t that Y t (ω) Y t (ω ) Cρ 1 (Lδ). By exchagg the roles of ω ad ω, we obta the requred estmate. It remas to prove (5.5). Deote ω s := ω s 1 [0,t)(s) + [ω t + αp (s t)]1 [t,t ] (s). Sce t < H(ω) H(ω ), we have ω t, ω t O. By the covexty of O, ths mples that ω s O for s [t, t δ], ad thus H t,ω (B t ) = (H t,ω ( B t δ) + δ) t 0, P -a.s. Therefore, E P [X t,ω ] = E P X τ (B t ) τ H t,ω H t,ω (B t ), ω t B t = E P X [τ( B t δ ) + δ] [H t,ω ( B t δ ) + δ] t 0, ω tδ B t δ δ = E P X [τ(b t ) + δ] [H t,ω (B t ) + δ] t 0, ω tδ B t δ, (5.6) whle E P [X t,ω τ H t,ω ] = E P X τ(b t ) H t,ω (B t ), ω t B t. Notce that, wheever τ(b t ) H t,ω (B t ) = [τ(b t ) + δ] [H t,ω (B t ) + δ] t 0, we have τ(b t ) H t,ω (B t ) = t 0. Ths excludes the exceptoal case (5.2). The t follows from (5.6)
ad (5.2) that I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3291 I E P ρ 0 δ + (ω t B t ) τ(b t ) H t,ω (B t ) ( ω tδ B t δ ) [τ(b t )+δ] [H t,ω (B t )+δ] t 0 t0. Note that, deotg θ := τ(b t ) H t,ω (B t ), (ω t B t ) τ(b t ) H t,ω (B t ) ( ω tδ B t δ ) [τ(bt )+δ] [H t,ω (B t )+δ] t 0 t0 ω t B t ω tδ B t δ t 0 + sup (ω t B t ) θ+r (ω t B t ) θ ω ω t 0 r δ sup ω t + Bs t ω s t s t δ + sup (ω t B t ) θ+r (ω t B t ) θ 0 r δ 2Lδ + B t tδ + sup Bs t Bt s δ + sup Bθ+r t Bt θ. t δ s t 0 Sce L 1, we have 0 r δ sup ω t + Bs t ω t δ Bs δ t t δ s t 0 I E P ρ 0 3δ + B t tδ + sup Bs t Bt s δ + sup Bθ+r t Bt θ. t δ s t 0 0 r δ If δ 1 8, the I 2C 0 Cρ 1 (Lδ). We the cotue assumg δ 8 1, ad thus 3δ + 1 4 δ 1 3 δ 1 3. Therefore, I ρ 0 (δ 3 1 ) + CP B t tδ + sup Bs t Bt s δ + sup Bθ+r t Bt θ 1 t δ s t 0 0 r δ 4 δ 1 3 ρ 0 (δ 3 1 ) + Cδ 8 3 E P B t 8 t δ + sup Bs t Bt s δ 8 + sup Bθ+r t Bt θ 8 t δ s t 0 ρ 0 (δ 1 3 ) + Cδ 4 3 + Cδ 8 3 E P sup t δ s t 0 B t s Bt s δ 8. 0 r δ Set t δ = s 0 < < s = t 0 such that δ s +1 s 2δ, = 0,..., 1. The E P sup t δ s t 0 B t s Bt s δ 8 = E P max 0 1 sup Bs t Bt s δ 8 s s s +1 1 E P sup [ Bs t Bt s δ + Bt s δ Bt s δ ]8 s s s +1 =0 1 C (s +1 s + δ) 4 Cδ 1 δ 4 = Cδ 3. =0 Thus I ρ 0 (δ 1 3 ) + Cδ 4 3 + Cδ 8 3 δ 3 ρ 0 (δ 1 3 ) + Cδ 1 3 lemma. Cρ 1 (Lδ), provg (5.5) ad hece the We ext show that the dyamc programmg prcple holds alog determstc tmes. Lemma 5.2. Let t 1 < H(ω) ad t 2 [t 1, t 0 ]. We have: Y t1 (ω) = sup E t1 X t 1,ω τ T t τ H t 1,ω 1 {τ H t 1,ω <t 2 } + Y t 1,ω t 2 1 {τ H t 1,ω t 2 }. 1
3292 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 Proof. Whe t 2 = t 0, the lemma cocdes wth the defto of Y. Wthout loss of geeralty we assume (t 1, ω) = (0, 0) ad t := t 2 < t 0. Frst, follow the argumets Lemma 4.1() Step 1, oe ca easly prove Y 0 sup E X τ H 1 {τ H<t} + Y t 1 {τ H t}. (5.7) τ T To show that equalty holds the above equalty, fx arbtrary P P ad τ T satsfyg τ H (otherwse reset τ as τ H), we shall prove E P X τ 1 {τ<t} + Y t 1 {τ t} Y 0. Sce Y H = X H, ths amouts to show that: E P X τ 1 {τ<t} {H t} + Y t 1 {τ t,h>t} Y 0. (5.8) We adapt the argumets Lemma 4.1() Step 2 to the preset stuato. Fx 0 < δ t 0 t. Let {E } 1 be a F t measurable partto of the evet {τ t, H > t} F t such that ω ω Lδ for all ω, ω E. Fx a ω E for each. By the defto of Y we have Y t (ω ) E P X t,ω + δ for some (τ, P ) T t P t. (5.9) τ H t,ω As Lemma 5.1, we set t δ := t + δ < t 0, B t δ s := B t s+δ Bt t δ for s t, ad τ (B t ) := [τ ( B t δ) + δ] t 0. The τ T t. Moreover by Assumpto 3.4 ad Property (P3), for each ω E, we may defe P,ω P t as follows: α P,ω := 1 δ (ω t ω t), β P,ω := 0 o [t, t δ ], ad the P,ω -dstrbuto of B t δ s equal to the P -dstrbuto of B t. By (5.5), we have E P [X t,ω τ H t,ω ] E P,ω [ X t,ω τ H t,ω ] Cρ 1 (Lδ). (5.10) The by Lemma 5.1 ad (5.9), (5.10) we have Y t (ω) Y t (ω ) + Cρ 1 (Lδ) E P,ω [X t,ω τ H t,ω ] + δ + Cρ 1 (Lδ), for all ω E. (5.11) We ext defe: τ := 1 {τ<t} {H t} τ + 1 E τ (B t ), ad the {τ < t} {H t} = { τ < t} {H t}. 1 Sce τ H, we see that {τ < t} {H t} = {τ < t} {τ = H = t}, ad thus t s clear that τ T. Moreover, we clam that there exsts P P such that P = P o F t ad the regular codtoal probablty dstrbuto (5.12) ( P) t,ω = P,ω for P-a.e. ω E, 1, ( P) t,ω = P t,ω for P-a.e. ω {τ < t} {H t}. The, by (5.11) we have Y t (ω) E ( P) t,ω X t,ω ( τ H) t,ω + δ + Cρ1 (Lδ), P-a.e. ω {τ t, H > t}, (5.13)
ad therefore: I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3293 E P X τ 1 {τ<t} {H t} + Y t 1 {τ t,h>t} E P X τ H 1 {τ<t} {H t} + X τ H 1 {τ t,h>t} + δ + Cρ 1 (Lδ) = E P X τ H + δ + Cρ 1 (Lδ) Y 0 + δ + Cρ 1 (Lδ), whch mples (5.8) by sedg δ 0. The the reverse equalty of (5.7) follows from the arbtraress of P ad τ. It remas to prove (5.12). For ay ε > 0 ad each 1, there exsts a partto {E j, j 1} of E such that ω ω t ε for ay ω, ω E j. Fx a ω j E j for each (, j). By Property (P3) we may defe P ε P by: P ε := P t P,ω j 1 E + P1 {τ<t} {H t}. j 1 j 1 By Property (P1), P s weakly compact. The P ε has a weak lmt P P as ε 0. We ow show that P satsfes all the requremets (5.12). Ideed, for ay partto 0 = s 0 < < s m = t < s m+1 < < s M = t δ < s M+1 < < s N = T ad ay bouded ad uformly cotuous fucto ϕ : R N d R, let ξ := ϕ B s1 B s0,..., B sn B sn 1. The, deotg s k := s k+1 s k, ω k := ω sk ω sk 1, we see that E P,ω [ξ t,ω ] = η t (ω), where: ηt (ω) := EP ϕ j EP,ω [ξ t,ω ] = η, j t (ω), ( ω k ) 1 k m, ω t ω t δ ( s k ) m+1 k M, (B sk δ B sk 1 δ) M+1 k N ; η, j t (ω) := E P ϕ ( ω k ) 1 k m, ω t ω j t ( s k ) m+1 k M, (B sk δ B sk 1 δ) M+1 k N. δ Let ρ deote the modulus of cotuty fucto of ϕ. The E P,ω j [ξ t,ω ] E P,ω [ξ t,ω ] ρ(ε) for all ω E j, ad thus E P ε [ξ] E P ξ1 {τ<t} {H t} + ηt 1 E 1 = EP E P,ω j [ξ t, ]1 E E P η j t 1 E j, j 1, j 1 E P E P,ω j [ξ t, ] E P, [ξ t, ] 1E j, j 1 E P ρ(ε)1 E ρ(ε). j, j 1
3294 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 By sedg ε 0, we obta E P [ξ] = E P ξ1 {τ<t} {H t} + 1 η t 1 E, whch proves (5.12) by the arbtraress of ξ. We ow prove the regularty the t-varable. Recall the ρ 2 defed (5.4). Lemma 5.3. Let 0 t 1 < H(ω 1 ), 0 t 2 < H(ω 2 ), ad t 1 t 2. The we have: Y t1 (ω 1 ) Y t2 (ω 2 1 ) C 1 + ρ d(ωt 1 1, O c 2 d (t1, ω 1 ), (t 2, ω 2 ). ) Proof. Wthout loss of geeralty we assume t 1 < t 2. Also, vew of the uform cotuty ω of Lemma 5.1, t suffces to prove the lemma the case ω 1 = ω 2 = ω. Deote δ := d (t1, ω), (t 2, ω) ad ε := d(ω t1, O c ). For δ 1 8, we have Y t1 (ω) Y t2 (ω) 2C 0 Cε 1 ρ 2 (δ). So we assume the rest of ths proof that δ < 8 1. Frst, by Assumpto 3.4, we may cosder the measure P P t1 such that αt P := 0, βt P := 0, t [t 1, t 2 ]. The, by settg τ := t 0 Lemma 5.2, we see that Y t1 (ω) E t1 [Y t 1,ω t 2 E P [Y t 1,ω t 2 ] = Y t2 (ω t1 ). Note that H(ω t1 ) = t 0 > t 2. Thus, by Lemma 5.1, Y t2 (ω) Y t1 (ω) Cρ 1 d (t2, ω t1 ), (t 2, ω) Cρ 1 (δ) Cρ 2 (δ). (5.14) Next, for arbtrary τ T t 1, otg that X Y we have I (τ) := E t1 X t 1,ω τ H t 1,ω 1 {τ H t 1,ω <t 2 } + Y t 1,ω t 2 1 {τ H t 1,ω t 2 } Y t2 (ω) = E t1 X t 1,ω τ 1 {τ<h t 1,ω t 2 } + X t 1,ω H t 1,ω 1 {H t 1,ω <t 2,H t 1,ω τ} + Y t 1,ω t 2 1 {τ H t 1,ω t 2 } E t1 X t 1,ω τ X t 1,ω H t 1,ω 1{τ<H t t 1,ω 2 t 2 } + Y t 1,ω E t1 X t 1,ω τ X t 1,ω H t 1,ω t 2 1{τ<H t 1,ω t 2 } + C C t1 H t 1,ω t 2. + E t1 H t 1,ω t 2 Y t2 (ω) Y t 1,ω t 2 Y t2 (ω) 1 {H t 1,ω >t 2 } By (5.3) ad Lemma 5.1 we have I (τ) E t1 ρ 0 d ((t 1, ω), (t 2, ω t1 B t 1 )) + CE t1 ρ 1 ω ω t1 B t 1 t2 + CC t1 B t 1 t2 ε E t1 ρ 0 δ + B t 1 t2 + CE t1 ρ 1 δ + B t 1 t2 + Cε 1 E t1 B t 1 t2 C[1 + ε 1 ]E t1 ρ 1 δ + B t 1 t2. Sce δ 1 8, followg the proof of (4.6) we have I (τ) C[1 + ε 1 ] ρ 1 (δ 1 1 3 ) + δ 3 C[1 + ε 1 ]ρ 2 (δ). ] Y t2 (ω) By the arbtraress of τ ad the dyamc programmg prcple of Theorem 5.4, we obta Y t1 (ω) Y t2 (ω) Cε 1 ρ 2 (δ), ad the proof s complete by (5.14). Applyg Lemmas 5.1 5.3, ad followg the same argumets as those of Theorem 4.3, we establsh the dyamc programmg prcple the preset cotext.
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3295 Theorem 5.4. Let t < H(ω) ad τ T t. The Y t (ω) = sup E t X t,ω τ T t τ H t,ω 1 { τ H t,ω <τ} + Y τ t,ω Cosequetly, Y s a E-supermartgale o [0, H]. 1 { τ H t,ω τ} By Lemma 5.3, Y s cotuous for t [0, H). Moreover, sce Ŷ s a E-supermartgale, we see that Ŷ H exsts. However, Example A.2 below shows that geeral Y may be dscotuous at H. Ths ssue s crucal for our purpose, ad we wll dscuss more Secto 5.4 below. 5.2. Cotuous approxmato of the httg tmes Smlar to the proof of Theorem 3.3, we eed to apply some lmtg argumets. We therefore assume wthout loss of geeralty that Y 0 > X 0 ad troduce the stoppg tmes: for ay m 1 ad > (Y 0 X 0 ) 1, Hm := f τ := f t 0 : d(ω t, O c ) 1 m t 0 : Y t X t 1. t 0 1, m. (5.15) Here we abuse the otato slghtly by usg the same otato τ as (4.7). Our ma task ths subsecto s to buld a approxmato of H m ad τ by cotuous radom varables. Ths wll be obtaed by a repeated use of Lemma 4.5. We start by a cotuous approxmato of the sequece (H m ) m 1 defed (5.15). Lemma 5.5. For all m 2: () H m 1 (ω) Hm(ω ) H m+1 (ω), wheever ω ω t0 () there exsts a ope subset Ω m 0 1 m(m+1), Ω, ad a uformly cotuous Ĥ m such that C (Ω m 0 )c < 2 m ad H m 1 2 m Ĥm H m+1 + 2 m o Ω m 0, () there exst δ m > 0 such that Ĥ m (ω) Ĥ m (ω ) 2 m wheever ω ω t0 δ m, ad: C ( ˆΩ m 0 )c 2 m where ˆΩ m 0 := {ω Ω m 0 : d(ω, [Ω m 0 ]c ) > δ m }. Proof. Notce that () s a drect cosequece of () obtaed by applyg Lemma 4.5 wth ε = 2 m. To prove (), we observe that for ω ω 1 t0 m(m+1) ad t < H m(ω ), we have d(ω t, O c ) d(ω t, Oc ) 1 m(m + 1) > 1 m 1 m(m + 1) = 1 m + 1. Ths shows that H m (ω ) H m+1 (ω) wheever ω ω t0 m(m+1). Smlarly, H m 1(ω) Hm(ω ) wheever ω ω t0, ad the equalty () follows. 1 m(m 1) It remas to prove (). The frst clam follows from the uform cotuty of Ĥm. For each δ > 0, defe h δ : [0, ) [0, 1] as follows: h δ (x) := 1 for x δ, h δ (x) = 0 for x 2δ, ad h δ s lear o [δ, 2δ]. (5.16) 1
3296 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 The the map ω ψ δ (ω) := h δ (d(ω, [Ω0 m]c )) s cotuous, ad ψ δ 1 [Ω m 0 ] c as δ 0. Applyg Proposto 2.6() we have lm E[ψ δ] = E 1 (Ω m δ 0 0 ) c = C (Ω m 0 ) c < 2 m. By defto of ˆΩ 0 m, otce that 1 ( ˆΩ 0 m ψ )c δm. The C ( ˆΩ 0 m)c E[ψ δm ], ad () holds true for suffcetly small δ m. We ext derve a cotuous approxmato of the sequeces τ m := τ Ĥ m, where τ ad Ĥm are defed (5.15) ad Lemma 5.5(), respectvely. Lemma 5.6. For all m 2, > (Y 0 X 0 ) 1, there exsts a ope subset Ω m uformly cotuous map ˆτ m such that τ 1 m 21 m 2 ˆτ m τ +1 m + 21 m + 2 o ˆΩ 0 m Ω m, ad C (Ω m )c 2. (5.17) Ω ad a Proof. Fx m, ad recall the modulus of cotuty ρ 1 troduced (5.4). For each, let 0 < δ m < δm such that (ρ 0 + Cρ 1 )(δ m) (+1) 1, where C s the costat Lemma 5.1. We shall prove (τ 1 Ĥm)(ω) 2 1 m (τ Ĥm)(ω ) (τ +1 Ĥm)(ω) + 2 1 m wheever ω ˆΩ m 0, ω ω t0 δ m. (5.18) The the requred statemet follows from Lemma 4.5 wth ε = 2. We shall prove oly the rght equalty of (5.18). The left oe ca be proved smlarly. Let ω, ω be as (5.18). Frst, by Lemma 5.5() we have ω Ω m 0 ad Ĥ m (ω ) Ĥ m (ω) + 2 m. (5.19) We ow prove the rght equalty of (5.18) three cases. Case 1. f τ +1 (ω) Ĥm(ω ) 2 m, the Ĥm(ω ) (τ +1 Ĥm)(ω) + 2 m ad thus the result s true. Case 2. If τ +1 (ω) = H(ω), the by Lemma 5.5() we have Ĥm(ω) H m+1 (ω) + 2 m τ +1 (ω)+2 m, ad thus Ĥ m (ω ) Ĥ m (ω)+2 m τ +1 (ω)+2 1 m. Ths, together wth (5.19), proves the desred equalty. Case 3. We ow assume τ +1 (ω) < Ĥ m (ω ) 2 m ad τ +1 (ω) < H(ω). By Lemma 5.5() we have τ +1 (ω) < H m+1 (ω ), ad thus τ +1 (ω) < H(ω ). The t follows from Lemma 5.1 that (Y X) τ+1 (ω)(ω ) (Y X) τ+1 (ω)(ω) + (ρ 0 + Cρ 1 )(δ m ) 1 + 1 + 1 ( + 1) = 1. That s, τ (ω ) τ +1 (ω). Ths, together wth (5.19), proves the desred equalty. For our fal approxmato result, we troduce the otatos: τ := τ H, θ 1 := ˆτ 1 23, θ +1 := ˆτ +1 + 21, (5.20)
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3297 ad Ω := ˆΩ 1 0 Ω 1 1 ˆΩ +1 0 Ω +1 +1. (5.21) Lemma 5.7. For all (Y 0 X 0 ) 1 2, θ, θ are uformly cotuous, ad θ τ θ o Ω. Proof. Ths s a drect combato of Lemmas 5.5 ad 5.6. 5.3. Proof of Theorem 3.6 We frst prove the E-martgale property uder a addtoal codto. Lemma 5.8. Let τ T such that τ τ ad E[Y τ ] = E[Y τ ] ( partcular f τ < H). The Y s a E-martgale o [0, τ]. Proof. If Y 0 = X 0, the τ = 0 ad obvously the statemet s true. We the assume Y 0 > X 0, ad prove the lemma several steps. Step 1. Let be suffcetly large so that 1 < Y 0 X 0. Follow the same argumets as that of Lemma 4.4, oe ca easly prove: Y s a E-martgale o [0, τ ]. (5.22) Step 2. Recall the sequece of stoppg tmes ( τ ) 1 troduced (5.20). By Step 1 we have Y 0 = E[Y τ ]. The for ay ε > 0, there exsts P P such that Y 0 ε < E P [Y τ ]. Sce P s weakly compact, there exsts subsequece { j } ad P P such that P j coverges weakly to P. Now for ay ad j, sce Y s a supermartgale uder each P j ad ( τ ) 1 s creasg, we have Y 0 ε < E P j Y τ j E P j Y τ. (5.23) Our ext objectve s to sed j, for fxed, ad use the weak covergece of P j towards P. To do ths, we eed to approxmate Y τ wth cotuous radom varables. Deote ψ (ω) := h f d(ω t, O c ) 0 t θ (ω) wth h (x) := 1 [( + 3)( + 4)x ( + 3)] +. (5.24) The ψ s cotuous ω, ad {ψ > 0} f d(ω t, O c ) > 1 {θ 0 t θ (ω) + 4 < H +4}. (5.25) I partcular, ths mples that Y θ ψ ad Y θ ψ are cotuous ω. We ow decompose the rght had-sde term of (5.23) to: Y 0 ε E P j Y θ + (Y τ Y θ )1 Ω ψ + (1 ψ ) + (Y τ Y θ )1 (Ω ) c. Note that θ τ θ o Ω. The Y 0 ε E P j Y θ + sup θ t θ (Y t Y θ ) ψ + CC[ψ < 1] + CC (Ω )c.
3298 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 Sed j, we obta Y 0 ε E P ψ Y θ + E P ψ sup (Y t Y θ ) θ t θ + CC[ψ < 1] + CC[(Ω )c ].(5.26) Step 3. I ths step we show that lm EP ψ sup (Y t Y θ ) = lm C[ψ < 1] = lm C[(Ω )c ] = 0. (5.27) θ t θ () Frst, by the defto of Ω (5.21) together wth Lemmas 5.5() ad 5.6, t follows that C (Ω )c C2 0 as. () Next, otce that {ψ < 1} = f d(ω t, O c ) < 1 {θ 0 t θ (ω) + 3 > H +3}. Moreover, by (5.20) ad Lemma 5.7, The θ = ˆτ +1 +1 + 21 = θ +2 + 22 τ +2 + 2 2 H +2 + 2 2, o Ω +2. {ψ < 1} (Ω+2 )c {H +3 < H +2 + 2 2 } (Ω+2 )c sup B t B 1 H H+2 t H+2+2 2 +2 ( + 2)( + 3) The oe ca easly see that C[ψ < 1] 0, as. () Fally, t s clear that θ τ, θ τ. Recall that Y τ exsts. By (5.25), we see that ψ sup θ t θ ( Y t Y θ ) 0, P -a.s. as. The by applyg the domated covergece theorem uder P we obta the frst covergece (5.27). Step 4. By the domated covergece theorem uder P we obta lm E P [ψ Y θ ] = E P [Y τ ]. Ths, together wth (5.26) ad (5.27), mples that Y 0 E P [Y τ ] + ε. Note that Y s a P -supermartgale ad τ τ, the Y 0 E P [Y τ ] + ε. Sce ε s arbtrary, we obta Y 0 E[Y τ ], ad thus by the assumpto E[Y τ ] = E[Y τ ] we have Y 0 E[Y τ ]. Ths, together wth the fact that Y s a E-supermartgale, mples that Y 0 = E[Y τ ]. Smlarly, oe ca prove Y t (ω) = E t [Y t,ω τ t,ω ] for t < τ(ω), ad thus Y τ s a E-martgale. I lght of Lemma 5.8, the followg result s obvously mportat for us. Proposto 5.9. It holds that E[Y τ ] = E[Y τ ].. (5.28) We recall aga that Y τ = Y τ wheever τ < H. So the oly possble dscotuty s at H. The proof of Proposto 5.9 s reported Secto 5.4 below. Let us frst show how t allows to complete the
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3299 Proof of Theorem 3.6. By Lemma 5.8 ad Proposto 5.9, Y s a E-martgale o [0,τ ]. Moreover, sce X τ = Y τ, the Y 0 = E[X τ ] ad thus τ s a optmal stoppg tme. Remark 5.10. Assume Assumpto 3.4() ad the codtos of Lemma A.1 below hold, by Remark 3.5() ad Lemma A.1 we see that Proposto 5.9 ad hece Theorem 3.6 hold. That s, ths case the Secto 5.4 below s ot eeded. 5.4. E-cotuty of Y at the radom maturty Ths subsecto s dedcated to the proof of Proposto 5.9. We frst reformulate some pathwse propertes establshed prevous subsectos. For that purpose, we troduce the followg addtoal otato: for ay P P, τ T, ad E F τ P(P, τ, E) := P P : P = P τ P 1 E + P1 E c, P(P, τ) := P(P, τ, Ω). (5.29) That s, P P(P, τ, E) meas P = P o F τ ad (P ) τ,ω = P τ,ω for P-a.e. ω E c. The frst result correspods to Theorem 5.4. Lemma 5.11. Let P P, τ 1, τ 2 T, ad E F τ1. Assume τ 1 τ 2 H, ad τ 1 < H o E. The for ay ε > 0, there exst P ε P(P, τ 1, E) ad τ ε T wth values [τ 1, τ 2 ], s.t. E P Y τ1 1 E E ε P X τε 1 {τε <τ 2 } + Y τ2 1 {τε =τ 2 } 1E + ε. Proof. Let τ1 be a sequece of stoppg tmes such that τ 1 τ ad each τ 1 takes oly ftely may values. Applyg Lemma 5.3 together wth the domated covergece theorem uder P, we see that lm E P Y τ 1 τ 2 Y τ1 = 0. Fx such that E P Y τ 1 τ 2 Y τ1 ε 2. (5.30) Assume τ1 takes values {t, = 1,..., m}, ad for each, deote E := E {τ1 = t < τ 2 } F t. By (5.13), there exsts τ T ad P P(P, t ) such that τ t o E ad Y t E P t X τ H + ε 2, P-a.s. o E. (5.31) Here E P t [ ] := E P [ F t ] deotes the codtoal expectato. Defe m m τ := τ 2 1 E c {τ 2 τ1 } + τ 1 E, P := P1 E c {τ 2 τ1 } + P 1 E. (5.32) =1 The oe ca check straghtforwardly that τ T ad τ τ 2 τ 1 ; (5.33) ad P P(P, τ 2 τ1, E) P(P, τ 1, E). Moreover, by (5.31) ad (5.32), E P Y τ2 τ1 1 m E = E P Y τ2 1 {τ2 τ1 } + Y t 1 E 1 E E P Y τ2 1 {τ2 τ 1 } + =1 =1 X τ H + 2 ε 1 {τ 1 <τ 2 } 1 E.
3300 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 Ths, together wth (5.30) ad (5.33), leads to E P Y τ1 X τ 1 { τ<τ2 } Y τ2 1 { τ τ2 } 1E ε + E P Y τ2 1 {τ2 τ1 } + X τ H 1 {τ 1 <τ 2 } X τ 1 { τ<τ2 } Y τ2 1 { τ τ2 } 1E = ε + E P X τ H Y τ2 1{τ 1 <τ 2 τ}1 E = ε + E P E P τ2 [X τ H ] Y τ2 1{τ 1 <τ 2 τ}1 E ε, where the last equalty follows from the defto of Y. The, by settg τ ε := τ τ 2 we prove the result. Next result correspods to Lemma 5.8. Lemma 5.12. Let P P, τ T, ad E F τ such that τ τ o E. The for all ε > 0: E P 1 E Y τ E P ε 1E Y τ + ε for some Pε P(P, τ, E). Proof. We proceed three steps. Step 1. We frst assume τ = t < τ o E. We shall prove the result followg the argumets Lemma 5.8. Recall the otatos Secto 5.2 ad the ψ defed (5.24), ad let ρ deote the modulus of cotuty fuctos of θ, θ, ad ψ. Deote τ := 0 for (Y 0 X 0 ) 1. For ay ad δ > 0, let {E,δ partto of E { τ 1 t < τ } such that ω ω t (, ), fx ω, := ω,δ, E,δ. By Lemma 5.8, Y 1 E,δ Y t (ω, ) = E t [Y t,ω, τ t,ω, Y t (ω, ) E P,δ Y t,ω, τ t,ω, Note that m=1 1 E m,δ P,δ := P t m=1 ], ad thus there exsts P,δ 1 P t such that, 1} F t be a δ for ay ω, ω E,δ. For each s a E-martgale o [t, τ ]. The + ε. (5.34) = E {t < τ }. Set P m,δ 1 E m,δ + P1 E c {t τ } Recall the h δ defed by (5.16). We clam that, for ay N, P(P, t, E). (5.35) E P [Y t 1 E ] E PN,δ [Y t θ ψ 1 E ] CE ρ 2 δ + ρ (δ) + 2η (δ) + Cρ (δ) + ε + C2 + CC(ψ < 1) + 2E PN,δ sup Y s Y θ ψ 1 E + CE h δ d ω, (Ω ) c, (5.36) θ s θ where η (δ) := sup Bs t 1 Bs t 2. t s 1 <s 2 t 0,s 2 s 1 ρ (δ)
I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 3301 Moreover, oe ca easly fd F t -measurable cotuous radom varables ϕ k such that ϕ k 1 ad lm k E P [ 1 E ϕ k ] = 0. The E P [Y t 1 E ] E PN,δ [Y t θ ψ ϕ k ] CE ρ 2 δ + ρ (δ) + 2η (δ) + Cρ (δ) + ε + C2 + CC(ψ < 1) + CE PN,δ sup Y s Y θ ψ ϕ k + CE θ s θ h δ d ω, (Ω ) c + CE P [ 1 E ϕ k ]. Sed δ 0. Frst ote that [δ + ρ (δ) + 2η (δ)] 0 ad h δ 1 {0}, the by Proposto 2.6() we have lm E ρ 2 δ + ρ (δ) + 2η (δ) = 0; δ 0 lm E h δ d ω, (Ω ) c = C δ 0 d ω, (Ω )c = 0 = C[(Ω )c ] C2. Moreover, for each N, by the weak compactess assumpto (P1) we see that P N,δ has a weak lmt P N P. It s straghtforward to check that P N P(P, t, E). Note that the radom varables Y t θ ψ ϕ k ad sup θ s θ Y s Y θ ψ ϕ k are cotuous. The E P [Y t 1 E ] E PN [Y t θ ψ ϕ k ] ε + C2 + CC(ψ < 1) + CE PN sup θ s θ Y s Y θ ψ ϕ k + CE P [ 1 E ϕ k ]. Aga by the weak compactess assumpto (P1), P N has a weak lmt P P(P, t, E) as N. Now sed N, by the cotuty of the radom varables we obta E P [Y t 1 E ] E P [Y t θ ψ ϕ k ] ε + C2 + CC(ψ < 1) + CE P sup θ s θ Sed k ad recall that P = P o F t, we have E P [Y t 1 E ] E P [Y t θ ψ 1 E ] ε + C2 + CC(ψ < 1) + 2E P sup θ s θ Y s Y θ ψ ϕ k + CE P [ 1 E ϕ k ]. Y s Y θ ψ 1 E. Fally sed, by (5.27) ad applyg the domated covergece theorem uder P ad P we have E P [Y t 1 E ] E P [Y τ 1 E ] ε. That s, P ε := P satsfes the requremet the case τ = t < τ o E. Step 2. We ow prove Clam (5.36). Ideed, for ay m ad ay ω E m,δ, by Lemma 5.1 we have Y t (ω) E Pm,δ Y t,ω τ t,ω = Y t (ω) Y t (ω m, ) + Y t (ω m, ) E Pm,δ Y t,ωm, τ t,ωm, + E Pm,δ Y t,ωm, τ t,ωm, Y t,ω τ t,ω
3302 I. Ekre et al. / Stochastc Processes ad ther Applcatos 124 (2014) 3277 3311 Cρ 1 (δ) + ε + E Pm,δ Y t,ωm, Y t,ω τ t,ωm, τ t,ω 1(Ω ψ t,ωm, ) t,ωm, (Ω )t,ω ψ t,ω + CP m,δ [(Ω )t,ωm, ] c [(Ω )t,ω ] c + CE Pm,δ 1 ψ t,ωm, + 1 ψ t,ω. (5.37) Note that E Pm,δ P m,δ 1 ψ t,ωm, + 1 ψ t,ω 2E Pm,δ [(Ω )t,ωm, ] c [(Ω )t,ω ] c 2P m,δ + P m,δ 1 ψ t,ω [(Ω )t,ω ] c + ρ (δ); [(Ω )t,ωm, ] c (Ω )t,ω 2P m,δ [(Ω )t,ω ] c + P m,δ 0 < d ω t B t, (Ω )c < δ (5.38) Moreover, o (Ω )t,ωm, (Ω )t,ω {ψ t,ωm, have The (θ )t,ωm, τ t,ωm, Y t,ωm, τ t,ωm, Y t,ω τ t,ω 2P m,δ [(Ω )t,ω ] c + E Pm,δ h δ d ω t B t, (Ω )c. > 0} {ψ t,ω > 0}, by Lemma 5.7 ad (5.25) we (θ )t,ωm, < H t,ωm, +4 ; (θ )t,ω τ t,ω (θ )t,ω < H t,ω +4. Y t,ωm, (θ )t,ωm, Y t,ω (θ )t,ω + sup Y s t,ωm, Y t,ωm, + sup Y (θ )t,ωm, s (θ (θ )t,ωm, )t,ωm, (θ )t,ω s (θ s t,ω Y t,ω (θ )t,ω )t,ω = Y t,ωm, Y t,ω (θ )t,ωm, (θ + 2 sup Y )t,ω (θ )t,ω s (θ s t,ω Y t,ω (θ )t,ω )t,ω + sup Y s t,ωm, Y t,ωm, sup Y (θ )t,ωm, s (θ (θ )t,ωm, )t,ωm, (θ )t,ω s (θ s t,ω Y t,ω (θ. )t,ω )t,ω Applyg Lemma 5.3 we get Y t,ωm, Y t,ω (θ )t,ωm, (θ Cρ )t,ω 2 d ((θ ) t,ωm,, ω m, t B t ), ((θ )t,ω, ω t B t ) Cρ 2 δ + ρ (δ) + 2 sup B (θ )t,ω ρ (δ) s (θ s t Bt (θ )t,ω +ρ (δ) )t,ω Cρ 2 δ + ρ (δ) + 2η (δ),