American options with multiple priors in continuous time

Transcription

1 Working Papers Insiue of Mahemaical Economics 448 April 2011 American opions wih muliple priors in coninuous ime Jörg Vorbrink IMW Bielefeld Universiy Posfach Bielefeld Germany hp:// imw/papers/showpaper.php?448 ISSN:

2 American opions wih muliple priors in coninuous ime Jörg Vorbrink Insiue of Mahemaical Economics Bielefeld Universiy April 25, 2011 Absrac We invesigae American opions in a muliple prior seing of coninuous ime and deermine opimal exercise sraegies form he perspecive of an ambiguiy averse buyer. The muliple prior seing relaxes he presumpion of a known disribuion of he sock price process and capures he idea of incomplee informaion of he marke daa leading o model uncerainy. Using he heory of (refleced backward sochasic differenial equaions we are able o solve he opimal sopping problem under muliple priors and idenify he paricular wors-case scenario in erms of he wors-case prior. By means of he analysis of exoic American opions we highligh he main difference o classical single prior models. This is characerized by a resuling endogenous dynamic srucure of he wors-case scenario generaed by model adjusmens of he agen due o paricular occurring evens ha change he agen s beliefs. Key words and phrases: opimal sopping for exoic American opions, uncerainy aversion, muliple priors, robusness, (refleced BSDEs JEL subjec classificaion: G13, D81, C61 The auhor hanks Prof. Frank Riedel for valuable advice and commens. Financial suppor hrough he German Research Foundaion (DFG and he Inernaional Research Training Group Sochasics and Real World Models Beijing Bielefeld are graefully acknowledged. 1

3 1 Inroducion This paper builds on a previous analysis of opimal sopping problems for American exoic opions under ambiguiy, Chudjakow and Vorbrink [5]. The moivaions and he economic relevance of his sudy are similar o before, alhough we move from discree o coninuous ime. In finance i is more appropriae o use coninuous ime models. Closedform soluions have he advanage of being easier o inerpre, and as such, end o predominae. They allow for comparaive saics ha would be oherwise difficul o inerpre. In our analysis coninuous ime also provides a direc relaionship o he famous Black-Scholes model, Black and Scholes [1]. We analyze American opions from he perspecive of an ambiguiy averse buyer in he sense of Ellsberg s paradox. The ask of he buyer holding he opion is o exercise i opimally realizing he highes possible uiliy. The valuaion reflecs he agen s personal uiliy as i depends on invesmen horizon, objecive, and on risk, as well as ambiguiy aiude. Generally his valuaion is no relaed o he marke value direcly. Given a classical sochasic model in coninuous ime such as he Black- Scholes model, one can solve he opimal sopping problem of he buyer using classical heory on opimal sopping, or he relaion o free-boundary problems. Despie he abundance of lieraure on he issue, e.g. Peskir and Shiryaev [20] or El Karoui [9], hese seings impose he assumpion of a unique probabiliy measure ha drives sock price processes. This assumpion migh be oo srong in many cases since i requires perfec undersanding of he marke and complee agreemen on one paricular model. To incorporae uncerainy we drop his assumpion. We consider a Black-Scholes-like marke whose sock price X = (X evolves according o dx = µx d + σx dw (1 where W = (W represens sandard Brownian moion under some reference measure P. 1 The various beliefs of he agen are refleced by a se of muliple priors (probabiliy measures P. Thus she considers he dynamics in (1 under each prior Q of he se P which provides a family of models ha come ino quesion o evaluae he claims. 1 Laer we change his poin of view slighly, cf. page 11. 2

4 As an example we have in mind a bank which holds an American claim in is rading book. The rading sraegy of he bank depends on he underlying model used by he bank. If he model specificaion is error-prone he bank faces model uncerainy. Being unable o compleely specify he model, raders raher use muliple priors model insead of choosing one paricular model. If he uncerainy canno be resolved and he accurae model specificaion is impossible, raders prefer more robus sraegies as hey perform well even if he model is specified slighly incorrec. Also, a risk conrolling uni assigning he porfolio value and riskiness uses raher a muliple priors model in order o es for model robusness and o measure model risk. Taking several models ino accoun, while performing porfolio disress ess, allows o check he sensiiviy of he porfolio o model misspecificaion. Again in a siuaion of model uncerainy more robus riskiness assignmen is desirable as i minimizes model risks. Similar reasoning can be applied o accouning issues. An invesmen funds manager making his annual valuaion is ineresed in he value of opions in he book ha are no seled ye. In case he company applies coheren risk measures as sandard risk evaluaion ool for fuure cash flows on he shor side, i is plausible o use a muliple priors model evaluaing long posiions. Finally, a privae invesor holding American claims in his depo migh exhibi ambiguiy aversion in he sense of Ellsberg paradox or Knighian uncerainy. Such behavior may arise from lack of experise or bad qualiy of informaion ha is available o he decision maker. Alhough for differen reasons, all he marke paricipans described above face problems ha should no be analyzed in a single prior model and need o be formulaed as muliple priors problems. As o he ambiguiy model, we use κ-ignorance, see Chen and Epsein [2]. I models uncerainy in he drif rae of he sock price. Under each prior, he sock price in (1 obains an addiional drif rae erm varying wihin he inerval [ κ, κ], where κ measures he degree of ambiguiy/uncerainy. As noed in Cheng and Riedel [4], i is essenial ha he addiional erms be allowed o be sochasic and ime-varying as his guaranees dynamic consisency. 2 Dynamic consisency allows he agen o adap he model according o changing beliefs induced by occurring evens. In his seing, he agen hold- 2 See Cheng and Riedel [4] and Delbaen [6] for a discussion of he concep of dynamic consisency in dynamic models. 3

5 ing an American opion who is uncerain abou he correc drif of he underlying sock price faces he opimizaion problem V := ess sup ess inf ( τ Q P EQ H τ γτ F 1. (2 To clarify, a he curren ime, he agen aims o opimize her expeced discouned payoff H τ γτ 1 in a wors-case scenario by exercising he claim prior o mauriy. In our analysis he opimizaion problem is solved by using he relaionship o refleced backward sochasic differenial equaions (RBSDEs. 3 To obain his relaion, he generaor of he (refleced BSDE should be chosen as f(, y, z = ry κ z where κ z describes he ambiguiy aversion and ry he discouning. This was firs esablished by Chen and Epsein [2] who used he generaor f(z = κ z for a BSDE o derive a generalized sochasic differenial uiliy. A similar BSDE framework o is used in El Karoui and Quenez [12] in he conex of pricing and hedging under consrains. BSDEs provide a powerful mehod for analyzing problems in mahemaical finance, (El Karoui, Peng, and Quenez [11] and Duffie and Epsein [8], or in sochasic conrol and differenial games (Hamadene and Lepelier [13] & Pham [21]. BSDEs, in conjuncion wih g-expecaions, play an imporan role in he heory of dynamic risk measures, (Peng [18] and dynamic convex risk measures, respecively, (Delbaen, Peng, and Gianin [7]. By means of reflecion, he soluion is mainained above a given sochasic process, in our case, he payoff process of he respecive American claim. We analyze he problem in (2 for several American opions exemplifying he effec of ambiguiy. As described in Chudjakow and Vorbrink [5] he effec of ambiguiy depends highly on he payoff srucure of he claim. If he payoff saisfies cerain monooniciy behavior as is he case for he American call and pu opion, he siuaion resembles he classical one wihou he emergence of ambiguiy. The agen s wors-case scenario is specified by he leas favorable drif rae of he sock price process ha affecs he performance of he agen s opion. This scenario is idenified by he wors-case prior. In he above described monoone case, he wors-case prior leads o he lowes possible drif rae for he sock price process in case of a call, and he highes possible drif rae in he case of a pu opion. 3 Anoher approach is he characerizaion of he value funcion (V by Cheng and Riedel [4] as he smalles righ-coninuous g-supermaringale ha dominaes he payoff from exercising he claim. 4

6 For opions wih more complex payoffs, he wors-case prior generaes a sochasic drif rae in (1 which is pah-dependen and produces endogenous dynamics in he model. These are induced by he ambiguiy averse agen and her reacion o he laes informaion by adjusing he model from ime o ime as necessary depending on her changing beliefs, or fears, respecively. As such, in he muliple prior seing, changing fears due o ranspired evens are aken ino accoun when American claims are evaluaed and early exercise sraegies are deermined. This cenral difference o classical models is exemplified wih he help of barrier opions and shou opions. In he laer case, he agen will change her beliefs direcly afer aking acion, when she fixes he srike price. In he case of barrier opions, here exemplified by means of an up-and-in pu opion, she adaps he model as a consequence of he rigger even when he underlying sock price reaches he barrier specified in he claim s conracual erms. From decision heoreical poin of view, our examples expose ha opimal sopping under ambiguiy aversion is behaviorally disinguishable from opimal sopping under subjecive expeced uiliy. For example, he holder of an American up-and-in pu behaves as wo readily disinguishable expeced uiliy maximizers. The paper is srucured as follows. The following secion inroduces he ambiguiy seup in coninuous ime and relaes he resuling muliple prior framework o he financial marke. Secion 3 presens he decision problem of an ambiguiy averse agen who holds an American opion. I conains a shor deour o refleced BSDEs and explains heir relaionship o he decision problem of he ambiguiy averse agen. This secion also provides he soluion o he opimal sopping problem for American opions feauring some monoone payoff srucure (see Secion 3.2. This secion builds he base for he subsequen analysis in Secion 4 concerning American claims wih more complex payoffs such as up-and-in pu opions or shou opions. Exensive proofs are given in he appendix, Secion 5 concludes. 2 The seing We inroduce he ambiguiy framework in coninuous ime. We focus on κ- ignorance, a paricular ambiguiy seing, as described by Chen and Epsein [2] who inroduced various ambiguiy models. Throughou his paper we 5

7 consider an ambiguiy framework for a fixed finie ime horizon T > 0. Firs, we depic he ambiguiy model κ-ignorance as in Chen and Epsein [2]. Second we inroduce he financial marke wihin his ambiguiy framework. Remark 2.1 Given an infinie ime horizon, one faces addiional echnical difficulies according o he underlying filraion arising from Girsanov s heorem and a Brownian moion environmen. 4 This leads o weaker assumpions on filraion. In paricular, he usual condiions on filraion should be relaxed. 5 This someimes causes echnical problems since he heory of sochasic calculus and backward sochasic differenial equaions is usually developed under hese condiions The ambiguiy model κ-ignorance Le W = (W be a sandard Brownian moion on he probabiliy space (Ω, F, P where F is he compleed Borel σ-algebra on Ω. We denoe by (F 0 T he filraion generaed by he process W and augmened wih respec o P. We have F T = F and he filraion saisfies he usual condiions. P serves as a reference measure in he ambiguiy model. As we shall see, under κ-ignorance all occurring probabiliy measures Q P are equivalen. So, P has he role of fixing he evens of measure zero. Hence, here will be no uncerainy abou he evens of measure zero. Remark 2.2 Throughou he analysis, unless saed oherwise, all equaliies and inequaliies will hold almos surely. The almos-sure-saemens are o be undersood wih respec o he reference measure P. Due o he equivalence of all priors Q P he saemens will also hold almos surely wih respec o any prior Q P. If we wrie E wihou any measure we will mean he expecaion wih respec o he reference measure P. Le us depic he consrucion of he ambiguiy model κ-ignorance, Chen and Epsein [2], Delbaen [6]. I relies heavily on Girsanov s heorem. We 4 See Remark 2.4 as an illusraion. 5 Usually he filraion is assumed o saisfy he usual condiions. This means ha he filraion is righ-coninuous and augmened, cf. Karazas and Shreve [14]. 6 The ineresed reader is referred o von Weizsäcker and Winkler [23] who develop sochasic calculus in paricular Iô calculus wihou assuming he usual condiions. 6

8 only focus on he one-dimensional case. The d-dimensional case works in a sraighforward way. Firs consider R-valued measurable, (F -adaped, and square-inegrable processes θ = (θ such ha he process z θ = (z θ defined by ha is, z θ = exp { 1 2 dz θ = θ z θ dw, z θ 0 = 1, θ 2 sds θ s dw s } [0, T ] (3 0 0 is a P -maringale. Given κ > 0 we define he se of densiy generaors Θ by Θ = {θ θ progressively measurable and θ κ, [0, T ]}. 7 (4 κ is called he degree of ambiguiy (uncerainy. Obviously, for each θ Θ he Novikov condiion E ( T exp{ 1 θ 2 2 sds} < is saisfied. Therefore, 0 E(zT θ = zθ 0 = 1 and zt θ is a P -densiy on F, Karazas and Shreve [14]. Consequenly, each θ Θ induces a probabiliy measure Q θ on (Ω, F ha is equivalen o P where Q θ is defined by Q θ (A := E(1 A z θ T A F. (5 In oher words, dq θ dp = z θ [0, T ]. F According o Girsanov s heorem (cf. Karazas and Shreve [14] we define he se of probabiliy measures P := P Θ on (Ω, F generaed by Θ by P Θ := {Q θ θ Θ and Q θ is defined by (5}. (6 7 Since we work in a Brownian moion environmen we do no need o require predicabiliy in (4 as in Delbaen [6], cf. Theorem in von Weizsäcker and Winkler [23]. 7

9 Noe ha we allow for sochasic and ime-varying Girsanov kernels θ. This is imporan o ensure he dynamic consisency. We oherwise lose his imporan propery. 8 Addiionally, by Girsanov s heorem, he process W θ = (W θ defined by W θ := W + 0 θ s ds [0, T ] (7 is a sandard Brownian moion on (Ω, F wih respec o he measure Q θ. Remark 2.3 κ-ignorance as an ambiguiy model has imporan properies. I allows for explici resuls when evaluaing financial claims since he range of values of he densiy processes θ does no change over ime as is he case for oher models like IID-ambiguiy in Chen and Epsein [2]. Consequenly we shall see ha he wors-case densiies become very simple in some examples, meaning wihou any formal difficulies. Furhermore, under κ-ignorance, he se of priors P possesses imporan properies like m-sabiliy or imeconsisency, Delbaen [6], and he exisence of wors-case priors, Chen and Epsein [2]. 9 Regarding Remark 2.1 he following remark illusraes he imporance of relaxing he usual condiions for filraion when κ-ignorance is consruced on an infinie ime horizon. Remark 2.4 (cf. Karazas and Shreve [14] Le P be Wiener measure on (Ω, F := (C ([0,, R, B (C([0,, R such ha he canonical process W = (W, W (ω := ω(, 0 <, ω Ω is a sandard Brownian moion. Denoe by (F W he (no augmened filraion generaed by W such ha F W = F. Le θ = (θ be a progressively measurable process wih corresponding filraion (F W, and square-inegrable for each T [0,. Assume ha he process z θ = (z θ defined as in (3 is a P -maringale. Then Girsanov s heorem for an infinie ime horizon 10 saes ha here exiss a probabiliy measure Q θ saisfying Q θ (A = E(z θ T 1 A, A F W T, T [0, (8 8 See Chen and Epsein [2] for deails. Also he examples in Secion 4 illusrae his fac. 9 See also Chudjakow and Vorbrink [5]. 10 See Corollary 5.2 in Karazas and Shreve [14]. 8

10 and he process W θ = (W θ defined as in Equaion (7 wih corresponding filraion (F W is a Brownian moion on (Ω, F, Q θ. I is essenial ha (F W be raw, unaugmened filraion. Therefore, κ- ignorance can only be consruced wih respec o a filraion ha does no fulfill he usual condiions. The difference o he finie ime horizon is ha now P and Q θ are only muually locally absoluely coninuous, i.e., equivalen on each FT W, T [0,. Viewed as probabiliy measures on F, P and Q θ are equivalen if and only if z θ is uniformly inegrable. To undersand why (8 is only required o hold for A FT W, T [0,, consider he following example. Example 2.5 Le µ > 0 and fix a process θ wih θ := µ [0,. For his θ consider he P -maringale z θ defined by z θ = exp{ 1 2 µ2 + µw } [0,. z θ is no uniformly inegrable. By Girsanov s heorem and he law of large numbers for Brownian moion, Karazas and Shreve [14] we obain for A := W { lim = µ} F Q θ (A = 1 and P (A = 0. Clearly, he P -null even A is in he augmened σ-field F T for every T [0,. This is he reason why (8 is only required o hold for all A FT W, T <. Oherwise P and Q θ were muually singular on F T for every T 0. Therefore, κ-ignorance in a Brownian moion environmen wih infinie ime horizon mus be se up on a filraion ha is no augmened by he P -null ses of F. 2.2 The financial marke under κ-ignorance Throughou his paper we consider a Black-Scholes-like marke consising of wo asses, a riskless bond γ and a risky sock X. Their prices evolve according o dγ = rγ d, γ 0 = 1, dx = µx d + σx dw, X 0 = x > 0 (9 9

11 where r is a consan ineres rae, µ a consan drif rae, and σ > 0 a consan volailiy rae for he sock price. 11 The dynamics in (9 are obviously free of ambiguiy. To incorporae ambiguiy, he decision maker considers Equaion (9 under muliple priors. She uses he se of priors P as defined in (6. As we shall see, by uilizing he se P she ries o capure her uncerainy abou he rue drif rae of he sock. Le Q P, if Q is equal o Q θ for θ Θ hen he sock price dynamics under Q become dx = µx d σx θ d + σx dw θ. This illusraes ha κ-ignorance jus models uncerainy abou he rue drif rae of he sock price. A his poin i is worhwhile menioning ha by changing he prior under consideraion, he sock price s volailiy rae remains compleely unchanged. Based on he equivalence of all priors and Girsanov s heorem, κ-ignorance canno be used o model volailiy uncerainy. This requires a se of muually singular priors. For a deailed sudy of his issue see Peng [19] or Vorbrink [24]. In he nex secion, we consider American coningen claims from he perspecive of an ambiguiy averse decision maker who holds a long posiion in he claims. The decision maker, a privae invesor or financial insiuion, for example, may seek o evaluae or liquidae heir posiion. Boh may happen wih respec o heir subjecive probabiliy disribuion. They may use heir subjecive probabiliy disribuion o evaluae he claim and o figure ou an opimal exercise sraegy due o he claim s American feaure. In addiion, in real opion invesmen decisions, he subjecive probabiliy law appears naurally when coming o a decision. 12 All decision problems are considered under Knighian uncerainy. We focus on a decision maker who is uncerain abou marke daa. As a consequence she does no believe compleely in he dynamics proposed in (9. For insance she is uncerain abou he sock s drif rae which in urn affecs he marke price of risk. Coningen claims in finance are ypically evaluaed wih respec o riskneural probabiliy measures. Therefore, we assume ha he agen will con- 11 As i is ofen possible we may also consider a price process wih non-consan and sochasic coefficiens. To avoid laer disincions of cases and missing he poin we assume consan coefficiens. 12 See McDonald and Siegel [17], for example. 10

12 sider he sock s dynamics in (9 under he risk-neural probabiliy measure. Since she does no compleely rus in he marke, nor all he daa, she allows for various marke prices of risk. 13 She akes ino accoun prices surfacing around µ r currenly observed a he marke. Expanding on his idea, if σ Q = Q θ for some θ defined by θ = µ r + ψ σ, [0, T ], wih ψ = (ψ Θ hen he dynamics in (9 become dx = µx d σx θ d + σx dw θ = rx d σx ψ d + σx dw θ. To say in he framework of κ-ignorance, as inroduced above, we need o change he reference measure. To avoid his sep, we prefer o model he sock price dynamics direcly under he risk-neural probabiliy measure, i.e., he agen sars wih he reference dynamics dx = rx d + σx dw. (10 Now, if she considers (10 under Q = Q θ become for some θ Θ he dynamics dx = rx d σx θ d + σx dw θ. (11 Throughou he paper, Equaion (11 for varying θ Θ represens he dynamics our decision maker will ake ino accoun when sudying opimal sopping problems under he ambiguiy aversion modeled by κ-ignorance. 3 American opions under ambiguiy aversion We focus on American coningen claims under ambiguiy aversion. 14 For his issue, we analyze opimal sopping problems under muliple priors. Formally, he opimal sopping problem under ambiguiy aversion is defined as V := ess sup ess inf ( τ Q P EQ H τ γτ F 1, [0, T ] (12 13 As menioned above he subjecive evaluaion appears naural. By he variey of considered models subjecive beliefs are neverheless conained. If one prefers he subjecive in place of he risk-neural probabiliy measure as a reference one may also use he model in (9 wih drif rae µ as a reference. 14 A deailed economic moivaion is given in Chudjakow and Vorbrink [5]. 11

13 where γτ 1 is he discouning from curren ime up o sopping ime τ when he claim is exercised. H = (H represens he payoff process. We only consider claims wih mauriy T. The ess inf accords wih ambiguiy aversion which leads o wors-case pricing. The ess sup imposes he goal of he agen o opimize he claim s payoff by finding an opimal exercise sraegy in he wors-case scenario. All sopping imes τ ha will come ino quesion in (12 are naurally bounded by he ime horizon and claim s mauriy T. Wihou ambiguiy, V represens he unique price for he claim a ime, see Peskir and Shiryaev [20] for example. We analyze American opions wrien on X. In general, he claim s payoff from exercising depends on he whole hisory of he price process. To ensure ha he value V, [0, T ] is well-defined, we impose he following assumpion on he claim s payoff process. Assumpion 3.1 Given an American coningen claim H, he payoff from exercising H = (H is an adaped, measurable, nonnegaive process wih coninuous sample pahs 15 saisfying E ( sup 0 T H 2 <. To solve he opimal sopping problem under muliple priors in (12 we uilize he mehodology of refleced backward sochasic differenial equaions (RBSDEs. 3.1 A deour: refleced backward sochasic differenial equaions A his poin we briefly inroduce he noion of RBSDEs and poin ou is relaionship o he opimal sopping problem under ambiguiy aversion. The proof can be found in Appendix A. The Markovian framework conains a very useful connecion o parial differenial equaions (PDEs, a generalizaion of he Feynman-Kac formula. As a reference for he paricular case of backward sochasic differenial equaions (BSDEs see El Karoui, Peng, and Quenez [11]. In Secion 3.2 we employ he resuls of Chen, Kulperger, and Wei [3] which srongly exploi he relaionship o PDEs. In his deour we use he same sochasic foundaion inroduced above. The inroducion is aken from El Karoui, Kapoudjian, Pardoux, Peng, and 15 I is possible o relax he assumpion, see Cheng and Riedel [4]. 12

14 Quenez [10]. 16 We also inroduce he following noaion, cf. Pham [21]: L 2 :={ξ ξ is an F-measurable random variable wih E( ξ 2 < }, { T } H 2 := (ϕ (ϕ is a progressively mb. process s.. E ϕ 2 d <, 0 { ( } S 2 := (ϕ (ϕ is a progressively mb. process s.. E sup ϕ 2 <. 0 T Given a progressively measurable process S = (S, inerpreed as an obsacle, he aim is o conrol a process Y = (Y such ha i remains above he obsacle and saisfies equaliy a erminal ime, i.e., Y T = S T. This is achieved by a RBSDE. We briefly sae he definiion. Le S = (S be a real-valued process in S 2, and a generaor f : Ω [0, T ] R R R such ha f(, y, z H 2 (y, z R R, and f(, y, z f(, y, z C( y y + z z [0, T ] for some consan C > 0 and all y, y R, z, z R. Definiion 3.2 The soluion of he RBSDE wih parameers (f, S is a riple (Y, Z, K = (Y, Z, K of (F -progressively measurable processes aking values in R, R, and R +, respecively, and saisfying: (i Y = S T + T (ii Y S, [0, T ] f(s, Y s, Z s ds + K T K T Z s dw s, [0, T ] (iii K = (K is coninuous, increasing, K 0 = 0, and T 0 (Y S dk = 0 (iv Z = (Z H 2, Y = (Y S 2, and K T L 2 The dynamics in (i are ofen expressed in differenial form. Tha is dy = f(, Y, Z d + dk Z dw, Y T = S T. (13 Inuiively, he process K pushes Y upwards such ha he consrain (ii is saisfied, bu minimally in he sense of condiion (iii. From (i and (iii 16 The framework is based on predicable processes. Bu he argumens rely only on progressive measurabiliy, cf. Pham [21]. Therefore we require he measurabiliy condiions as in Pham [21]. 13

15 i follows ha (Y is coninuous. El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10] proved he exisence and uniqueness of a soluion o he RBSDE as defined here. Le us consider equaion (12 for a fixed probabiliy measure Q omiing he operaor ess inf. If Q = Q θ P hen he process Y θ defined as he unique soluion of he refleced BSDE wih obsacle S = H 17 Y θ = H T + T ( ry θ s θ s Z θ s d + K θ T K θ T Z θ s dw s, [0, T ] also solves Equaion (12 wihou ambiguiy under he single prior Q = Q θ. Hence Y θ = V Q wih V Q := ess sup E ( Q H τ γτ F 1, [0, T ]. τ This follows by Proposiion 7.1 in El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10] ogeher wih Girsanov s heorem. I illusraes ha for each θ Θ he decision maker faces a RBSDE induced by he parameers (f θ, H wih f θ (, y, z = ry θ z [0, T ]. The following heorem esablishes he link o he opimal sopping problem defined in (12. I presens he key o solving he opimal sopping problem under ambiguiy aversion. Theorem 3.3 (Dualiy Given a payoff process H, define f θ (, y, z := ry θ z for each [0, T ] and consider he unique soluion (Y θ, Z θ, K θ o he RBSDE associaed wih (f θ, H for each θ Θ. Le (Y, Z, K denoe he soluion of he RBSDE wih parameers (f, H where f(, y, z := ess inf θ Θ f θ (, y, z [0, T ], y, z R. Then here exiss θ Θ such ha Hence, f(, Y, Z := ess inf θ Θ f θ (, Y, Z = f θ (, Y, Z = ry max θ Θ θ Z = ry κ Z (Y, Z, K = (Y θ Y = ess inf θ Θ, Z θ, K θ [0, T ] a.s. and Y θ = ess inf Q P V Q d P a.e. [0, T ] a.s. 17 Since we assumed H = (H o be adaped, measurable, and coninuous i is progressively measurable, cf. Proposiion 1.13 in Karazas and Shreve [14]. 14

16 Furhermore, Y = ess inf Q P ess sup E Q (H τ γτ F 1 = ess sup τ τ ess inf Q P EQ (H τ γ 1 τ F = V Hence, Y also solves he opimal sopping problem of he ambiguiy averse decision maker in (12. In paricular we have max min τ 0 Q P EQ (H τ γτ 1 = min max Q P τ 0 EQ (H τ γτ 1. An opimal sopping rule is given by τ := inf{s V s = H s } [0, T ]. a.s. The subscrip indicaes ha τ a ime. is an opimal sopping ime when we begin Proof: The proof is mosly given in El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10], Theorem 7.2. Since i is no direcly relaed o muliple priors under κ-ignorance, we presen he main ideas in Appendix A. Remark 3.4 The infimum above is an infimum of random variables. Therefore i mus be seen as an essenial infimum. For ime zero here is no ambiguiy in he definiions since he σ-algebra F 0 is rivial. By inerpreing he heorem, he ambiguiy averse agen solves he opimal sopping problem under a wors-case prior Q := Q θ P. Tha is, she firs deermines he wors-case scenario and hen solves a classical opimal sopping problem wih respec o his scenario. The heorem saes he relevance of RBSDEs for solving he opimal sopping problem under ambiguiy aversion. As indicaed in Theorem 3.3, from his poin on, he payoff process of he claim H will represen he obsacle for he associaed RBSDEs. We are ineresed in he soluion of he RBSDE associaed wih he parameers (f, H. In paricular, we arge undersanding he process θ ha induces he wors-case measure. 3.2 Opions wih monoone payoffs We focus on American claims whose curren payoff can be expressed by a funcion only depending on he curren sock price of he claim s underlying. 15

17 We assume H = Φ (X for each [0, T ]. 18 In his case he RBSDE wih parameers (f, H becomes a refleced forward backward sochasic differenial equaion (RFBSDE, cf. El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10]. The soluion for (12 is given by he process Y deermined as he soluion for dx = rx d + σx dw, X 0 = x dy = min ( ry θ Z d + dk Z dw, Y T = Φ T (X T (14 θ Θ }{{} = ry κ Z =f(,y,z wih obsacle H = Φ (X [0, T ]. From his poin forward, he mapping (, x Φ (x is assumed o be joinly coninuous for all (, x [0, T ] R +, and Φ (X L 2 (Ω, F, P [0, T ]. The laer is for insance rue if each Φ is of polynomial growh (see for example Malliavin [15], p. 6. Remark 3.5 If he payoff is zero for each [0, T, i.e., he obsacle only consiss of he erminal condiion Y T = Φ(X T he process K is se equal o zero and (14 jus becomes a forward BSDE wihou reflecion. In his case, he soluion Y of (14 solves he opimal sopping problem under ambiguiy aversion for a European coningen claim. In order o solve he opimal sopping problem in (12 we focus on he RFB- SDE in (14. The characerisic of his seing is ha he generaor and he obsacle are deerminisic. The only randomness of he parameers (f, H comes from he sae of he forward SDE X, a Markov process. We will make use of his observaion in he nex resuls. Firs we derive a resul which characerizes he process Z of he soluion o (14. Lemma 3.6 Consider he RFBSDE in (14 wih obsacle H = Φ (X [0, T ]. Le (Y, Z, K be he unique soluion. (i If Φ is increasing for all [0, T ], we have Z 0 d P a.e. 18 Since i is assumed ha H = (H has coninuous sample pahs he mapping (, x Φ (x has o be joinly coninuous for all (, x [0, T ] R +. 16

18 (ii If Φ is decreasing for all [0, T ], we have Z 0 d P a.e. Proof: We only prove (i; (ii follows analogously. Wihou he obsacle requiremen in (14, and jus he erminal condiion Y T = Φ T (X T, i follows from a resul in Chen, Kulperger, and Wei [3] 19 ha Z 0 d P a.e. To achieve he passage o refleced BSDEs we employ a penalizaion mehod. 20 Le n N, and (Y (n, Z (n be he unique soluion of he penalized BSDE wih dynamics Y (n = Φ T (X T + T [f(s, Y s (n, Z s (n + n(y s (n Φ s (X s ]ds }{{} =: f(s,x s,y s,z s T Z (n s dw s, [0, T ], (x := max{ x, 0}, and f(, y, z = ry κ z as above. f saisfies he assumpions of a generaor for a BSDE as saed in he deour for (refleced BSDEs. 21 In Chen, Kulperger, and Wei [3] he generaor of he BSDE considered does no depend on X. Forunaely, he map x f(, x, y, z is increasing for all [0, T ], y, z R if and only if x Φ (x is increasing for all [0, T ]. Thus, a larger x leads o larger generaor f and larger erminal payoff. This monooniciy behavior is compaible wih he applicaion of he comparison heorem for BSDEs which is necessary o derive he resul in Chen, Kulperger, and Wei [3]. Thus, he resul in Chen, Kulperger, and Wei [3] can also be derived for his penalized BSDE. Hence, Z (n 0 d P a.e. Now we le n go o infiniy. Then Z (n converges o Z in L 2 (d P, cf. Secion 6 in El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10]. By 19 See Theorem 2 in Chen, Kulperger, and Wei [3]. I is proved by a generalizaion of he Feynman-Kac formula for BSDEs in connecion wih he comparison heorem for BSDEs, cf. Peng [18]. 20 Approximaion via penalizaion is a sandard mehod o ransfer resuls on BSDEs o RBSDEs, see El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10]. 21 The addiional dependence on X in erms of he funcion Φ does no exhibi any furher difficuly here, cf. El Karoui, Peng, and Quenez [11]. 17

19 sandard subsequence argumen we also obain Z 0 d P a.e. Using he lemma we can prove he following heorem. Theorem 3.7 (Claims wih monoone payoffs Consider an American claim H wih payoff a curren ime given by H = Φ (X [0, T ]. The value of he opimal sopping problem under ambiguiy aversion in (12 is given by ( V = ess sup E Q Φτ (X τ γτ F 1, [0, T ]. τ The wors-case prior Q can be specified by is Girsanov densiy z θ T. (i If Φ is increasing for all [0, T ], we have Q = Q κ, z θ T = zκ T wih z κ T = exp{ 1 2 κ2 T κw T }. (ii If Φ is decreasing for all [0, T ], we have Q = Q κ, z θ T z κ T = exp{ 1 2 κ2 T + κw T }. = z κ T wih In boh cases, an opimal sopping ime is given by τ := inf{s [, T ] V s = Φ s (X s }. Proof: Le (Y, Z, K be he unique ( soluion of (14. For [0, T ] we have V = Y = Y θ = ess sup τ E Q Φτ (X τ γτ F 1 by dualiy, see Theorem 3.3. This also verifies he saemen abou an opimal sopping ime. In case (i, by Lemma 3.6 we know ha Z 0 d P a.e. Hence, f(, Y, Z = ry κz d P a.e. which implies f(, Y, Z = f θ (, Y, Z d P a.e. 18

20 for θ = (κ Θ. So, he wors-case prior is given by Q = Q κ where Q κ is idenified by is Girsanov densiy z κ T = exp{ 1 2 κ2 T κw T }. In case (ii, f(, Y, Z = ry + κz d P a.e. Therefore we idenify Q = Q κ as he wors-case prior. The preceding heorem s proof relies heavily on he close relaionship beween opimal sopping problems and RBSDEs, he comparison heorem for (refleced BSDEs, and he Markovian framework which is essenial for Lemma 3.6. In discree ime, he corresponding heorem has been proven by a generalized backward inducion and firs-order sochasic dominance, Riedel [22]. As a direc applicaion, we quickly collec he conclusions for he American call and pu opion. Corollary 3.8 (American call Given L > 0, le he payoff from exercising he claim be H := (X L + for all [0, T ]. Then Q κ is he wors-case measure. Thus, a risk-neural buyer of an American call opion deermines an opimal sopping rule under he prior Q κ. Corollary 3.9 (American pu Given L > 0, le H := (L X + for all [0, T ]. Then Q κ is he wors-case measure and a risk-neural buyer of an American pu opion uilizes an opimal sopping rule for he prior Q κ. The inerpreaion of hese resuls is as follows. Exacly as in he corresponding discree ime seing, he ambiguiy averse buyer uses for her valuaion of a call opion for example he prior under which he underlying sock price possesses he lowes possible drif rae among all priors of he se. Tha is, under he wors-case prior Q κ, he sock evolves according o he dynamics of dx = (r σκx d + σx dw κ. In he case of an American pu opion she assumes he highes possible drif rae corresponding o he following sochasic evoluion of he sock wih respec o Q κ dx = (r + σκx d + σx dw κ. 19

21 Since X is a Markov process, we wrie Xs,x, s o indicae he price of he sock a ime s under he presumpion ha i is equal o x a ime, i.e., = x. As discussed above, by he Markovian srucure of (14 and X as he only randomness, we also wrie (Ys,x, Zs,x, Ks,x s [,T ] for he soluion of X,x (14 o indicae he Markovian framework. Tha is, he soluion Y can be wrien as a funcion of ime and sae X, (see Secion 4 in El Karoui, Peng, and Quenez [11] or Secion 8 in El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10]. Using he Markovian srucure, he value funcion V, [0, T ] in Theorem 3.7 simplifies o a funcion depending solely on he presen ime and presen sock price. Tha is, under he assumpion of X = x a ime he value of he opimal sopping problem under ambiguiy aversion in (12 reduces o V = Y,x = sup τ = ess sup ess inf ( τ Q P EQ Φ τ (X τ γτ X 1 = x ( E Q Φτ (Xτ,x γτ 1 =: u(, x. Remark 3.10 The value in (12 is sricly a funcion in he above seing, i.e. u of he presen ime and he presen sock price X. Noe ha we did no assume his o deermine he wors-case prior. In paricular we did no assume ha he value funcion u(, x is differenial wih respec o x and increasing in x, decreasing, respecively, an assumpion ofen made. The proofs of Lemma 3.6 and Theorem 3.7 do no require hese assumpions, see also Chen, Kulperger, and Wei [3]. Besides, he monooniciy of x u(, x follows direcly by comparison heorem. In case (i of Theorem 3.7 for insance, he mapping x Φ s (Xs,x increases because x Xs,x increases 22 for each s [, T ]. Then, by comparison heorem for RBSDEs, we obain ha u(, x is monoone increasing in x. The usual characerizaion of Markovian processes yields he following resul concerning he remaining mauriy of an American pu opion. The opion s American syle as well as he fac ha he payoff from exercising is jus a funcion depending on he curren sock price is essenial for his resul. 22 See he comparison resul for forward SDEs in for example Karazas and Shreve [14]. 20

22 Lemma 3.11 Consider an American pu opion wih srike price L. Given [0, T ], he soluion of he opimal sopping problem under ambiguiy aversion a ime V decreases in. Proof: Le (, x [0, T ] R + and (Ys,x, Zs,x, Ks,x be he unique soluion of he RFBSDE in (14 wih obsacle H s = (L Xs,x + s [, T ]. The Markov propery of X and Y, Corollary 3.9, and Theorem 3.3 yield ( Y,x = sup E Q κ (L X 0,x τ + γτ 1. 0 τ T Now le ε > 0 wih + ε T. Again, ( Y +ε,x +ε = sup E Q κ (L X 0,x τ + γτ 1. 0 τ T ε Hence, Y +ε,x +ε Y,x and he claim follows by dualiy, cf. Theorem 3.3. For laer use le us denoe for [0, T ] he value in (12 for an American pu opion wih srike price L under he assumpion of X = x by Y,x = sup τ 4 Exoic opions ( E Q κ (L X,x τ + γτ 1. (15 In his secion we leave he world of Markovian claims wih monoone payoffs in he curren sock price. We move on o consider he problem in (12 for exoic American claims. Wih he help of wo paricular examples, we analyze he effec of ambiguiy aversion on he opimal sopping behavior in his more involved siuaion. Examples are a shou opion and an American barrier opion in erms of an up-and-in pu. Similar o he discree ime seing in Chudjakow and Vorbrink [5], he analysis of hese examples demonsraes one of he main differences o he classical siuaion wihou ambiguiy. Even hough muliple priors lead o a more complex evaluaion, he approach is more appropriae in he sense of invesmen evaluaion for accouning and risk measuremen. We will see ha dynamical model adjusmens occur. Wih hese adjusmens he agen akes ino accoun changing beliefs based on realized evens wihin he evaluaion period. As such, he muliple priors seing 21

23 induces paricular endogenous dynamics. The agen evaluaes her sopping behavior under he wors-case scenario, he wors-case prior. This prior will depend crucially on he payoff process as well as on evens occurring during he lifeime of he claim under consideraion. 4.1 American up-and-in pu opion An American up-and-in pu presens is owner he righ o sell a specified underlying sock a a predeermined srike price under he condiion ha he underlying sock firs has o rise above a given barrier level. Formally, he payoff from exercising he opion a ime [0, T ] is defined as H := (L X + 1 {τh } (16 where τ H := inf{0 s T X s H} T denoes he knock-in ime a which he opion becomes valuable. This is he firs ime ha he underlying reaches he barrier. L defines he srike price and H he barrier. We assume H > L o focus on he mos ineresing case. We hope no o confuse he reader by he ambiguous use of he leer H denoing he barrier and he claim s payoff process a he same ime. Using previous resuls and firs-order sochasic dominance, we obain he following evaluaion scheme for he American up-and-in pu opion. Theorem 4.1 (Up-and-in pu Consider an American up-and-in pu wih payoff as defined in (16. The funcion ( V = ess sup E Q Hτ γτ F 1 τ solves he opimal sopping problem under ambiguiy aversion in (12 whereas he wors-case prior Q = Q θ is specified by he Girsanov densiy { zt θ := exp 1 T T } (θ 2 s 2 ds θsdw s wih θ defined as θ := 0 { κ, if < τ H κ, if τ H T. 0 22

24 An opimal sopping ime is given by τ := inf { τ H s T V s = (L X s +}. The heorem saes ha he agen considers he sopping problem under he measure Q θ. I is he pasing of he measures Q κ and Q κ a he ime of knock-in. Thus, she assumes he sock o evolve according o he leas favorable drif rae r σκ a he beginning of he conrac. During he conrac s lifeime, she changes her beliefs and assumes he highes possible drif rae r + σκ for he underlying. Tha is, she adaps her beliefs based on ranspired evens corresponding o her pessimisic poin of view. So a τ H, he poin in ime when he opion knocks in he agen s beliefs or fears change abruply. From a decision heoreical poin of view, his resul illusraes ha opimal sopping under ambiguiy aversion is behaviorally disinguishable from opimal sopping under expeced uiliy. The buyer of an American up-and-in pu for example behaves as wo readily disinguishable expeced uiliy maximizers. This is so because he wors-case measure ˆP depends on he payoff process. Proof: In his secion we provide an overview of he main ideas. More deails can be found in Appendix B. Given he even {τ H } ( he claim equals he usual American pu opion. Hence, V = ess sup τ E Q κ (L Xτ + γτ F 1. On {τ H > } we have V = ess inf Q P E ( Q V τh γτ 1 H F, (see he appendix for more deails. V τh represens he value of he opimal sopping problem under ambiguiy aversion a he specific ime of knock-in. Le us wrie g(s := Y s,h s where Y s,h s is he value of he American pu opion under ambiguiy aversion, see (15. By Lemma 3.11 he funcion s g(s decreases, as is s γs. 1 In he appendix we show ha τ H is sochasically larges under Q κ in he se of all priors P. Tha is, for all, s wih < s T, we have on {τ H > } and for all θ Θ Q κ (τ H s F Q θ (τ H s F. Then he usual characerizaion of firs-order sochasic dominance, Mas- Colell, Whinson, and Green [16], yields on {τ H > } E Qκ ( g(τh γ 1 τ H F E Q θ ( g(τh γ 1 τ H F. 23

25 Thus he wors-case prior Q is equal o Q κ on {τ H > }. Seing boh ogeher, Q is given by Q θ wih θ as defined in he heorem. Since θ is righ-coninuous, i is progressively measurable, per Proposiion 1.13 in Karazas and Shreve [14]. Hence θ Θ, which finishes he proof. An analogous resul holds for he American down-and-in call opion. In ha case, he agen solves he sopping problem under he wors-case scenario Q = Q θ where θ is now defined as { θ κ, if < τ H := κ, if τ H T. Here, τ H denoes he iniial ime when he underlying sock price breaks from above hrough he barrier H. 4.2 Shou opion A shou opion gives is owner he righ o deermine he srike price of a corresponding call or pu opion. We focus on he European pu opion version. Tha is, we consider a shou opion ha gives is buyer he righ o freeze he asse price a any ime τ S before mauriy o insure herself agains laer losses. A mauriy he buyer obains he payoff { X H T = τ S X T, if X T < X τ S. (17 0, else The value of he opimal sopping problem under ambiguiy aversion for a shou opion a ime τ S T is defined as V = ess sup ess inf ( τ S Q P EQ (X τ S X T + γ 1 T F. (18 We only consider he problem for imes τ S. This is he mos ineresing case since he owner has no fixed he srike price ye. She sill faces he opimal sopping decision which is he decision of shouing. To evaluae his conrac under ambiguiy aversion, we firs menion he following observaion already made in he discree ime seing, Chudjakow and Vorbrink [5]. This opion is equivalen o he following: upon shouing he owner receives a European pu opion (a he money wih srike X τ S 24

26 and remaining ime o mauriy T τ S. We obain he following evaluaion scheme. Theorem 4.2 (Shou opion Consider a shou opion a is saring ime zero wih a payoff as defined in (17. The soluion of (18 a ime zero simplifies o ( V 0 = sup E Q (Xτ S X T + γ 1 T τ S 0 where he wors-case prior Q = Q θ is specified by he Girsanov densiy z θ wih θ defined by { θ κ, if < τ S := κ, if τ S T. An opimal shouing ime is given by { ( τ S := inf 0 T V = E Q κ (X X T + γ 1 T F }. So in his case he ambiguiy averse agen changes her beliefs afer aking acion. Before shouing she assumes he lowes drif rae (r σκ, and he highes rae (r + σκ aferwards. Boh raes correspond o he respecive leas favorable rae, see also Chudjakow and Vorbrink [5]. Similarly o he up-and-in pu, her pessimisic perspecive leads o fearing he lowes possible reurns of he risky asse before shouing and he highes possible reurns hence. Proof: in (18 is As noed above, a he ime of shouing, he value of he conrac ( ess inf Q P EQ (X τ S X T + γ 1 F T τ S τ S. This is a European ype of monoone problem. The payoff a mauriy T is Φ T (x := (X τ S x + which is monoone decreasing in x. As a special case of Theorem 3.7 we derive he value a he ime of acion as ( ess inf Q P EQ (X τ S X T + γ 1 F T τ S τ S = g(τ S, X τ S 25

27 ( where g(, x := E Q κ (x X,x T + γ 1 T F. To deermine he value before shouing, consider he following refleced FBSDE wih obsacle (g(, X [0,T ] { dx = rx d + σx dw, X 0 = x. (19 dy = ry κ Z d + dk Z dw, Y T = g(t, X T A his poin i is imporan o noe ha he funcion g(, X saisfies he assumpions for presening an obsacle for a refleced BSDE. The join coninuiy in (, x follows by he properies of soluions o (refleced BSDEs. 23 Since g can be rewrien in he following form ( ( + g(, x = xe Q κ 1 exp{(r σ2 2 (T + σ(w T W } γ 1 T F we deduce ha he funcion x g(, x is increasing for all [0, T ]. Using Theorem 3.7 we conclude ( ( V 0 = Y 0 = sup E Qκ g (τ, Xτ γτ 1 = sup E Q (Xτ X T + γ 1 T. τ 0 The las equaliy follows from he law of ieraed expecaion. Addiionally we obain an opimal shouing ime τ S. I is deermined as he firs ime ha value V is equal o g(, X, he value of he European pu under ambiguiy aversion. This proves he heorem. θ Θ since i is righ-coninuous, again implying progressive measurabiliy. τ 0 5 Conclusion The paper sudies he opimal sopping problem of he buyer of various American opions in a framework of model uncerainy in coninuous ime. Model uncerainy induced by imprecise informaion is mirrored in a se of muliple probabiliy measures. Each measure corresponds o a specific drif rae for he sock price process in he respecive marke model. The agen hen is allowed o adap 23 The value for he European pu opion is obained as he soluion of a BSDE. Due o he European version of he pu opion g even belongs o C 1,2 ([0, T ] R +, cf. El Karoui, Peng, and Quenez [11]. 26

28 he model she uses o assign a value o he claim according o he wors possible model due o her ambiguiy averse aiude. We characerize he wors possible model by deermining a wors-case measure ha drives he processes wihin his model. We esablished a link o he calculus of refleced BSDEs o solve he opimal sopping problem from arising given he opions American syle under muliple priors. While he soluion for plain vanilla opions is sraighforward, he siuaion differs if he payoff of he opion is more complex. The buyer of such opion adaps her beliefs o he sae of he world, and o he overall effec of Knighian uncerainy. This leads o dynamical srucure of he wors-case measure highlighing he srucural differences beween sandard models in finance and he muliple priors models. The characerisics are exemplified by solving he problem explicily for an American barrier opion and a shou opion. Paricularly wih regard o risk managemen objecives, hese models are more appropriae since he valuaion becomes less sensiive in erms of varying model daa and provides more robus exercise sraegies. A Proof of Theorem This proof is a sligh modificaion of he proof of Theorem 7.2 in El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10]. Since heir formulaion of he heorem is no direcly relaed o muliple priors, we presen he main ideas here. Le (H define he obsacle and H T he erminal payoff of all regarded RBSDEs. Consider he unique soluion (Y 0, Z 0, K 0 of he RBSDE wih dynamics Then for each [0, T ] dy 0 = ry 0 0 d + dk Z 0 dw }{{}. =f 0 (,Y 0,Z0 Y 0 ( τ = ess sup E τ ry 0 s ds + H τ F, see Proposiion 2.3 in El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10]. Analogously for any θ Θ, he soluion (Y θ, Z θ, K θ of he RBSDE 27

29 wih dynamics dy θ = ( ry θ θ Z θ θ d + dk Z θ dw }{{} =f θ (,Y θ,zθ saisfies for [0, T ] Y θ = ess sup E τ ( τ ( ry θ s θ s Z θ s ds + H τ F. (20 Now consider for [0, T ] and any probabiliy measure Q he equaion Y Q ( τ = ess sup E Q rys Q ds + H τ F. (21 τ If Q = Q θ for some θ Θ hen he soluion (Y Q, Z Q, K Q of he RBSDE wih dynamics dy Q = ry Q d + dk Q Z Q dw θ saisfies Equaion (21. Using Girsanov s heorem, W θ = W + 0 θ sds, we can rewrie he dynamics as dy Q = ( ry Q θ Z Q d + dk Q Z Q dw. Thus, by uniqueness, we obain Y Q = Y θ, and as a consequence ess inf Q P Y Q = ess inf θ Θ Since f(, y, z f θ (, y, z y, z R, θ Θ, we obain by comparison for RBSDEs, Theorem 4.1 in El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10] ha Y Y θ θ Θ. Since Θ is weakly compac in L 1 ([0, T ] Ω, 24 for any real-valued measurable process Z here exiss θ Θ such ha θ Z = max θ Θ θ Z = 24 See Chen and Epsein [2]. This again induces he weak compacness of P which is ha induced by he se of bounded measurable funcions. Y θ. 28

30 κ Z [0, T ], by Lemma B.1 in Chen and Epsein [2]. Hence f(, y, z = f θ (, y, z d P a.e. y, z R and Y = Y θ ess inf θ Θ Y θ, [0, T ]. In brief, Y = ess inf Q P = ess inf θ Θ = ess inf θ Θ ( τ ess sup E Q ry s ds + H τ F τ ( τ ess sup E ry s θ s Z s ds + H τ F τ Y θ. Using Proposiion 7.1 in El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10] and Bayes rule (Lemma 5.3 in Karazas and Shreve [14] we obain for each θ Θ Hence, Y θ Y = ess inf θ Θ ( = ess sup E H τ γ 1 τ = ess sup E τ ess sup τ ( H τ γ 1 τ exp{ zτ θ τ z θ ( E H τ γ 1 zτ θ τ z θ τ θ s dw s 1 2 F = ess sup τ F = ess inf Q P τ θ 2 sds} F E Qθ ( Hτ γ 1 τ F. ess sup τ E Q ( H τ γ 1 τ F. We clearly have ( Y ess sup ess inf E H τ γτ 1 τ θ Θ zτ θ z θ F. To obain he oher inequaliy, we use he sopping ime D θ := inf{s [, T ] Ys θ = H s } which is opimal in Equaion (20 for each fixed θ Θ, see 29

31 El Karoui, Kapoudjian, Pardoux, Peng, and Quenez [10], Theorem 7.2. Then ( Y = E Qθ H D θ γ 1 F D θ ( Dθ = ess inf E H D θ θ Θ γ 1 exp{ θ D θ s db s 1 D θ θ 2 2 sds} F ( ess sup ess inf E H τ γ 1 z θ τ τ F τ θ Θ = ess sup ess inf ( τ Q P EQ H τ γτ F 1. This proves for [0, T ] z θ Y = ess sup ess inf ( τ Q P EQ H τ γτ F 1 = V. By a coninuiy argumen Y = V [0, T ] a.s., 25 and τ is opimal for V. Since he minimum for f is aained we conclude he claim for = 0. B Proof of Theorem We sar wih a lemma yielding ha τ H is sochasically larges under Q κ in he se of priors P in he following sense. Lemma B.1 On {τ H > } we have for all, s wih < s T and all θ Θ Q κ (τ H s F Q θ (τ H s F. Proof: Throughou his proof, all resuls are condiioned on he even {τ H > }. Consider for any u (, s] he se {X u H} and define M u := 1 [ln H σ X (r σ2 (u ]. Le θ Θ. By definiion and consrucion of Qθ 2 and W θ by means of Girsanov s heorem we have X u = X exp{(r σ2 2 (u + σ(w u W } = X exp{(r σ2 2 (u + σ(w θ u W θ σ u θ s ds} 25 Cheng and Riedel [4] showed ha here exiss a version of (V ha is righ-coninuous. Using his version we can deduce he claim. 30