Optimal Stopping with Expectation Constraint
aa r X i v : . [ m a t h . O C ] N ov Optimal Stopping with Expectation Constraints
Erhan Bayraktar ∗† , Song Yao ‡§ Abstract
We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation con-straints in a general non-Markovian framework. We show that the optimal stopping problem with expectationconstraints in a concrete probability setting is equivalent to the constrained problem in weak formulation (opti-mization over joint laws of stopping rules with Brownian motion and state dynamics on an enlarged canonicalspace). Thus the value of the optimal stopping problem with expectation constraints is independent of the spe-cific probabilistic setting. Using a martingale-problem formulation, we make an equivalent characterization ofthe probability class in the weak formulation, which implies that the value function of the constrained optimalstopping problem is upper semi-analytic. Then we exploit a measurable selection argument to establish a dy-namic programming principle (DPP) in the weak formulation for the value of the optimal stopping problem withexpectation constraints, in which the conditional expected integrals of constraint functions act as additional states.
MSC 2020:
Keywords:
Optimal stopping with expectation constraints, equivalence of different formulations, characteriza-tion via martingale-problem formulation, dynamic programming principle, measurable selection.
In this article we analyze a continuous-time optimal stopping problem with a series of inequality-type and equality-type expectations constraints in a general non-Markovian framework.Let ( Q , F , p ) be a generic probability space equipped with a Brownian motion B . Given a historical path x | [0 ,t ] ,the state process (cid:8) X t, x s (cid:9) s ∈ [ t, ∞ ) evolves along the following dynamics: X s = x ( t ) + Z st b ( r, X r ∧· ) dr + Z st σ ( r, X r ∧· ) dB r , ∀ s ∈ [ t, ∞ ) , (1.1)where drift coefficient b and diffusion coefficient σ depend on the past trajectory of the solution.The player decides a relative exercise time τ after t to exit the game when she will receive a running reward R t + τt ( f + g ) (cid:0) r, X t, x r ∧· (cid:1) dr plus a terminal reward π (cid:0) t + τ, X t, x ( t + τ ) ∧· (cid:1) with a running cost R t + τt g (cid:0) r, X t, x r ∧· (cid:1) dr . Subject to aseries of expectation constraints E p h Z t + τt g i ( r, X t, x r ∧· ) dr i ≤ y i , E p h Z t + τt h i ( r, X t, x r ∧· ) dr i = z i , ∀ i ∈ N , ∀ ( y i , z i ) ∈ ( −∞ , ∞ ] × [ −∞ , ∞ ] . the player wants to maximize her expected profit by choosing an approximate τ . Set ( y, z ) = (cid:0) { y i } i ∈ N , { z i } i ∈ N (cid:1) . Thevalue of our optimal stopping problem with expectation constraints is V ( t, x , y, z ) := sup τ ∈S t, x ( y,z ) E p (cid:20) Z t + τt f (cid:0) r, X t, x r ∧· (cid:1) dr + π (cid:0) t + τ, X t, x ( t + τ ) ∧· (cid:1)(cid:21) , ∗ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109; email: [email protected] . † E. Bayraktar is supported in part by the National Science Foundation under DMS-1613170, and in part by the Susan M. SmithProfessorship. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and donot necessarily reflect the views of the National Science Foundation. ‡ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260; email: [email protected] . § S. Yao is supported in part by the National Science Foundation under DMS-1613208. ptimal Stopping with Expectation Constraints 2as long as S t, x ( y, z ) := (cid:8) τ ∈ S t : E p (cid:2) R t + τt g i ( r, X t, x r ∧· ) dr (cid:3) ≤ y i , E p (cid:2) R t + τt h i ( r, X t, x r ∧· ) dr (cid:3) = z i , ∀ i ∈ N (cid:9) is not empty. Suchkind of constrained optimal stopping problem has applications in various economic, engineering and financial areassuch as travel problem with fuel constraint, pricing American options, quickest detection problem and etc.Since a large number of traded assets and contingent claims are not continuous in time and state, we aim tostudy the measurability of the value function V ( t, x , y, z ) and establish a form of dynamic programming principlefor V without imposing any continuity condition on the reward/cost functions f, π, g i , h i .Inspired by [28]’s idea, we transfer the optimal stopping problem with expectation constraints to an enlargedcanonical space Ω := Ω × Ω X × [0 , ∞ ] using the map Q ∋ ω ( B · ( ω ) , X · ( ω ) , τ ( ω )) ∈ Ω and regarding the joint laws of( B · , X · , τ ) as a form of new controls on Ω. In this weak formulation, the optimal stopping problem with expectationconstraints E P h R Tt g i ( r, X r ∧· ) dr i ≤ y i , E P h R Tt h i ( r, X r ∧· ) dr i = z i , ∀ i ∈ N has a value V ( t, x , y, z ) := sup P ∈P t, x ( y,z ) E P (cid:20) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + π (cid:0) T , X T ∧· (cid:1)(cid:21) . Here P t, x ( y, z ) := (cid:8) P ∈ P t, x : E P (cid:2) R Tt g i ( r, X r ∧· ) dr (cid:3) ≤ y i , E P (cid:2) R Tt h i ( r, X r ∧· ) dr (cid:3) = z i , ∀ i ∈ N (cid:9) , and P t, x denotes aclass of of probabilities P on Ω under which the first canonical coordinator W is a Brownian motion, the secondcoordinator X satisfies a similar SDE to (1.1) driven by W and the third coordinator T terminates the game.One of our achievements is to show that the value V of the optimal stopping problem with expectation constraintson ( Q , F , p ) is equal to its value V in the weak formulation if F supports an η ∼ unif (0 ,
1) independent of B .Via the induced probability P := p ◦ ( B · , X · , τ ) − it is relatively easy to obtain V ( t, x , y, z ) ≤ V ( t, x , y, z ) whilethe inverse inequality is more technically involved: Given a P ∈ P t, x ( y, z ), we consider the process ϑ s (cid:0) W t · (cid:1) := E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t s i , s ∈ [0 , ∞ ), where W t is the increments of W after t and { ϑ s } s ∈ [0 , ∞ ) is a process on Ω .Setting B ts := B t + s − B t , s ∈ [0 , ∞ ), we assign the hitting time of process ϑ s ( B t · ) − η above level 0 as the correspondingstopping strategy τ on Q . By exploiting an equality E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t s i = E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t ∞ i , P − a.s. (knownas Property (K), see [61]), applying the “change-of-variable” formula (cid:0) see (3.14) (cid:1) and using some other techniques ofstochastic analysis, we deduce that V ( t, x , y, z ) ≤ V ( t, x , y, z ). This equivalence result indicates that the value of theoptimal stopping problem with expectation constraints is actually an robust value independent of specific probabilitymodels.A dynamic programming principle of a stochastic control problem allows people to optimize the problem stageby stage in a backward recursive way. The wellposedness of the DPP requires the problem value to be a measurablefunction so that one can do optimization at an intermediate horizon first. To show the measurability of the valueof the optimal stopping problem with expectation constraint, we utilize the martingale-problem formulation of [60]to describe the probability class P t, x ( y, z ) as a series of probabilistic tests on the stochastic performances of somequadratic form of the canonical coordinators. By such a characterization of P t, x ( y, z ), we demonstrate that the valuefunction V = V is upper semi-analytic and we thus establish a DPP for V in the weak formulation of our constrainedoptimal stopping problem: V ( t, x , y, z ) = sup P ∈P t, x ( y,z ) E P (cid:20) { T Lagrange multiplier method to reformulate a discrete-time optimal stopping problem with first-moment constraint toa minimax problem and showed that the optimal value of the dual problem is equal to that of the primal problem.Since then, the Lagrangian technique has been prevailing in research of optimal stopping problems with expectationconstraints (see e.g. [50, 40, 3, 63, 41, 62]), and has been applied to various economic and financial problems such asMarkov decision processes with constrained stopping times [32, 31], mean-variance optimal control/stopping problem[46, 47], quickest detection problem [48] and etc.Recently, Ankirchner et al. [1] and Miller [42] took a different approach to optimal stopping problems for diffusionprocesses with expectation constraints by transforming them to stochastic optimization problems with martingalecontrols. The former characterizes the value function in terms of a Hamilton-Jacobi-Bellman equation and obtainsa verification theorem, while the latter analyzes the optimal stopping problem with first-moment constraint thatis embedded in a time-inconsistent (unconstrained) stopping problem. However, they only postulate a dynamicprogramming principle for the optimal stopping with expectation constraint. In contrast, the main contributionof this paper is to establish a dynamic programming principle with rigorous proof for the optimal stopping withexpectation constraints in a general non-Markovian setting.A closely related topic to our research is optimal stopping with constraint on the distribution of stopping time.Bayraktar and Miller [6] studied the problem of optimally stopping a Brownian motion with the restriction that thedistribution of the stopping time must equal to an given measure with finitely many atoms, and obtained a dynamicprogramming result which relates each of the sequential optimal control problems. K¨allblad [33] used measure-valuedmartingales to transform the distribution-constrained optimal stopping problem to a stochastic control problem andderived a dynamic programming principle by measurable selection arguments. From the perspective of optimaltransport, Beiglb¨ock et al. [12] gave a geometric description of optimal stopping times of a Brownian motion withdistribution constraint.In their study of a continuous-time stochastic control problem for diffusion processes, El Karoui, Huu Nguyenand Jeanblanc-Picqu´e [28] used a martingale-problem formulation and regarded the joint laws of control and stateprocesses as relaxed controls on a canonical path space. The authors then employed the measurable selection theoremin the descriptive-set theory to establish a DPP without assuming any regularity conditions on the coefficients. Thisapproach was later generalized by [29, 30, 64] in abstract frameworks, and developed by [43, 44, 51] for tower propertyof sub-linear expectations.As to the stochastic control theory with expectation constraint, Yu et al. [23] used the measurable selectionargument to obtain a DPP result and applied it to quantitative finance problems with various expectation constraints.Pfeiffer et al. [49] took a Lagrange relaxation approach to study a continuous-time stochastic control problem withboth inequality-type and equality-type expectation constraints and obtained a duality result by the knowledge ofconvex analysis. Moreover, for stochastic control problems with state constraints, the stochastic target problems withcontrolled losses and the related geometric dynamic programming principle, see [16, 17, 19, 56, 57, 58, 20, 14, 18, 15]and etc.The rest of the paper is organized as follows: Section 2 sets up the optimal stopping problem with expectationconstraints in a generic probabilistic setting. In Section 3, we show that the constrained optimal stopping problemcan be equivalently embedded into an enlarged canonical space or the optimal stopping problem with expectationconstraints in the strong formulation has the same value as that in the weak formulation. In Section 4, we use amartingale-problem formulation to make a characterization of the probability class in the weak formulation and thusshow that the value function of the constrained optimal stopping problem is upper semi-analytic. Then in Section 5,we utilize a measurable selection argument to establish a dynamic programming principle in the weak formulationfor the value of the optimal stopping problem with expectation constraints. The appendix contains some technicallemmata necessary for the proofs of the main results.We close this section by introducing our notation and standing assumptions on drift/diffusion coefficients andreward/constraint functions.ptimal Stopping with Expectation Constraints 4 For any s ∈ [0 , ∞ ), set Q s := (cid:0) Q ∩ [0 , s ) (cid:1) ∪{ s } . For a generic topological space (cid:0) X , T ( X ) (cid:1) , we denote its Borel sigma-field by B ( X ) and let P ( X ) collection all probabilities on (cid:0) X , B ( X ) (cid:1) . Given n ∈ N , let (cid:8) b O i (cid:9) i ∈ N be a countablesubbase of T ( R n ). The collection O ( R n ) of all finite intersections in (cid:8) b O i (cid:9) i ∈ N as well as ∅ , R n (cid:16) i.e. O ( R n ) := n n ∩ i =1 b O k i : { k i } ni =1 ⊂ N o ∪ {∅ , R n } (cid:17) forms a countable base of T ( R n ) and thus B ( R n ) = σ (cid:0) O ( R n ) (cid:1) . We also set b O ( R n ) := ∪ k ∈ N (cid:0)(cid:0) Q ∩ [0 , ∞ ) (cid:1) × O ( R n ) × O ( R n ) (cid:1) k .Fix d, l ∈ N . Let Ω = (cid:8) ω ∈ C ([0 , ∞ ); R d ) : ω (0) = 0 (cid:9) be the space of all R d − valued continuous paths on [0 , ∞ ) thatstart from 0. It is a Polish space under the topology of locally uniform convergence. Let P be the Wiener measureof (cid:0) Ω , B (Ω ) (cid:1) , under which the canonical process W = { W t } t ∈ [0 , ∞ ) of Ω is a d − dimensional standard Brownianmotion. Also, let Ω X = C ([0 , ∞ ); R l ) be the space of all R l − valued continuous paths on [0 , ∞ ) endowed with thetopology of locally uniform convergence.Let b : (0 , ∞ ) × Ω X R l and σ : (0 , ∞ ) × Ω X R l × d be two Borel-measurable functions such that for any t ∈ (0 , ∞ ) (cid:12)(cid:12) b ( t, x ) − b ( t, x ′ ) (cid:12)(cid:12) + (cid:12)(cid:12) σ ( t, x ) − σ ( t, x ′ ) (cid:12)(cid:12) ≤ κ ( t ) k x − x ′ k , ∀ x , x ′ ∈ Ω X and Z t (cid:0) | b ( r, ) | + | σ ( r, ) | (cid:1) dr < ∞ , (1.2)where κ : (0 , ∞ ) (0 , ∞ ) is some non-decreasing function and k x − x ′ k := sup s ∈ [0 , ∞ ) (cid:12)(cid:12) x ( s ) − x ′ ( s ) (cid:12)(cid:12) .Also, let f, g i , h i : (0 , ∞ ) × Ω X [ −∞ , ∞ ] be Borel-measurable functions for all i ∈ N and let π : [0 , ∞ ) × Ω X R be a Borel-measurable function.We make some notations on a general measurable space ( Q , F ): • For any measure m on ( Q , F ) and for any [ −∞ , ∞ ] − valued F− measurable random variable ξ on Q , define theintegral R Ω ξ ( ω ) m ( dω ) := R Ω ξ + ( ω ) m ( dω ) − R Ω ξ − ( ω ) m ( dω ) with the convention (+ ∞ )+( −∞ ) = ( −∞ )+(+ ∞ ) = −∞ . • For any process X = { X t } t ∈ [0 , ∞ ) on ( Q , F ), let F X = (cid:8) F Xt := σ ( X s ; s ∈ [0 , t ]) (cid:9) t ∈ [0 , ∞ ) be the raw filtration of X . • Let p be a probability on ( Q , F ). For any sub-sigma-field G of F , we set N p ( G ) := (cid:8) N ⊂ Q : N ⊂ A for some A ∈G with p ( A ) = 0 (cid:9) . For any filtration F = {F t } t ∈ [0 , ∞ ) on ( Q , F , p ), let F p = (cid:8) F p t := σ (cid:0) F t ∪ N p ( F ∞ ) (cid:1)(cid:9) t ∈ [0 , ∞ ) be the p − augmentation of F with F ∞ := σ (cid:16) ∪ t ∈ [0 , ∞ ) F t (cid:17) .Moreover, we set R := ( −∞ , ∞ ] ∞ × [ −∞ , ∞ ] ∞ and define φ ( t, E ) := (cid:0) πt (cid:1) − d/ R z ∈E e − z t dz , ∀ ( t, E ) ∈ (0 , ∞ ) × B ( R d ). Let ( Q , F , p ) be a generic probability space equipped with a d − dimensional standard Brownian motion B = { B t } t ∈ [0 , ∞ ) and an F− measurable unif (0 , 1) random variable η that is independent of B under p .Let t ∈ [0 , ∞ ). The evolution of B after time t , B ts = B t + s − B t , s ∈ [0 , ∞ ), is also a standard Brownian motion on Q under p . We can view s as the relative time after t .Given x ∈ Ω X , (1.2) ensures that the following SDE on ( Q , F , p ) X s = x ( t ) + Z st b ( r, X r ∧· ) dr + Z st σ ( r, X r ∧· ) dB r , ∀ s ∈ [ t, ∞ ) (2.1)with initial condition X s = x ( s ), ∀ s ∈ [0 , t ] admits a unique solution (cid:8) X t, x s (cid:9) s ∈ [0 , ∞ ) (cid:0) In particular, (cid:8) X t, x t + s (cid:9) s ∈ [0 , ∞ ) is an F B t , p − adapted process with all continuous paths satisfying p (cid:8) X t, x t + s = x ( t ) + R s b ( t + r, X t, x ( t + r ) ∧· ) dr + R s σ ( t + r, X t, x ( t + r ) ∧· ) dB tr , ∀ s ∈ [0 , ∞ ) (cid:9) = 1 (cid:1) . We define a filtration F B t ,η = (cid:8) F B t ,ηs := σ (cid:0) F B t s ∪ σ ( η ) (cid:1)(cid:9) s ∈ [0 , ∞ ) and denote by S t the set of all [0 , ∞ ] − valued F B t ,η, p − stopping times.Given a historical path x | [0 ,t ] , the state then evolves along process X t, x . The player decides a relative exercisetime τ ∈ S t after t to cease the game when she receives a running reward R t + τt f (cid:0) r, X t, x r ∧· (cid:1) dr plus a terminal reward π (cid:0) t + τ, X t, x ( t + τ ) ∧· (cid:1) . The player would like to maximize the expectation of her total wealth, but her choice of τ issubject to a series of expectation constraints E p h Z t + τt g i ( r, X t, x r ∧· ) dr i ≤ y i , E p h Z t + τt h i ( r, X t, x r ∧· ) dr i = z i , ∀ i ∈ N , ∀ ( y i , z i ) ∈ ( −∞ , ∞ ] × [ −∞ , ∞ ] . (2.2) . Weak Formulation y, z ) = (cid:0) { y i } i ∈ N , { z i } i ∈ N (cid:1) ∈ R such that S t, x ( y, z ) := (cid:8) τ ∈ S t : E p (cid:2) R t + τt g i ( r, X t, x r ∧· ) dr (cid:3) ≤ y i , E p (cid:2) R t + τt h i ( r,X t, x r ∧· ) dr (cid:3) = z i , ∀ i ∈ N (cid:9) is not empty, the value of the optimal stopping problem with expectation constraints (2.2) is V ( t, x , y, z ) := sup τ ∈S t, x ( y,z ) E p h Z t + τt f (cid:0) r, X t, x r ∧· (cid:1) dr + { τ< ∞} π (cid:0) t + τ, X t, x ( t + τ ) ∧· (cid:1)i . (2.3) Remark 2.1. ) ( finitely many constraints ) For some i ∈ N , the constraint E p (cid:2) R t + τt g i ( r, X t, x r ∧· ) dr (cid:3) ≤ y i is trivial if y i = ∞ , while the constraint E p (cid:2) R t + τt h i ( r, X t, x r ∧· ) dr (cid:3) = z i is trivial if ( h i , z i ) = (0 , .1a ) If we take ( y i , h i , z i ) = ( ∞ , , , ∀ i ∈ N , there is no constraint at all.1b ) If one takes y i = ∞ , ∀ i ≥ and ( h i , z i ) = (0 , ∀ i ∈ N , (2.2) degenerates to a single constraint E p (cid:2) R t + τt g ( r, X t, x r ∧· ) dr (cid:3) ≤ y . In addition, if g ≥ and y ≥ , then ∈ S t, x ( y, z ) and S t, x ( y, z ) is thus non-empty.1c ) If we take y i = ∞ , ∀ i ∈ N and ( h i , z i ) = (0 , ∀ i ≥ , (2.2) reduces to a single constraint E p (cid:2) R t + τt h ( r, X t, x r ∧· ) dr (cid:3) = z .1d ) If one takes ( y i , h i , z i ) = ( ∞ , , , ∀ i ≥ , (2.2) becomes a couple of constraints E p (cid:2) R t + τt g ( r, X t, x r ∧· ) dr (cid:3) ≤ y & E p (cid:2) R t + τt h ( r, X t, x r ∧· ) dr (cid:3) = z .2 ) ( moment constraints ) Let i ∈ N , a ∈ (0 , ∞ ) , q ∈ (1 , ∞ ) and λ ∈ [0 , ∞ ) . If g i ( t, x ) = aqt q − + λ , ∀ ( t, x ) ∈ (0 , ∞ ) × Ω X (cid:0) or h i ( t, x ) = aqt q − + λ , ∀ ( t, x ) ∈ (0 , ∞ ) × Ω X (cid:1) , then the expectation constraints E p (cid:2) R t + τt g i ( r, X t, x r ∧· ) dr (cid:3) ≤ y i (cid:0) or E p (cid:2) R t + τt h i ( r, X t, x r ∧· ) dr (cid:3) = z i (cid:1) specify as a moment constraint E p (cid:2) a (cid:0) ( t + τ ) q − t q (cid:1) + λτ (cid:3) ≤ y i (cid:0) or E p (cid:2) a (cid:0) ( t + τ ) q − t q (cid:1) + λτ (cid:3) = z i (cid:1) . To study the measurability of the value function V and derive a dynamic programming principle for V withoutimposing any continuity condition on reward/constraint functions f, π, g i , h i , we would embed the stopping rulestogether with the Brownian/state information into an enlarged canonical space and consider their joint distributionas a form of new controls: In this section, we study the optimal stopping problem with expectation constraints in a weak formulation or overan enlarged canonical space Ω := Ω × Ω X × [0 , ∞ ] . Clearly, Ω is a Borel space under the product topology. Let P (Ω) be the space of all probabilities on (cid:0) Ω , B (Ω) (cid:1) equipped with the topology of weak convergence, which is also a Borel space (see e.g. Corollary 7.25.1 of [13]).Define the canonical coordinates on Ω by W s ( ω ) := ω ( s ) , X s ( ω ) := ω X ( s ) , ∀ s ∈ [0 , ∞ ) and T ( ω ) := t , ∀ ω = (cid:0) ω , ω X , t (cid:1) ∈ Ω , in which one can regard W as a canonical coordinate for Brownian motion, X as a canonical coordinate about stateprocess, and T as a canonical coordinate for stopping rules.Let t ∈ [0 , ∞ ). We also define shifted canonical processes Ξ t = ( W t , X t ) on Ω by (cid:0) W ts , X ts (cid:1) ( ω ) := (cid:0) W t + s ( ω ) − W t ( ω ) , X t + s ( ω ) (cid:1) ∀ ( s, ω ) ∈ [0 , ∞ ) × Ω , and set the filtrations F t = n F ts := σ (cid:16) W tr , (cid:8) T ∈ [ t, t + r ] (cid:9) ; r ∈ [0 , s ] (cid:17)o s ∈ [0 , ∞ ) , G t = n G ts := σ (cid:16) Ξ tr , (cid:8) T ∈ [ t, t + r ] (cid:9) ; r ∈ [0 , s ] (cid:17)o s ∈ [0 , ∞ ) . For any s ∈ [0 , ∞ ), F ts can be countably generated by the Pi system C ts := (cid:26) k ∩ i =1 (cid:16)(cid:2) ( W ts i ∧ s ) − ( O i ) ∩ (cid:8) T ∈ [ t, t + s i ∧ s ] (cid:9)(cid:3) ∪ (cid:2) ( W ts i ∧ s ) − ( O ′ i ) ∩ (cid:8) T ∈ [ t, t + s i ∧ s ] c (cid:9)(cid:3)(cid:17) : ∀ (cid:8) ( s i , O i , O ′ i ) (cid:9) ki =1 ∈ b O ( R d ) (cid:27) , (3.1)and G ts can be countably generated by the Pi system C ts := (cid:26) k ∩ i =1 (cid:16)(cid:2) (Ξ ts i ∧ s ) − ( O i ) ∩ (cid:8) T ∈ [ t, t + s i ∧ s ] (cid:9)(cid:3) ∪ (cid:2) (Ξ ts i ∧ s ) − ( O ′ i ) ∩ (cid:8) T ∈ [ t, t + s i ∧ s ] c (cid:9)(cid:3)(cid:17) : ∀ (cid:8) ( s i , O i , O ′ i ) (cid:9) ki =1 ∈ b O ( R d + l ) (cid:27) . (3.2)ptimal Stopping with Expectation Constraints 6For any P ∈ P (Ω), we set B P (Ω) := σ (cid:16) B (Ω) ∪ N P (cid:0) B (Ω) (cid:1)(cid:17) . The P − augmentation of F t is F t,P = n F t,Ps := σ (cid:16) F ts ∪ N P (cid:0) F t ∞ (cid:1)(cid:17)o s ∈ [0 , ∞ ) and the P − augmentation of G t is G t,P = n G t,Ps := σ (cid:16) G ts ∪ N P (cid:0) G t ∞ (cid:1)(cid:17)o s ∈ [0 , ∞ ) .The weak formulation of our optimal stopping problem with expectation constraints relies on the followingprobability classes of P (Ω): Definition 3.1. For any ( t, x ) ∈ [0 , ∞ ) × Ω X , let P t, x be the collection of all probabilities P ∈ P (Ω) satisfying:i ) On (cid:0) Ω , B (Ω) , P (cid:1) , the process W t is a d − dimensional standard Brownian motion with respect to filtration F t .ii ) P (cid:8) X s = X t, x s , ∀ s ∈ [0 , ∞ ) (cid:9) = 1 , where (cid:8) X t, x s (cid:9) s ∈ [0 , ∞ ) uniquely solves the following SDE on (cid:0) Ω , B (cid:0) Ω (cid:1) , P (cid:1) : X s = x ( t ) + Z st b (cid:0) r, X r ∧· (cid:1) dr + Z st σ (cid:0) r, X r ∧· (cid:1) dW r , s ∈ [ t, ∞ ) (3.3) with initial condition X s = x ( s ) , ∀ s ∈ [0 , t ] (cid:0) In particular, (cid:8) X t, x t + s (cid:9) s ∈ [0 , ∞ ) is an F W t ,P − adapted process with allcontinuous paths satisfying P (cid:8) X t, x t + s = x ( t )+ R s b ( t + r, X t, x ( t + r ) ∧· ) dr + R s σ ( t + r, X t, x ( t + r ) ∧· ) dW tr , ∀ s ∈ [0 , ∞ ) (cid:9) = 1 (cid:1) .iii ) P { T ≥ t } = 1 . Given a historical path of the state x | [0 ,t ] , for any ( y, z ) = (cid:0) { y i } i ∈ N , { z i } i ∈ N (cid:1) ∈ R such that P t, x ( y, z ) := (cid:8) P ∈ P t, x : E P (cid:2) R Tt g i ( r, X r ∧· ) dr (cid:3) ≤ y i , E P (cid:2) R Tt h i ( r, X r ∧· ) dr (cid:3) = z i , ∀ i ∈ N (cid:9) is not empty, V ( t, x , y, z ) := sup P ∈P t, x ( y,z ) E P (cid:20) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + { T< ∞} π (cid:0) T , X T ∧· (cid:1)(cid:21) is the value of the optimal stopping problem with expectation constraints E P h Z Tt g i ( r, X r ∧· ) dr i ≤ y i , E P h Z Tt h i ( r, X r ∧· ) dr i = z i , ∀ i ∈ N (3.4)in the weak formulation.We can consider another value of the constrained optimal stopping problem in the weak formulation: Let ( t, w , x ) ∈ [0 , ∞ ) × Ω × Ω X and define P t, w , x := (cid:8) P ∈ P t, x : P (cid:8) W s = w ( s ) , ∀ s ∈ [0 , t ] (cid:9) = 1 (cid:9) as the probability class given theBrownian and state history ( w , x ) (cid:12)(cid:12) [0 ,t ] . For any ( y, z ) = (cid:0) { y i } i ∈ N , { z i } i ∈ N (cid:1) ∈ R such that P t, w , x ( y, z ) := (cid:8) P ∈ P t, w , x : E P (cid:2) R Tt g i ( r, X r ∧· ) dr (cid:3) ≤ y i , E P (cid:2) R Tt h i ( r, X r ∧· ) dr (cid:3) = z i , ∀ i ∈ N (cid:9) is not empty, the value of the optimal stoppingproblem with expectation constraints (3.4) given ( w , x ) (cid:12)(cid:12) [0 ,t ] is V ( t, w , x , y, z ) := sup P ∈P t, w , x ( y,z ) E P (cid:20) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + { T < ∞} π (cid:0) T , X T ∧· (cid:1)(cid:21) in the weak formulation.Set D := { ( t, x , y, z ) ∈ [0 , ∞ ) × Ω X ×R : P t, x ( y, z ) = ∅} and D := { ( t, x , y, z ) ∈ [0 , ∞ ) × Ω X ×R : P t, w , x ( y, z ) = ∅} .One of our main results in the next theorem exposes that the value V ( t, x , y, z ) in (2.3) coincides with the value V ( t, x , y, z ) in the weak formulation, and is even equal to the value V ( t, w , x , y, z ). To wit, the value of the optimalstopping with expectation constraints is independent of a specific probabilistic setup and is also indifferent to theBrownian history. Theorem 3.1. For any ( t, w , x , y, z ) ∈ [0 , ∞ ) × Ω × Ω X ×R , S t, x ( y, z ) = ∅ ⇔ P t, x ( y, z ) = ∅ ⇔ P t, w , x ( y, z ) = ∅ , andone has V ( t, x , y, z ) = V ( t, x , y, z ) = V ( t, w , x , y, z ) in this situation. By Remark 2.1 (1b) and Theorem 3.1, Proj (cid:0) D (cid:1) := { ( t, x ) ∈ [0 , ∞ ) × Ω X : ( t, x , y, z ) ∈ D for some ( y, z ) ∈ R} =[0 , ∞ ) × Ω X and Proj (cid:0) D (cid:1) := { ( t, w , x ) ∈ [0 , ∞ ) × Ω × Ω X : ( t, w , x , y, z ) ∈ D for some ( y, z ) ∈ R} = [0 , ∞ ) × Ω × Ω X . Remark 3.1. Theorem 3.1 indicates that our optimal stopping problem with expectation constraints is actuallyindependent of a particular probabilistic setting. It even allows us to deal with the robust case.Let (cid:8) ( Q α , F α , p α ) (cid:9) α ∈ A be a family of probability spaces, where A is a countable or uncountable index set ( e.g. itcan be a non-dominated class of probabilities ) . Given α ∈ A , let B α be a d − dimensional standard Brownian motion. Weak Formulation on ( Q α , F α , p α ) and let η α be an F α − measurable unif (0 , random variable that is independent of B α under p α .For any ( t, x ) ∈ [0 , ∞ ) × Ω X , let X α,t, x be the unique solution of the SDE on ( Q α , F α , p α ) X αs = x ( t ) + Z st b ( r, X αr ∧· ) dr + Z st σ ( r, X αr ∧· ) dB αr , s ∈ [ t, ∞ ) with initial condition X αs = x ( s ) , ∀ s ∈ [0 , t ] . Also, let S tα denote the set of all [0 , ∞ ] − valued, F B α,t ,η α , p α − stoppingtimes, where B α,ts := B αt + s − B αt , s ∈ [0 , ∞ ) and F B α,t ,η α = (cid:8) F B α,t ,η α s := σ (cid:0) F B α,t s ∪ σ ( η α ) (cid:1)(cid:9) s ∈ [0 , ∞ ) .According to Theorem 3.1, for any ( t, x ) ∈ [0 , ∞ ) × Ω X and ( y, z ) = (cid:0) { y i } i ∈ N , { z i } i ∈ N (cid:1) ∈ R such that P t, x ( y, z ) = ∅ , V ( t, x , y, z ) = sup α ∈ A sup τ α ∈S αt, x ( y,z ) E p α (cid:20) Z t + τ α t f (cid:0) r, X α,t, x r ∧· (cid:1) dr + { τ α < ∞} π (cid:0) t + τ α , X α,t, x ( t + τ α ) ∧· (cid:1)(cid:21) , where S αt, x ( y, z ) := (cid:8) τ α ∈ S tα : E p α (cid:2) R t + τ α t g i ( r, X α,t, x r ∧· ) dr (cid:3) ≤ y i , E p α (cid:2) R t + τ α t h i ( r, X α,t, x r ∧· ) dr (cid:3) = z i , ∀ i ∈ N (cid:9) is not emptyfor all α ∈ A . Proof of Theorem 3.1 (Part I): Fix ( t, w , x ) ∈ [0 , ∞ ) × Ω × Ω X and ( y, z ) = (cid:0) { y i } i ∈ N , { z i } i ∈ N (cid:1) ∈ R such that S t, x ( y, z ) = ∅ . We show in this part that P t, w , x ( y, z ) = ∅ and V ( t, x , y, z ) ≤ V ( t, w , x , y, z ) ≤ V ( t, x , y, z ).Fix τ ∈ S t, x ( y, z ). We define processes ˘ B s ( ω ) := w ( s ∧ t ) + B s ∨ t ( ω ) − B t ( ω ), ∀ ( s, ω ) ∈ [0 , ∞ ) × Q and define amapping Φ : Q 7→ Ω by Φ( ω ) := (cid:0) ˘ B · ( ω ) , X t, x · ( ω ) , t + τ ( ω ) (cid:1) ∈ Ω, ∀ ω ∈ Q . So it holds for any ω ∈ Q that( W s , X s ) (cid:0) Φ( ω ) (cid:1) = (cid:0) w ( s ) , x ( s ) (cid:1) , ∀ s ∈ [0 , t ] and Ξ t · (cid:0) Φ( ω ) (cid:1) = (cid:0) B t · ( ω ) , X t, x t + · ( ω ) (cid:1) , T (cid:0) Φ( ω ) (cid:1) = t + τ ( ω ) . (3.5)The mapping Φ induces a probability P ∈ P (Ω) by P (cid:0) A (cid:1) := p (cid:0) Φ − (cid:0) A (cid:1)(cid:1) , ∀ A ∈ B (Ω). For s ∈ [0 , ∞ ], since it holds forany r ∈ [0 , s ] ∩ R and E ∈ B ( R d ) that Φ − (cid:0) ( W tr ) − ( E ) (cid:1) = (cid:8) W tr (Φ) ∈ E (cid:9) = (cid:8) B tr ∈ E (cid:9) ∈ F B t s , one can deduce thatΦ − (cid:0) F W t ,Ps (cid:1) ⊂ F B t , p s , ∀ s ∈ [0 , ∞ ] . (3.6)Let 0 ≤ s < r < ∞ . For any E ∈ B ( R d ), one clearly has P (cid:0) ( W tr − W ts ) − ( E ) (cid:1) = p (cid:8) ( W tr − W ts )(Φ) ∈ E (cid:9) = p (cid:8) B tr − B ts ∈E (cid:9) = φ ( r − s, E ). Also, let (cid:8) ( s i , E i , E ′ i ) (cid:9) ni =1 ⊂ [0 , s ] × B ( R d + l ) × B ( R d + l ). Since the Brownian motion B t is independentof η under p and since { τ ≤ s i } ∈ F B t ,η, p s i for i = 1 , · · · , n , P n ( W tr − W ts ) − ( E ) ∩ (cid:16) n ∩ i =1 (cid:16)(cid:2) ( W ts i ) − ( E i ) ∩ (cid:8) T ∈ [ t, t + s i ] (cid:9)(cid:3) ∪ (cid:2) ( W ts i ) − ( E ′ i ) ∩ (cid:8) T ∈ [ t, t + s i ] c (cid:9)(cid:3)(cid:17)(cid:17)o = p n(cid:8) ( W tr − W ts )(Φ) ∈ E (cid:9) ∩ (cid:16) n ∩ i =1 (cid:16)n W ts i (Φ) ∈ E i , T (Φ) ∈ [ t, t + s i ] o ∪ n W ts i (Φ) ∈ E ′ i , T (Φ) ∈ [ t, t + s i ] c o(cid:17)(cid:17)o = p (cid:8) B tr − B ts ∈ E (cid:9) × p n n ∩ i =1 (cid:16)n B ts i ∈ E i , τ ∈ [0 , s i ] o ∪ n B ts i ∈ E ′ i , τ ∈ ( s i , ∞ ) o(cid:17)o = P (cid:8) ( W tr − W ts ) − ( E ) (cid:9) × P n n ∩ i =1 (cid:16)(cid:2) ( W ts i ) − ( E i ) ∩ (cid:8) T ∈ [ t, t + s i ] (cid:9)(cid:3) ∪ (cid:2) ( W ts i ) − ( E ′ i ) ∩ (cid:8) T ∈ [ t, t + s i ] c (cid:9)(cid:3)(cid:17)o . An application of Dynkin’s Theorem and (3.1) yield that P (cid:0) ( W tr − W ts ) − ( E ) ∩ A (cid:1) = P (cid:0) ( W tr − W ts ) − ( E ) (cid:1) P (cid:0) A (cid:1) , ∀ A ∈ F ts . So W t is a d − dimensional standard Brownian motion with respect to the filtration F t on (cid:0) Ω , B (Ω) , P (cid:1) ,or P satisfies Definition 3.1 (i).As (cid:8) X t, x t + s (cid:9) s ∈ [0 , ∞ ) is an F W t ,P − adapted process, M s := R s σ ( t + r, X t, x ( t + r ) ∧· ) dW tr , s ∈ [0 , ∞ ) defines an (cid:0) F W t ,P , P (cid:1) − martingale. There is a sequence of R l × d − valued, F W t ,P − simple processes n H ns = P ℓ n i =1 ξ ni { s ∈ ( s ni ,s ni +1 ] } , s ∈ [0 , ∞ ) o n ∈ N (cid:0) with 0 = s n < · · · < s nℓ n +1 < ∞ and ξ ni ∈ F W t ,Ps ni , i = 1 , · · · , ℓ n (cid:1) such that P − lim n →∞ Z ∞ (cid:12)(cid:12) H nr − σ (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1)(cid:12)(cid:12) dr = 0 and P − lim n →∞ sup s ∈ [0 , ∞ ) (cid:12)(cid:12) M ns − M s (cid:12)(cid:12) = 0 , (3.7)where M ns := R s H nr dW tr . It follows that p − lim n →∞ Z ∞ (cid:12)(cid:12) H nr (Φ) − σ (cid:0) t + r, X t, x ( t + r ) ∧· (Φ) (cid:1)(cid:12)(cid:12) dr = 0 and p − lim n →∞ sup s ∈ [0 , ∞ ) (cid:12)(cid:12) ( M ns − M s )(Φ) (cid:12)(cid:12) = 0 . (3.8)ptimal Stopping with Expectation Constraints 8For n ∈ N , since ξ ni (Φ) ∈ F B t , p s ni for i = 1 , · · · , ℓ n by (3.6), applying Proposition 3.2.26 of [34] and using the first limitin (3.8) yield that p − lim n →∞ sup s ∈ [ s, ∞ ) (cid:12)(cid:12)(cid:12) R s H nr (Φ) dB tr − R s σ (cid:0) t + r, X t, x ( t + r ) ∧· (Φ) (cid:1) dB tr (cid:12)(cid:12)(cid:12) = 0, which together with the secondlimit in (3.8) renders that p (cid:26)(cid:16) Z s σ ( t + r, X t, x ( t + r ) ∧· ) dW tr (cid:17) (Φ) = Z s σ (cid:0) t + r, X t, x ( t + r ) ∧· (Φ) (cid:1) dB tr , s ∈ [0 , ∞ ) (cid:27) = 1 . (3.9)Then we can deduce that p − a.s., X t, x s (Φ) = x ( t )+ Z s − t b (cid:0) t + r, X t, x ( t + r ) ∧· (Φ) (cid:1) dr + (cid:16) Z s − t σ ( t + r, X t, x ( t + r ) ∧· ) dW tr (cid:17) (Φ)= x ( t )+ Z s − t b (cid:0) t + r, X t, x ( t + r ) ∧· (Φ) (cid:1) dr + Z s − t σ (cid:0) t + r, X t, x ( t + r ) ∧· (Φ) (cid:1) dB tr = x ( t )+ Z st b (cid:0) r, X t, x r ∧· (Φ) (cid:1) dr + Z st σ (cid:0) r, X t, x r ∧· (Φ) (cid:1) dB r , ∀ s ∈ [ t, ∞ ) . So (cid:8) X t, x s (Φ) (cid:9) s ∈ [0 , ∞ ) also solves (2.1). Consequently, P (cid:8) X s = X t, x s , ∀ s ∈ [0 , ∞ ) (cid:9) = p (cid:8) X s (Φ) = X t, x s (Φ) , ∀ s ∈ [0 , ∞ ) (cid:9) = p (cid:8) X t, x s = X t, x s (Φ) , ∀ s ∈ [0 , ∞ ) (cid:9) = 1. Namely, P satisfies Definition 3.1 (ii). Since P { T ≥ t } = p (cid:8) T (Φ) ≥ t (cid:9) = p (cid:8) τ ≥ (cid:9) = 1, we see from (3.5) that P ∈ P t, w , x .For any i ∈ N , one has E P (cid:20) Z Tt g i (cid:0) r, X r ∧· (cid:1) dr (cid:21) = E p h Z T (Φ) t g i (cid:0) r, X r ∧· (Φ) (cid:1) dr i = E p h Z t + τt g i ( r, X t, x r ∧· ) dr i ≤ y i (3.10)and similarly E P (cid:2) R Tt h i (cid:0) r, X r ∧· (cid:1) dr (cid:3) = E p (cid:2) R t + τt h i ( r, X t, x r ∧· ) dr (cid:3) = z i , which shows that P ∈ P t, w , x ( y, z ) ⊂ P t, x ( y, z ). Byan analogy to (3.10), E p h Z t + τt f ( r, X t, x r ∧· ) dr + { τ< ∞} π (cid:0) t + τ, X t, x ( t + τ ) ∧· (cid:1)i = E P (cid:20) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + { T< ∞} π (cid:0) T , X T ∧· (cid:1)(cid:21) ≤ V ( t, w , x , y, z ) . Letting τ run through S t, x ( y, z ) yields that V ( t, x , y, z ) ≤ V ( t, w , x , y, z ) ≤ V ( t, x , y, z ). (cid:3) To demonstrate the inequality V ( t, x , y, z ) ≤ V ( t, x , y, z ) in Theorem 3.1, we need to introduce an auxiliary value e V ( t, x , y, z ) over another enlarged canonical space e Ω := Ω × Ω X × [0 , . Analogous to P (Ω), the space P (cid:0)e Ω (cid:1) of all probabilities on (cid:0)e Ω , B (cid:0)e Ω (cid:1)(cid:1) equipped with the topology of weak convergenceis a Borel space. Define the canonical coordinates on e Ω by f W s ( e ω ) := ω ( s ) , e X s ( e ω ) := ω X ( s ) , ∀ s ∈ [0 , ∞ ) and e η ( e ω ) := λ, ∀ e ω = (cid:0) ω , ω X , λ (cid:1) ∈ e Ω . Also, for t ∈ [0 , ∞ ) we define shifted canonical processes e Ξ t = ( f W t , e X t ) on e Ω by (cid:0)f W ts , e X ts (cid:1) ( e ω ) := (cid:0)f W t + s ( e ω ) − f W t ( e ω ) , e X t + s ( e ω ) (cid:1) , ∀ ( s, e ω ) ∈ [0 , ∞ ) × e Ω , and set the filtration F f W t , e η = n F f W t , e ηs := σ (cid:16) F f W t s ∪ σ (cid:0)e η (cid:1)(cid:17)o s ∈ [0 , ∞ ) . For any e P ∈ P (cid:0)e Ω (cid:1) , let T e Pt denote the set of all[0 , ∞ ] − valued, F f W t , e η, e P − stopping times.We have the following counterpart of P t, x ’s in P (cid:0)e Ω (cid:1) : Definition 3.2. For any ( t, x ) ∈ [0 , ∞ ) × Ω X , let e P t, x be the collection of all probabilities e P ∈ P (cid:0)e Ω (cid:1) satisfying:i ) On (cid:0)e Ω , B (cid:0)e Ω (cid:1) , e P (cid:1) , the process f W t is a d − dimensional standard Brownian motion and e η is a unif (0 , randomvariable independent of f W t .. Weak Formulation ii ) e P (cid:8) e X s = X t, x s , ∀ s ∈ [0 , ∞ ) (cid:9) = 1 , where (cid:8) X t, x s (cid:9) s ∈ [0 , ∞ ) uniquely solves the following SDE on (cid:0)e Ω , B (cid:0)e Ω (cid:1) , e P (cid:1) : X s = x ( t ) + Z st b (cid:0) r, X r ∧· (cid:1) dr + Z st σ (cid:0) r, X r ∧· (cid:1) d f W r , s ∈ [0 , ∞ ) with initial condition X s = x ( s ) , ∀ s ∈ [0 , t ] (cid:0) In particular, (cid:8) X t, x t + s (cid:9) s ∈ [0 , ∞ ) is an F f W t , e P − adapted process with allcontinuous paths satisfying e P (cid:8) X t, x t + s = x ( t )+ R s b ( t + r, X t, x ( t + r ) ∧· ) dr + R s σ ( t + r, X t, x ( t + r ) ∧· ) d f W tr , ∀ s ∈ [0 , ∞ ) (cid:9) = 1 (cid:1) . For any ( t, x ) ∈ [0 , ∞ ) × Ω X and any ( y, z ) = (cid:0) { y i } i ∈ N , { z i } i ∈ N (cid:1) ∈ R such that T e Pt, x ( y, z ) := (cid:8)e τ ∈ T e Pt : E e P (cid:2) R t + e τt g i ( r, e X r ∧· ) dr (cid:3) ≤ y i , E e P (cid:2) R t + e τt h i ( r, e X r ∧· ) dr (cid:3) = z i , ∀ i ∈ N (cid:9) is not empty for some e P ∈ e P t, x , we define the auxiliary value e V ( t, x , y, z ) := sup e P ∈ e P t, x ( y,z ) sup e τ ∈T e Pt, x ( y,z ) E e P (cid:20) Z t + e τt f (cid:0) r, e X r ∧· (cid:1) dr + { e τ< ∞} π (cid:0) t + e τ , e X ( t + e τ ) ∧· (cid:1)(cid:21) , where e P t, x ( y, z ) := (cid:8) e P ∈ e P t, x : T e Pt, x ( y, z ) = ∅ (cid:9) . Proof of Theorem 3.1 (Part II): Fix ( t, x ) ∈ [0 , ∞ ) × Ω X and ( y, z ) = (cid:0) { y i } i ∈ N , { z i } i ∈ N (cid:1) ∈ R such that P t, x ( y, z ) = ∅ .In this part, we demonstrate that S t, x ( y, z ) = ∅ and V ( t, x , y, z ) ≤ V ( t, x , y, z ). (II.a) To show that e P t, x ( y, z ) = ∅ and V ( t, x , y, z ) ≤ e V ( t, x , y, z ), we fix P ∈ P t, x ( y, z ).Define a probability e P of P (cid:0)e Ω (cid:1) by e P := P | Ω × Ω X ⊗ dλ (cid:0) i.e., e P is the product measure between the projection of P on Ω × Ω X and the Lebesgue measure on [0 , (cid:1) . Clearly, e η is a unif (0 , 1) random variable independent of ( f W , e X )under e P and the joint e P − distribution of ( f W , e X ) is equal to the joint P − distribution of ( W , X ). As conditions (i),(ii)of Definition 3.1 hold for P , the probability e P correspondingly satisfies Definition 3.2, namely, e P ∈ e P t, x . (II.a.1) We next construct a e τ ∈ T e Pt, x ( y, z ).For any s ∈ [0 , ∞ ), there is F Ws − measurable random variable ϑ s on Ω such that ϑ s (cid:0) W t · ( ω ) (cid:1) = E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t s i ( ω ) , ∀ ω ∈ Ω . (3.11)Since the filtration F W,P is right-continuous and since E P [ ϑ s ] = E P (cid:2) ϑ s ( W t · ) (cid:3) = E P (cid:2) { T ∈ [ t,t + s ] } (cid:3) is right-continuousin s ∈ [0 , ∞ ), the process { ϑ s } s ∈ [0 , ∞ ) admits a c`adl`ag modification (cid:8) b ϑ s (cid:9) s ∈ [0 , ∞ ) on Ω .Let s ∈ [0 , ∞ ) and A ∈ F W t ∞ . Since W t is a Brownian motion with respect to filtration F t (cid:0) and thus with respectto F W t (cid:1) under P , the Markov property implies that E P (cid:2) A (cid:12)(cid:12) F ts (cid:3) = E P (cid:2) A (cid:12)(cid:12) F W t s (cid:3) , P − a.s. By the tower property , E P (cid:2) A { T ∈ [ t,t + s ] } (cid:3) = E P h { T ∈ [ t,t + s ] } E P (cid:2) A (cid:12)(cid:12) F ts (cid:3)i = E P h { T ∈ [ t,t + s ] } E P (cid:2) A (cid:12)(cid:12) F W t s (cid:3)i = E P h E P (cid:2) { T ∈ [ t,t + s ] } E P [ A |F W t s ] (cid:12)(cid:12) F W t s (cid:3)i = E P h E P (cid:2) A (cid:12)(cid:12) F W t s (cid:3) E P (cid:2) { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t s (cid:3)i = E P h E P h A E P (cid:2) { T ∈ [ t,t + s ] } |F W t s ] (cid:12)(cid:12) F W t s ii = E P h A E P (cid:2) { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t s (cid:3)i . Letting the set A run through F W t ∞ , we obtain E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t ∞ i = E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t s i , P − a.s. (3.12)For 0 ≤ s < r < ∞ , (3.11) and (3.12) show that0 ≤ ϑ s (cid:0) W t · ( ω ) (cid:1) = E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t ∞ i ( ω ) ≤ E P h { T ∈ [ t,t + r ] } (cid:12)(cid:12) F W t ∞ i ( ω ) = ϑ r (cid:0) W t · ( ω ) (cid:1) ≤ P − a.s. ω ∈ Ω,which implies that 0 ≤ b ϑ s = ϑ s ≤ ϑ r = b ϑ r ≤ P − a.s. So b ϑ is a [0 , − valued, F W,P − adapted c`adl`ag increasingprocess on Ω .Define a mapping ̺ : Ω × [0 , [0 , ∞ ] by ̺ ( ω , λ ) := inf (cid:8) s ∈ [0 , ∞ ) : b ϑ s ( ω ) > λ (cid:9) , ∀ ( ω , λ ) ∈ Ω × [0 , . In particular, e τ := ̺ (cid:0)f W t · , e η (cid:1) can be viewed as the hitting time of the F f W t , e η, e P − adapted c`adl`ag increasing process (cid:8) b ϑ s ( f W t · ) − e η (cid:9) s ∈ [0 , ∞ ) above level 0. Since f W t is a Brownian motion and since e η ∼ unif (0 , 1) is independent of f W t ptimal Stopping with Expectation Constraints 10under e P , the e P − augmentation F f W t , e η, e P of F f W t , e η is a right-continuous filtration (see e.g. Proposition 2.7.7 of [34]).Then the d´ebut Theorem renders that e τ = ̺ (cid:0)f W t · , e η (cid:1) is an F f W t , e η, e P − stopping time or e τ ∈ T e Pt . (II.a.2) Let Φ : [0 , ∞ ) × Ω X [ −∞ , ∞ ] and Υ : Ω X [ −∞ , ∞ ] be two Borel-measurable functions. We claim that E P h { T < ∞} Φ (cid:0) T , X · (cid:1) + { T = ∞} Υ (cid:0) X · (cid:1)i = E e P h { e τ< ∞} Φ (cid:0) t + e τ, e X · (cid:1) + { e τ = ∞} Υ (cid:0) e X · (cid:1)i , (3.13)To see this, we first assume that Υ ≡ s ∈ [0 , ∞ ) and A ∈ B (Ω X ),Definition 3.1 (ii), (3.12) and (3.11) imply that P − a.s. E P h { T ∈ [ t,t + s ] } { X · ∈ A } (cid:12)(cid:12) F W t ,P ∞ i = E P h { T ∈ [ t,t + s ] } { X t, x · ∈ A } (cid:12)(cid:12) F W t ,P ∞ i = { X t, x · ∈ A } E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t ,P ∞ i = { X · ∈ A } E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t ∞ i = { X · ∈ A } E P h { T ∈ [ t,t + s ] } (cid:12)(cid:12) F W t s i = { X · ∈ A } b ϑ s ( W t · ) = Z [0 , ∞ ) { r ≤ s } { X · ∈ A } b ϑ ( dr, W t · ) , where b ϑ ( ds, ω ) denotes the measure on [0 , ∞ ) with cumulative function b ϑ ([0 , s ] , ω ) = b ϑ s ( ω ), ∀ ( s, ω ) ∈ [0 , ∞ ) × Ω .So all measurable rectangles of B [0 , ∞ ) ⊗ B (Ω X ) are included in the Lambda systemΛ := n D ∈ B [ t, ∞ ) ⊗ B (Ω X ) : E P h { ( T ,X · ) ∈D} (cid:12)(cid:12) F W t ,P ∞ i = Z [0 , ∞ ) { ( t + r,X · ) ∈D} b ϑ ( dr, W t · ) , P − a.s. o . Then Dynkin’s Theorem shows that Λ = B [0 , ∞ ) ⊗ B (Ω X ). Using the standard approximation argument and the“change-of-variable” formula (see e.g. Proposition 0.4.9 of [52]) yield that P − a.s. E P h { T < ∞} Φ( T , X · ) (cid:12)(cid:12) F W t ,P ∞ i = Z [0 , ∞ ) Φ( t + r, X · ) b ϑ ( dr, W t · ) = Z { ̺ ( W t · ,λ ) < ∞} Φ( t + ̺ ( W t · , λ ) , X · ) dλ. (3.14)Since the joint P − distribution of ( W · , X · ) is equal to the joint e P − distribution of (cid:0)f W , e X (cid:1) and since e η ∼ unif (0 , F f W t , e P ∞ under e P , taking the expectation E P [ · ] in (3.14), we see from Fubini Theorem that E P h { T < ∞} Φ (cid:0) T , X · (cid:1)i = Z E P h { ̺ ( W t · ,λ ) < ∞} Φ (cid:0) t + ̺ ( W t · , λ ) , X · (cid:1)i dλ = Z E e P h { ̺ ( f W t · ,λ ) < ∞} Φ (cid:0) t + ̺ ( f W t · , λ ) , e X · (cid:1)i dλ = Z E e P h (cid:8) ̺ ( f W t · , e η ) < ∞ (cid:9) Φ (cid:0) t + ̺ ( f W t · , e η ) , e X · (cid:1)(cid:12)(cid:12)(cid:12)e η = λ i dλ = E e P h (cid:8) ̺ ( f W t · , e η ) < ∞ (cid:9) Φ (cid:0) t + ̺ ( f W t · , e η ) , e X · (cid:1)i . (3.15)Next, let Φ be a general [ −∞ , ∞ ] − valued, Borel-measurable function on [0 , ∞ ) × Ω X and let Υ be a gen-eral [ −∞ , ∞ ] − valued, Borel-measurable function on Ω X . We set ξ := { T < ∞} Φ (cid:0) T , X · (cid:1) + { T = ∞} Υ (cid:0) X · (cid:1) and e ξ := { e τ< ∞} Φ (cid:0) t + e τ , e X · (cid:1) + { e τ = ∞} Υ (cid:0) e X · (cid:1) . Given n ∈ N , applying (3.15) to Φ ± ∧ n and to Υ ± ∧ n respectively yields that E P h { T < ∞} n ∧ Φ ± (cid:0) T , X · (cid:1)i = E e P h { e τ< ∞} n ∧ Φ ± (cid:0) t + e τ , e X · (cid:1)i and E P h { T < ∞} n ∧ Υ ± (cid:0) X · (cid:1)i = E e P h { e τ< ∞} n ∧ Υ ± (cid:0) e X · (cid:1)i . Subtracting the latter from E P (cid:2) n ∧ Υ ± ( X · ) (cid:3) = E e P (cid:2) n ∧ Υ ± ( e X · ) (cid:3) and then adding to the former render that E P (cid:2) ξ ± ∧ n (cid:3) = E e P (cid:2) e ξ ± ∧ n (cid:3) . As n → ∞ , the monotone convergence theorem gives that E P (cid:2) ξ ± (cid:3) = E e P (cid:2) e ξ ± (cid:3) and (3.13) thus holds. (II.a.3) Let i ∈ N . Since function l ( s, ω X ) := ω X ( s ∧ · ) is continuous in ( s, ω X ) ∈ [0 , ∞ ) × Ω X , the measurabilityof functions f, π, g i , h i implies that ( f , g i , h i )( r, ω X ) := { r ≥ t } ( f, g i , h i ) (cid:0) r, l ( r, ω X ) (cid:1) = { r ≥ t } ( f, g i , h i ) (cid:0) r, ω X ( r ∧· ) (cid:1) are[ −∞ , ∞ ] − valued Borel-measurable functions and ̟ ( r, ω X ) := π (cid:0) r, ω X ( r ∧· ) (cid:1) , ∀ ( r, ω X ) ∈ [0 , ∞ ) × Ω X is a R − valuedBorel-measurable function. Then (cid:0)b f , b g i , b h i (cid:1) ( s, r, ω X ) := { r ≤ s } ( f , g i , h i )( r, ω X ), ∀ ( s, r, ω X ) ∈ [0 , ∞ ) × [0 , ∞ ) × Ω X arealso [ −∞ , ∞ ] − valued Borel-measurable functions and it follows that (cid:0) Υ f , Υ g i , Υ h i (cid:1) ( ω X ) := Z ∞ t ( f, g i , h i ) (cid:0) r, ω X ( r ∧· ) (cid:1) dr, ∀ ω ∈ Ω X , Φ f,π ( s, ω X ) := Z s ∨ tt f (cid:0) r, ω X ( r ∧· ) (cid:1) dr + π (cid:0) s, ω X ( s ∧· ) (cid:1) , ∀ ( s, ω X ) ∈ [0 , ∞ ) × Ω X , (cid:0) Φ g i , Φ h i (cid:1) ( s, ω X ) := Z s ∨ tt ( g i , h i ) (cid:0) r, ω X ( r ∧· ) (cid:1) dr, ∀ ( s, ω X ) ∈ [0 , ∞ ) × Ω X . Weak Formulation −∞ , ∞ ] − valued Borel-measurable functions.Now, taking (Φ , Υ) = (Φ g i , Υ g i ) and (Φ , Υ) = (Φ h i , Υ h i ) respectively in (3.13) and using the integration conventionyield that E e P h Z t + e τt g i ( r, e X r ∧· ) dr i = E P h Z Tt g i ( r, X r ∧· ) dr i ≤ y i , E e P h Z t + e τt h i ( r, e X r ∧· ) dr i = E P h Z Tt h i ( r, X r ∧· ) dr i = z i , which means e τ ∈ T e Pt, x ( y, z ). To wit, e P ∈ e P t, x ( y, z ). Similarly, applying (3.13) to (Φ , Υ) = (Φ f,π , Υ f ), we obtain E P (cid:20) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + { T< ∞} π (cid:0) T , X T ∧· (cid:1)(cid:21) = E e P (cid:20) Z t + e τt f (cid:0) r, e X r ∧· (cid:1) dr + { e τ< ∞} π (cid:0) t + e τ, e X ( t + e τ ) ∧· (cid:1)(cid:21) ≤ e V ( t, x , y, z ) . Taking supremum over P ∈ P t, x ( y, z ) eventually leads to V ( t, x , y, z ) ≤ e V ( t, x , y, z ). (II.b) It remains to show that e V ( t, x , y, z ) ≤ V ( t, x , y, z ): Fix e P ∈ e P t, x ( y, z ) and e τ ∈ T e Pt, x ( y, z ).Define a mapping Ψ : Q 7→ e Ω by Ψ( ω ) := (cid:0) B t ∨· ( ω ) − B t ( ω ) , X t, x · ( ω ) , η ( ω ) (cid:1) ∈ e Ω, ∀ ω ∈ Q . So for any ω ∈ Q , e X s (cid:0) Ψ( ω ) (cid:1) = x ( s ) , ∀ s ∈ [0 , t ] and e Ξ t · (cid:0) Ψ( ω ) (cid:1) = (cid:0) B t · ( ω ) , X t, x t + · ( ω ) (cid:1) , e η (cid:0) Ψ( ω ) (cid:1) = η ( ω ) . For s ∈ [0 , ∞ ], since it holds for any r ∈ [0 , s ] ∩ R , E ∈ B ( R d ) and E ′ ∈ B [0 , 1] that Ψ − (cid:0) ( f W tr ) − ( E ) ∩ e η − ( E ′ ) (cid:1) = (cid:8)f W tr (Ψ) ∈ E (cid:9) ∩ (cid:8)e η (Ψ) ∈ E ′ (cid:9) = (cid:8) B tr ∈ E (cid:9) ∩ (cid:8) η ∈ E ′ (cid:9) ∈ F B t ,ηs , one can deduce that Ψ − (cid:0) F f W t s (cid:1) ⊂ F B t s and Ψ − (cid:0) F f W t , e ηs (cid:1) ⊂F B t ,ηs . As the p − joint distribution of ( B t , η ) is equal to the e P − joint distribution of ( f W t , e η ), we see that for (cid:8) ( s i , E i ) (cid:9) ni =1 ⊂ [0 , ∞ ) × B ( R d ) and E ′ ∈ B [0 , (cid:0) p ◦ Ψ − (cid:1)(cid:0) ( f W ts ) − ( E ) ∩· · ·∩ ( f W ts n ) − ( E n ) ∩ e η − ( E ′ ) (cid:1) = p (cid:8) B ts ∈ E , · · · , B ts n ∈ E n , η ∈ E ′ (cid:9) = e P nf W ts ∈ E , · · · , f W ts n ∈ E n , e η ∈ E ′ o . An application of Dynkin’s Theorem yields that (cid:0) p ◦ Ψ − (cid:1)(cid:0) e A (cid:1) = e P (cid:0) e A (cid:1) , ∀ e A ∈ F f W t , e η ∞ , then one can further deduce thatΨ − (cid:0) F f W t , e Ps (cid:1) ⊂ F B t , p s , Ψ − (cid:0) F f W t , e η, e Ps (cid:1) ⊂ F B t ,η, p s , ∀ s ∈ [0 , ∞ ] and (cid:0) p ◦ Ψ − (cid:1)(cid:0) e A (cid:1) = e P (cid:0) e A (cid:1) , ∀ e A ∈ F f W t , e η, e P ∞ . (3.16)It follows that τ ( ω ) := e τ (cid:0) Ψ( ω ) (cid:1) , ω ∈ Q is [0 , ∞ ] − valued, F B t ,η, p − stopping time or τ ∈ S t .Using similar arguments to those leading to (3.9), we can derive from (3.16) that p (cid:8)(cid:0) R s σ ( t + r, X t, x ( t + r ) ∧· ) d f W tr (cid:1) (Ψ) = R s σ (cid:0) t + r, X t, x ( t + r ) ∧· (Ψ) (cid:1) dB tr , s ∈ [0 , ∞ ) (cid:9) = 1 and thus that p − a.s. X t, x s (Ψ) = x ( t )+ Z s − t b (cid:0) t + r, X t, x ( t + r ) ∧· (Ψ) (cid:1) dr + (cid:16) Z s − t σ ( t + r, X t, x ( t + r ) ∧· ) d f W tr (cid:17) (Ψ)= x ( t )+ Z s − t b (cid:0) t + r, X t, x ( t + r ) ∧· (Ψ) (cid:1) dr + Z s − t σ (cid:0) t + r, X t, x ( t + r ) ∧· (Ψ) (cid:1) dB tr = x ( t )+ Z st b (cid:0) r, X t, x r ∧· (Ψ) (cid:1) dr + Z st σ (cid:0) r, X t, x r ∧· (Ψ) (cid:1) dB r , ∀ s ∈ [ t, ∞ ) . So (cid:8) X t, x s (Ψ) (cid:9) s ∈ [0 , ∞ ) is the unique solution of (2.1) or p (cid:8) X t, x s = X t, x s (Ψ) , ∀ s ∈ [0 , ∞ ) (cid:9) = 1.For any i ∈ N , (3.16) implies that E p h Z t + τt g i ( r, X t, x r ∧· ) dr i = E p h Z t + e τ (Ψ) t g i (cid:0) r, X t, x r ∧· (Ψ) (cid:1) dr i = E e P h Z t + e τt g i ( r, X t, x r ∧· ) dr i = E e P h Z t + e τt g i ( r, e X r ∧· ) dr i ≤ y i (3.17)and similarly E p (cid:2) R t + τt h i ( r, X t, x r ∧· ) dr (cid:3) = E e P (cid:2) R t + e τt h i ( r, e X r ∧· ) dr (cid:3) = z i , which shows τ ∈ S t, x ( y, z ). By an analogy to(3.17), E e P (cid:20) Z t + e τt f (cid:0) r, e X r ∧· (cid:1) dr + { e τ< ∞} π (cid:0) t + e τ, e X ( t + e τ ) ∧· (cid:1)(cid:21) = E p h Z t + τt f (cid:0) r, X t, x r ∧· (cid:1) dr + { τ< ∞} π (cid:0) t + τ, X t, x ( t + τ ) ∧· (cid:1)i ≤ V ( t, x , y, z ) . Letting e τ vary over T e Pt, x ( y, z ) and then letting e P run through e P t, x ( y, z ) yield e V ( t, x , y, z ) ≤ V ( t, x , y, z ). (cid:3) ptimal Stopping with Expectation Constraints 12 V In this section, using the martingale-problem formulation of stochastic differential equations (see Stroock & Varad-han [60]) we first characterize the probability class P t, x via the stochastic behaviors of the canonical coordinators( W , X, T ). This will enable us to analyze the measurability of the value function V of our optimal stopping problemwith expectation constraints and we can thus study the dynamic programming principle of V in the next section.Set Q ,< := (cid:8) ( s, r ) ∈ Q : 0 ≤ s < r (cid:9) and denote by P ( R d + l ) the set of all polynomial functions on R d + l with Q − valued coefficients. Let t ∈ [0 , ∞ ). For any ϕ ∈ P ( R d + l ), we define process M ts ( ϕ ) := ϕ (cid:0) Ξ ts (cid:1) − Z s b (cid:0) t + r, X ( t + r ) ∧· (cid:1) · Dϕ (cid:0) Ξ tr (cid:1) dr − Z s σ σ T (cid:0) t + r, X ( t + r ) ∧· (cid:1) : D ϕ (cid:0) Ξ tr (cid:1) dr, s ∈ [0 , ∞ ) , where b ( r, x ) := (cid:18) b ( r, x ) (cid:19) ∈ R d + l and σ ( r, x ) := (cid:18) I d × d σ ( r, x ) (cid:19) ∈ R d × d + l × d , ∀ ( r, x ) ∈ (0 , ∞ ) × Ω X . For any n ∈ N , let usalso set τ tn := inf (cid:8) s ∈ [0 , ∞ ) : | Ξ ts | ≥ n (cid:9) ∧ n , which is an F Ξ t − stopping time.We can characterize the probability class P t, x by a martingale-problem formulation as follows. Proposition 4.1. For any ( t, x ) ∈ [0 , ∞ ) × Ω X , the probability class P t, x is the intersection of the following twosubsets of P (Ω) :1 ) P t := n P ∈ P (Ω) : E P h(cid:16) M tτ tn ∧ r ( ϕ ) − M tτ tn ∧ s ( ϕ ) (cid:17) k Q i =1 (cid:0) { Ξ tsi ∈O i }∩{ T ≤ t + s i } + { Ξ tsi ∈O ′ i }∩{ T>t + s i } (cid:1)i = 0 , ∀ ϕ ∈ P ( R d + l ); ∀ n ∈ N ; ∀ ( s, r ) ∈ Q ,< ; ∀ { ( s i , O i , O ′ i ) } ki =1 ⊂ (cid:0) Q ∩ [0 , s ] (cid:1) × O ( R d + l ) × O ( R d + l ) o .2 ) P t, x := (cid:8) P ∈ P (Ω) : P { T ≥ t ; X s = x ( s ) , ∀ s ∈ [0 , t ] } = 1 (cid:9) . Proof of Proposition 4.1: Fix ( t, x ) ∈ [0 , ∞ ) × Ω X . Let P ∈ P t, x , which is clearly of P t, x . For any ϕ ∈ P ( R d + l ), n ∈ N , ( s, r ) ∈ Q ,< and { ( s i , O i , O ′ i ) } ki =1 ⊂ (cid:0) Q ∩ [0 , s ] (cid:1) × O ( R d + l ) × O ( R d + l ), an application of Lemma A.1 shows that E P h(cid:0) M tτ tn ∧ r ( ϕ ) − M tτ tn ∧ s ( ϕ ) (cid:1) k Q i =1 (cid:0) { Ξ tsi ∈O i }∩{ T ≤ t + s i } + { Ξ tsi ∈O ′ i }∩{ T>t + s i } (cid:1)i = 0. Namely, P is also of P t . Next, let P ∈ P t ∩P t, x . It is clear that P { T ≥ t } = 1 or P satisfies Definition 3.1 (iii). Set N X := { ω ∈ Ω : X s ( ω ) = x ( s ) for some s ∈ [0 , t ] } ∈ N P (cid:0) F Xt (cid:1) . Let ϕ ∈ P ( R d + l ) and n ∈ N . We define X t, x s := { s ∈ [0 ,t ] } x ( s ) + { s ∈ ( t, ∞ ) } (cid:0) X s − X t + x ( t ) (cid:1) , s ∈ [0 , ∞ ). It is acontinuous process such that (cid:8) X t, x t + s (cid:9) s ∈ [0 , ∞ ) is F X t − adapted. Then Ξ t, x s := (cid:0) W ts , X t, x t + s (cid:1) , s ∈ [0 , ∞ ) and M t, x s ( ϕ ) := ϕ (cid:0) Ξ t, x s (cid:1) − Z s b (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) · Dϕ (cid:0) Ξ t, x r (cid:1) dr − Z s σ σ T (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) : D ϕ (cid:0) Ξ t, x r (cid:1) dr, s ∈ [0 , ∞ ) (4.1)are F Ξ t − adapted continuous processes, and τ t, x n := inf (cid:8) s ∈ [0 , ∞ ) : (cid:12)(cid:12) Ξ t, x s (cid:12)(cid:12) ≥ n (cid:9) ∧ n is an F Ξ t − stopping time. Clearly, X t, x s ( ω ) = X s ( ω ) , M t, x s ( ϕ )( ω ) = M ts ( ϕ )( ω ) and τ t, x n ( ω ) = τ tn ( ω ) , ∀ ( s, ω ) ∈ [0 , ∞ ) × (cid:0) N X (cid:1) c . (4.2)For any ( s, r ) ∈ Q ,< and { ( s i , O i , O ′ i ) } ki =1 ⊂ (cid:0) Q ∩ [0 , s ] (cid:1) × O ( R d + l ) × O ( R d + l ), we see from P ∈ P t and (4.2) that E P h(cid:0) M t, x τ t, x n ∧ r ( ϕ ) − M t, x τ t, x n ∧ s ( ϕ ) (cid:1) k Y i =1 (cid:0) { Ξ tsi ∈O i }∩{ T ∈ [ t,t + s i ] } + { Ξ tsi ∈O ′ i }∩{ T ∈ [ t,t + s i ] c } (cid:1)i = E P h(cid:0) M tτ tn ∧ r ( ϕ ) − M tτ tn ∧ s ( ϕ ) (cid:1) k Y i =1 (cid:0) { Ξ tsi ∈O i }∩{ T ∈ [ t,t + s i ] } + { Ξ tsi ∈O ′ i }∩{ T ∈ [ t,t + s i ] c } (cid:1)i = 0 . Dynkin’s Theorem and (3.2) render that E P h(cid:16) M t, x τ t, x n ∧ r ( ϕ ) − M t, x τ t, x n ∧ s ( ϕ ) (cid:17) A i = 0, ∀ A ∈ G ts , which further im-plies that (cid:8) M t, x τ t, x n ∧ s ( ϕ ) (cid:9) s ∈ [0 , ∞ ) is an (cid:0) G t,P , P (cid:1) − martingale. Since lim n →∞ ↑ τ t, x n = ∞ , (cid:8) M t, x s ( ϕ ) (cid:9) s ∈ [0 , ∞ ) is actually an (cid:0) G t,P , P (cid:1) − local martingale. . Martingale-problem Formulation and Measurability of V Given i, j = 1 , · · · , d , set φ i ( w, x ) := w i and φ ij ( w, x ) := w i w j for any w = ( w , · · · , w d ) ∈ R d and x ∈ R l . ByPart (2a), the processes M t, x s ( φ i ) = ( W ts ) ( i ) , s ∈ [0 , ∞ ) and M t, x s ( φ ij ) = ( W ts ) ( i ) ( W ts ) ( j ) − δ ij s , s ∈ [0 , ∞ ) are all (cid:0) G t,P , P (cid:1) − local martingales for i, j = 1 , · · · , d , where W ts = (cid:0) ( W ts ) (1) , · · · , ( W ts ) ( d ) (cid:1) . L´evy’s characterization theoremshows that W t is a d − dimensional standard Brownian motion with respect to the filtration G t,P and is thus aBrownian motion with respect to the filtration F t . So P satisfies Definition 3.1 (i).On the other hand, we simply denote b t, x s := b (cid:0) t + s, X t, x ( t + s ) ∧· (cid:1) and α t, x s := σ σ T (cid:0) t + s, X t, x ( t + s ) ∧· (cid:1) , s ∈ [0 , ∞ ). Given i, j = 1 , · · · , d + l , we set ψ i ( ) := ( i ) and ψ ij ( ) := ( i ) ( j ) for any = ( , · · · , d + l ) ∈ R d + l . By Part (2a), theprocesses M t, x s ( ψ i ) = (cid:0) Ξ t, x s (cid:1) ( i ) − R s (cid:0) b t, x r (cid:1) ( i ) dr , s ∈ [0 , ∞ ) and M t, x s ( ψ ij ) = (cid:0) Ξ t, x s (cid:1) ( i ) (cid:0) Ξ t, x s (cid:1) ( j ) − R s (cid:0) b t, x r (cid:1) ( i ) (cid:0) Ξ t, x r (cid:1) ( j ) dr − R s (cid:0) b t, x r (cid:1) ( j ) (cid:0) Ξ t, x r (cid:1) ( i ) dr − R s (cid:0) α t, x r (cid:1) ij dr , s ∈ [0 , ∞ ) are (cid:0) G t,P , P (cid:1) − local martingales and thus (cid:0) F Ξ t ,P , P (cid:1) − local martin-gales. Since the integration by part formula renders that P − a.s. (cid:0) Ξ t, x s (cid:1) ( i ) (cid:0) Ξ t, x s (cid:1) ( j ) − M t, x s ( ψ i ) M t, x s ( ψ j ) = M t, x s ( ψ i ) Z s (cid:0) b t, x r (cid:1) ( j ) dr + M t, x s ( ψ j ) Z s (cid:0) b t, x r (cid:1) ( i ) dr + Z s (cid:0) b t, x r (cid:1) ( i ) dr Z s (cid:0) b t, x r (cid:1) ( j ) dr = Z s M t, x r ( ψ i ) (cid:0) b t, x r (cid:1) ( j ) dr + Z s (cid:16) Z r (cid:0) b t, x r ′ (cid:1) ( j ) dr ′ (cid:17) dM t, x r ( ψ i )+ Z s M t, x r ( ψ j ) (cid:0) b t, x r (cid:1) ( i ) dr + Z s (cid:16) Z r (cid:0) b t, x r ′ (cid:1) ( i ) dr ′ (cid:17) dM t, x r ( ψ j )+ Z s (cid:16) Z r (cid:0) b t, x r ′ (cid:1) ( i ) dr ′ (cid:17)(cid:0) b t, x r (cid:1) ( j ) dr + Z s (cid:16) Z r (cid:0) b t, x r ′ (cid:1) ( j ) dr ′ (cid:17)(cid:0) b t, x r (cid:1) ( i ) dr = Z s h(cid:0) Ξ t, x s (cid:1) ( i ) (cid:0) b t, x r (cid:1) ( j ) + (cid:0) Ξ t, x s (cid:1) ( j ) (cid:0) b t, x r (cid:1) ( i ) i dr + Z s (cid:16) Z r (cid:0) b t, x r ′ (cid:1) ( j ) dr ′ (cid:17) dM t, x r ( ψ i )+ Z s (cid:16) Z r (cid:0) b t, x r ′ (cid:1) ( i ) dr ′ (cid:17) dM t, x r ( ψ j ) ,s ∈ [0 , ∞ ), we obtain that M t, x s ( ψ i ) M t, x s ( ψ j ) − Z s (cid:0) α t, x r (cid:1) ij dr = M t, x s ( ψ ij ) − Z s (cid:16) Z r (cid:0) b t, x r ′ (cid:1) ( j ) dr ′ (cid:17) dM t, x r ( ψ i ) − Z s (cid:16) Z r (cid:0) b t, x r ′ (cid:1) ( i ) dr ′ (cid:17) dM t, x r ( ψ j ) ,s ∈ [0 , ∞ ) is also an (cid:0) F Ξ t ,P , P (cid:1) − local martingale. Thus the quadratic variation of the (cid:0) F Ξ t ,P , P (cid:1) − local martingale M t, x s := (cid:0) M t, x s ( ψ ) , · · · , M t, x s ( ψ d + l ) (cid:1) = Ξ t, x s − R s b t, x r dr , s ∈ [0 , ∞ ) is (cid:10) M t, x s , M t, x s (cid:11) = R s α t, x r dr, s ∈ [0 , ∞ ).Let n ∈ N , a ∈ R l and set H s := (cid:18) − { s ≤ τ t, x n } σ T (cid:0) t + s, X t, x ( t + s ) ∧· (cid:1) aa (cid:19) , s ∈ [0 , ∞ ). The stochastic exponential of the (cid:0) F Ξ t ,P , P (cid:1) − martingale (cid:8) R τ t, x n ∧ s H r · d M t, x r (cid:9) s ∈ [0 , ∞ ) isexp (cid:26) Z τ t, x n ∧ s H r · d M t, x r − Z τ t, x n ∧ s H Tr α t, x r H r dr (cid:27) = exp (cid:26) Z τ t, x n ∧ s H r · d Ξ t, x r − Z τ t, x n ∧ s H r · b t, x r dr (cid:27) = exp (cid:26) a · (cid:18) X t, x t + τ t, x n ∧ s − X t, x t − Z τ t, x n ∧ s b (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) dr − Z τ t, x n ∧ s σ (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) dW tr (cid:19)(cid:27) , s ∈ [0 , ∞ ) . Letting a varies over R l yields that P − a.s., X t, x t + τ t, x n ∧ s = x ( t )+ R τ t, x n ∧ s b (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) dr + R τ t, x n ∧ s σ (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) dW tr , s ∈ [0 , ∞ ). Sending n → ∞ then shows that (cid:8) X t, x s (cid:9) s ∈ [0 , ∞ ) solves the SDE (3.3). As X t, x coincides with X on (cid:0) N X (cid:1) c by (4.2), we see that P satisfies Definition 3.1 (ii). (cid:3) Based on the countable decomposition of the probability class P t, x by Proposition 4.1, the next result shows thatthe graph of the probability class {P t, x } ( t, x ) ∈ [0 , ∞ ) × Ω X is a Borel subset of [0 , ∞ ) × Ω X × P (Ω), which is crucial forthe measurability of the value function V . Proposition 4.2. The graph (cid:2)(cid:2) P (cid:3)(cid:3) := (cid:8)(cid:0) t, x , y, z, P (cid:1) ∈ D × P (Ω) : P ∈ P t, x ( y, z ) (cid:9) is a Borel subset of D × P (Ω) andthe graph (cid:8)(cid:8) P (cid:9)(cid:9) := (cid:8)(cid:0) t, w , x , y, z, P (cid:1) ∈ D× P (Ω) : P ∈ P t, w , x ( y, z ) (cid:9) is a Borel subset of D× P (Ω) . Proof of Proposition 4.2: 1) We first show that the graph (cid:10)(cid:10) P (cid:11)(cid:11) := (cid:8)(cid:0) t, x , P (cid:1) ∈ [0 , ∞ ) × Ω X × P (Ω) : P ∈ P t, x (cid:9) is aBorel subset of [0 , ∞ ) × Ω X × P (Ω).According to Proposition 4.1, (cid:10)(cid:10) P (cid:11)(cid:11) is the intersection of (cid:10)(cid:10) P (cid:11)(cid:11) := (cid:8)(cid:0) t, x , P (cid:1) ∈ [0 , ∞ ) × Ω X × P (Ω) : P ∈ P t (cid:9) and (cid:10)(cid:10) P (cid:11)(cid:11) := (cid:8)(cid:0) t, x , P (cid:1) ∈ [0 , ∞ ) × Ω X × P (Ω) : P ∈ P t, x (cid:9) .ptimal Stopping with Expectation Constraints 14 Since the function W ( s, ω ) := ω ( s ) is continuous in ( s, ω ) ∈ [0 , ∞ ) × Ω , and the function W X ( s, ω X ) := ω X ( s ) iscontinuous in ( s, ω X ) ∈ [0 , ∞ ) × Ω X , Ξ( t, r, ω , ω X ) := (cid:0) W ( t + r, ω ) − W ( t, ω ) , W X ( t + r, ω X ) (cid:1) ∈ R d + l , ∀ ( t, r, ω , ω X ) ∈ [0 , ∞ ) × (0 , ∞ ) × Ω × Ω X is Borel-measurable.Let ϕ ∈ P ( R d + l ) and n ∈ N . As the function l ( s, ω X ) := ω X ( s ∧ · ) is continuous in ( s, ω X ) ∈ [0 , ∞ ) × Ω X , themeasurability of the functions b, σ and the mappings l , Ξ imply that the mapping H ϕ ( t, s, r, ω , ω X ) := { r ≤ s } n b (cid:0) t + r, l ( t + r, ω X ) (cid:1) · Dϕ (cid:0) Ξ( t, r, ω , ω X ) (cid:1) + 12 σσ T (cid:0) t + r, l ( t + r, ω X ) (cid:1) : D ϕ (cid:0) Ξ( t, r, ω , ω X ) (cid:1)o , ∀ ( t, s, r, ω , ω X ) ∈ [0 , ∞ ) × [0 , ∞ ) × (0 , ∞ ) × Ω × Ω X is Borel-measurable, and I ϕ ( t, s, ω , ω X ) := R ∞ H ϕ ( t, s, r, ω , ω X ) dr , ∀ ( t, s, ω , ω X ) ∈ [0 , ∞ ) × [0 , ∞ ) × Ω × Ω X is thus Borel-measurable. It follows that M ϕ ( t, s, ω ) := (cid:0) M t ( ϕ ) (cid:1) ( s, ω ) = ϕ (cid:16) Ξ (cid:0) t, s, W ( ω ) , X ( ω ) (cid:1)(cid:17) −I ϕ (cid:0) t, s, W ( ω ) , X ( ω ) (cid:1) , ∀ ( t, s, ω ) ∈ [0 , ∞ ) × [0 , ∞ ) × Ω (4.3)is also B [0 , ∞ ) ⊗ B [0 , ∞ ) ⊗ B (Ω) − measurable.For any s ∈ [0 , ∞ ), since D ns := (cid:8) ( t, ω ) ∈ [0 , ∞ ) × Ω: τ tn ( ω ) > s (cid:9) = (cid:8) ( t, ω ) ∈ [0 , ∞ ) × Ω : | Ξ tr ( ω ) | < n, ∀ r ∈ [0 , s ] (cid:9) , we candeduce from the continuity of processes ( W , X ) and the topology of locally uniform convergence on (Ω , Ω X ) that D ns is an open subset of [0 , ∞ ) × Ω and the process T n ( t, ω ) := τ tn ( ω ), ∀ ( t, ω ) ∈ [0 , ∞ ) × Ω is thus B [0 , ∞ ) ⊗ B (Ω) − measurable.This measurability together with (4.3) further shows that for any s ∈ [0 , ∞ ) M ϕ,ns ( t, ω ) := (cid:0) M t ( ϕ ) (cid:1)(cid:0) τ tn ( ω ) ∧ s, ω (cid:1) = M ϕ (cid:0) t, T n ( t, ω ) ∧ s, ω (cid:1) , ∀ ( t, ω ) ∈ [0 , ∞ ) × Ω (4.4)is also a B [0 , ∞ ) ⊗ B (Ω) − measurable process. Let θ := (cid:0) ϕ, n, ( s, r ) , { ( s i , O i , O ′ i ) } ki =1 (cid:1) ∈ P ( R d + l ) × N × Q ,< × b O ( R d + l ). As the process ( t, ω ) → T ( ω ) − t is B [0 , ∞ ) ⊗ B (Ω) − measurable, we see from (4.4) that f θ ( t, ω ) := (cid:0) M ϕ,nr ( t, ω ) − M ϕ,ns ( t, ω ) (cid:1) k Y i =1 (cid:16) { Ξ( t,s i ∧ s,W ( ω ) ,X ( ω )) ∈O i }∩{ T ( ω ) ≤ t + s i ∧ s } + { Ξ( t,s i ∧ s,W ( ω ) ,X ( ω )) ∈O ′ i }∩{ T ( ω ) >t + s i ∧ s } (cid:17) , ∀ ( t, ω ) ∈ [0 , ∞ ) × Ω is B [0 , ∞ ) ⊗ B (Ω) − measurable. Applying Lemma A.2 yields that the mapping ( t, P ) R ω ∈ Ω f θ ( t, ω ) P ( d ω ) is B [0 , ∞ ) ⊗ B (cid:0) P (Ω) (cid:1) − measurable and the set (cid:26) ( t, x , P ) ∈ [0 , ∞ ) × Ω X × P (Ω) : E P h(cid:16) M tτ tn ∧ r ( ϕ ) − M tτ tn ∧ s ( ϕ ) (cid:17) k Y i =1 (cid:0) { Ξ tsi ∧ s ∈O i }∩{ T ≤ t + s i ∧ s } + { Ξ tsi ∧ s ∈O ′ i }∩{ T >t + s i ∧ s } (cid:1)i = 0 (cid:27) is thus Borel-measurable. Letting θ run through P ( R d + l ) × N × Q ,< × b O ( R d + l ) shows that (cid:10)(cid:10) P (cid:11)(cid:11) is a Borel subsetof [0 , ∞ ) × Ω X × P (Ω).Since the mapping ( t, x , ω ) Q r ∈ Q ∩ [0 , ∞ ) { X ( r ∧ t,ω )= x ( r ∧ t ) } is B [0 , ∞ ) ⊗ B (Ω X ) ⊗ B (Ω) − measurable, Lemma A.2again renders that the mapping ( t, x , P ) R ω ∈ Ω { T ( ω ) − t ≥ } (cid:16) Q r ∈ Q ∩ [0 , ∞ ) { X ( r ∧ t,ω )= x ( r ∧ t ) } (cid:17) P ( d ω ) = P (cid:8) T ≥ t ; X s = x ( s ) , ∀ s ∈ [0 , t ] (cid:9) is B [0 , ∞ ) ⊗ B (Ω X ) ⊗ B (cid:0) P (Ω) (cid:1) − measurable and thus (cid:10)(cid:10) P (cid:11)(cid:11) = (cid:8) ( t, x , P ) ∈ [0 , ∞ ) × Ω X × P (Ω) : P { T ≥ t ; X s = x ( s ) , ∀ s ∈ [0 , t ] } = 1 (cid:9) is a Borel subset of [0 , ∞ ) × Ω X × P (Ω). Totally, (cid:10)(cid:10) P (cid:11)(cid:11) = (cid:10)(cid:10) P (cid:11)(cid:11) ∩ (cid:10)(cid:10) P (cid:11)(cid:11) is also aBorel subset of [0 , ∞ ) × Ω X × P (Ω). Let i ∈ N . By the measurability of functions l , g i and h i , the mapping (cid:0) g i , h i (cid:1) ( t, s, r, ω X ) := { t ≤ r ≤ s } ( g i , h i ) (cid:0) r, l ( r, ω X ) (cid:1) , ∀ ( t, s, r, ω X ) ∈ [0 , ∞ ) × [0 , ∞ ) × (0 , ∞ ) × Ω X is Borel-measurable. It follows that (cid:0) I g i , I h i (cid:1) ( t, s, ω X ) := R ∞ (cid:0) g i , h i (cid:1) ( t, s, r, ω X ) dr = R s ∨ tt ( g i , h i ) (cid:0) r, l ( r, ω X ) (cid:1) dr , ∀ ( t, s, ω X ) ∈ [0 , ∞ ) × [0 , ∞ ) × Ω X is Borel-measurable and thus the process (cid:0)b I g i , b I h i (cid:1) ( t, ω ) := (cid:0) I g i , I h i (cid:1)(cid:0) t, T ( ω ) , X ( ω ) (cid:1) = Z T ( ω ) ∨ tt ( g i , h i ) (cid:0) r, X ( r ∧· , ω ) (cid:1) dr, ∀ ( t, ω ) ∈ [0 , ∞ ) × Ω (4.5)is B [0 , ∞ ) ⊗ B (Ω) − measurable. An application of Lemma A.2 yields that the mapping (cid:0) Φ g i , Φ h i (cid:1) ( t, P ) := R ω ∈ Ω (cid:0)b I g i , b I h i (cid:1) ( t, ω ) P ( d ω ) = E P (cid:2) R Tt ( g i , h i ) (cid:0) r, X r ∧· (cid:1) dr (cid:3) , ∀ ( t, P ) ∈ [0 , ∞ ) × P (Ω) is B [0 , ∞ ) ⊗ B (cid:0) P (Ω) (cid:1) − measurable. Then D := (cid:8) ( t, x , y, z, P ) ∈ [0 , ∞ ) × Ω X ×R× P (Ω) : Φ g i ( t, P ) ≤ y i , Φ h i ( t, P ) = z i , ∀ i ∈ N (cid:9) . Dynamic Programming Principle , ∞ ) × Ω X ×R× P (Ω) and thus a Borel subset of D × P (Ω). It follows that (cid:2)(cid:2) P (cid:3)(cid:3) = n ( t, x , y, z, P ) ∈ [0 , ∞ ) × Ω X ×R× P (Ω) : P ∈ P t, x ; E P (cid:2) R Tt g i (cid:0) r, X r ∧· (cid:1) dr (cid:3) ≤ y i , E P (cid:2) R Tt h i (cid:0) r, X r ∧· (cid:1) dr (cid:3) = z i , ∀ i ∈ N o = (cid:0)(cid:10)(cid:10) P (cid:11)(cid:11) ×R (cid:1) ∩ D is a Borel subset of D × P (Ω). Since the mapping φ W ( t, w , ω ) := Q r ∈ Q ∩ [0 , ∞ ) { W ( r ∧ t,ω )= w ( r ∧ t ) } , ( t, w , ω ) ∈ [0 , ∞ ) × Ω × Ω is B [0 , ∞ ) ⊗ B (Ω ) ⊗ B (Ω) − measurable, applying Lemma A.2 again shows that the mapping Φ( t, w , P ) := E P (cid:2) φ W ( t, w , ω ) (cid:3) = P (cid:8) W s = w ( s ), ∀ s ∈ [0 , t ] (cid:9) , ∀ ( t, w , P ) ∈ [0 , ∞ ) × Ω × P (Ω) is B [0 , ∞ ) ⊗ B (Ω ) ⊗ B (cid:0) P (Ω) (cid:1) − measurable. By the projectionsΠ ( t, w , x , y, z, P ) := (cid:0) t, x , P (cid:1) , Π ( t, w , x , y, z, P ) := (cid:0) t, x , y, z, P (cid:1) , Π ( t, w , x , y, z, P ) := ( t, w , x , P ) , we can derive that (cid:8)(cid:8) P (cid:9)(cid:9) = n ( t, w , x , y, z, P ) ∈ [0 , ∞ ) × Ω × Ω X ×R× P (Ω) : P ∈ P t, x o ∩ n ( t, w , x , y, z, P ) ∈ [0 , ∞ ) × Ω × Ω X ×R× P (Ω) : P (cid:8) W s = w ( s ) , ∀ s ∈ [0 , t ] (cid:9) = 1 o ∩ (cid:26) ( t, w , x , y, z, P ) ∈ [0 , ∞ ) × Ω × Ω X ×R× P (Ω) : E P h Z Tt g i (cid:0) r, X r ∧· (cid:1) dr i ≤ y i , E P h Z Tt h i (cid:0) r, X r ∧· (cid:1) dr i = z i , ∀ i ∈ N (cid:27) = Π − (cid:0)(cid:10)(cid:10) P (cid:11)(cid:11)(cid:1) ∩ Π − (cid:0) Φ − (1) (cid:1) ∩ Π − ( D )is a Borel subset of D× P (Ω). (cid:3) By Proposition 4.2, the value function V is upper semi-analytic and thus universally measurable. Theorem 4.1. The value function V is upper semi-analytic in ( t, x , y, z ) ∈ D and the value function V is uppersemi-analytic in ( t, w , x , y, z ) ∈ D . Proof of Theorem 4.1: Similar to (4.5), b I f ( t, ω ) := R T ( ω ) ∨ tt f (cid:0) r, X ( r ∧ · , ω ) (cid:1) dr is B [0 , ∞ ) ⊗ B (Ω) − measurable.Since the measurability of functions l and π implies that ( s, ω ) π (cid:0) s, l ( s, X ( ω )) (cid:1) = π (cid:0) s, X ( s ∧ · , ω ) (cid:1) is B (0 , ∞ ) ⊗ B (Ω) − measurable, the mapping φ π ( ω ) := { T ( ω ) < ∞} π (cid:0) T ( ω ) , X ( T ( ω ) ∧ · , ω ) (cid:1) , ω ∈ Ω is B (Ω) / B ( R ) − measurable.According to Lemma A.2, V ( t, P ) := Z ω ∈ Ω (cid:0)b I f ( t, ω )+ φ π ( ω ) (cid:1) P ( d ω ) = E P (cid:20) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + { T< ∞} π (cid:0) T , X T ∧· (cid:1)(cid:21) , ( t, P ) ∈ [0 , ∞ ) × P (Ω) is B [0 , ∞ ) ⊗ B (cid:0) P (Ω) (cid:1) − measurable. Then Proposition 4.2 and Proposition 7.47 of [13] yield that V ( t, x , y, z ) = sup P ∈P t, x ( y,z ) V ( t, P ) = sup ( t, x ,y,z,P ) ∈ [[ P ]] V ( t, P ) is upper semi-analytic in ( t, x , y, z ) ∈ D and V ( t, w , x , y, z ) =sup P ∈P t, w , x ( y,z ) V ( t, P ) = sup ( t, w , x ,y,z,P ) ∈ V ( t, P ) is upper semi-analytic in ( t, w , x , y, z ) ∈ D . (cid:3) In this section, we explore a dynamic programming principle (DPP) for the value V in the weak formulation, whichtakes the conditional expected integrals of constraint functions as additional states.Given t ∈ [0 , ∞ ), let τ be a [0 , ∞ ) − valued F W t − stopping time and set A ( τ ) := (cid:8) T ∈ [ t, t + τ ) (cid:9) ∈ F tτ . We denoteby F ( τ ) the sigma field of Ω generated by F W t τ and the set A ( τ ), which consists of sets (cid:0) A ∩ A ( τ ) (cid:1) ∪ (cid:0) A ∩ A c ( τ ) (cid:1) , ∀ A , A ∈ F W t τ . Since F W t τ is countably generated (see e.g. Lemma 1.3.3 of [60]), F ( τ ) is also countably generated.Let P ∈ P (Ω). According to Theorem 1.1.8 of [60], there is a family (cid:8) P tτ,ω (cid:9) ω ∈ Ω of probabilities in P (Ω), calledthe regular conditional probability distribution (r.c.p.d.) of P with respect to F ( τ ), such that(R1) for any A ∈ B (Ω), the mapping ω P tτ,ω (cid:0) A (cid:1) is F ( τ ) − measurable;(R2) for any [ −∞ , ∞ ] − valued, B P (Ω) − measurable random variable ξ , it holds for all ω ∈ Ω except on a N ξ ∈ N P (cid:0) F ( τ ) (cid:1) that ξ is B P tτ,ω (Ω) − measurable and E P tτ,ω (cid:2) ξ (cid:3) = E P (cid:2) ξ (cid:12)(cid:12) F ( τ ) (cid:3) ( ω );(R3) for some N ∈ N P (cid:0) F ( τ ) (cid:1) , P tτ,ω (cid:0) A (cid:1) = { ω ∈ A } , ∀ (cid:0) ω, A (cid:1) ∈ N c ×F ( τ ).ptimal Stopping with Expectation Constraints 16For any ω ∈ Ω, W tτ,ω := (cid:8) ω ′ ∈ Ω : W tr ( ω ′ ) = W tr ( ω ) , ∀ r ∈ [0 , τ ( ω )] (cid:9) is an F W t τ − measurable set including ω . Weknow from Galmarino’s test that τ ( W tτ,ω ) = τ ( ω ) , ∀ ω ∈ Ω , (5.1)and (R3) shows that P tτ,ω (cid:0) W tτ,ω (cid:1) = (cid:8) ω ∈ W tτ,ω (cid:9) = 1 , ∀ ω ∈ N c . (5.2)In term of the r.c.p.d. (cid:8) P tτ,ω (cid:9) ω ∈ Ω , we have the following flow property for SDE (3.3): Simply speaking, if process X satisfies the SDE under P , then for P − a.s. ω ∈ Ω, it satisfies the SDE with initial condition (cid:0) t + τ ( ω ) , X ( t + τ ) ∧· ( ω ) (cid:1) under P tτ,ω . Proposition 5.1. Given ( t, x ) ∈ [0 , ∞ ) × Ω X , let τ be a (0 , ∞ ) − valued F W t − stopping time. If P ∈ P (Ω) satisfiesDefinition 3.1 ( i ) and ( ii ) , there is a P − null set N such that for any ω ∈ N c , ( ) W t ω is a standard Brownian motion with respect to filtration F t ω under P tτ,ω , where t ω := t + τ ( ω ) ; ( ) P tτ,ω (cid:8) X s = X ωs , ∀ s ∈ [0 , ∞ ) (cid:9) = 1 , where (cid:8) X ωs (cid:9) s ∈ [0 , ∞ ) uniquely solves the following SDE on (cid:0) Ω , B (cid:0) Ω (cid:1) , P tτ,ω (cid:1) : X s = X ( t ω , ω ) + Z st ω b (cid:0) r, X r ∧· (cid:1) dr + Z st ω σ (cid:0) r, X r ∧· (cid:1) dW r , s ∈ [ t ω , ∞ ) (5.3) with initial condition X s = X s ( ω ) , ∀ s ∈ [0 , t ω ] (cid:0) In particular, (cid:8) X ωs (cid:9) s ∈ [0 , ∞ ) is an F W tω ,P tτ,ω − adapted process withall continuous paths satisfying P tτ,ω (cid:8) X ωt ω + s = X ( t ω , ω )+ R s b ( t ω + r, X ω ( t ω + r ) ∧· ) dr + R s σ ( t ω + r, X ω ( t ω + r ) ∧· ) dW t ω r , ∀ s ∈ [0 , ∞ ) (cid:9) = 1 (cid:1) . Proof of Proposition 5.1: 1a) Let θ = (cid:0) ( s, r ) , O , { ( s i , O i , O ′ i ) } ni =1 (cid:1) ∈ Q ,< × O ( R d ) × b O ( R d ). Set A s (cid:0) { ( s i , O i , O ′ i ) } ni =1 (cid:1) := n ∩ i =1 (cid:16)(cid:2)(cid:0) W tτ + s i ∧ s − W tτ (cid:1) − ( O i ) ∩{ T ∈ [ t + τ , t + τ + s i ∧ s ] } (cid:3) ∪ (cid:2)(cid:0) W tτ + s i ∧ s − W tτ (cid:1) − ( O ′ i ) ∩{ T ∈ [ t + τ , t + τ + s i ∧ s ] c } (cid:3)(cid:17) ∈ F tτ + s .Since W tτ + r − W tτ + s ∼ Normal (0 , r − s ) is independent of F tτ + s under P and since F ( τ ) ⊂ F tτ ⊂ F tτ + s , we can deducefrom (R2) that for any ω ∈ Ω except on a N θ ∈ N P (cid:0) F ( τ ) (cid:1) E P tτ,ω h ( W tτ + r − W tτ + s ) − ( O ) A s ( { ( s i , O i , O ′ i ) } ni =1 ) i = E P h ( W tτ + r − W tτ + s ) − ( O ) A s ( { ( s i , O i , O ′ i ) } ni =1 ) (cid:12)(cid:12)(cid:12) F ( τ ) i ( ω )= E P h ( W tτ + r − W tτ + s ) − ( O ) i E P h A s ( { ( s i , O i , O ′ i ) } ni =1 ) (cid:12)(cid:12)(cid:12) F ( τ ) i ( ω ) = φ ( r − s, O ) E P tτ,ω h A s ( { ( s i , O i , O ′ i ) } ni =1 ) i . (5.4)As Q ,< × O ( R d ) × b O ( R d ) is a countable set, N := ∪ (cid:8) N θ : θ ∈ Q ,< × O ( R d ) × b O ( R d ) (cid:9) still belongs to N P (cid:0) F ( τ ) (cid:1) .Fix ω ∈ N c ∩N c . Given (cid:0) ( s, r ) , O , { ( s i , O i , O ′ i ) } ni =1 (cid:1) ∈ Q ,< × O ( R d ) × b O ( R d ), since (5.1) shows W tτ,ω ∩ (cid:0) W t ω r − W t ω s (cid:1) − ( O ) ∩ (cid:26) n ∩ i =1 (cid:16)(cid:2) ( W t ω s i ∧ s ) − ( O i ) ∩ (cid:8) T ∈ [ t ω , t ω + s i ∧ s ] (cid:9)(cid:3) ∪ (cid:2) ( W t ω s i ∧ s ) − ( O ′ i ) ∩ (cid:8) T ∈ [ t ω , t ω + s i ∧ s ] c (cid:9)(cid:3)(cid:17)(cid:27) = W tτ,ω ∩ (cid:8) ω ′ ∈ Ω : W ( t + τ ( ω ′ )+ r, ω ′ ) − W ( t + τ ( ω ′ )+ s, ω ′ ) ∈ O (cid:9) ∩ (cid:18) n ∩ i =1 (cid:16)(cid:8) ω ′ ∈ Ω : (cid:0) W ( t + τ ( ω ′ )+ s i ∧ s, ω ′ ) − W ( t + τ ( ω ′ ) , ω ′ ) (cid:1) ∈ O i , T ( ω ′ ) ∈ [ t + τ ( ω ′ ) , t + τ ( ω ′ )+ s i ∧ s ] (cid:9) ∪ (cid:8) ω ′ ∈ Ω : (cid:0) W ( t + τ ( ω ′ )+ s i ∧ s, ω ′ ) − W ( t + τ ( ω ′ ) , ω ′ ) (cid:1) ∈ O ′ i , T ( ω ′ ) ∈ [ t + τ ( ω ′ ) , t + τ ( ω ′ )+ s i ∧ s ] c (cid:9)(cid:17)(cid:19) = W tτ,ω ∩ (cid:0) W tτ + r − W tτ + s (cid:1) − ( O ) ∩ A s (cid:0) { ( s i , O i , O ′ i ) } ni =1 (cid:1) , (5.2) and (5.4) imply that P tτ,ω (cid:26)(cid:0) W t ω r − W t ω s (cid:1) − ( O ) ∩ (cid:18) n ∩ i =1 (cid:16)(cid:2) ( W t ω s i ∧ s ) − ( O i ) ∩ (cid:8) T ∈ [ t ω , t ω + s i ∧ s ] (cid:9)(cid:3) ∪ (cid:2) ( W t ω s i ∧ s ) − ( O ′ i ) ∩ (cid:8) T ∈ [ t ω , t ω + s i ∧ s ] c (cid:9)(cid:3)(cid:17)(cid:19)(cid:27) = P tτ,ω n(cid:0) W tτ + r − W tτ + s (cid:1) − ( O ) ∩ A s (cid:0) { ( s i , O i , O ′ i ) } ni =1 (cid:1)o = φ ( r − s, O ) P tτ,ω n A s (cid:0) { ( s i , O i , O ′ i ) } ni =1 (cid:1)o = φ ( r − s, O ) P tτ,ω (cid:26) n ∩ i =1 (cid:16)(cid:2) ( W t ω s i ∧ s ) − ( O i ) ∩ (cid:8) T ∈ [ t ω , t ω + s i ∧ s ] (cid:9)(cid:3) ∪ (cid:2) ( W t ω s i ∧ s ) − ( O ′ i ) ∩ (cid:8) T ∈ [ t ω , t ω + s i ∧ s ] c (cid:9)(cid:3)(cid:17)(cid:27) . . Dynamic Programming Principle P tτ,ω (cid:8)(cid:0) W t ω r − W t ω s (cid:1) − ( E ) ∩ A (cid:9) = φ ( r − s, E ) P tτ,ω (cid:0) A (cid:1) , ∀ ( s, r ) ∈ Q ,< , E ∈ B ( R d ) , A ∈ F t ω s . (5.5) Let 0 ≤ s < r < ∞ , A ∈ F t ω s and let O be an open subset of R d . For any n ∈ N with n > n := − log ( r − s ), we set( s n , r n ) := (cid:16) ⌈ n s ⌉ n , ⌈ n r ⌉ n (cid:17) ∈ Q ,< . By the continuity of W t ω , (cid:8) W t ω r − W t ω s ∈ O (cid:9) ⊂ ∞ ∪ n = n +1 ∩ k ≥ n (cid:8) W t ω r k − W t ω s k ∈ O (cid:9) . So(5.5) implies that P tτ,ω n(cid:0) W t ω r − W t ω s (cid:1) − ( O ) ∩ A o ≤ P tτ,ω n ∞ ∪ n = n +1 ∩ k ≥ n (cid:16)(cid:0) W t ω r k − W t ω s k (cid:1) − ( O ) ∩ A (cid:17)o = lim n →∞ ↑ P tτ,ω n ∩ k ≥ n (cid:16)(cid:0) W t ω r k − W t ω s k (cid:1) − ( O ) ∩ A (cid:17)o ≤ lim n →∞ P tτ,ω n(cid:0) W t ω r n − W t ω s n (cid:1) − ( O ) ∩ A o = P tτ,ω (cid:0) A (cid:1) lim n →∞ φ ( r n − s n , O ) = P tτ,ω (cid:0) A (cid:1) φ ( r − s, O ) . (5.6)On the other hand, we let ℓ ∈ N and define a closed set E ℓ = { x ∈ R d : dist( x , O c ) ≥ /ℓ } . Since the continuity of W t ω also shows that (cid:0) W t ω r − W t ω s (cid:1) − ( E ℓ ) ⊃ ∞ ∩ n = n +1 ∪ k ≥ n (cid:0) W t ω r k − W t ω s k (cid:1) − ( E ℓ ), (5.5) again renders that P tτ,ω n(cid:0) W t ω r − W t ω s (cid:1) − ( E ℓ ) ∩ A o ≥ P tτ,ω n ∞ ∩ n = n +1 ∪ k ≥ n (cid:16)(cid:0) W t ω r k − W t ω s k (cid:1) − ( E ℓ ) ∩ A (cid:17)o = lim n →∞ ↓ P tτ,ω n ∪ k ≥ n (cid:16)(cid:0) W t ω r k − W t ω s k (cid:1) − ( E ℓ ) ∩ A (cid:17)o ≥ lim n →∞ P tτ,ω n(cid:0) W t ω r n − W t ω s n (cid:1) − ( E ℓ ) ∩ A o = P tτ,ω (cid:0) A (cid:1) lim n →∞ φ ( r n − s n , E ℓ ) = P tτ,ω (cid:0) A (cid:1) φ ( r − s, E ℓ ) . As O = ∪ ℓ ∈ N E ℓ , it follows that P tτ,ω n(cid:0) W t ω r − W t ω s (cid:1) − ( O ) ∩ A o = lim ℓ →∞ ↑ P tτ,ω n(cid:0) W t ω r − W t ω s (cid:1) − ( E ℓ ) ∩ A o ≥ P tτ,ω (cid:0) A (cid:1) lim ℓ →∞ ↑ φ ( r − s, E ℓ ) = P tτ,ω (cid:0) A (cid:1) φ ( r − s, O ), which together with (5.6) means that the Lambda system n E ∈ B ( R d ) : P tτ,ω (cid:8)(cid:0) W t ω r − W t ω s (cid:1) − ( E ) ∩ A (cid:9) = P tτ,ω (cid:0) A (cid:1) φ ( r − s, E ) o contains all open sets and is thus equal to B ( R d ). To wit, P tτ,ω (cid:8)(cid:0) W t ω r − W t ω s (cid:1) − ( E ) ∩ A (cid:9) = P tτ,ω (cid:0) A (cid:1) φ ( r − s, E ) , ∀ ≤ s < r < ∞ , E ∈ B ( R d ) , A ∈ F t ω s . Thus for any ω ∈ N c ∩N c , W t ω is a standard Brownian motion with respect to filtration F t ω under P tτ,ω . Set N X := (cid:8) ω ∈ Ω : X s ( ω ) = x ( s ) for some s ∈ [0 , t ] (cid:9) ∈ N P (cid:0) F Xt (cid:1) and N X := (cid:8) ω ∈ Ω : X ts ( ω ) = X t, x t + s ( ω ) for some s ∈ [0 , ∞ ) (cid:9) ∈ N P (cid:0) F Ξ t ∞ (cid:1) . As (cid:8) X t, x t + s (cid:9) s ∈ [0 , ∞ ) is an F W t ,P − adapted continuous process, there is an F W t − predictableprocess (cid:8) K ts (cid:9) s ∈ [0 , ∞ ) such that N K := (cid:8) ω ∈ Ω : K ts ( ω ) = X t, x t + s ( ω ) for some s ∈ [0 , ∞ ) (cid:9) ∈ N P (cid:0) F W t ∞ (cid:1) (see e.g. Lemma2.4 of [59]). Since K ω := ∩ r ∈ Q ∩ [0 , ∞ ) (cid:8) ω ′ ∈ Ω : K tτ ∧ r ( ω ′ ) = X tτ ∧ r ( ω ) (cid:9) is F W t τ − measurable set including ω , (R3) showsthat P tτ,ω ( K ω ) = 1, ∀ ω ∈ N c (cid:0) cf. (5.2) (cid:1) . We see from (5.1) that W tτ,ω ∩ (cid:0) N X ∪N K (cid:1) c ∩ K ω = W tτ,ω ∩ (cid:0) N X ∪N K (cid:1) c ∩ (cid:8) ω ′ ∈ Ω : K t ( τ ( ω ) ∧ r, ω ′ ) = X t ( τ ( ω ) ∧ r, ω ) , ∀ r ∈ Q ∩ [0 , ∞ ) (cid:9) ⊂ W tτ,ω ∩ (cid:0) N X ∪N K (cid:1) c ∩ (cid:8) ω ′ ∈ Ω : X t ( τ ( ω ) ∧ r, ω ′ ) = X t ( τ ( ω ) ∧ r, ω ) , ∀ r ∈ Q ∩ [0 , ∞ ) (cid:9) = W tτ,ω ∩ (cid:0) N X ∪N K (cid:1) c ∩ (cid:8) ω ′ ∈ Ω : X t, x r ( ω ′ ) = X r ( ω ′ ) = X r ( ω ) , ∀ r ∈ (cid:2) t, t ω (cid:3)(cid:9) . (5.7)Define M s := R s σ ( t + r, X t, x ( t + r ) ∧· ) dW tr , s ∈ [0 , ∞ ). There is a sequence of R l × d − valued, F W t ,P − simple processes n H ns = P ℓ n i =1 ξ ni { s ∈ ( s ni ,s ni +1 ] } , s ∈ [0 , ∞ ) o n ∈ N (cid:0) with 0 = s n < · · · < s nℓ n +1 < ∞ and ξ ni ∈ F W t ,Ps ni , i = 1 , · · · , ℓ n (cid:1) such that(3.7) holds. For any n ∈ N and i = 1 , · · · , ℓ n , one can find a R l × d − valued b ξ ni ∈ F W t s ni such that N n,iξ := (cid:8) ω ∈ Ω : b ξ ni ( ω ) = ξ ni ( ω ) (cid:9) ∈ N P (cid:0) F W t ∞ (cid:1) . Set N ξ := ∪ n ∈ N ℓ n ∪ i =1 N n,iξ ∈ N P (cid:0) F W t ∞ (cid:1) . We know from (R2) that for some N ∈ N P (cid:0) F ( τ ) (cid:1) P tτ,ω (cid:0) N X ∪N X ∪N K ∪N ξ (cid:1) = 0 , ∀ ω ∈ N c . (5.8)Since β n := R ∞ (cid:12)(cid:12) H nr − σ (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1)(cid:12)(cid:12) dr + sup s ∈ [0 , ∞ ) (cid:12)(cid:12) M ns − M s (cid:12)(cid:12) ∈ F W t ,P ∞ converges to 0 in probability P by (3.7),Lemma A.3 shows that for some subsequence { n j } j ∈ N of N and some N β ∈ N P (cid:0) F ( τ ) (cid:1) P tτ,ω − lim j →∞ β n j = 0 , ∀ ω ∈ N cβ . (5.9)ptimal Stopping with Expectation Constraints 18 Define N ♯ := (cid:16) ∪ i =0 N i (cid:17) ∪N β ∪N X and fix ω ∈ N c♯ .For any r ∈ [0 , ∞ ), applying Lemma A.4 with s = τ ( ω ) yields that K tτ ( ω )+ r ∈ F W t τ ( ω )+ r coincides with some R l − valued X ωr ∈ F W tω r on W tτ,ω . So it holds for any ( r, ω ′ ) ∈ [0 , ∞ ) × (cid:16) W tτ,ω ∩N cK (cid:17) that X ωr ( ω ′ ) = K t (cid:0) τ ( ω )+ r, ω ′ (cid:1) = X t, x (cid:0) t + τ ( ω )+ r, ω ′ (cid:1) . (5.10)Since (5.1) implies that X ω ( ω ′ ) = K t (cid:0) τ ( ω ) , ω ′ (cid:1) = K t (cid:0) τ ( ω ′ ) , ω ′ (cid:1) = X t (cid:0) τ ( ω ) , ω (cid:1) = X ( t ω , ω ) , ∀ ∀ ω ′ ∈ W tτ,ω ∩ K ω , (5.11)we see from (5.10), (5.2) and (5.8) that X ω ( s, ω ′ ) := { s ∈ [0 ,t ω ] } X s ( ω )+ { s ∈ ( t ω , ∞ ) } (cid:2) W tτ,ω ∩ K ω ∩N cK X ω (cid:0) s − t ω , ω ′ (cid:1) + W tτ,ωc ∪ K cω ∪N K X ( t ω , ω ) (cid:3) , ∀ ( s, ω ′ ) ∈ [0 , ∞ ) × Ωis a process with all continuous paths such that (cid:8) X ω ( t ω + s, ω ′ ) (cid:9) ( s,ω ′ ) ∈ [0 , ∞ ) × Ω is F W tω ,P tτ,ω − adapted.Given ω ′ ∈ W tτ,ω ∩ K ω ∩ (cid:0) N X ∪N X ∪N K (cid:1) c , as ω ∈ (cid:0) N X (cid:1) c , (5.7) renders that for any r ∈ [0 , t ω ], X r ( ω ) = { r ∈ [0 ,t ) } x ( r )+ { r ∈ [ t,t ω ] } X t, x r ( ω ′ ) = X t, x r ( ω ′ ). For any s ∈ [0 , ∞ ), it follows from (5.7) and (5.10) that X ωs ( ω ′ ) = { s ∈ [0 ,t ) } x ( s )+ { s ∈ [ t,t ω ] } X s ( ω ′ )+ { s ∈ ( t ω , ∞ ) } X t, x s ( ω ′ ) = X s ( ω ′ ) , and (5.12) X ω (cid:0) ( t ω + s ) ∧· , ω ′ (cid:1) = (cid:16) sup r ∈ [0 ,t ω ] X t, x r ( ω ′ ) (cid:17) ∨ (cid:16) sup r ∈ ( t ω ,t ω + s ] X t, x r ( ω ′ ) (cid:17) = X t, x (cid:0) ( t ω + s ) ∧· , ω ′ (cid:1) . (5.13)Let n ∈ N and set s ni ( ω ) := τ ( ω ) ∨ s ni − τ ( ω ), i = 1 , · · · , ℓ n + 1 (cid:0) in particular, s n ( ω ) = 0 (cid:1) . For any i = 1 , · · · , ℓ n , as b ξ ni ∈ F W t s ni ⊂ F W t τ ( ω )+ s ni ( ω ) , Lemma A.4 shows that b ξ ni coincides with some R l × d − valued b ξ ωn,i ∈ F W tω s ni ( ω ) on W tτ,ω . Forthe R l × d − valued, F W tω − simple process H ω,nr := P ℓ n i =1 b ξ ωn,i { r ∈ ( s ni ( ω ) , s ni +1 ( ω )] } , r ∈ [0 , ∞ ), set M ω,ns := R s H ω,nr dW t ω r = P ℓ n i =1 b ξ ωn,i (cid:16) W t ω s ∧ s ni +1 ( ω ) − W t ω s ∧ s ni ( ω ) (cid:17) , ∀ s ∈ [0 , ∞ ).Next, let ω ′ ∈ Ω ω := W tτ,ω ∩ K ω ∩ (cid:0) N X ∪N X ∪N K ∪N ξ (cid:1) c . Since H ω,nr ( ω ′ ) = ℓ n X i =1 b ξ ni ( ω ′ ) { τ ( ω )+ r ∈ ( τ ( ω ) ∨ s ni ,τ ( ω ) ∨ s ni +1 ] } = ℓ n X i =1 ξ ni ( ω ′ ) { τ ( ω )+ r ∈ ( s ni ,s ni +1 ] } = H n (cid:0) τ ( ω )+ r, ω ′ (cid:1) , and M ω,ns ( ω ′ ) = ℓ n X i =1 b ξ ni ( ω ′ ) (cid:16) W (cid:0) t ω + s ∧ s ni +1 ( ω ) , ω ′ (cid:1) − W (cid:0) t ω + s ∧ s ni ( ω ) , ω ′ (cid:1)(cid:17) = ℓ n X i =1 ξ ni ( ω ′ ) (cid:16) W t (cid:0) ( τ ( ω )+ s ) ∧ ( s ni +1 ∨ τ ( ω )) , ω ′ (cid:1) − W t (cid:0) ( τ ( ω )+ s ) ∧ ( s ni ∨ τ ( ω )) , ω ′ (cid:1)(cid:17) = M n (cid:0) τ ( ω )+ s, ω ′ (cid:1) − M n (cid:0) τ ( ω ) , ω ′ (cid:1) , ∀ s ∈ [0 , ∞ ) , we see from (5.13) that Z ∞ (cid:12)(cid:12)(cid:12) H ω,nr ( ω ′ ) − σ (cid:0) t ω + r, X ω (cid:0) ( t ω + r ) ∧· , ω ′ (cid:1)(cid:1)(cid:12)(cid:12)(cid:12) dr + sup s ∈ [0 , ∞ ) (cid:12)(cid:12)(cid:12) M ω,ns ( ω ′ ) − M (cid:0) τ ( ω )+ s, ω ′ (cid:1) + M (cid:0) τ ( ω ) , ω ′ (cid:1)(cid:12)(cid:12)(cid:12) ≤ Z ∞ τ ( ω ) (cid:12)(cid:12)(cid:12) H n ( r ′ , ω ′ ) − σ (cid:0) t + r ′ , X t, x (cid:0) ( t + r ′ ) ∧· , ω ′ (cid:1)(cid:1)(cid:12)(cid:12)(cid:12) dr ′ +2 sup s ∈ [0 , ∞ ) (cid:12)(cid:12) M ns ( ω ′ ) − M s ( ω ′ ) (cid:12)(cid:12) ≤ β n ( ω ′ ) . Then (5.2), (5.8) and (5.9) imply that P tτ,ω − lim j →∞ Z ∞ (cid:12)(cid:12)(cid:12) H ω,n j r − σ (cid:0) t ω + r, X ω ( t ω + r ) ∧· (cid:1)(cid:12)(cid:12)(cid:12) dr = 0 and P tτ,ω − lim j →∞ sup s ∈ [0 , ∞ ] (cid:12)(cid:12)(cid:12) M ω,n j s − M τ ( ω )+ s + M τ ( ω ) (cid:12)(cid:12)(cid:12) = 0 . (5.14)As W t ω is a Brownian motion with respect to the filtration F t ω under P tτ,ω by Part (1), applying Proposition 3.2.26 of[34] and using the first limit of (5.14) yield that 0 = P tτ,ω − lim j →∞ sup s ∈ [0 , ∞ ] (cid:12)(cid:12)(cid:12) R s H ω,n j r dW t ω r − R s σ (cid:0) t ω + r, X ω ( t ω + r ) ∧· (cid:1) dW t ω r (cid:12)(cid:12)(cid:12) ,which together with the second limit of (5.14) indicates that except on a P tτ,ω − null set N ω Z s σ (cid:0) t ω + r, X ω ( t ω + r ) ∧· (cid:1) dW t ω r = M τ ( ω )+ s − M τ ( ω ) = Z τ ( ω )+ sτ ( ω ) σ ( t + r, X t, x ( t + r ) ∧· ) dW tr , ∀ s ∈ [0 , ∞ ) . (5.15) . Dynamic Programming Principle ω ′ ∈ Ω ω ∩N cω and let s ∈ [0 , ∞ ). We can deduce from (5.15), (5.10), (5.11) and (5.13) that for any s ∈ [0 , ∞ ) (cid:16) Z s σ (cid:0) t ω + r, X ω ( t ω + r ) ∧· (cid:1) dW t ω r (cid:17) ( ω ′ ) = X t, x (cid:0) t + τ ( ω )+ s, ω ′ (cid:1) − X t, x (cid:0) t + τ ( ω ) , ω ′ (cid:1) − Z τ ( ω )+ sτ ( ω ) b (cid:0) t + r, X t, x ( t + r ) ∧· ( ω ′ ) (cid:1) dr = X ωs ( ω ′ ) − K t (cid:0) τ ( ω ) , ω ′ (cid:1) − Z s b (cid:0) t + τ ( ω )+ r ′ , X t, x (cid:0) ( t + τ ( ω )+ r ′ ) ∧· , ω ′ (cid:1)(cid:1) dr ′ = X ω ( t ω + s, ω ′ ) − X t (cid:0) τ ( ω ) , ω (cid:1) − Z s b (cid:0) t ω + r, X ω (cid:0) ( t ω + r ) ∧· , ω ′ (cid:1)(cid:1) dr. As P tτ,ω (cid:0) Ω ω ∩N cω (cid:1) = 1 by (5.2) and (5.8), it holds equivalently that P tτ,ω − a.s. X ωs = X ( t ω , ω )+ Z st ω b (cid:0) r, X ωr ∧· (cid:1) dr + Z st ω σ (cid:0) r, X ωr ∧· (cid:1) dW r , ∀ s ∈ [ t ω , ∞ ) . Namely, (cid:8) X ωs (cid:9) s ∈ [0 , ∞ ) solves the SDE (5.3). We also see from (5.12) that P tτ,ω (cid:8) X ωs = X s , ∀ s ∈ [0 , ∞ ) (cid:9) = 1. Hence,for any ω ∈ N c♯ , P tτ,ω satisfies the second statement of this Proposition. (cid:3) Let t ∈ [0 , ∞ ), P ∈ P (Ω) and let τ be a [0 , ∞ ) − valued F W t − stopping time. We define Y iP ( τ ) := E P (cid:20) Z TT ∧ ( t + τ ) g i ( r, X r ∧· ) dr (cid:12)(cid:12)(cid:12) F ( τ ) (cid:21) , Z iP ( τ ) := E P (cid:20) Z TT ∧ ( t + τ ) h i ( r, X r ∧· ) dr (cid:12)(cid:12)(cid:12) F ( τ ) (cid:21) , ∀ i ∈ N , and set (cid:0) Y P ( τ ) , Z P ( τ ) (cid:1) := (cid:16)(cid:8) Y iP ( τ ) (cid:9) i ∈ N , (cid:8) Z iP ( τ ) (cid:9) i ∈ N (cid:17) . Corollary 5.1. Given ( t, x ) ∈ [0 , ∞ ) × Ω X , let τ be a (0 , ∞ ) − valued F W t − stopping time. If P ∈ P (Ω) satisfiesDefinition 3.1 ( i ) and ( ii ) , there is a P − null set N such that P tτ,ω ∈ P t + τ ( ω ) ,X ( t + τ ) ∧· ( ω ) (cid:0) Y P ( τ )( ω ) , Z P ( τ )( ω ) (cid:1) , ∀ ω ∈ A c ( τ ) ∩N c . Proof of Corollary 5.1: We have seen from the proof of Proposition 5.1: there is a P − null set N ♯ such that forany ω ∈ N c♯ , P tτ,ω satisfies Definition 3.1 (i), (ii) with ( t, x ) = (cid:0) t + τ ( ω ) , X ( t + τ ) ∧· ( ω ) (cid:1) .By (R2), there is a N T ,g,h ∈ N P (cid:0) F ( τ ) (cid:1) such that for any ω ∈ N cT ,g,h P tτ,ω { T ≥ t + τ } = E P tτ,ω (cid:2) { T ≥ t + τ } (cid:3) = E P (cid:2) { T ≥ t + τ } |F ( τ ) (cid:3) ( ω ) = E P (cid:2) A c ( τ ) |F ( τ ) (cid:3) ( ω ) = { ω ∈ A c ( τ ) } and E P tτ,ω h Z TT ∧ ( t + τ ) ( g i , h i )( r, X r ∧· ) dr i = E P h Z TT ∧ ( t + τ ) ( g i , h i )( r, X r ∧· ) dr (cid:12)(cid:12) F ( τ ) i ( ω ) = (cid:0) Y iP ( τ )( ω ) , Z iP ( τ )( ω ) (cid:1) , ∀ i ∈ N . For any ω ∈ A c ( τ ) ∩ (cid:16) N ∪ N T ,g,h (cid:17) c , (5.2) and (5.1) imply that 1 = P tτ,ω { T ≥ t + τ } = P tτ,ω (cid:0) W tτ,ω ∩ { T ≥ t + τ } (cid:1) = P tτ,ω (cid:0) W tτ,ω ∩{ T ≥ t + τ ( ω ) } (cid:1) = P tτ,ω { T ≥ t + τ ( ω ) } and (cid:0) Y iP ( τ )( ω ) , Z iP ( τ )( ω ) (cid:1) = E P tτ,ω h Z TT ∧ ( t + τ ) ( g i , h i )( r, X r ∧· ) dr i = E P tτ,ω (cid:20) W tτ,ω Z TT ∧ ( t + τ ( ω )) ( g i , h i )( r, X r ∧· ) dr (cid:21) = E P tτ,ω (cid:20) Z Tt + τ ( ω ) ( g i , h i )( r, X r ∧· ) dr (cid:21) , ∀ i ∈ N . (5.16)Hence, P tτ,ω ∈ P t + τ ( ω ) ,X ( t + τ ) ∧· ( ω ) (cid:0) Y P ( τ )( ω ) , Z P ( τ )( ω ) (cid:1) for any ω ∈ A c ( τ ) ∩ (cid:16) N ♯ ∪N T ,g,h (cid:17) c . (cid:3) Corollary 5.1 implies that the probability class (cid:8) P t, x ( y, z ) : ( t, x , y, z ) ∈ D (cid:9) is stable under conditioning . It willplay an important role in deriving the sub-solution part of the DPP for V .Now, we are ready to present a dynamic programming principle in the weak formulation for the value of theoptimal stopping with expectation constraints, in which (cid:0) Y P ( τ ) , Z P ( τ ) (cid:1) acts as additional states at the intermediatehorizon τ .ptimal Stopping with Expectation Constraints 20 Theorem 5.1. Let ( t, x ) ∈ [0 , ∞ ) × Ω X and let ( y, z ) ∈ R such that P t, x ( y, z ) = ∅ . Let (cid:8) τ P (cid:9) P ∈P t, x ( y,z ) be a family of (0 , ∞ ) − valued F W t − stopping times. Then V ( t, x , y, z ) = sup P ∈P t, x ( y,z ) E P (cid:20) { T We set ζ := { T< ∞} π (cid:0) T , X T ∧· (cid:1) . (I) (sub-solution side) Fix P ∈ P t, x ( y, z ) and simply denote τ P by τ . According to Corollary 5.1, there is a P − nullset N ∗ such that P tτ,ω ∈ P t + τ ( ω ) ,X ( t + τ ) ∧· ( ω ) (cid:0) Y P ( τ )( ω ) , Z P ( τ )( ω ) (cid:1) , ∀ ω ∈ A c ( τ ) ∩N c ∗ . (5.17)And (R2) shows that for some N f,π ∈ N P (cid:0) F ( τ ) (cid:1) E P tτ,ω (cid:20) Z TT ∧ ( t + τ ) f ( r, X r ∧· ) dr + ζ (cid:21) = E P (cid:20) Z TT ∧ ( t + τ ) f ( r, X r ∧· ) dr + ζ (cid:12)(cid:12)(cid:12) F ( τ ) (cid:21) ( ω ) , ∀ ω ∈ N cf,π . (5.18)Fix ω ∈ A c ( τ ) ∩ (cid:0) N ∗ ∪ N f,π (cid:1) c . Since E P tτ,ω (cid:2) R TT ∧ ( t + τ ) f ( r, X r ∧· ) dr + ζ (cid:3) = E P tτ,ω (cid:2) R Tt + τ ( ω ) f ( r, X r ∧· ) dr + ζ (cid:3) by ananalogy to (5.16), we see from (5.18) and (5.17) that E P (cid:20) Z TT ∧ ( t + τ ) f ( r, X r ∧· ) dr + ζ (cid:12)(cid:12)(cid:12) F ( τ ) (cid:21) ( ω ) = E P tτ,ω (cid:20) Z Tt + τ ( ω ) f ( r, X r ∧· ) dr + ζ (cid:21) ≤ V (cid:16) t + τ ( ω ) , X ( t + τ ) ∧· ( ω ) , Y P ( τ )( ω ) , Z P ( τ )( ω ) (cid:17) . It follows that E P h { T ≥ t + τ } V (cid:0) t + τ , X ( t + τ ) ∧· , Y P ( τ ) , Z P ( τ )( ω ) (cid:1)i = E P h A c ( τ ) V (cid:0) t + τ , X ( t + τ ) ∧· , Y P ( τ ) , Z P ( τ )( ω ) (cid:1)i ≥ E P (cid:20) A c ( τ ) E P h Z TT ∧ ( t + τ ) f ( r, X r ∧· ) dr + ζ (cid:12)(cid:12)(cid:12) F ( τ ) i(cid:21) = E P (cid:20) E P h A c ( τ ) (cid:16) Z TT ∧ ( t + τ ) f ( r, X r ∧· ) dr + ζ (cid:17)(cid:12)(cid:12)(cid:12) F ( τ ) i(cid:21) = E P (cid:20) A c ( τ ) (cid:16) Z TT ∧ ( t + τ ) f ( r, X r ∧· ) dr + ζ (cid:17)(cid:21) = E P (cid:20) { T ≥ t + τ } (cid:16) Z Tt + τ f ( r, X r ∧· ) dr + ζ (cid:17)(cid:21) and thus E P (cid:20) Z Tt f ( r, X r ∧· ) dr + ζ (cid:21) = E P (cid:20) { T As in Part (2a) of the Proof of Proposition 5.1, we still define Set N X := (cid:8) ω ∈ Ω : X s ( ω ) = x ( s ) for some s ∈ [0 , t ] (cid:9) ∈ N P (cid:0) F Xt (cid:1) and N X := (cid:8) ω ∈ Ω : X ts ( ω ) = X t, x t + s ( ω ) for some s ∈ [0 , ∞ ) (cid:9) ∈ N P (cid:0) F Ξ t ∞ (cid:1) . There is an F W t − predictableprocess (cid:8) K ts (cid:9) s ∈ [0 , ∞ ) such that N K := (cid:8) ω ∈ Ω : K ts ( ω ) = X t, x t + s ( ω ) for some s ∈ [0 , ∞ ) (cid:9) ∈ N P (cid:0) F W t ∞ (cid:1) .Define path random variables (cid:0) W t,τ , X t,τ (cid:1) : Ω Ω × Ω X by (cid:0) W t,τr , X t,τr (cid:1) ( ω ) := (cid:0) W (cid:0) ( r ∨ t ) ∧ ( t + τ ( ω )) , ω (cid:1) − W ( t, ω ) , X (cid:0) r ∧ ( t + τ ( ω )) , ω (cid:1)(cid:1) , ∀ ( r, ω ) ∈ [0 , ∞ ) × Ω . Clearly, W t,τ is F W t τ (cid:14) B (Ω ) − measurable. As X t · coincides with K t · on (cid:0) N X ∪N K (cid:1) c , one can deduce that X t,τ is σ (cid:16) F W t τ ∪F Xt ∪ N P (cid:0) F Ξ t ∞ (cid:1)(cid:17). B (Ω X ) − measurable. For any i ∈ N , since (cid:0) Y iP ( τ ) , Z iP ( τ ) (cid:1) is F ( τ ) − measurable, there are( −∞ , ∞ ] × [ −∞ , ∞ ] − valued F W t τ − measurable random variables (cid:0) Y i , Z i (cid:1) , (cid:0) Y i , Z i (cid:1) such that (cid:0) Y iP ( τ ) , Z iP ( τ ) (cid:1) ( ω ) = (cid:0) Y i ( ω ) , Z i ( ω ) (cid:1) { ω ∈ A ( τ ) } + (cid:0) Y i ( ω ) , Z i ( ω ) (cid:1) { ω ∈ A c ( τ ) } , ∀ ω ∈ Ω . (5.22)We set (cid:0) Y , Z (cid:1) := (cid:0)(cid:8) Y i } i ∈ N , (cid:8) Z i } i ∈ N (cid:1) and (cid:0) Y , Z (cid:1) := (cid:0)(cid:8) Y i } i ∈ N , (cid:8) Z i } i ∈ N (cid:1) .Let ¨Ω := [0 , ∞ ) × Ω × Ω X × [0 , ∞ ]. Putting the above measurability together shows that the mappingΨ τ ( ω ) := (cid:16) t + τ ( ω ) , W t,τ ( ω ) , X t,τ ( ω ) , Y ( ω ) , Z ( ω ) (cid:17) ∈ ¨Ω , ∀ ω ∈ Ωis σ (cid:16) F W t τ ∪F Xt ∪ N P (cid:0) F Ξ t ∞ (cid:1)(cid:17). B (cid:0) ¨Ω (cid:1) − measurable, which induces a probability ¨ P τ := P ◦ Ψ − τ on (cid:0) ¨Ω , B ( ¨Ω) (cid:1) . Then Ψ τ is further σ (cid:16) F W t τ ∪F Xt ∪ N P (cid:0) F W t ∞ ∨F X ∞ (cid:1)(cid:17). σ (cid:16) B ( ¨Ω) ∪ N ¨ P τ (cid:0) B ( ¨Ω) (cid:1)(cid:17) − measurable. Since the universally measurablefunction V is σ (cid:16) B ( ¨Ω) ∪ N ¨ P τ (cid:0) B ( ¨Ω) (cid:1)(cid:17). B ( −∞ , ∞ ] by Theorem 4.1, we see that V (cid:0) Ψ τ ( ω ) (cid:1) = V (cid:16) t + τ ( ω ) , W t,τ ( ω ) , X t,τ ( ω ) , Y ( ω ) , Z ( ω ) (cid:17) ∈ ( −∞ , ∞ ] , ∀ ω ∈ Ω (5.23)is σ (cid:16) F W t τ ∪F Xt ∪ N P (cid:0) F W t ∞ ∨F X ∞ (cid:1)(cid:17). B ( −∞ , ∞ ] − measurable. II.b) Fix ε ∈ (0 , 1) through Part (II.e).The analytically measurable function Q ε is also universally measurable. In particular, it is σ (cid:0) B ( ¨Ω) ∪ N ¨ P τ (cid:0) B ( ¨Ω) (cid:1)(cid:1)(cid:14) B (cid:0) P (Ω) (cid:1) − measurable and Q ωε := Q ε (cid:0) Ψ τ ( ω ) (cid:1) = Q ε (cid:16) t + τ ( ω ) , W t,τ ( ω ) , X t,τ ( ω ) , Y ( ω ) , Z ( ω ) (cid:17) ∈ P t + τ ( ω ) ,W t,τ ( ω ) ,X t,τ ( ω ) (cid:0) Y ( ω ) , Z ( ω ) (cid:1) , ∀ ω ∈ Ω (5.24)is thus σ (cid:16) F W t τ ∪F Xt ∪ N P (cid:0) F W t ∞ ∨F X ∞ (cid:1)(cid:17). B (cid:0) P (Ω) (cid:1) − measurable.Let ω ∈ Ω and set Ω tτ,ω := n ω ′ ∈ Ω : ( W s , X s )( ω ′ ) = (cid:0) W t,τs , X t,τs (cid:1) ( ω ) , ∀ s ∈ (cid:2) , t + τ ( ω ) (cid:3)o . By (5.24), Q ωε (cid:0) Ω tτ,ω (cid:1) = 1 and Q ωε (cid:8) T ≥ t + τ ( ω ) (cid:9) = Q ωε (cid:8) ω ′ ∈ Ω : T ( ω ′ ) ≥ t + τ ( ω ) (cid:9) = 1 . (5.25)Since the set Ξ tτ,ω := (cid:8) ω ′ ∈ Ω : Ξ tr ( ω ′ ) = Ξ tr ( ω ) , ∀ r ∈ (cid:2) , τ ( ω ) (cid:3)(cid:9) satisfiesΩ tτ,ω ⊂ (cid:8) ω ′ ∈ Ω : W r ( ω ′ ) = 0 , ∀ r ∈ [0 , t ]; W r ( ω ′ ) = W r ( ω ) − W t ( ω ) , ∀ r ∈ ( t, t + τ ( ω )]; X r ( ω ′ ) = X r ( ω ) , ∀ r ∈ [0 , t + τ ( ω )] (cid:9) ⊂ (cid:8) ω ′ ∈ Ω : W r ( ω ′ ) − W t ( ω ′ ) = W r ( ω ) − W t ( ω ) , ∀ r ∈ [ t, t + τ ( ω )]; X tr ( ω ′ ) = X tr ( ω ) , ∀ r ∈ [0 , τ ( ω )] (cid:9) = Ξ tτ,ω ⊂ W tτ,ω , we further have Q ωε (cid:0) W tτ,ω (cid:1) = Q ωε (cid:0) Ξ tτ,ω (cid:1) = 1 , ∀ ω ∈ Ω . (5.26)ptimal Stopping with Expectation Constraints 22Then (5.1) and (5.25) render that for any ω ∈ Ω, Q ωε { T ≥ t + τ } = Q ωε (cid:0) W tτ,ω ∩{ T ≥ t + τ ( ω ) } (cid:1) = 1. So Q ωε (cid:0) A ( τ ) (cid:1) = 0 and Q ωε (cid:0) A c ( τ ) (cid:1) = 1 , ∀ ω ∈ Ω . (5.27)The following SDE on (cid:0) Ω , B (Ω) , Q ωε (cid:1) X s = X ( t ω , ω )+ Z st ω b ( r, X r ∧· ) dr + Z st ω σ ( r, X r ∧· ) dW r , s ∈ [ t ω , ∞ )with initial condition X s = X s ( ω ), ∀ s ∈ [0 , t ω ] admits a unique solution n X ωs = X t ω ,X ( t + τ ) ∧· ( ω ) s o s ∈ [0 , ∞ ) (cid:0) In particular, (cid:8) X ωs (cid:9) s ∈ [0 , ∞ ) is an F W tω ,Q ωε − adapted process with all continuous paths satisfying Q ωε (cid:8) X ωt ω + s = X ( t ω , ω )+ R s b ( t ω + r, X ω ( t ω + r ) ∧· ) dr + R s σ ( t ω + r, X ω ( t ω + r ) ∧· ) dW t ω r , ∀ s ∈ [0 , ∞ ) (cid:9) = 1 (cid:1) . By (5.24) and Definition 3.1 (i), N ωX, := (cid:8) ω ′ ∈ Ω : X t ω s ( ω ′ ) = X ω ( t ω + s, ω ′ ) for some s ∈ [0 , ∞ ) (cid:9) ∈ N Q ωε (cid:16) F Ξ tω ∞ (cid:17) . Also, there is an F W tω − predictable process (cid:8) K ωs (cid:9) s ∈ [0 , ∞ ) such that N ωK := (cid:8) ω ′ ∈ Ω : K ωs ( ω ′ ) = X ω ( t ω + s, ω ′ ) for some s ∈ [0 , ∞ ) (cid:9) ∈ N Q ωε (cid:16) F W tω ∞ (cid:17) .Given A ∈ B (Ω), we claim that Q ωε ( A ∩A ) = { ω ∈ A } Q ωε ( A ) , ∀ A ∈ G ( τ ) , ∀ ω ∈ A c ( τ ) . (5.28)To see this, we let A , A ∈ F Ξ t τ . When ω ∈ A , set s := τ ( ω ). Since A ∩{ τ ≤ s } is a F Ξ t s − measurable set including ω ,Ξ tτ,ω = (cid:8) ω ′ ∈ Ω : Ξ tr ( ω ′ ) = Ξ tr ( ω ) , ∀ r ∈ [0 , s ] (cid:9) is also contained in A ∩{ τ ≤ s } . (5.29)By (5.26), Q ωε (cid:0) A (cid:1) = 1 and thus Q ωε (cid:0) A ∩ A (cid:1) = Q ωε (cid:0) A (cid:1) . Similarly, for any ω ∈ A c one has Q ωε ( A c ) = 1 and thus Q ωε (cid:0) A ∩A (cid:1) = 0. Then (5.27) implies that for A = (cid:0) A ∩ A ( τ ) (cid:1) ∪ (cid:0) A ∩ A c ( τ ) (cid:1) ∈ G ( τ ) Q ωε (cid:0) A ∩A (cid:1) = Q ωε (cid:0) A ∩ A ( τ ) ∩A (cid:1) + Q ωε (cid:0) A ∩ A c ( τ ) ∩A (cid:1) = Q ωε (cid:0) A ∩A (cid:1) = { ω ∈ A } Q ωε (cid:0) A (cid:1) = { ω ∈ A } Q ωε (cid:0) A (cid:1) , ∀ ω ∈ A c ( τ ) . Let us consider a pasted probability P ε ∈ P (Ω): P ε ( A ) := P (cid:0) A ( τ ) ∩ A (cid:1) + Z ω ∈ A c ( τ ) Q ωε ( A ) P ( dω ) , ∀ A ∈ B (Ω) . (5.30)In particular, taking A = Ω in (5.28) renders that P ε ( A ) = P (cid:0) A ( τ ) ∩ A (cid:1) + Z ω ∈ A c ( τ ) { ω ∈ A } P ( dω ) = P ( A ) , ∀ A ∈ G ( τ ) . (5.31)We shall verify in next steps that the pasted probability P ε also belongs to P t, x ( y, z ), or the probability class P t, x ( y, z ) is stable under the pasting (5.30). II.c) In this step, we demonstrate that W t is a d − dimensional standard Brownian motion with respect to filtration G t under P ε . Let 0 ≤ s < r < ∞ and E ∈ B ( R d ). We need to show that P ε (cid:8)(cid:0) W tr − W ts (cid:1) − ( E ) ∩ A (cid:9) = φ ( r − s, E ) P ε (cid:0) A (cid:1) , ∀ A ∈ G ts . (5.32)To verify (5.32), we let (cid:8) ( s i , O i , O ′ i ) (cid:9) ni =1 ⊂ Q s × O ( R d + l ) × O ( R d + l ) and set A := n ∩ i =1 (cid:16)(cid:2)(cid:0) Ξ ts i (cid:1) − ( O i ) ∩{ T ∈ [ t, t + s i ] } (cid:3) ∪ (cid:2)(cid:0) Ξ ts i (cid:1) − ( O ′ i ) ∩{ T ∈ [ t, t + s i ] c } (cid:3)(cid:17) ∈ C ts . II.c.1) Let ω ∈ { τ > s } ∩ A ∩{ T ≥ t + τ } . For i = 1 , · · · , n , since T ( ω ) ≥ t + τ ( ω ) > t + s ≥ t + s i and since ω ∈ A , wesee that Ξ ts i ( ω ) ∈ O ′ i . It follows that Ξ tτ,ω ⊂ (cid:8) ω ′ ∈ Ω : Ξ ts i ( ω ′ ) = Ξ ts i ( ω ) ∈ O ′ i , i = 1 , · · · , n (cid:9) ⊂ n ∩ i =1 (cid:0) Ξ ts i (cid:1) − ( O ′ i ), and thusΞ tτ,ω ∩ (cid:8) T ≥ t + τ ( ω ) (cid:9) ⊂ n ∩ i =1 (cid:16)(cid:0) Ξ ts i (cid:1) − ( O ′ i ) ∩{ T ∈ [ t, t + s i ] c } (cid:17) ⊂ A . By (5.25) and (5.26), Q ωε ( A ) = 1 , ∀ ω ∈ { τ > s }∩ A ∩{ T ≥ t + τ } . (5.33) . Dynamic Programming Principle ω ∈ { τ > s }∩ A c ∩{ T ≥ t + τ } . As ω ∈ A c , there is j ∈ { , · · · , n } such that ω ∈ (cid:2)(cid:0) Ξ ts j (cid:1) − ( O cj ) ∩{ T ∈ [ t, t + s j ] } (cid:3) ∪ (cid:2)(cid:0) Ξ ts j (cid:1) − (cid:0) ( O ′ j ) c (cid:1) ∩{ T ∈ [ t, t + s j ] c } (cid:3) . In particular, Ξ ts j ( ω ) ∈ ( O ′ j ) c . Then Ξ tτ,ω ⊂ (cid:8) ω ′ ∈ Ω : Ξ ts j ( ω ′ ) = Ξ ts j ( ω ) ∈ ( O ′ j ) c (cid:9) ⊂ (cid:0) Ξ ts j (cid:1) − (cid:0) ( O ′ j ) c (cid:1) and thus Ξ tτ,ω ∩ (cid:8) T ≥ t + τ ( ω ) (cid:9) ⊂ (cid:0) Ξ ts j (cid:1) − (cid:0) ( O ′ j ) c (cid:1) ∩{ T ∈ [ t, t + s j ] c } ⊂ A c . By (5.25) and (5.26) again, Q ωε (cid:0) A c (cid:1) = 1 or Q ωε ( A ) = 0 , ∀ ω ∈ { τ > s }∩ A c ∩{ T ≥ t + τ } . (5.34)Since it holds for any ω ∈ { τ ≥ r } that W tτ,ω ⊂ { τ ≥ r } by (5.1), (5.26) shows that Z ω ∈ A c ( τ ) { τ ( ω ) ≥ r } Q ωε (cid:0) ( W tr − W ts ) − ( E ) ∩ A (cid:1) P ( dω ) = Z ω ∈ A c ( τ ) { τ ( ω ) ≥ r } Q ωε (cid:0) { τ ≥ r }∩ ( W tr − W ts ) − ( E ) ∩ A (cid:1) P ( dω ) . As { τ ≥ r }∩ ( W tr − W ts ) − ( E ) = { τ ≥ r }∩ ( W tτ ∧ r − W tτ ∧ s ) − ( E ) ∈ F W t τ ∧ r ⊂ F Ξ t τ , (5.28), (5.33) and (5.34) imply that Z ω ∈ A c ( τ ) { τ ( ω ) ≥ r } Q ωε (cid:0) ( W tr − W ts ) − ( E ) ∩ A (cid:1) P ( dω ) = Z ω ∈ A c ( τ ) { τ ( ω ) ≥ r }∩{ ( W tr − W ts )( ω ) ∈E} Q ωε (cid:0) A (cid:1) P ( dω )= Z ω ∈ Ω { τ ( ω ) ≥ r }∩{ ( W tr − W ts )( ω ) ∈E}∩{ T ( ω ) ≥ t + τ ( ω ) } Q ωε ( A ) P ( dω ) = P (cid:0) { τ ≥ r }∩ ( W tr − W ts ) − ( E ) ∩ A ∩{ T ≥ t + τ } (cid:1) . (5.35) II.c.2) Define A := n ∩ i =1 (cid:0) Ξ tτ ∧ s i (cid:1) − ( O ′ i ) ∈ F Ξ t τ and let ω ∈ { s < τ < r } . By (5.1), A ∩ (cid:8) T ≥ t + τ ( ω ) (cid:9) ∩ W tτ,ω = A ∩ (cid:8) T ≥ t + τ > t + s (cid:9) ∩ W tτ,ω = (cid:16) n ∩ i =1 (cid:8) Ξ ts i ∈ O ′ i (cid:9)(cid:17) ∩ (cid:8) T ≥ t + τ > t + s (cid:9) ∩ W tτ,ω = (cid:16) n ∩ i =1 (cid:8) Ξ tτ ∧ s i ∈ O ′ i (cid:9)(cid:17) ∩ (cid:8) T ≥ t + τ > t + s (cid:9) ∩ W tτ,ω = A∩ (cid:8) T ≥ t + τ ( ω ) (cid:9) ∩ W tτ,ω . (5.36)Set E ω := (cid:8) x − W tτ ( ω )+ W ts ( ω ) : x ∈ E (cid:9) ∈ B ( R d ). Given ω ′ ∈ W tτ,ω , ( W tr − W ts )( ω ′ ) ∈ E if and only if W t ω (cid:0) r − τ ( ω ) , ω ′ (cid:1) = W ( t + r, ω ′ ) − W (cid:0) t + τ ( ω ) , ω ′ (cid:1) = W tr ( ω ′ ) − W t (cid:0) τ ( ω ) , ω ′ (cid:1) = W tr ( ω ′ ) − W ts ( ω ′ ) − W t (cid:0) τ ( ω ) , ω (cid:1) + W ts ( ω ) ∈ E ω . So( W tr − W ts ) − ( E ) ∩ W tτ,ω = (cid:16) W t ω r − τ ( ω ) (cid:17) − (cid:0) E ω (cid:1) ∩ W tτ,ω . (5.37)As W t ω is a (cid:0) F t ω , Q ωε (cid:1) − Brownian motion by (5.24), we can deduce from (5.36), (5.25), (5.26) and (5.28) that Q ωε (cid:0) ( W tr − W ts ) − ( E ) ∩ A (cid:1) = Q ωε (cid:0) ( W tr − W ts ) − ( E ) ∩A (cid:1) = { ω ∈A} Q ωε (cid:0) ( W tr − W ts ) − ( E ) (cid:1) = { ω ∈A} Q ωε n(cid:16) W t ω r − τ ( ω ) (cid:17) − (cid:0) E ω (cid:1)o = { ω ∈A} φ (cid:0) r − τ ( ω ) , E ω (cid:1) , ∀ ω ∈ { s < τ < r }∩ A c ( τ ) , thus Z ω ∈ A c ( τ ) { s<τ ( ω ) Next, we suppose s > s < s < · · · < s n − < s n = s with n ≥ i = 1 , · · · , n − ω ∈ { s i < τ ≤ s i +1 } . As τ ( ω ) > s j for j = 1 , · · · , i , one can deduce that A ∩ (cid:8) T ≥ t + τ ( ω ) (cid:9) = (cid:26) i ∩ j =1 (cid:16)(cid:0) Ξ ts j (cid:1) − ( O ′ j ) ∩ (cid:8) T ≥ t + τ ( ω ) (cid:9)(cid:17)(cid:27) ∩ (cid:26) n ∩ j = i +1 (cid:16)(cid:2)(cid:0) Ξ ts j (cid:1) − ( O j ) ∩{ T ∈ [ t + τ ( ω ) , t + s j ] } (cid:3) ∪ (cid:2)(cid:0) Ξ ts j (cid:1) − ( O ′ j ) ∩{ T > t + s j } (cid:3)(cid:17)(cid:27) . (5.43)Define A i := i ∩ j =1 (cid:0) Ξ tτ ∧ s j (cid:1) − ( O ′ j ) ∈ F Ξ t τ . Since W tτ,ω ⊂ { s i < τ } by (5.1), W tτ,ω ∩ (cid:16) i ∩ j =1 (cid:0) Ξ ts j (cid:1) − ( O ′ j ) (cid:17) = W tτ,ω ∩ (cid:16) i ∩ j =1 (cid:8) Ξ tτ ∧ s j ∈ O ′ j (cid:9)(cid:17) = W tτ,ω ∩A i . (5.44)Set a ω := (cid:0) W tτ ( ω ) , (cid:1) ∈ R d + l and define A ωi := n ∩ j = i +1 (cid:18)h(cid:16) W t ω s j − τ ( ω ) , K ωs j − τ ( ω ) (cid:17) − (cid:0) O j,ω (cid:1) ∩ (cid:8) T ∈ [ t ω , t + s j ] (cid:9)i ∪ h(cid:16) W t ω s j − τ ( ω ) ,K ωs j − τ ( ω ) (cid:17) − (cid:0) O ′ j,ω (cid:1) ∩ (cid:8) T ∈ [ t ω , t + s j ] c (cid:9)i(cid:19) ∈ F t ω s − τ ( ω ) , where O j,ω := (cid:8) x − a ω : x ∈ O j (cid:9) ∈ B ( R d + l ) and O ′ j,ω := (cid:8) x − a ω : x ∈O ′ j (cid:9) ∈ B ( R d + l ).Given j = i +1 , · · · , n and ω ′ ∈ W tτ,ω ∩ (cid:0) N ωX, ∪N ωK (cid:1) c , we can derive that Ξ ts j ( ω ′ ) ∈ O j if and only if (cid:0) W t ω , K ω (cid:1)(cid:0) s j − τ ( ω ) , ω ′ (cid:1) = (cid:0) W ( t + s j , ω ′ ) − W (cid:0) t + τ ( ω ) , ω ′ (cid:1) , X ω (cid:0) t + s j , ω ′ (cid:1)(cid:1) = (cid:0) W ts j ( ω ′ ) − W t (cid:0) τ ( ω ) , ω ′ (cid:1) , X t ω (cid:0) s j − τ ( ω ) , ω ′ (cid:1)(cid:1) = Ξ ts j ( ω ′ ) − (cid:0) W tτ ( ω ) , (cid:1) ∈ O j,ω . Similarly, Ξ ts j ( ω ′ ) ∈ O ′ j if and only if (cid:0) W t ω , K ω (cid:1)(cid:0) s j − τ ( ω ) , ω ′ (cid:1) ∈ O ′ j,ω . Putting them with (5.43)and (5.44) renders that A ∩ (cid:8) T ≥ t + τ ( ω ) (cid:9) ∩ W tτ,ω ∩ (cid:0) N ωX, ∪N ωK (cid:1) c = W tτ,ω ∩ (cid:0) N ωX, ∪N ωK (cid:1) c ∩ (cid:16) A i ∩ (cid:8) T ≥ t + τ ( ω ) (cid:9)(cid:17) ∩ (cid:26) n ∩ j = i +1 (cid:18)h(cid:16) W t ω s j − τ ( ω ) , K ωs j − τ ( ω ) (cid:17) − (cid:0) O j,ω (cid:1) ∩ (cid:8) T ∈ [ t ω , t + s j ] (cid:9)i ∪ h(cid:16) W t ω s j − τ ( ω ) , K ωs j − τ ( ω ) (cid:17) − (cid:0) O ′ j,ω (cid:1) ∩ (cid:8) T > t + s j (cid:9)i(cid:19)(cid:27) = A i ∩ A ωi ∩ (cid:8) T ≥ t + τ ( ω ) (cid:9) ∩ W tτ,ω ∩ (cid:0) N ωX, ∪N ωK (cid:1) c . . Dynamic Programming Principle W t ω is a standard Brownian motion with respect to filtration F t ω under Q ωε by (5.24), we then see from (5.25),(5.26) and (5.28) that for any ω ∈ { s i < τ ≤ s i +1 }∩ A c ( τ ) Q ωε (cid:16) ( W tr − W ts ) − ( E ) ∩ A (cid:17) = Q ωε n A i ∩ A ωi ∩ ( W tr − W ts ) − ( E ) o = { ω ∈A i } Q ωε n A ωi ∩ (cid:16) W t ω r − τ ( ω ) − W t ω s − τ ( ω ) (cid:17) − ( E ) o = { ω ∈A i } Q ωε (cid:0) A ωi (cid:1) × Q ωε n(cid:16) W t ω r − τ ( ω ) − W t ω s − τ ( ω ) (cid:17) − ( E ) o = Q ωε (cid:0) A i ∩ A ωi (cid:1) φ ( r − s, E ) = Q ωε (cid:0) A (cid:1) φ ( r − s, E ) , and thus R ω ∈ A c ( τ ) { s i <τ ( ω ) ≤ s i +1 } Q ωε (cid:0) ( W tr − W ts ) − ( E ) ∩ A (cid:1) P ( dω ) = φ ( r − s, E ) × R ω ∈ A c ( τ ) { s i <τ ( ω ) ≤ s i +1 } Q ωε (cid:0) A (cid:1) P ( dω ).Taking summation of this equality from i = 1 through i = n − Z ω ∈ A c ( τ ) Q ωε (cid:0) { τ ≤ s }∩ ( W tr − W ts ) − ( E ) ∩ A (cid:1) P ( dω ) = Z ω ∈ A c ( τ ) { τ ( ω ) ≤ s } Q ωε (cid:0) ( W tr − W ts ) − ( E ) ∩ A (cid:1) P ( dω )= φ ( r − s, E ) Z ω ∈ A c ( τ ) { τ ( ω ) ≤ s } Q ωε (cid:0) A (cid:1) P ( dω ) = φ ( r − s, E ) Z ω ∈ A c ( τ ) Q ωε (cid:0) { τ ≤ s }∩ A (cid:1) P ( dω ) . Since A ( τ ) ∈ F tτ and since W t is a standard Brownian motion with respect to filtration F t under P , P ε (cid:0) { τ ≤ s }∩ ( W tr − W ts ) − ( E ) ∩ A (cid:1) = P (cid:0) { τ ≤ s }∩ ( W tr − W ts ) − ( E ) ∩ b A∩ A ( τ ) (cid:1) + φ ( r − s, E ) Z ω ∈ A c ( τ ) Q ωε (cid:0) { τ ≤ s }∩ A (cid:1) P ( dω )= P (cid:0) ( W tr − W ts ) − ( E ) (cid:1) × P (cid:0) { τ ≤ s }∩ b A∩ A ( τ ) (cid:1) + φ ( r − s, E ) Z ω ∈ A c ( τ ) Q ωε (cid:0) { τ ≤ s }∩ A (cid:1) P ( dω )= φ ( r − s, E ) n P (cid:0) { τ ≤ s }∩ A ∩ A ( τ ) (cid:1) + Z ω ∈ A c ( τ ) Q ωε (cid:0) { τ ≤ s }∩ A (cid:1) P ( dω ) o = φ ( r − s, E ) P ε (cid:0) { τ ≤ s }∩ A (cid:1) , which together with (5.41) shows that P ε (cid:0) ( W tr − W ts ) − ( E ) ∩ A (cid:1) = φ ( r − s, E ) P ε (cid:0) A (cid:1) . Combining it with (5.42), we canderive (5.32) from Dynkin Theorem and (3.2). Hence, P ε satisfies Definition 3.1 (i). II.d) In this step, we demonstrate that Definition 3.1 (ii) holds for P ε . Let ϕ ∈ C ( R d + l ) and n ∈ N . We still defineΞ t, x , M t, x ( ϕ ), τ t, x n as in (4.1) and set M t, x ,ns := M t, x τ t, x n ∧ s , s ∈ [0 , ∞ ). We also let 0 ≤ s < r < ∞ . II.d.1) Let (cid:8) ( s i , E i ) (cid:9) ki =1 ⊂ [0 , s ] × B ( R d + l ). For any ω ∈ Ω, set r ω := (cid:0) r − τ ( ω ) (cid:1) + , s ω := (cid:0) s − τ ( ω ) (cid:1) + and s i,ω := (cid:0) s i − τ ( ω ) (cid:1) + for i = 1 , · · · , k . We claim that E P ε (cid:20) { τ> s } (cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) k ∩ i =1 (cid:8) Ξ tsi ∈E i (cid:9)(cid:21) = 0 . (5.45)Since { τ > s } ∈ F W t τ and since k ∩ i =1 (cid:8) Ξ tτ ∧ s i ∈ E i (cid:9) ∈ F Ξ t τ , (5.28) implies that E P ε (cid:20) { τ> s } (cid:16) M t, x ,n r ( ϕ ) − M t, x ,nτ ∧ r ( ϕ ) (cid:17) k ∩ i =1 (cid:8) Ξ tsi ∈E i (cid:9)(cid:21) = E P ε (cid:20) { τ> s } (cid:16) M t, x ,nτ ∨ r ( ϕ ) − M t, x ,nτ ( ϕ ) (cid:17) k ∩ i =1 (cid:8) Ξ tτ ∧ si ∈E i (cid:9)(cid:21) = E P (cid:20) A ( τ ) ∩{ τ> s } (cid:16) M t, x ,nτ ∨ r ( ϕ ) − M t, x ,nτ ( ϕ ) (cid:17) k ∩ i =1 (cid:8) Ξ tτ ∧ si ∈E i (cid:9)(cid:21) + Z ω ∈ A c ( τ ) { τ ( ω ) > s } k ∩ i =1 (cid:8) Ξ tτ ∧ si ( ω ) ∈E i (cid:9) E Q ωε h M t, x ,nτ ∨ r ( ϕ ) − M t, x ,nτ ( ϕ ) i P ( dω ) . (5.46)Since M t, x ,nτ ∧ r ( ϕ ) − M t, x ,nτ ∧ s ( ϕ ) ∈ F Ξ t τ t, x n ∧ τ ∧ r ⊂ F Ξ t τ and since { τ > s }∩ (cid:16) k ∩ i =1 (cid:8) Ξ tτ ∧ s i ∈ E i (cid:9)(cid:17) ∈ F Ξ t τ ∧ s ⊂ G tτ ∧ s , applying LemmaA.1 with (cid:0) ζ , ζ (cid:1) = (cid:0) τ ∧ s , τ ∧ r (cid:1) , we see from (5.31) and (4.2) that E P ε h { τ> s } (cid:0) M t, x ,nτ ∧ r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:1) k ∩ i =1 (cid:8) Ξ tsi ∈E i (cid:9)i = E P h { τ> s } (cid:0) M t, x ,nτ ∧ r ( ϕ ) − M t, x ,nτ ∧ s ( ϕ ) (cid:1) k ∩ i =1 (cid:8) Ξ tτ ∧ si ∈E i (cid:9)i = E P h { τ> s } (cid:0) M tτ tn ∧ τ ∧ r ( ϕ ) − M tτ tn ∧ τ ∧ s ( ϕ ) (cid:1) k ∩ i =1 (cid:8) Ξ tτ ∧ si ∈E i (cid:9)i = 0 . (5.47)ptimal Stopping with Expectation Constraints 26Also, taking (cid:0) ζ , ζ (cid:1) = (cid:0) τ , τ ∨ r (cid:1) in Lemma A.1 and using (4.2) yield that E P (cid:20) A ( τ ) ∩{ τ> s } (cid:16) M t, x ,nτ ∨ r ( ϕ ) − M t, x ,nτ ( ϕ ) (cid:17) k ∩ i =1 (cid:8) Ξ tτ ∧ si ∈E i (cid:9)(cid:21) = E P (cid:20) A ( τ ) ∩{ τ> s } (cid:16) M tτ tn ∧ ( τ ∨ r ) ( ϕ ) − M tτ tn ∧ τ ( ϕ ) (cid:17) k ∩ i =1 (cid:8) Ξ tτ ∧ si ∈E i (cid:9)(cid:21) = 0 . (5.48)Fix ω ∈ A c ( τ ) ∩ (cid:0) N X (cid:1) c . As X r ( ω ) = x ( r ), ∀ r ∈ [0 , t ], one has Ω tτ,ω ⊂ (cid:8) ω ′ ∈ Ω : X r ( ω ′ ) = X r ( ω ) , ∀ r ∈ [0 , t ] (cid:9) = (cid:8) ω ′ ∈ Ω : X r ( ω ′ ) = x ( r ) , ∀ r ∈ [0 , t ] (cid:9) = (cid:0) N X (cid:1) c . By (5.25), Q ωε (cid:16)(cid:0) N X (cid:1) c (cid:17) = 1 and we thus see from (4.2) that E Q ωε h M t, x ,nτ ∨ r ( ϕ ) − M t, x ,nτ ( ϕ ) i = E Q ωε h M tτ tn ∧ ( τ ∨ r ) ( ϕ ) − M tτ tn ∧ τ ( ϕ ) i . (5.49)Set n ω := n − τ ( ω ) > 0. We define a C function ϕ ω ( w, x ) := ϕ (cid:16) w + W tτ ( ω ) , x (cid:17) , ( w, x ) ∈ R d + l . For i = 1 , , 3, since D i ϕ ω ( w, x ) = D i ϕ (cid:0) w + W t (cid:0) τ ( ω ) , ω (cid:1) , x (cid:1) , ∀ ( w, x ) ∈ R d + l (with D ϕ := ϕ ), D i ϕ (cid:16) Ξ t ( r, ω ′ ) (cid:17) = D i ϕ (cid:16) W t ( r, ω ′ ) − W t (cid:0) τ ( ω ) , ω ′ (cid:1) + W t (cid:0) τ ( ω ) , ω (cid:1) , X t ( r, ω ′ ) (cid:17) = D i ϕ ω (cid:16) W t ω (cid:0) r − τ ( ω ) , ω ′ (cid:1) , X t ω (cid:0) r − τ ( ω ) , ω ′ (cid:1)(cid:17) = D i ϕ ω (cid:16) Ξ t ω (cid:0) r − τ ( ω ) , ω ′ (cid:1)(cid:17) , ∀ r ∈ (cid:2) τ ( ω ) , ∞ (cid:1) . (5.50)We also define an F Ξ tω − stopping time by γ nω ( ω ′ ) := inf (cid:8) s ∈ [0 , ∞ ) : (cid:12)(cid:12) Ξ t ω s ( ω ′ ) + (cid:0) W tτ ( ω ) , (cid:1)(cid:12)(cid:12) ≥ n (cid:9) , ω ′ ∈ Ω. By ananalogy to Lemma A.1, E Q ωε (cid:20) M t ω γ nω ∧ n ω ∧ r ω (cid:0) ϕ ω (cid:1) − M t ω (cid:0) ϕ ω (cid:1)(cid:21) = E Q ωε (cid:20) M t ω γ nω ∧ n ω ∧ r ω (cid:0) ϕ ω (cid:1) − M t ω γ nω ∧ n ω ∧ (cid:0) ϕ ω (cid:1)(cid:21) = 0 . (5.51)As ω ∈ (cid:8) ω ′ ∈ Ω : τ tn ( ω ′ ) > τ ( ω ) (cid:9) ∈ F Ξ t τ ( ω ) , an analogy to (5.29) shows that Ξ tτ,ω ⊂ (cid:8) ω ′ ∈ Ω : τ tn ( ω ′ ) > τ ( ω ) (cid:9) .Let ω ′ ∈ Ξ tτ,ω . Since inf (cid:8) s ∈ [0 , ∞ ) : | Ξ ts ( ω ′ ) | ≥ n (cid:9) ≥ τ tn ( ω ′ ) > τ ( ω ), one has | Ξ ts ( ω ′ ) | < n , ∀ s ∈ [0 , τ ( ω )] and thusinf (cid:8) s ∈ [0 , ∞ ) : | Ξ ts ( ω ′ ) | ≥ n (cid:9) = inf (cid:8) s ∈ [ τ ( ω ) , ∞ ) : | Ξ ts ( ω ′ ) | ≥ n (cid:9) = inf n s ∈ [ τ ( ω ) , ∞ ) : (cid:12)(cid:12) Ξ ts ( ω ′ ) − (cid:0) W t ( τ ( ω ) , ω ′ ) , (cid:1) + (cid:0) W tτ ( ω ) , (cid:1)(cid:12)(cid:12) ≥ n o = inf n s ∈ [ τ ( ω ) , ∞ ) : (cid:12)(cid:12) Ξ t ω ( s − τ ( ω ) , ω ′ )+ (cid:0) W tτ ( ω ) , (cid:1)(cid:12)(cid:12) ≥ n o = γ nω ( ω ′ )+ τ ( ω ) . It follows that τ tn ( ω ′ ) = (cid:0) γ nω ( ω ′ )+ τ ( ω ) (cid:1) ∧ n = γ nω ( ω ′ ) ∧ n ω + τ ( ω ). We can then deduce from (5.1) and (5.50) that (cid:0) M t ( ϕ ) (cid:1)(cid:0) τ tn ( ω ′ ) ∧ (cid:0) τ ( ω ′ ) ∨ r (cid:1) , ω ′ (cid:1) − (cid:0) M t ( ϕ ) (cid:1)(cid:0) τ ( ω ′ ) , ω ′ (cid:1) = (cid:0) M t ( ϕ ) (cid:1)(cid:0) τ tn ( ω ′ ) ∧ (cid:0) τ ( ω ) ∨ r (cid:1) , ω ′ (cid:1) − (cid:0) M t ( ϕ ) (cid:1)(cid:0) τ ( ω ) , ω ′ (cid:1) = ϕ (cid:16) Ξ t (cid:0) τ tn ( ω ′ ) ∧ (cid:0) τ ( ω ) ∨ r (cid:1) , ω ′ (cid:1)(cid:17) − ϕ (cid:0) Ξ t (cid:0) τ ( ω ) , ω ′ (cid:1)(cid:1) − Z τ tn ( ω ′ ) ∧ ( τ ( ω ) ∨ r ) τ ( ω ) b (cid:0) t + r, X (cid:0) ( t + r ) ∧· , ω ′ (cid:1)(cid:1) · Dϕ (cid:0) Ξ t ( r, ω ′ ) (cid:1) dr − Z τ tn ( ω ′ ) ∧ ( τ ( ω ) ∨ r ) τ ( ω ) σ σ T (cid:0) t + r, X (cid:0) ( t + r ) ∧· , ω ′ (cid:1) : D ϕ (cid:0) Ξ t ( r, ω ′ ) (cid:1) dr = ϕ ω (cid:16) Ξ t ω (cid:0) γ nω ( ω ′ ) ∧ n ω ∧ r ω , ω ′ (cid:1)(cid:17) − ϕ ω (cid:0) Ξ t ω (0 , ω ′ ) (cid:1) − Z γ nω ( ω ′ ) ∧ n ω ∧ r ω b (cid:16) t ω + r ′ , X (cid:0) ( t ω + r ′ ) ∧· , ω ′ (cid:1)(cid:17) · Dϕ ω (cid:16) Ξ t ω ( r ′ , ω ′ ) (cid:17) dr ′ − Z γ nω ( ω ′ ) ∧ n ω ∧ r ω σ σ T (cid:16) t ω + r ′ , X (cid:0) ( t ω + r ′ ) ∧· , ω ′ (cid:1)(cid:17) : D ϕ ω (cid:16) Ξ t ω ( r ′ , ω ′ ) (cid:17) dr ′ (cid:0) by setting r ′ = r − τ ( ω ) (cid:1) = (cid:0) M t ω ( ϕ ω ) (cid:1)(cid:16) γ nω ( ω ′ ) ∧ n ω ∧ r ω , ω ′ (cid:17) − (cid:0) M t ω ( ϕ ω ) (cid:1)(cid:0) , ω ′ (cid:1) . (5.52)As { τ tn > τ } ∈ F Ξ t τ tn ∧ τ ⊂ F Ξ t τ , (5.49), (5.26), (5.52), (5.28) and (5.51) imply that E Q ωε h M t, x ,nτ ∨ r ( ϕ ) − M t, x ,nτ ( ϕ ) i = E Q ωε h { τ tn >τ } (cid:0) M tτ tn ∧ ( τ ∨ r ) ( ϕ ) − M tτ ( ϕ ) (cid:1)i = E Q ωε h { τ tn >τ } (cid:16) M t ω γ nω ∧ n ω ∧ r ω ( ϕ ω ) − M t ω ( ϕ ω ) (cid:17)i = (cid:8) τ tn ( ω ) >τ ( ω ) (cid:9) E Q ωε h M t ω γ nω ∧ n ω ∧ r ω ( ϕ ω ) − M t ω ( ϕ ω ) i = 0 , ∀ ω ∈ A c ( τ ) ∩ (cid:0) N X (cid:1) c , . Dynamic Programming Principle R ω ∈ A c ( τ ) { τ ( ω ) > s } k ∩ i =1 (cid:8) Ξ tτ ∧ si ( ω ) ∈E i (cid:9) E Q ωε h M t, x ,nτ ∨ r ( ϕ ) − M t, x ,nτ ( ϕ ) i P ( dω ) = 0, which together with (5.46) − (5.48)leads to (5.45). II.d.2) If 0 = s < r < ∞ , as { τ > } = Ω, we have seen from (5.45) that E P ε (cid:20)(cid:16) M t, x ,n r ( ϕ ) − M t, x ( ϕ ) (cid:17) (cid:8) Ξ t ∈E (cid:9)(cid:21) = 0 , ∀ E ∈ B ( R d + l ) . (5.53)Assume next that 0 < s < r < ∞ . Let 0 = s < s < · · · < s k − < s k = s with k ≥ {E i } ki =1 ⊂ B ( R d + l ). As A ( τ ) ∈ F tτ ⊂ G tτ , one has A ( τ ) ∩{ τ ≤ s } ∈ G t s . Lemma A.1 and (4.2) yield that E P (cid:20) A ( τ ) ∩{ τ ≤ s } (cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) k ∩ i =1 { Ξ tsi ∈E i } (cid:21) = E P (cid:20) A ( τ ) ∩{ τ ≤ s } (cid:16) M tτ tn ∧ r ( ϕ ) − M tτ tn ∧ s ( ϕ ) (cid:17) k ∩ i =1 { Ξ tsi ∈E i } (cid:21) = 0 . (5.54)Let i ∈ { , · · · , k − } and set ˘ A i := i ∩ j =1 (cid:0) Ξ tτ ∧ s j (cid:1) − ( E j ) ∈ F Ξ t τ . We fix ω ∈ (cid:8) s i < τ ≤ s i +1 (cid:9) ∩ A c ( τ ). Analogous to (5.44), W tτ,ω ∩ (cid:16) i ∩ j =1 (cid:0) Ξ ts j (cid:1) − ( E j ) (cid:17) = W tτ,ω ∩ ˘ A i . (5.55)Define ˘ A ωi := k ∩ j = i +1 (cid:16) Ξ t ω s j,ω (cid:17) − (cid:0) ˘ E j,ω (cid:1) ∈ F Ξ tω s ω ⊂ G t ω s ω , with ˘ E j,ω := (cid:8) x − ( W tτ ( ω ) , ) : x ∈ E j (cid:9) . Similar to Lemma A.1, E Q ωε (cid:20)(cid:16) M t ω γ nω ∧ n ω ∧ r ω (cid:0) ϕ ω (cid:1) − M t ω γ nω ∧ n ω ∧ s ω (cid:0) ϕ ω (cid:1)(cid:17) ˘ A ωi (cid:21) = 0 . (5.56)For any j = i +1 , · · · , k and ω ′ ∈ W tτ,ω , Ξ ts j ( ω ′ ) ∈ E j if and only if Ξ t ω (cid:0) s j − τ ( ω ) , ω ′ (cid:1) = (cid:16) W ( t + s j , ω ′ ) − W (cid:0) t + τ ( ω ) , ω ′ (cid:1) , X ( t + s j , ω ′ ) (cid:17) = (cid:16) W ts j ( ω ′ ) − W t (cid:0) τ ( ω ) , ω ′ (cid:1) , X ts j ( ω ′ ) (cid:17) = Ξ ts j ( ω ′ ) − (cid:0) W tτ ( ω ) , (cid:1) ∈ ˘ E j,ω . So W tτ,ω ∩ (cid:16) k ∩ j = i +1 (cid:0) Ξ ts j (cid:1) − ( E j ) (cid:17) = W tτ,ω ∩ ˘ A ωi . (5.57)Let ω ′ ∈ Ξ tτ,ω . As τ ( ω ) ≤ s i +1 ≤ s , following a similar argument to those in (5.52) yields that (cid:0) M t ( ϕ ) (cid:1)(cid:0) τ tn ( ω ′ ) ∧ r , ω ′ (cid:1) − (cid:0) M t ( ϕ ) (cid:1)(cid:0) τ tn ( ω ′ ) ∧ s , ω ′ (cid:1) = (cid:0) M t ω ( ϕ ω ) (cid:1)(cid:0) γ nω ( ω ′ ) ∧ n ω ∧ r ω , ω ′ (cid:1) − (cid:0) M t ω ( ϕ ω ) (cid:1)(cid:0) γ nω ( ω ′ ) ∧ n ω ∧ s ω , ω ′ (cid:1) . By an analogy to(5.49), one can then deduce from (5.26), (5.55), (5.57), (5.28) and (5.56) that E Q ωε (cid:20)(cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) k ∩ j =1 (cid:8) Ξ tsj ∈E j (cid:9)(cid:21) = E Q ωε (cid:20) ˘ A i ∩ ˘ A ωi (cid:16) M tτ tn ∧ r ( ϕ ) − M tτ tn ∧ s ( ϕ ) (cid:17)(cid:21) = { ω ∈ ˘ A i } E Q ωε (cid:20) ˘ A ωi (cid:16) M tτ tn ∧ r ( ϕ ) − M tτ tn ∧ s ( ϕ ) (cid:17)(cid:21) = { ω ∈ ˘ A i } E Q ωε (cid:20)(cid:16) M t ω γ nω ∧ n ω ∧ r ω (cid:0) ϕ ω (cid:1) − M t ω γ nω ∧ n ω ∧ s ω (cid:0) ϕ ω (cid:1)(cid:17) ˘ A ωi (cid:21) = 0 , ∀ ω ∈ { s i < τ ≤ s i +1 }∩ A c ( τ ) ∩ (cid:0) N X (cid:1) c , and thus R ω ∈ A c ( τ ) { s i <τ ( ω ) ≤ s i +1 } E Q ωε (cid:20)(cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) k ∩ j =1 (cid:8) Ξ tsj ∈E j (cid:9)(cid:21) P ( dω ) = 0. Taking summation from i = 1 through i = k − 1, we obtain from (5.54) and (5.28) that E P ε (cid:20) { τ ≤ s } (cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) k ∩ i =1 { Ξ tsi ∈E i } (cid:21) = E P (cid:20) A ( τ ) ∩{ τ ≤ s } (cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) k ∩ i =1 { Ξ tsi ∈E i } (cid:21) + Z ω ∈ A c ( τ ) { τ ( ω ) ≤ s } E Q ωε (cid:20)(cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) k ∩ j =1 (cid:8) Ξ tsj ∈E j (cid:9)(cid:21) P ( dω ) = 0 . Adding it to (5.45) shows that E P ε h(cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) k ∩ i =1 { Ξ tsi ∈E i } i = 0, which together with (5.53) and Dynkin’sTheorem imply that for any 0 ≤ s < r < ∞ , E P ε h(cid:16) M t, x ,n r ( ϕ ) − M t, x ,n s ( ϕ ) (cid:17) A i = 0, ∀ A ∈ F Ξ t s . So (cid:8) M t, x τ t, x n ∧ s ( ϕ ) (cid:9) s ∈ [0 , ∞ ) is an (cid:0) F Ξ t ,P , P ε (cid:1) − martingale. As lim n →∞ ↑ τ t, x n = ∞ , (cid:8) M t, x s ( ϕ ) (cid:9) s ∈ [0 , ∞ ) is an (cid:0) F Ξ t ,P , P ε (cid:1) − local martingale for any ϕ ∈ C ( R d + l ).ptimal Stopping with Expectation Constraints 28Then using similar arguments to those in Part (2b) of the proof of Proposition 4.1, we can derive that (cid:8) X t, x (cid:9) s ∈ [0 , ∞ ) is a solution of SDE (3.3) under P ε . As X t, x coincides with X on (cid:0) N X (cid:1) c by (4.2), P ε (cid:8) X s = X t, x s , ∀ s ∈ [0 , ∞ ) (cid:9) ≥ P ε (cid:16)(cid:0) N X (cid:1) c (cid:17) = P (cid:16) A ( τ ) ∩ (cid:0) N X (cid:1) c (cid:17) + Z ω ∈ A c ( τ ) Q ωε (cid:16)(cid:0) N X (cid:1) c (cid:17) P ( dω ) = P (cid:0) A ( τ ) (cid:1) + P (cid:0) A c ( τ ) (cid:1) = 1 . Hence, P ε satisfies Definition 3.1 (ii). II.e) As Q ωε { T ≥ t } ≥ Q ωε { T ≥ t + τ ( ω ) } = 1 for any ω ∈ A c ( τ ) by (5.25), one has P ε { T ≥ t } = P (cid:0) A ( τ ) ∩ { T ≥ t } (cid:1) + R ω ∈ Ω { ω ∈ A c ( τ ) } Q ωε { T ≥ t } P ( dω ) = P (cid:0) A ( τ ) (cid:1) + P (cid:0) A c ( τ ) (cid:1) = 1. So P ε ∈ P t, x .Set K t, x s := { s ∈ [0 ,t ] } x ( s )+ { s ∈ ( t, ∞ ) } (cid:0) K ts − t − K t + x ( t ) (cid:1) , s ∈ [0 , ∞ ). It is a continuous process such that (cid:8) K t, x t + s (cid:9) s ∈ [0 , ∞ ) is F W t − predictable and that K t, x s ( ω ) = X t, x s ( ω ) = X s ( ω ) , ∀ ( s, ω ) ∈ [0 , ∞ ) × (cid:0) N X ∪N X ∪N K (cid:1) c . (5.58)Let ω ∈ A c ( τ ) ∩ (cid:0) N X ∪N X ∪N K (cid:1) c and i ∈ N , By (5.24) and (5.22), one has E Q ωε (cid:2) R Tt + τ ( ω ) g i (cid:0) r, X r ∧· (cid:1) dr (cid:3) ≤ Y i ( ω ) = (cid:0) Y iP ( τ ) (cid:1) ( ω ) and E Q ωε (cid:2) R Tt + τ ( ω ) h i (cid:0) r, X r ∧· (cid:1) dr (cid:3) = Z i ( ω ) = (cid:0) Z iP ( τ ) (cid:1) ( ω ). Since Ω tτ,ω ⊂ (cid:8) ω ′ ∈ Ω : X s ( ω ′ ) = X s ( ω ) , ∀ s ∈ (cid:2) , t + τ ( ω ) (cid:3)(cid:9) we see from (5.25) and (5.58) that E Q ωε h Z Tt g i (cid:0) r, X r ∧· (cid:1) dr i = Z t + τ ( ω ) t g i (cid:0) r, X r ∧· ( ω ) (cid:1) dr + E Q ωε h Z Tt + τ ( ω ) g i (cid:0) r, X r ∧· (cid:1) dr i (5.59) ≤ Z τ ( ω )0 g i (cid:0) t + r, K t, x ( t + r ) ∧· ( ω ) (cid:1) dr + E P (cid:20) Z TT ∧ ( t + τ ) g i ( r, X r ∧· ) dr (cid:12)(cid:12)(cid:12) F ( τ ) (cid:21) ( ω ) , and similarly E Q ωε h Z Tt h i (cid:0) r, X r ∧· (cid:1) dr i = Z τ ( ω )0 h i (cid:0) t + r, K t, x ( t + r ) ∧· ( ω ) (cid:1) dr + E P (cid:20) Z TT ∧ ( t + τ ) h i ( r, X r ∧· ) dr (cid:12)(cid:12)(cid:12) F ( τ ) (cid:21) ( ω ) . As R τ ( g i , h i ) (cid:0) t + r, K t, x ( t + r ) ∧· (cid:1) dr ∈ F W t τ ⊂ F ( τ ), one can further deduce from (5.58) that Z ω ∈ A c ( τ ) E Q ωε (cid:20) Z Tt g i (cid:0) r, X r ∧· (cid:1) dr (cid:21) P ( dω ) ≤ E P (cid:20) A c ( τ ) (cid:18) Z τ g i (cid:0) t + r, K t, x ( t + r ) ∧· (cid:1) dr + E P h Z TT ∧ ( t + τ ) g i ( r, X r ∧· ) dr (cid:12)(cid:12)(cid:12) F ( τ ) i(cid:19)(cid:21) = E P (cid:20) E P (cid:20) A c ( τ ) (cid:16) Z τ g i (cid:0) t + r, K t, x ( t + r ) ∧· (cid:1) dr + Z TT ∧ ( t + τ ) g i ( r, X r ∧· ) dr (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) F ( τ ) (cid:21)(cid:21) = E P (cid:20) { T ≥ t + τ } (cid:16) Z τ g i (cid:0) t + r, X ( t + r ) ∧· (cid:1) dr + Z Tt + τ g i ( r, X r ∧· ) dr (cid:17)(cid:21) = E P (cid:20) A c ( τ ) Z Tt g i (cid:0) r, X r ∧· (cid:1) dr (cid:21) . It follows that E P ε (cid:2) R Tt g i (cid:0) r, X r ∧· (cid:1) dr (cid:3) ≤ E P (cid:2) R Tt g i (cid:0) r, X r ∧· (cid:1) dr (cid:3) ≤ y i . Similarly, one has E P ε (cid:2) R Tt h i (cid:0) r, X r ∧· (cid:1) dr (cid:3) = E P (cid:2) R Tt h i (cid:0) r, X r ∧· (cid:1) dr (cid:3) = z i . Hence, P ε belongs to P t, x ( y, z ). II.f ) By (5.23), D ∞ := (cid:8) ω ∈ Ω : V (cid:0) Ψ τ ( ω ) (cid:1) = ∞ (cid:9) is a σ (cid:0) F W t τ ∪F Xt ∪ N P (cid:0) F W t ∞ ∨F X ∞ (cid:1)(cid:1) − measurable set.Let ε ∈ (0 , ω ∈ A c ( τ ) E Q ωε (cid:20) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + ζ (cid:21) = Z t + τ ( ω ) t f (cid:0) r, X r ∧· ( ω ) (cid:1) dr + E Q ε (Ψ τ ( ω )) (cid:20) Z Tt + τ ( ω ) f (cid:0) r, X r ∧· (cid:1) dr + ζ (cid:21) ≥ Z t + τ ( ω ) t f (cid:0) r, X r ∧· ( ω ) (cid:1) dr + { ω ∈D c ∞ } (cid:16) V (cid:0) Ψ τ ( ω ) (cid:1) − ε (cid:17) + { ω ∈D ∞ } ε . As P ε ∈ P t, x ( y, z ), we then have V ( t, x , y, z ) ≥ E P ε (cid:20) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + ζ (cid:21) ≥ E P (cid:20) A ( τ ) (cid:16) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + ζ (cid:17)(cid:21) + E P (cid:20) A c ( τ ) (cid:16) Z t + τt f (cid:0) r, X r ∧· (cid:1) dr + { ω ∈D c ∞ } (cid:16) V (cid:0) Ψ τ (cid:1) − ε (cid:17) + { ω ∈D ∞ } ε (cid:17)(cid:21) . (5.60) . Appendix P (cid:0) A c ( τ ) ∩D ∞ (cid:1) = 0, this equality together with Theorem 3.1 and (5.22) shows that for any ε ∈ (0 , V ( t, x , y, z ) ≥ E P (cid:20) A ( τ ) (cid:16) Z Tt f (cid:0) r, X r ∧· (cid:1) dr + ζ (cid:1)(cid:17) + A c ( τ ) (cid:18) Z t + τt f (cid:0) r, X r ∧· (cid:1) dr + V (cid:16) t + τ , X ( t + τ ) ∧· , Y , Z (cid:17)(cid:19)(cid:21) − ε = E P (cid:20) { T 0. For any ε ∈ (0 , V ( t, x , y, z ) ≥ E P (cid:20) A ( τ ) (cid:16) − Z Tt f − (cid:0) r, X r ∧· (cid:1) dr − ζ − (cid:17) + A c ( τ ) (cid:16) − Z t + τt f − (cid:0) r, X r ∧· (cid:1) dr − V − (cid:0) t + τ, X ( t + τ ) ∧· , Y P ( τ ) , Z P ( τ ) (cid:1)(cid:17)(cid:21) − ε + 1 ε P (cid:0) A c ( τ ) ∩ D ∞ (cid:1) . Sending ε → V ( t, x , y, z ) = ∞ . Hence (5.20) still holds. Taking supremum over P ∈ P t, x ( y, z )reaches the super-solution part of the DPP for V . (cid:3) A Appendix Lemma A.1. Let ( t, x ) ∈ [0 , ∞ ) × Ω X , P ∈ P t, x , ϕ ∈ C ( R d + l ) and n ∈ N . For any two F W t − stopping times ζ , ζ with ζ ≤ ζ , P − a.s., we have E P h(cid:16) M tτ tn ∧ ζ ( ϕ ) − M tτ tn ∧ ζ ( ϕ ) (cid:17) A i = 0 , ∀ A ∈ G tζ . Proof: As P ∈ P t, x , N X := (cid:8) ω ∈ Ω : X s ( ω ) = x ( s ) for some s ∈ [0 , t ] (cid:9) ∈ N P (cid:0) F Xt (cid:1) and N X := (cid:8) ω ∈ Ω : X ts ( ω ) = X t, x t + s ( ω ) for some s ∈ [0 , ∞ ) (cid:9) is of N P (cid:0) F W t ∞ (cid:1) . Define an F W t ,P − adapted process b Ξ t, x s := (cid:0) W ts , X t, x t + s (cid:1) , s ∈ [0 , ∞ ) anddefine an F W t ,P − stopping time by γ tn := inf (cid:8) s ∈ [0 , ∞ ) : (cid:12)(cid:12)b Ξ t, x s (cid:12)(cid:12) ≥ n (cid:9) ∧ n .For any s ∈ [0 , ∞ ), since Ξ t coincides with b Ξ t, x on (cid:0) N X (cid:1) c , the sigma-field of Ω, S s := n A ∈ B (Ω) : (cid:0) N X (cid:1) c ∩ A = (cid:0) N X (cid:1) c ∩A for some A ∈ F t,Ps o , includes F Ξ t s as well as the sets (cid:8) T ∈ [ t, t + r ] (cid:9) , r ∈ [0 , s ]. So G ts ⊂ S s .Applying Itˆo’s formula yields that P − a.s. c M t, x s ( ϕ ) := ϕ (cid:0)b Ξ t, x s (cid:1) − Z s b (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) · Dϕ (cid:0)b Ξ t, x r (cid:1) dr − Z s σ σ T (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) : D ϕ (cid:0)b Ξ t, x r (cid:1) dr = ϕ (cid:0) , x ( t ) (cid:1) + Z s Dϕ (cid:0)b Ξ t, x r (cid:1) · σ (cid:0) t + r, X t, x ( t + r ) ∧· (cid:1) dW tr , s ∈ [0 , ∞ ) , which shows that (cid:8) c M t, x γ tn ∧ s ( ϕ ) (cid:9) s ∈ [0 , ∞ ) is a bounded (cid:0) F t,P , P (cid:1) − martingale.Let ζ , ζ be two F W t − stopping times with ζ ≤ ζ , P − a.s. We may assume first that ζ takes values in acountable subset { s i } i ∈ N of [0 , ∞ ). For any A ∈ F t,Pζ , since (cid:0) N X (cid:1) c ⊂ (cid:8) τ tn = γ tn (cid:9) and since (cid:0) N X ∪N X (cid:1) c ⊂ (cid:8) M ts ( ϕ ) = c M t, x s ( ϕ ) , ∀ s ∈ [0 , ∞ ) (cid:9) , we can deduce from the optional sampling theorem that E P h M tτ tn ∧ ζ ( ϕ ) A i = E P (cid:20) c M t, x γ tn ∧ ζ ( ϕ ) A (cid:21) = E P (cid:20) E P h c M t, x γ tn ∧ ζ ( ϕ ) (cid:12)(cid:12)(cid:12) F t,Pζ i A (cid:21) = E P (cid:20) c M t, x γ tn ∧ ζ ( ϕ ) A (cid:21) = E P (cid:20) M tτ tn ∧ ζ ( ϕ ) A (cid:21) . (A.1)Let A ∈ G tζ . For any i ∈ N , as G ts i ⊂ S s i there is an A i ∈ F t,Ps i such that (cid:0) N X (cid:1) c ∩ A ∩ (cid:8) ζ = s i (cid:9) = (cid:0) N X (cid:1) c ∩A i . Then A := ∪ i ∈ N (cid:16)(cid:8) ζ = s i (cid:9) ∩A i (cid:17) ∈ F t,Pζ satisfies that (cid:0) N X (cid:1) c ∩ A = ∪ i ∈ N (cid:16)(cid:0) N X (cid:1) c ∩A i ∩ (cid:8) ζ = s i (cid:9)(cid:17) = (cid:0) N X (cid:1) c ∩A , and (A.1) becomes E P h(cid:16) M tτ tn ∧ ζ ( ϕ ) − M tτ tn ∧ ζ ( ϕ ) (cid:17) A i = 0.For general F W t − stopping time ζ , we can approximate ζ i from above by ζ ki := P j ∈ N { ( j − − k ≤ ζ i 2, and then use the continuity of the bounded process (cid:8) M tτ tn ∧ s ( ϕ ) (cid:9) s ∈ [0 , ∞ ) to reach E P h(cid:16) M tτ tn ∧ ζ ( ϕ ) − M tτ tn ∧ ζ ( ϕ ) (cid:17) A i = lim k →∞ E P h(cid:16) M tτ tn ∧ ζ k ( ϕ ) − M tτ tn ∧ ζ k ( ϕ ) (cid:17) A i = 0 . (cid:3) ptimal Stopping with Expectation Constraints 30 Lemma A.2. Let X be a topological space and let Y be a Borel space. If f : X × Y [ −∞ , ∞ ] is a B ( X ) ⊗ B ( Y ) − measurable function, then φ f ( x, P ) := R y ∈ Y f ( x, y ) P ( dy ) , ( x, P ) ∈ X × P ( Y ) is B ( X ) ⊗ B (cid:0) P ( Y ) (cid:1) − measurable. Proof: Let E ∈ B ( X ) and A ∈ B ( Y ). Since Proposition 7.25 of [13] shows that the function φ A ( P ) := E P [ A ], P ∈ P ( Y ) is B (cid:0) P ( Y ) (cid:1) − measurable, the mapping φ E ,A ( x, P ) := R y ∈ Y E ( x ) A ( y, z ) P ( dy ), ( x, P ) ∈ X × P ( Y ) is B ( X ) ⊗ B (cid:0) P ( Y ) (cid:1) − measurable. So all measurable rectangles of B ( X ) ⊗ B ( Y ) are included in the Lambda systemΛ := n D ∈ B ( X ) ⊗ B ( Y ) (cid:12)(cid:12)(cid:12) φ D ( x, P ) := Z y ∈ Y D ( x, y ) P ( dy ) , ( x, P ) ∈ X × P ( Y ) is B ( X ) ⊗ B (cid:0) P ( Y ) (cid:1) − measurable o . An application of Dynkin’s Theorem yields that B ( X ) ⊗ B ( Y ) = Λ.Next, let f : X × Y R be a B ( X ) ⊗ B ( Y ) − measuable function taking values in a finite subset { a < · · · < a N } of R and define D i := { f = a i } ∈ B ( X ) ⊗ B ( Y ) for i = 1 , · · · , N . By the equality B ( X ) ⊗ B ( Y ) = S , we see that φ f ( x, P ) = Z y ∈ Y f ( x, y ) P ( dy ) = N X i =1 a i Z y ∈ Y { ( x,y ) ∈ D i } P ( dy ) = N X i =1 a i φ D i ( x, P ) , ( x, P ) ∈ X × P ( Y )is B ( X ) ⊗ B (cid:0) P ( Y ) (cid:1) − measurable. Then we can use the standard approximation to extend this measurability for ageneral [ −∞ , ∞ ] − valued, B ( X ) ⊗ B ( Y ) − measuable function. (cid:3) Lemma A.3. Given t ∈ [0 , ∞ ) and P ∈ P (Ω) , let τ be a [0 , ∞ ) − valued F W t − stopping time and let (cid:8) P tτ,ω (cid:9) ω ∈ Ω be ther.c.p.d. of P with respect to F ( τ ) . If (cid:8) β n (cid:9) n ∈ N is a sequence of [0 , ∞ ] − valued, B P (Ω) − measurable random variablesthat converge to in probability P , there are a subsequence { n j } j ∈ N of N and a N β ∈ N P (cid:0) F ( τ ) (cid:1) such that (cid:8) β n j (cid:9) j ∈ N converge to in probability P tτ,ω for any ω ∈ N cβ . Proof: Let [0 , ∞ ] − valued, B P (Ω) − measurable random variables (cid:8) β n (cid:9) n ∈ N converge to 0 in probability P , i.e.lim n →∞ E P (cid:2) {| β n | > /k } (cid:3) = lim n →∞ P (cid:8) | β n | > /k (cid:9) = 0 , ∀ k ∈ N . (A.2)In particular, lim n →∞ E P (cid:2) {| β n | > } (cid:3) = 0 allows us to extract a subsequence S := (cid:8) n (1) j (cid:9) j ∈ N from N such thatlim j →∞ (cid:8)(cid:12)(cid:12) β n (1) j (cid:12)(cid:12) > (cid:9) = 0, P − a.s. Clearly, (cid:8) β n (1) j (cid:9) j ∈ N also satisfies (A.2). Then via lim j →∞ E P h (cid:8)(cid:12)(cid:12) β n (1) j (cid:12)(cid:12) > / (cid:9)i = 0,we can find a subsequence S := (cid:8) n (2) j (cid:9) j ∈ N of S such that lim j →∞ (cid:8)(cid:12)(cid:12) β n (2) j (cid:12)(cid:12) > / (cid:9) = 0, P − a.s. Inductively, for each ℓ ∈ N we can select a subsequence S ℓ +1 := (cid:8) n ( ℓ +1) j (cid:9) j ∈ N of S ℓ such that lim j →∞ (cid:8)(cid:12)(cid:12) β n ( ℓ +1) j (cid:12)(cid:12) > / ( ℓ +1) (cid:9) = 0, P − a.s.For any j ∈ N , we set n j := n ( j ) j , which belongs to S ℓ for ℓ = 1 , · · · , j .Let ℓ ∈ N . Since { n j } ∞ j = ℓ ⊂ S ℓ , it holds P − a.s. that lim j →∞ (cid:8)(cid:12)(cid:12) β n j (cid:12)(cid:12) > /ℓ (cid:9) = 0. Then the bound convergence theoremand (R2) imply that for some N ℓβ ∈ N P (cid:0) F ( τ ) (cid:1) ,0 = lim j →∞ E P h (cid:8)(cid:12)(cid:12) β n j (cid:12)(cid:12) > /ℓ (cid:9)(cid:12)(cid:12) F ( τ ) i ( ω ) = lim j →∞ E P tτ,ω h (cid:8)(cid:12)(cid:12) β n j (cid:12)(cid:12) > /ℓ (cid:9)i = lim j →∞ P tτ,ω (cid:8)(cid:12)(cid:12) β n j (cid:12)(cid:12) > /ℓ (cid:9) , ∀ ω ∈ (cid:0) N ℓβ (cid:1) c . (A.3)Set N β := ∪ ℓ ∈ N N ℓβ ∈ N P (cid:0) F ( τ ) (cid:1) . For any ω ∈ N cβ , (A.3) means that (cid:8) β n j (cid:9) j ∈ N converge to 0 in probability P tτ,ω . (cid:3) Lemma A.4. Let m ∈ N and t, s, r ∈ [0 , ∞ ) . If ξ is an R m − valued F W t s + r − measurable random variable, then for any ω ∈ Ω there is an R m − valued F W t + s r − meaurable random variable ξ ω such that ξ ( ω ′ ) = ξ ω ( ω ′ ) for any ω ′ ∈ W ts,ω := (cid:8) ω ′ ∈ Ω : W tr ( ω ′ ) = W tr ( ω ) , ∀ r ∈ [0 , s ] (cid:9) . Proof: 1) Let ω ∈ Ω and define S ω := (cid:8) A ⊂ Ω : ∃ A ∈ F W t + s r s.t. W ts,ω ∩ A = W ts,ω ∩A (cid:9) , which is a sigma-field of Ω.Let a ∈ [0 , s + r ] and E ∈ B ( R d ). We discuss by three cases that ( W ta ) − ( E ) ∈ S ω :(i) If a ∈ [0 , s ] and W ta ( ω ) ∈ E , then W ts,ω ∩ ( W ta ) − ( E ) = W ts,ω ∩{ ω ′ ∈ Ω : W ta ( ω ′ ) ∈ E} = W ts,ω = W ts,ω ∩ Ω. eferences a ∈ [0 , s ] and W ta ( ω ) / ∈ E , then W ts,ω ∩ ( W ta ) − ( E ) = W ts,ω ∩{ ω ′ ∈ Ω : W ta ( ω ′ ) ∈ E} = ∅ = W ts,ω ∩∅ .(iii) If a ∈ ( s, s + r ], we set E ω := (cid:8) x − W ts ( ω ) : ∀ x ∈ E} ∈ B ( R d ) then ( W t + sa − s ) − ( E ω ) ∈ F W t + s r satisfies that W ts,ω ∩ ( W ta ) − ( E ) = W ts,ω ∩{ ω ′ ∈ Ω : W ta ( ω ′ ) ∈ E} = W ts,ω ∩{ ω ′ ∈ Ω : W ta ( ω ′ ) − W ts ( ω ′ ) ∈ E ω } = W ts,ω ∩{ ω ′ ∈ Ω : W t + sa − s ( ω ′ ) ∈ E ω } = W ts,ω ∩ ( W t + sa − s ) − ( E ω ) . Hence, F W t s + r ⊂ S ω . Let ξ is an R m − valued F W t s + r − measurable random variable. It suffices to assume that m = 1 and ξ is nonnegative.Fix ω ∈ Ω and let k, i ∈ N . As A ki := { ( i − − k ≤ ξ < i − k } ∈ F W t s + r . Part (1) shows that for some A ωk,i ∈ F W t + s r , W ts,ω ∩ A ki = W ts,ω ∩A ωk,i . Set ξ kω := P ki =1 ( i − − k A ωk,i ∈ F W t + s r . For any ω ′ ∈ Ω, one can deduce that { ω ′ ∈ W ts,ω } ξ ( ω ′ ) = lim k →∞ ↑ (cid:16) { ω ′ ∈ W ts,ω } k X i =1 ( i − − k { ω ′ ∈ A ki } (cid:17) = lim k →∞ ↑ (cid:16) k X i =1 ( i − − k { ω ′ ∈ W ts,ω ∩A ωk,i } (cid:17) = { ω ′ ∈ W ts,ω } lim k →∞ ↑ ξ kω ( ω ′ ) . So ξ ω := lim k →∞ ξ kω { lim k →∞ ξ kω < ∞} is a [0 , ∞ ) − valued F W t + s r − measurable random variable satisfying that for any ω ′ ∈ W ts,ω , lim k →∞ ξ kω ( ω ′ ) = lim k →∞ ↑ ξ kω ( ω ′ ) = ξ ( ω ′ ) and thus ξ ω ( ω ′ ) = lim k →∞ ξ kω ( ω ′ ) = ξ ( ω ′ ). (cid:3) Acknowledgments We are grateful to Xiaolu Tan for helpful comments. References [1] S. Ankirchner, M. Klein, and T. Kruse , A verification theorem for optimal stopping problems with expec-tation constraints , Appl. Math. Optim., 79 (2019), pp. 145–177.[2] K. J. Arrow, D. Blackwell, and M. A. 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