A First-Order BSPDE for Swing Option Pricing: Classical Solutions
aa r X i v : . [ q -f i n . P R ] N ov A First-Order BSPDE for Swing Option Pricing:Classical Solutions
Christian Bender , Nikolai Dokuchaev Submitted: February 26, 2014. Revised: November 20, 2014
Abstract
In Bender and Dokuchaev (2014) we studied a control problem related to swing optionpricing in a general non-Markovian setting. The main result there shows that the valueprocess of this control problem can be uniquely characterized in terms of a first order backwardSPDE and a pathwise differential inclusion. In the present paper we additionally assume thatthe cashflow process of the swing option is left-continuous in expectation (LCE). Under thisassumption we show that the value process is continuously differentiable in the space variablethat represents the volume which the holder of the option can still exercise until maturity.This gives rise to an existence and uniqueness result for the corresponding backward SPDE ina classical sense. We also explicitly represent the space derivative of the value process in termsof a nonstandard optimal stopping problem over a subset of predictable stopping times. Thisrepresentation can be applied to derive a dual minimization problem in terms of martingales.
Keywords:
Backward SPDE, optimal stopping, stochastic optimal control, swing options.
AMS classification:
Motivated by the pricing problem for swing options, we consider the following optimal controlproblem. The investor’s aim is to maximize the expected reward of exercising an adapted cashflowprocess X , i.e. she wishes to maximize E (cid:20)Z T u ( s ) X ( s ) ds (cid:21) (1.1) Saarland University, Department of Mathematics, Postfach 151150, D-66041 Saarbr¨ucken, Germany, [email protected] . Department of Mathematics & Statistics, Curtin University, GPO Box U1987, Perth, 6845 Western Australia,Australia,
[email protected] u with values in [0 , L ] which satisfy the condition R T u ( s ) ds ≤
1. Here L is a local constraint, which restricts the maximal rate at which the holder of the option canexercise the cashflow process X . Moreover, the global constraint (which is a finite fuel constraint)imposes that the total volume spent by the holder is bounded by one. We refer to Keppo (2004)for the modelling of swing options as continuous time optimal control problems, and note that theabove problem and related problems were recently investigated in a Markovian diffusion settingby Benth et al. (2011); Dokuchaev (2013); Basei et al. (2014).In our companion paper (Bender and Dokuchaev, 2014), to which we also refer for furtherreferences on swing option pricing, we studied the above optimal control problem in a generalnon-Markovian setting under the following mild assumptions: ( X ( t ) , ≤ t ≤ T ) is a nonneg-ative, right-continuous, F -adapted stochastic process on a filtered probability space (Ω , F , F , P )satisfying the usual conditions such that E [ sup ≤ t ≤ T X ( t ) p ] < ∞ (1.2)for some p >
1. We consider these conditions as standing assumptions for the rest of the paper.Then, a dynamic formulation of (1.1) reads as follows: For any [0 , T ]-valued stopping time τ and F τ -measurable, ( −∞ , Y denote by U ( τ, Y ) the set of all F -adaptedprocesses with values in [0 , L ] such that R Tτ u ( s ) ds ≤ − Y . Hence, the investor enters thecontract at time τ and can spend a remaining total volume of 1 − Y up to maturity T . Thecorresponding value of the optimization problem is¯ J ( τ, Y ) := esssup u ∈ U ( τ,Y ) E (cid:20) Z Tτ u ( s ) X ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) F τ (cid:21) . The main result in Bender and Dokuchaev (2014) states (roughly speaking) that a good ver-sion ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , J ( t, y ) = E (cid:20) L Z Tt ( X ( s ) + D − y J ( s, y )) + ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ,J ( t,
1) = 0 , which is smooth enough to ensure that the differential inclusion u ( s ) ∈ { } , X ( s ) + D − y J ( s, y + R st u ( r ) dr ) < { L } , X ( s ) + D − y J ( s, y + R st u ( r ) dr ) > , L ] , X ( s ) + D − y J ( s, y + R st u ( r ) dr ) = 0 . u ∈ U ( t, y ). Here, D − y denotes the lefthand side derivative in the y -variable and( · ) + denotes the positive part.The main purpose of the present paper is to study regularity of (the good version J ( t, y ) of)the value process in the y -variable and to replace the above smoothness condition in terms of thedifferential inclusion by a classical differentiability condition. To this end we shall assume that X is additionally left-continuous in expectation (LCE), i.e. for every [0 , T ]-valued stopping time σ and every nondecreasing sequence of [0 , T ]-valued stopping times ( σ n ) n ∈ N with limit σ it holdsthat lim n →∞ E [ X ( σ n )] = E [ X ( σ )] . (1.3)Intuitively this means that the jumps of X occur at total surprise and cannot be predicted. Underthis assumption we are going to prove the following theorem: Theorem 1.1.
Suppose the standing assumptions and that X is left-continuous in expectation.For every t ∈ [0 , T ] denote ∆ t := (1 − L ( T − t ) , , ¯∆ t := [1 − L ( T − t ) , . Then,(i) There is a measurable version ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ¯∆ t ) of ( ¯ J ( t, y ) , t ∈ [0 , T ] , y ∈ ¯∆ t ) whichfulfills:a) There is a set ¯Ω ∈ F with P ( ¯Ω) = 1 such that D − y J ( t, ω, y ) exists for every t ∈ [0 , T ] , y ∈ ¯∆ t ,and ω ∈ ¯Ω and is left-continuous in y . Moreover, J is Lipschitz in y in the following sense:There is an integrable random variable C satisfying | J ( t, ω, y ) − J ( t, ω, y ) | ≤ C ( ω ) | y − y | for every t ∈ [0 , T ] , ω ∈ ¯Ω and y , y ∈ ¯∆ t .b) For every t ∈ [0 , T ] , there is a set Ω t of full P -measure such that, for every ω ∈ Ω t , themapping y J ( t, ω, y ) is continuously differentiable on ∆ t .c) For every t ∈ [0 , T ] and y ∈ ∆ t , ∂∂y J ( s, ω, y ) exists for λ [ t,T − − yL ] ⊗ P -almost every ( s, ω ) , J ( t, y ) = E " Z TT − − yL LX ( s ) ds + L Z T − − yL t ( X ( s ) + ∂∂y J ( s, y )) + ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t (1.4)3 olds P -almost surely and the boundary conditions J ( t, − L ( T − t )) = E (cid:20) Z Tt LX ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , J ( t,
1) = 0 (1.5) are satisfied.(ii) Conversely, if ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ¯∆ t ) is a measurable random field satisfying a), b), andc), then it is a version of ¯ J , i.e. for every t ∈ [0 , T ] , y ∈ ¯∆ t J ( t, y ) = ¯ J ( t, y ) P -almost surely. In the above theorem and for the remainder of the paper λ denotes the Lebesgue measure and λ [ a,b ] its restriction to the interval [ a, b ].We note that the above theorem characterizes the value process on the set { ( t, y ); 0 ≤ t ≤ T, − L ( T − t ) ≤ y ≤ } . For y ≤ − L ( T − t ) the optimization becomes trivial, because theremaining volume 1 − y is at least as large as the maximal volume L ( T − t ) which one can spendwhen exercising at the maximal rate of L . Hence, L [ t,T ] is an optimal strategy and¯ J ( t, y ) = E (cid:20) Z Tt LX ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) . (1.6)This also explains the boundary condition (1.5) at y = 1 − L ( T − t ).We emphasize that condition b) in Theorem 1.1 is a classical C -condition on the solution of theBSPDE. Hence, we can interpret this theorem as an existence and uniqueness result of a classicalsolution for the BSPDE (1.4)–(1.5). Taking into account that this BSPDE is a non-Markovianversion of a Hamilton-Jacobi-Bellman equation, we think that existence of a classical solution is astriking feature. Indeed, recent studies of the corresponding HJB equation for stochastic controlproblems with integral constraints in the Markovian diffusion case such as Basei et al. (2014) onlydiscuss the HJB equation in the framework of viscosity solutions.The paper is organized as follows: In Section 2 we give a recap of some of the results inBender and Dokuchaev (2014), to which we refer as [BD] from now on. The proof of Theorem 1.1is divided into two parts. In Section 3 we prove the uniqueness part, i.e. we show that everyadapted random field which satisfies a), b), and c) coincides necessarily with the value process¯ J ( t, y ). It will turn out that the LCE assumption is not required for this part of Theorem 1.1. Itis however crucial for the smoothness part which is proved in Section 3. Here we show that a goodversion of the value process is indeed continuously differentiable in the sense of b). The derivativein the space variable is additionally represented via some nonstandard optimal stopping problemswhich can be linked to the interpretation of the derivative as the marginal value of the underlying4ontrol problem. Finally, in Section 4 we derive a dual minimization over martingales and relatethe minimizing martingale to the derivative of the value process. In this section we state some results from [BD] for handy reference. We recall that the standingassumptions are in force without further mention.The first result, Proposition 3.5 in [BD], provides a good version of the value process ¯ J . Proposition 2.1.
There is an adapted random field ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , such that J ( τ, Y ) = ¯ J ( τ, Y ) P − a.s. for every [0 , T ] -valued stopping time τ and every F τ -measurable, ( −∞ , -valued random variable Y . Moreover, J satisfies the following:There is a set ¯Ω ∈ F with P ( ¯Ω) = 1 such that the following properties hold on ¯Ω :1. For every y ∈ ( −∞ , , the mapping t J ( t, y ) is RCLL.2. For every t ∈ [0 , T ] and y , y ∈ ( −∞ , | J ( t, y ) − J ( t, y ) | ≤ sup r ∈ [0 ,T ] Z ( r ) ! | y − y | where Z ( t ) is a RCLL modification of E [sup r ∈ [0 ,T ] X ( r ) | F t ] which satisfies sup r ∈ [0 ,T ] Z ( r ) < ∞ on ¯Ω .3. For every t ∈ [0 , T ] , the mapping y J ( t, y ) is concave. The main theorem of [BD] characterizes the value process. It does not require the LCEassumption.
Theorem 2.2. (i) The version ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , of ( ¯ J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , constructed in Proposition 2.1 satisfies:a’) There is a set ¯Ω ∈ F with P ( ¯Ω) = 1 such that D − y J ( t, ω, y ) exists for every t ∈ [0 , T ] , y ∈ ( −∞ , and ω ∈ ¯Ω and is left-continuous in y . Moreover, J is Lipschitz in y in the followingsense: There is an integrable random variable C satisfying | J ( t, ω, y ) − J ( t, ω, y ) | ≤ C ( ω ) | y − y | or every t ∈ [0 , T ] , ω ∈ ¯Ω and y , y ∈ ( −∞ , .b’) For every ( t, y ) ∈ [0 , T ] × ( −∞ , , there is a control u t,y ∈ U ( t, y ) such that the differentialinclusion u t,y ( s ) ∈ { } , X ( s ) + D − y J ( s, y + R st u t,y ( r ) dr ) < { L } , X ( s ) + D − y J ( s, y + R st u t,y ( r ) dr ) > , L ] , X ( s ) + D − y J ( s, y + R st u t,y ( r ) dr ) = 0 . (2.1) is satisfied λ [ t,T ] ⊗ P -almost surely.c’) For every ( t, y ) ∈ [0 , T ] × ( −∞ , , J ( t, y ) = E (cid:20) L Z Tt ( X ( s ) + D − y J ( s, y )) + ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ,J ( t,
1) = 0 ,P -almost surely.(ii) Conversely, if ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , is a measurable random field satisfying a’),b’), and c’), then it is a version of ¯ J , i.e. for every ( t, y ) ∈ [0 , T ] × ( −∞ , J ( t, y ) = ¯ J ( t, y ) P -almost surely. In this case, u t,y ∈ U ( t, y ) is optimal for ¯ J ( t, y ) , if and only if (2.1) is satisfied. Theorem 2.2 includes an existence result for optimal controls. One can even choose an optimalcontrol with some additional properties which turns out to be useful later.
Proposition 2.3.
For every [0 , T ] -valued stopping time τ and every F τ -measurable, ( −∞ , -valued random variable Y , there is an optimal strategy u τ ,Y ∈ U ( τ , Y ) for ¯ J ( τ , Y ) such that u τ ,Y ( r ) = L on { L ( T − r ) ≤ − ( Y + Z rτ u τ ,Y ( s ) ds ) } . (2.2) and Z Tτ u τ ,Y ( s ) ds = 1 − Y on { L ( T − τ ) ≥ − Y } (2.3)The above proposition actually is a direct consequence of Proposition 3.2 in [BD].As a corollary to Theorem 2.2 we observe that it is optimal to exercise a submartingale aslate as possible and a supermartingale as early as possible, which is as expected. Corollary 2.4.
Suppose X satisfies the standing assumptions, τ is a [0 , T ] -valued stopping timeand Y is a F τ -measurable, ( −∞ , -valued random variable.(i) If X is an RCLL submartingale, then u τ ,Y = L [( T − (1 − Y ) /L ) ∨ τ ,T ] is optimal for ¯ J ( τ , Y ) .(ii) If X is an RCLL supermartingale, then u τ ,Y = L [ τ , ( τ +(1 − Y ) /L ) ∧ T ] is optimal for ¯ J ( τ , Y ) . Proof.
We only prove the submartingale case, as the supermartingale case is similar. For any[0 , T ]-valued stopping time τ and any F τ -measurable, ( −∞ , Y wedefine ¯ V ( τ , Y ) = E (cid:20) Z Tτ u τ ,Y ( s ) X ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = E " Z Tτ ∨ ( T − (1 − Y ) /L ) LX ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t . Denote the good version of the value process constructed in Proposition 2.1 by J . Then, J ( τ , Y ) ≥ ¯ V ( τ , Y ) P -almost surely. Hence, u τ ,Y is optimal if and only if0 = E [ J ( τ , Y ) − ¯ V ( τ , Y )] = E " J ( τ , Y ) − Z Tτ ∨ ( T − (1 − Y ) /L ) LX ( s ) ds . (2.4)Notice that, if (2.4) holds for all deterministic pairs ( τ , Y ), then it is also true for general pairs.Indeed, J ( τ , Y ) = ¯ V ( τ , Y ) then holds P -almost surely for pairs ( τ , Y ) which take at mostcountably many values. By the continuity properties of J one can then pass to the limit to obtain(2.4) for general pairs ( τ , Y ).It is hence sufficient to show optimality of u t,y for deterministic t ∈ [0 , T ] and y ∈ ( −∞ , V ( t, y ) of ¯ V ( t, y ) which satisfies a’) in Theorem 2.2. By the definition of ¯ V it is straightforwardthat D − y V ( t, y ) = ( − E [ X (( T − (1 − y ) /L ) − ) | F t ] , t < T − (1 − y ) /L , t ≥ T − (1 − y ) /L, where X ( s − ) denotes the left limit of X at s . Hence, by the submartingale property, X ( s ) + D − y V ( s, y ) ≤ s < T − (1 − y ) /L and X ( s ) + D − y V ( s, y ) = X ( s ) ≥ s ≥ T − (1 − y ) /L .Thus, V solves the BSPDE in c’) of Theorem 2.2. It is also straightforward to see that u t,y = L [ t ∨ ( T − (1 − y ) /L ) ,T ] solves the differential inclusion in b’) of Theorem 2.2 with V in place of Y ,because X ( s )+ D − y V ( s, y + Z st u t,y ( r ) dr ) = ( X ( s ) − E [ X (( T − (1 − y ) /L ) − ) | F s ] ≤ , s < T − (1 − y ) /LX ( s ) ≥ , s ≥ T − (1 − y ) /L. Consequently, by Theorem 2.2, u t,y is optimal. 7 Uniqueness of classical solutions
This section is devoted to the proof of the uniqueness part of Theorem 1.1. It relies on Theorem2.2, (ii). This is what we are actually going to show:
Theorem 3.1.
Under the standing assumption, suppose that ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ¯∆ t ) is ameasurable random field satisfying a), b), and c) of Theorem 1.1. Define J ( t, y ) := J ( t, − L ( T − t )) , t ∈ [0 , T ] , y < − L ( T − t ) . Then ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , fulfills conditions a’), b’), and c’) of Theorem 2.2. Inparticular, it is a version of ¯ J , i.e. for every ( t, y ) ∈ [0 , T ] × ( −∞ , J ( t, y ) = ¯ J ( t, y ) P -almost surely. For the remainder of this Section we assume that ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ¯∆ t ) is a measurablerandom field satisfying a), b), and c) of Theorem 1.1, and that it is extended to [0 , T ] × ( −∞ , Lemma 3.2. J satisfies a’) and c’) in Theorem 2.2.Proof. Property a’) is a direct consequence of a) in Theorem 1.1 and the constant extrapolationof J . Property c’) holds for t ∈ [0 , T ] and 1 − L ( T − t ) < y < D − y J ( s, y ) = 0 for s ≥ T − (1 − y ) /L by the constant extrapolation. It then extendsto y = 1 by the continuity properties in a’). Finally, for y ≤ − L ( T − t ), J ( t, y ) = J ( t, − L ( T − t )) = E (cid:20) Z Tt LX ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = E (cid:20) L Z Tt ( X ( s ) + D − y J ( s, y )) + ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , where we used the boundary condition (1.5) and again the fact that D − y J ( t, y ) = 0 for y ≤ − L ( T − t ).In order to prove that the differential inclusion (2.1) has a solution, and hence, J satisfiescondition b’) in Theorem 2.2, we denoteΓ + ( t, ω ) = { y ∈ (1 − L ( T − t ) , X ( t, ω ) + D − y J ( t, ω, y ) > } Γ − ( t, ω ) = { y ∈ (1 − L ( T − t ) , X ( t, ω ) + D − y J ( t, ω, y ) < } Γ ( t, ω ) = R \ (Γ + ( t, ω ) ∪ Γ − ( t, ω )) . As a preparation we first prove the following lemma.8 emma 3.3.
For every ( t , y ) ∈ [0 , T ] × R there is an adapted process ˆ u such that ˆ u ( s ) ∈ { } , y + R st ˆ u ( r ) dr ∈ Γ − ( s ) { L } , y + R st ˆ u ( r ) dr ∈ Γ + ( s )[0 , L ] , y + R st ˆ u ( r ) dr ∈ Γ ( s ) (3.1) λ [ t ,T ] ⊗ P -almost surely.Proof. We apply a somewhat standard technique approximating the differential inclusion by a se-quence of differential equations with Lipschitz coefficients, see e.g. the textbook by Aubin and Cellina(1984). The crucial observation is that we can construct the sequence of approximating differ-ential equations in a monotonic way, which leads to almost sure convergence (while withoutmonotonicity one obtains weak L -convergence only).To this end define for m ∈ N Γ + ,m ( t, ω ) = { y ∈ (1 − L ( T − t ) , X ( t, ω ) + D − y J ( t, ω, y ) ≥ /m } , and for n ∈ N , s ∈ [0 , T ], and y ∈ R F n ( s, ω, y ) = 2 n Z yy − − n L { v ; ∃ m ( ω ) ∈ N ∀ η ∈ [0 , − n ] v + η ∈ Γ + ,m ( ω ) ( s,ω ) } dv. Notice that due to the leftcontinuity assumption on D − y J ∃ m ( ω ) ∈ N ∀ η ∈ [0 , − n ] v + η ∈ Γ + ,m ( ω ) ( s, ω ) ⇔ v ∈ (1 − L ( T − s ) , − − n ) and inf η ∈ [0 , − n ] ∩ Q X ( s, ω ) + D − y J ( s, ω, v + η ) > . Hence, the integrand in the definition of F n is measurable (as function in ( s, ω, v )) and, in par-ticular, F n ( · , y ) is an ( F s ) s ∈ [ t ,T ] -adapted process for every y ∈ R . Moreover, by construction, F n is Lipschitz in y with constant L n (independent of s, ω ). Hence there is a unique (up toindistinguishability) continuous and adapted process y n which satisfies y n ( s ) = y + Z st F n ( r, y n ( r )) dr, s ∈ [ t , T ] . We define the sequence of [0 , L ]-valued adapted process ( u n ) n ∈ N via u n ( s ) = F n ( s, y n ( s )) , s ∈ [ t , T ] . This sequence belongs to the set of adapted and [0 , L ]-valued processes, which as a subset of L ([ t , T ] × Ω) is bounded, closed and convex, and, thus, weakly compact. Consequently, there is9n adapted [0 , L ]-valued process ˆ u and a subsequence ( n k ) such that u n k → ˆ u, k → ∞ weakly in L ([ t , T ] × Ω). We claim that ˆ u satisfies (3.1). In order to see this, fix s ∈ [ t , T ] andchoose an arbitrary ξ ∈ L (Ω). Then, ξ [ t ,s ] ∈ L ([ t , T ] × Ω). Hence, E [ ξ ( y n k ( s ) − y )] = E [ Z Tt ξ [ t ,s ] ( r ) u n k ( r ) dr ] → E [ Z Tt ξ [ t ,s ] ( r )ˆ u ( r ) dr ] = E [ ξ Z st ˆ u ( r ) dr ] , i.e. y n k ( s ) converges to y + R st ˆ u ( r ) dr weakly in L (Ω). We shall now show that this convergenceholds almost surely, indeed. To this end we first observe that F n ( s, y ) ≤ F n +1 ( s, y ) for every pair( s, y ), because F n ( s, y ) = 2 n Z yy − − ( n +1) L { v ; ∃ m ∈ N ∀ η ∈ [0 , − n ] v − − ( n +1) + η ∈ Γ + ,m ( s ) } dv +2 n Z yy − − ( n +1) L { v ; ∃ m ∈ N ∀ η ∈ [0 , − n ] v + η ∈ Γ + ,m ( s ) } dv ≤ · n Z yy − − ( n +1) L { v ; ∃ m ∈ N ∀ η ∈ [0 , − ( n +1) ] v + η ∈ Γ + ,m ( s ) } dv = F n +1 ( s, y ) . As y n +1 − y n is (up to indistinguishability) the unique continuous solution of the linear differentialequation y n +1 ( s ) − y n ( s ) = Z st ( b n ( r )( y n +1 ( r ) − y n ( r )) + c n ( r )) dr for b n ( r ) = { y n +1 ( r ) = y n ( r ) } F n +1 ( r, y n +1 ( r )) − F n +1 ( r, y n ( r )) y n +1 ( r ) − y n ( r ) c n ( r ) = F n +1 ( r, y n ( r )) − F n ( r, y n ( r )) ≥ P -measure independent of n, s , y n +1 ( s ) − y n ( s ) = Z st c n ( r ) exp { Z sr b n ( u ) du } dr ≥ , i.e. the sequence y n ( s ) is nondecreasing. As it is bounded by y + L ( T − t ), the P -a.s. limit y ( s ) = lim n →∞ y n ( s )exists for every s ∈ [ t , T ]. By dominated convergence ( y n ( s )) converges to y ( s ) strongly in L (Ω),10nd in view of the weak convergence obtained above we conclude that for every s ∈ [ t , T ] y ( s ) = y + Z st ˆ u ( r ) dr, P -a.s.We next introduce the B [ t ,T ] ⊗ F -measurable set A = { ( s, ω ); y ( s, ω ) ∈ Γ + ( s, ω ) and inf η ∈ [ − − n , − n ] ∩ Q X ( s, ω ) + D − y J ( s, ω, y n ( s, ω ) + η ) ≤ } , where ‘i.o.’ means that the event occurs for infinitely many n ’s. Denote its s -section for s ∈ [ t , T ]by A s = { ω ; ( s, ω ) ∈ A } . As y n ( s ) converges to y ( s ), we observe that A s ⊂ Ω cs , where Ω s is the set of full P -measurein property b) of Theorem 1.1, on which y D − y J ( s, y ) is continuous. Hence, P ( A s ) = 0.Consequently, by weak convergence of ( u n k ) to ˆ u , E [ Z Tt ( L − ˆ u ( s )) { y ( s ) ∈ Γ + ( s ) } ds ] = lim k →∞ E [ Z Tt ( L − u n k ( s )) { y ( s ) ∈ Γ + ( s ) } A cs ds ] . (3.2)We now notice that for every s ∈ [ t , T ] (using convergence of y n ( s ) to y ( s ) for the second identityto conclude that y n ( s ) ∈ (1 − L ( T − s ) + 2 − n , − − n ) for sufficiently large n ) { y ( s ) ∈ Γ + ( s ) } ∩ A cs = { y ( s ) ∈ Γ + ( s ) } ∩ [ N ∈ N \ n ≥ N (cid:26) inf η ∈ [ − − n , − n ] ∩ Q X ( s ) + D − y J ( s, y n ( s ) + η ) > (cid:27) = { y ( s ) ∈ Γ + ( s ) } ∩ [ N ∈ N \ n ≥ N (cid:8) ∃ m ∈ N ∀ η ∈ [ − − n , − n ] y n ( s ) + η ∈ Γ + ,m ( s ) (cid:9) ⊂ { y ( s ) ∈ Γ + ( s ) } ∩ [ N ∈ N \ n ≥ N { F n ( s, y n ( s )) = L } = { y ( s ) ∈ Γ + ( s ) } ∩ [ N ∈ N \ n ≥ N { u n ( s ) = L } Thus, by dominated convergence, the right-hand side of (3.2) converges to zero. An analogousargument shows E [ Z Tt ˆ u ( s ) { y ( s ) ∈ Γ − ( s ) } ds ] = 0 . As ˆ u is [0 , L ]-valued these two identities imply that ˆ u solves the differential inclusion (3.1). Proposition 3.4. J fulfills b’) of Theorem 2.2, i.e. for every ( t, y ) ∈ [0 , T ] × ( −∞ , , there is a t,y ∈ U ( t, y ) which satisfies (2.1).Proof. We first denote by ˆ u the solution of (3.1) with ( t , y ) = ( t, y ) constructed in the previouslemma. Define ¯ σ U := inf { r ≥ t ; y + Z rt ˆ u ( s ) ds ≥ } ∧ T, ¯ σ L := inf { r ≥ t ; y + Z rt ˆ u ( s ) ds ≤ − L ( T − t ) } ∧ T, and u t,y ( r ) := ( ˆ u ( r ) , r ∈ [ t, ¯ σ U ∧ ¯ σ L ) L { ¯ σ L < ¯ σ U } r ∈ [¯ σ U ∧ ¯ σ L , T ]Let y t,y ( r ) = y + R rt u t,y ( s ) ds and y ( r ) = y + R rt ˆ u ( s ) ds . We wish to show that u t,y solves (2.1),for which it suffices to verify that E (cid:20)Z Tt | u t,y ( s ) | { X ( s )+ D − y J ( s,y t,y ( s )) < } + | u t,y ( s ) − L | { X ( s )+ D − y J ( s,y t,y ( s )) > } ds (cid:21) = 0 . (3.3)We decompose E (cid:20)Z Tt | u t,y ( s ) | { X ( s )+ D − y J ( s,y t,y ( s )) < } + | u t,y ( s ) − L | { X ( s )+ D − y J ( s,y t,y ( s )) > } ds (cid:21) = E (cid:20)Z Tt | ˆ u ( s ) | { X ( s )+ D − y J ( s,y ( s )) < } { s< ¯ σ U ∧ ¯ σ L } ds (cid:21) + E (cid:20)Z Tt L { X ( s )+ D − y J ( s,y t,y ( s )) < } { s ≥ ¯ σ U ∧ ¯ σ L } { ¯ σ L < ¯ σ U } ds (cid:21) + E (cid:20)Z Tt | ˆ u ( s ) − L | { X ( s )+ D − y J ( s,y ( s )) > } { s< ¯ σ U ∧ ¯ σ L } ds (cid:21) + E (cid:20)Z Tt L { X ( s )+ D − y J ( s,y t,y ( s )) > } { s ≥ ¯ σ U ∧ ¯ σ L } { ¯ σ L ≥ ¯ σ U } ds (cid:21) = ( I ) + ( II ) + ( III ) + ( IV ) . The previous lemma implies that the first and the third term vanish. For the second term wenotice that y t,y ( s ) ≤ − L ( T − s ) on { s ≥ ¯ σ U ∧ ¯ σ L } ∩ { ¯ σ L < ¯ σ U } by the definition of ¯ σ L and u t,y .But then D − y J ( s, y t,y ( s )) = 0 by the constant extrapolation of J . Hence,( II ) ≤ E (cid:20)Z Tt L { X ( s ) < } ds (cid:21) = 0 , because of the nonnegativivity of X . Similarly one can treat the fourth term. We first observeby c’) of Theorem 2.2 that0 = J ( t,
1) = E (cid:20) L Z Tt ( X ( s ) + D − y J ( s, + ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , X ( s ) + D − y J ( s, ≤ , λ [ t,T ] ⊗ P -a.s.However, we have y t,y ( s ) = 1 on { s ≥ ¯ σ U ∧ ¯ σ L } ∩ { ¯ σ L ≥ ¯ σ U } by the definition of ¯ σ U and u t,y .Hence, ( IV ) ≤ E (cid:20)Z Tt L { X ( s )+ D − y J ( s, > } ds (cid:21) = 0 . We finally note that Z Tt u t,y ( r ) dr = y t,y ( T ) − y ≤ − y. Thus, u t,y belongs to U ( t, y ).In view of Theorem 2.2, (ii), and Lemma 3.2, the above proposition concludes the proof ofTheorem 3.1, and hence of the uniqueness part (ii) of Theorem 1.1. In this section we study regularity of the ‘good’ version J of the value process in the y -variable.For the remainder of this section we always assume that J is the random field constructed inProposition 2.1. Notice that, by concavity, the one-sided derivatives D ± y J ( t, y ) exist. In view ofTheorem 2.2 and (1.6) we observe that J satisfies the requirements of Theorem 1.1, (i), once weestablish the following result. Theorem 4.1.
Suppose that X satisfies the standing assumptions and it is LCE. Then, for every t ∈ [0 , T ] , there is a set Ω t of full P -measure such that, for every ω ∈ Ω t , the mapping y J ( t, ω, y ) is continuously differentiable on (1 − L ( T − t ) , . Moreover, for every t ∈ [0 , T ] and y ∈ (1 − L ( T − t ) , , ∂∂y J ( s, ω, y ) exists for λ [ t,T − − yL ] ⊗ P -almost every ( s, ω ) .Remark . (i) The LCE assumption is crucial for Theorem 4.1 to hold. Example 4.5 in [BD]provides a counterexample to the assertion of this theorem for a process X which fails to be LCE.(ii) Theorem 4.7 below implies that, under the assumptions of Theorem 4.1, the following strongerregularity assertion holds, if and only if X ( T ) = 0 P -almost surely: For every t ∈ [0 , T ], there isa set Ω t of full P -measure such that, for every ω ∈ Ω t , the mapping y J ( t, ω, y )is continuously differentiable on ( −∞ , X ( T ) = 0, see Bender (2011b).Recall that the derivatives − D ± y J ( t, y ) correspond to the marginal value of the control prob-lem. We first choose an optimal control u t,y for ¯ J ( t, y ). Then heuristically, in order to calculate − D − y J ( t, y ) ≈ J ( t,y − ∆ y ) − J ( t,y )∆ y one would like to spend optimally an inifinitesimal additional vol-ume of ∆ y at a time σ where exercise is still possible, i.e. where u t,y ( σ ) < L . This intuitivelyleads to the following optimal stopping problem: maximize E [ X ( σ ) | F t ] over stopping times σ ( ω )which take values in the set { s > t ; u t,y ( s, ω ) < L } . More precisely, the additional volume ∆ y cannot be spent at a single time point σ , but rather in ‘small’ neighborhoods around σ . Thereforewe have to restrict the optimal stopping problem to time points σ , such that, given the strategy u t,y , it is still possible ‘to exercise in small neighborhoods of σ ’. This is how we make this heuristicidea precise.Fix a stopping time τ with values in [0 , T ], an F τ measurable, ( −∞ , Y , and an optimal control u τ ,Y ∈ U ( τ , Y ) which satisfies the properties in Proposition 2.3. Wedenote M ( τ , Y ) := { L ( T − τ ) > − Y > } ∈ F τ , and define A ( τ , Y ) := { t ∈ ( τ , T ]; ∀ ǫ > λ ( { s ∈ [ τ ∨ ( t − ǫ ) , ( t + ǫ ) ∧ T ]; u τ ,Y ( s ) < L } ) > } . This set A ( τ , Y ) is our way to make precise the set of time points t , such that given u τ ,Y , onecan still exercise in small neighborhoods of t . We also introduce B ( τ , Y ) := { t ∈ ( τ , T ]; ∀ ǫ > λ ( { s ∈ [ τ ∨ ( t − ǫ ) , ( t + ǫ ) ∧ T ]; u τ ,Y ( s ) > } ) > } , which corresponds to those points, where one can take away some marginal volume from theoptimal control u τ ,Y .We denote by S pτ + the set of predictable stopping times σ with values in ( τ , T ] ∪ { T } ,by S pA ( τ ,Y ) the set of stopping times σ ∈ S pτ + such that σ takes values in A ( τ , Y ) on the set M ( τ , Y ), and by S pB ( τ ,Y ) the set of stopping times σ ∈ S pτ + such that σ takes values in B ( τ , Y )on the set M ( τ , Y ).The proof of Theorem 4.1 is prepared by several lemmas. We first show that the set S pA ( τ ,Y ) ∩ S pB ( τ ,Y ) is nonempty. Lemma 4.3.
Suppose X satisfies the standing assumptions and ( τ , Y ) are as above. Define ¯ σ = ( inf { t ≥ τ ; R tτ u τ ,Y ( s ) ds / ∈ (1 − Y − L ( T − t ) , − Y ) } , ω ∈ M ( τ , Y ) T, otherwise . (4.1)14 hen, ¯ σ ∈ S pA ( τ ,Y ) ∩ S pB ( τ ,Y ) .Proof. On the set M ( τ , Y ) we have Z Tτ u τ ,Y ( s ) ds = 1 − Y . by Proposition 2.3. Moreover, as 0 ∈ (1 − Y − L ( T − t ) , − Y ) on M ( τ , Y ), we get τ < ¯ σ ≤ T on M ( τ , Y ). Hence, the stopping time ¯ σ takes values in ( τ , T ] ∪ { T } . The sequence (¯ σ n ) definedby¯ σ n = inf { t ≥ τ ; R tτ u τ ,Y ( s ) ds / ∈ (1 − Y − L ( T − t ) + 1 /n, − Y − /n ) } , ω ∈ M ( τ , Y )and τ ≤ T − Tn T − Tn , otherwise . announces ¯ σ , because t R tτ u τ ,Y ( s ) ds is continuous and τ < T on M ( τ , Y ). Here, we usethe convention ( a, b ) = ∅ for a ≥ b . Therefore, ¯ σ is predictable. It, thus, remains to show that ¯ σ takes values in A ( τ , Y ) ∩ B ( τ , Y ) on the set M ( τ , Y ). We define¯ σ U = inf { t ∈ [ τ , T ]; Z tτ u τ ,Y ( s ) ds ≥ − Y } , ¯ σ L = inf { t ∈ [ τ , T ]; Z tτ u τ ,Y ( s ) ds ≤ − Y − L ( T − t ) } , (with the usual convention that the infimum of the empty set is + ∞ ). Then ¯ σ = ¯ σ U ∧ ¯ σ L on M ( τ , Y ). First note that by definition of ¯ σ U and ¯ σ L we obtain on M ( τ , Y ) ∀ ǫ > λ ( { s ∈ [ τ ∨ (¯ σ U − ǫ ) , ¯ σ U ]; u τ ,Y ( s ) > } ) > , on { ¯ σ U ≤ T } , ∀ ǫ > λ ( { s ∈ [ τ ∨ (¯ σ L − ǫ ) , ¯ σ L ]; u τ ,Y ( s ) < L } ) > , on { ¯ σ L ≤ T } . Hence, ¯ σ U ∈ B ( τ , Y ) on { ¯ σ U ≤ T } and ¯ σ L ∈ A ( τ , Y ) on { ¯ σ L ≤ T } . In particular, we have¯ σ ∈ A ( τ , Y ) ∩ B ( τ , Y ) on { ¯ σ U = ¯ σ L } . It now suffices to show that ¯ σ U ∈ A ( τ , Y ) on { ¯ σ U < ¯ σ L } and ¯ σ L ∈ B ( τ , Y ) on { ¯ σ L < ¯ σ U } . Obviously, u τ ,Y ( r ) = 0 < L almost everywhere on [¯ σ U , T ].However, ¯ σ U < T on { ¯ σ U < ¯ σ L } . Therefore, ¯ σ U ∈ A ( τ , Y ) on { ¯ σ U < ¯ σ L } . Now, by (2.2), we get u τ ,Y ( r ) = L > σ L , T ]. This implies ¯ σ L ∈ B ( τ , Y ) on { ¯ σ L < ¯ σ U } .The next lemma relates the left-hand side derivative D − y J ( τ , Y ) to stopping times in S pA ( τ ,Y ) .We recall that X is said to be LCE at a stopping time σ , if (1.3) holds for every nondecreasingsequence of [0 , T ]-valued stopping times ( σ n ) n ∈ N with limit σ .15 emma 4.4. Suppose X satisfies the standing assumptions, ( τ , Y ) are as above, and σ ∈ S pA ( τ ,Y ) . Then, there is a sequence of stopping times ( ρ n ) , taking values in [ τ , T ] , which nonde-creasingly converges to σ and such that − D − y J ( τ , Y ) ≥ E [ X ( ρ n ) | F τ ] on M ( τ , Y ) for every n ∈ N . Moreover, − E [ D − y J ( τ , Y ) M ( τ ,Y ) ] ≥ E [ X ( σ ) M ( τ ,Y ) ] , if X is LCE at σ .Proof. As σ is predictable, there is a sequence of stopping times (˜ σ n ) which announces σ . Thenthe sequence ( σ n ) = (˜ σ n ∨ τ ) nondecreasingly converges to σ and satisfies τ ≤ σ n < σ on M ( τ , Y ) ⊂ { τ < T } . We define˜ ρ n = inf { t ≥ σ n ; Z tσ n ( L − u τ ,Y ( s )) ds > } ∧ T and for h > ρ n,h = inf { t ≥ σ n ; Z tσ n ( L − u τ ,Y ( s )) ds ≥ h } ∧ T. Then, ˜ ρ n,h converges to ˜ ρ n as h ↓
0. As Z ˜ ρ n σ n ( L − u τ ,Y ( s )) ds = 0 , on M ( τ , Y ), we can conclude that u τ ,Y ( s ) = L for almost every ( s, ω ) such that s ∈ [ σ n ( ω ) , ˜ ρ n ( ω )]and ω ∈ M ( τ , Y ). Now, taking into account that σ ∈ A ( τ , Y ) and σ n < σ on M ( τ , Y ), weobserve that σ n ≤ ˜ ρ n ≤ σ on M ( τ , Y ). We now define ρ n = ( ˜ ρ n , ω ∈ M ( τ , Y ) σ, otherwiseThen, ( ρ n ) nondecreasingly converges to σ . Let u n,h ( t ) = ( L − u τ ,Y ( t )) [˜ ρ n , ˜ ρ n,h ] ( t ) . Then, u τ ,Y + u n,h ∈ U ( τ , Y − h ). Hence, J ( τ , Y − h ) − J ( τ , Y ) h ≥ h E [ Z ˜ ρ n,h ˜ ρ n X ( t )( L − u τ ,Y ( t )) dt | F τ ] .
16n the set M ( τ , Y ) we have1 h E [ Z ˜ ρ n,h ˜ ρ n X ( t )( L − u τ ,Y ( t )) dt | F τ ]= E [ X ( ρ n ) h − Z ˜ ρ n,h ρ n ( L − u τ ,Y ( t )) dt | F τ ]+ 1 h E [ Z ˜ ρ n,h ρ n ( X ( t ) − X ( ρ n ))( L − u τ ,Y ( t )) dt | F τ ] (4.2)and lim h ↓ h − Z ˜ ρ n,h ρ n ( L − u τ ,Y ( t )) dt = 1 . So, the first term on the righthand side of (4.2) converges to E [ X ( ρ n ) | F τ ] as h ↓
0. The secondterm on the righthand side of (4.2) converges to zero by right-continuity of X . Consequently, − D − y J ( τ , Y ) ≥ E [ X ( ρ n ) | F τ ]on M ( τ , Y ). As ρ n = σ on the complement of M ( τ , Y ), we then obtain − E [ D − y J ( τ , Y ) M ( τ ,Y ) ] ≥ E [ X ( ρ n )] − E [ X ( σ )(1 − M ( τ ,Y ) )]If X is LCE at σ , the right-hand side converges to E [ X ( σ ) M ( τ ,Y ) ], which completes the proof.The corresponding result for the right-hand side derivative reads as follows. Lemma 4.5.
Suppose X satisfies the standing assumptions, ( τ , Y ) are as above, and σ ∈ S pB ( τ ,Y ) . Then, there is a sequence of stopping times ( ρ n ) , taking values in [ τ , T ] , which nonde-creasingly converges to σ and such that − D + y J ( τ , Y ) ≤ E [ X ( ρ n ) | F τ ] on M ( τ , Y ) for every n ∈ N . Moreover, − E [ D + y J ( τ , Y ) M ( τ ,Y ) ] ≤ E [ X ( σ ) M ( τ ,Y ) ] , if X is LCE at σ . roof. The proof is similar to the proof of the previous lemma, starting from˜ ρ n = inf { t ≥ σ n ; Z tσ n u τ ,Y ( s ) ds > } ∧ T ˜ ρ n,h = inf { t ≥ σ n ; Z tσ n u τ ,Y ( s ) ds ≥ h } ∧ T, where the sequence ( σ n ) again nondecreasingly converges to σ and satisfies τ ≤ σ n < σ on M ( τ , Y ). It has one additional complication, namely that u τ ,Y − u τ ,Y [˜ ρ n , ˜ ρ n,h ] does in generalnot belong to U ( τ , Y + h ). As a remedy we fix some arbitrary m ∈ N and assume h < /m . Weintroduce the set M m ( τ , Y ) := { L ( T − τ ) > − Y ≥ /m } ∈ F τ and the stopping times¯ σ U,h := inf { t ≥ τ ; Y + Z tτ u τ ,Y ( s ) ds ≥ − h } ∧ T ¯ σ U := inf { t ≥ τ ; Y + Z tτ u τ ,Y ( s ) ds ≥ } ∧ T Let u n,h ( t ) = u τ ,Y ( t ) [˜ ρ n ∧ ¯ σ U,h , ˜ ρ n,h ∧ ¯ σ U ] ( t ) M m ( τ ,Y ) . As R Tτ u τ ,Y ( r ) dr = 1 − Y on M m ( τ , Y ) ⊂ M ( τ , Y ) by Proposition 2.3, we conclude that Z Tτ ( u τ ,Y ( r ) − u n,h ( r )) dr = 1 − Y − h on M m ( τ , Y ). This implies that u τ ,Y − u n,h ∈ U ( τ , Y + h M m ( τ ,Y ) ). Consequently, on M m ( τ , Y ), − J ( τ , Y + h ) − J ( τ , Y ) h ≤ h E [ Z Tτ u n,h ( t ) X ( t ) dt | F τ ] ≤ h E [ Z ˜ ρ n,h ˜ ρ n u τ ,Y ( t ) X ( t ) dt | F τ ] + E [ sup r ∈ [0 ,T ] X ( r ) { ¯ σ U,h < ˜ ρ n } | F τ ] (4.3)Note that by the definition of ˜ ρ n , we have1 − Y − Z ˜ ρ n τ u τ ,Y ( s ) ds = 1 − Y − Z σ n τ u τ ,Y ( s ) ds on M m ( τ , Y ). Moreover, this expression is strictly positive on M m ( τ , Y ), because σ ∈ S pB ( τ ,Y ) and σ n < σ . Hence, ˜ ρ n ≤ ¯ σ U,h for h sufficiently small (depending on ω ). This shows that the18econd term in (4.3) tends to zero as h goes to zero and thatlim h ↓ h − Z ˜ ρ n,h ˜ ρ n u τ ,Y ( t ) dt = 1 . Now the same argument as in (4.2) can be applied to the first term in (4.3). We hence concludethat − D + y J ( τ , Y ) ≤ E [ X (˜ ρ n ) | F τ ]on M m ( τ , Y ) for every m ∈ N , and thus, on M ( τ , Y ) = ∪ m ∈ N M m ( τ , Y ). Defining ρ n = ( ˜ ρ n , ω ∈ M ( τ , Y ) σ, otherwise,the rest of the proof is identical to the one of the previous lemma.As a consequence of the previous three lemmas we get the following criterion for the left-handside derivative and the right-hand side derivative of J to coincide. Proposition 4.6.
Suppose X satisfies the standing assumptions and ( τ , Y ) are as above. If X is LCE at ¯ σ , defined in (4.1), then D − y J ( τ , Y ) = D + y J ( τ , Y ) on { > Y = 1 − L ( T − τ ) } , P -almost surely. Moreover, D − y J ( τ , Y ) = 0 on { Y ≤ − L ( T − τ ) } .Proof. The previous three lemmas imply that E [ D + y J ( τ , Y ) M ( τ ,Y ) ] ≥ − E [ X (¯ σ ) M ( τ ,Y ) ] ≥ E [ D − y J ( τ , Y ) M ( τ ,Y ) ] . As D − y J ( τ , Y ) ≥ D + y J ( τ , Y ) by concavity, we conclude that D − y J ( τ , Y ) = D + y J ( τ , Y )on M ( τ , Y ). On the set { Y ≤ − L ( T − τ ) } , we have J ( τ , Y ) = E [ Z Tτ LX ( s ) ds | F τ ] . Hence, D − y J ( τ , Y ) = 0 on { Y ≤ − L ( T − τ ) } and D + y J ( τ , Y ) = 0 on { Y < − L ( T − τ ) } .We are now in the position to give the proof of Theorem 4.1, which at the same time finishesthe proof of Theorem 1.1. 19 roof of Theorem 4.1. The case t = T is trivial, because J ( T, y ) = 0 for every y ∈ ( −∞ , t ∈ [0 , T ). Then, for every ω ∈ ¯Ω (which is the set of full measure introduced inProposition 2.1), the mapping y D − y J ( t, ω, y )is nonincreasing and left-continuous on (1 − L ( T − t ) ,
1) by concavity. Hence, there is a countablefamily of F t -measurable random variables ( Y n ) n ∈ N with values in [1 − L ( T − t ) ,
1] such that thejumps of y D − y J ( t, y ), restricted to (1 − L ( T − t ) , Y n ) n ∈ N , More precisely, D ( ω ) := { y ∈ (1 − L ( T − t ) , D − y J ( t, ω, y ) = lim η ↓ y D − y J ( t, ω, η ) } = { y ∈ (1 − L ( T − t ) , D − y J ( t, ω, y ) = D + y J ( t, ω, y ) }⊂ { Y ( ω ) , Y ( ω ) , . . . } . Here the first identity follows again by concavity. By the previous proposition there is a setΩ t ⊂ ¯Ω of full P -measure such that, for every n ∈ N and ω ∈ Ω t , Y n ( ω ) ∈ (1 − L ( T − t ) ,
1) = ⇒ D − y J ( t, ω, Y n ( ω )) = D + y J ( t, ω, Y n ( ω ))This implies that D ( ω ) = ∅ for ω ∈ Ω t and, hence, y J ( t, ω, y )is continuously differentiable on (1 − L ( T − t ) ,
1) for ω ∈ Ω t .In order to prove the second assertion, we introduce the set C := (cid:8) ( t, ω, y ) ∈ [0 , T ] × Ω × ( −∞ , D + y J ( t, ω, y ) = D − y J ( t, ω, y ) (cid:9) ∈ B [0 ,T ] ⊗ F ⊗ B ( −∞ , . We fix t ∈ [0 , T ]. For y ∈ (1 − L ( T − t ) ,
1) and s ∈ [ t, T − − yL ] we consider the cuts C y = { ( r, ω ) ∈ [ t, T − − yL ] × Ω; ( r, ω, y ) ∈ C } , C ( s,y ) = { ω ∈ Ω; ( s, ω, y ) ∈ C } . Then, for every y ∈ (1 − L ( T − t ) ,
1) and s ∈ [ t, T − − yL )Ω s ⊂ (cid:8) ω ∈ Ω; ∀ η ∈ (1 − L ( T − s ) , D + y J ( s, ω, η ) = D − y J ( s, ω, η ) (cid:9) ⊂ C ( s,y ) , where Ω s was constructed in the first part of the proof. Consequently, P ( C ( s,y ) ) = 1 for every y ∈ (1 − L ( T − t ) ,
1) and s ∈ [ t, T − − yL ). An application of Fubini’s theorem then yields for20very y ∈ (1 − L ( T − t ) , λ [ t,T − − yL ] ⊗ P ( C y ) = Z T − − yL t P ( C ( s,y ) ) ds = T − − yL − t, which finishes the proof.The following theorem relates the derivative of J explicitly to optimal stopping problems, if X is LCE. It can be considered as the main result of this section. Theorem 4.7.
Suppose X satisfies the standing assumptions and it is LCE. Then, for every [0 , T ] -valued stopping time τ and every F τ -measurable, ( −∞ , -valued random variable Y thefollowing holds:(i) On the set { L ( T − τ ) > − Y > }− D − y J ( τ , Y ) = − D + y J ( τ , Y ) = esssup σ ∈ S pA ( τ ,Y E [ X ( σ ) | F τ ] = essinf ρ ∈ S pB ( τ ,Y E [ X ( ρ ) | F τ ] . Moreover, every stopping time from the nonempty set S pA ( τ ,Y ) ∩ S pB ( τ ,Y ) is optimal for bothoptimal stopping problems.(ii) On the set { Y < − L ( T − τ ) }− D − y J ( τ , Y ) = − D + y J ( τ , Y ) = 0 . (iii) On the set { Y = 1 − L ( T − τ ) } ∩ { τ < T }− D − y J ( τ , Y ) = 0 , − D + y J ( τ , Y ) = essinf σ ∈ S τ E [ X ( σ ) | F τ ] , where S τ denotes the set of stopping times with values in [ τ , T ] .(iv) On the set { Y = 1 } ∩ { τ < T }− D − y J ( τ , Y ) = esssup σ ∈ S τ E [ X ( σ ) | F τ ] . Remark . A related representation for the marginal value of a discrete time multiple stoppingproblem is derived in Theorem 2.2 of Bender (2011a). We also note that differentiability of thevalue process for some class of finite fuel problems related to the monotone follower problem canbe shown by expressing the derivative explicitly in terms of (standard) optimal stopping problems,see e.g. Karatzas (1985); Karatzas and Shreve (1986).Before we provide the proof, we note that the two stopping problems in Theorem 4.7, (i), makethe intuition at the beginning of this section rigorous. The marginal value can be calculated by21dding some marginal volume at the best time where exercise is still possible. It can also becalculated by removing some marginal volume at the cheapest time, where this is possible. Theinteresting aspect is that here ‘best time’ and ‘cheapest time’ refer to predictable stopping timesonly. The next example shows that this restriction is essential.
Example . Suppose ξ is a binary trial with P ( { ξ = 1 } ) = P ( { ξ = − } ). Define X ( t ) = 1 + ξ (2 − t ) [1 , ( t ) , t ∈ [0 , , and consider the filtration ( F t ) t ∈ [0 , generated by X . Then X satisfies the standing assumptionson the time horizon [0 ,
3] and is LCE as the sum of the martingale 1+ ξ [1 , ( t ) and the continuousprocess ξ (1 − t ) [1 , ( t ). We assume L = 1. For t ∈ [0 ,
1] and y ∈ [0 ,
1] it is then straightforwardto see that J ( t, y ) = 2 − (1 + y ) u t,y ( r ) = [1 , − y ] ( r ) { X (1)=2 } + [2+ y, ( r ) { X (1)=0 } is optimal. In particular, A (0 , /
2) = ( (0 , / , X (1) = 0;(0 , ∪ [3 / , , X (1) = 2 . Define the (non-predictable) stopping time τ = inf { r ≥ X ( r ) ≥ / } = { X (1)=2 } + 5 / { X (1)=0 } , which takes values in A (0 , / E [ X ( τ )] = 7 / > / − D − y J (0 , /
2) = sup σ ∈ S pA (0 , / E [ X ( σ )] , where the last identity is due to Theorem 4.7. However, in view of Lemma 4.3 and Theorem 4.7,an optimal stopping time in S pA (0 , / is given by¯ σ = inf { r ≥ Z r u , / ( s ) ds / ∈ ( t − / , / } = 3 / { X (1)=2 } + 5 / { X (1)=0 } = inf { r ≥ X ( r ) = 3 / } . This shows that the restriction to predictable stopping times cannot be avoided in the optimalstopping characterization of the y -derivative of J .We now give the proof of Theorem 4.7. 22 roof of Theorem 4.7. (i) Choose some stopping time σ ∈ S pA ( τ ,Y ) . By Lemma 4.4, there is asequence of stopping times ( ρ n ) with values in [ τ , T ], which nondecreasingly converges to σ andsatisfies − D − y J ( τ , Y ) ≥ E [ X ( ρ n ) | F τ ]on M ( τ , Y ). By the integrability property of X in (1.2) and left-continuity in expectation oneeasily obtains lim n →∞ E [ X ( ρ n ) | F τ ] = E [ X ( σ ) | F τ ] , because ρ n ≥ τ . Hence, − D − y J ( τ , Y ) ≥ E [ X ( σ ) | F τ ]on M ( τ , Y ). Analogously we obtain − D + y J ( τ , Y ) ≤ E [ X ( ρ ) | F τ ]on M ( τ , Y ) for ρ ∈ S pB ( τ ,Y ) making use of Lemma 4.5. Hence, choosing ˜ σ from the nonemptyset S pA ( τ ,Y ) ∩ S pB ( τ ,Y ) (by Lemma 4.3), we get − D − y J ( τ , Y ) ≥ esssup σ ∈ S pA ( τ ,Y E [ X ( σ ) | F τ ] ≥ E [ X (˜ σ ) | F τ ] ≥ essinf ρ ∈ S pB ( τ ,Y E [ X ( ρ ) | F τ ] ≥ − D + y J ( τ , Y )on M ( τ , Y ). As D − y J ( τ , Y ) ≥ D + y J ( τ , Y ) by concavity, the assertion follows.(iii) Define Y n := Y + (1 − Y ) /n . Denote by u τ ,Y n an optimal control for ¯ J ( τ , Y n ). Then, Y n > Y on { Y = 1 − L ( T − τ ) } ∩ { τ < T } . Hence, on this set, − D + y J ( τ , Y ) = lim n →∞ − J ( τ , Y n ) + J ( τ , − L ( T − τ )) Y n − Y = lim n →∞ E h R Tτ ( L − u τ ,Y n ( s )) X ( s ) (cid:12)(cid:12)(cid:12) F τ i (1 − Y ) /n . Denoting by Y ∗ ( s ) an RCLL version of the submartingale s essinf σ ∈ S s E [ X ( σ ) | F s ] , we, hence,23btain thanks to Corollary 2.4, − D + y J ( τ , Y ) ≥ lim sup n →∞ E h R Tτ ( L − u τ ,Y n ( s )) Y ∗ ( s ) (cid:12)(cid:12)(cid:12) F τ i (1 − Y ) /n ≥ lim sup n →∞ E h R Tτ ( L − L [ τ ∨ ( T − (1 − Y n ) /L ) ,T ] ( s )) Y ∗ ( s ) (cid:12)(cid:12)(cid:12) F τ i (1 − Y ) /n = lim sup n →∞ E h R τ +(1 − Y ) / ( Ln ) τ Y ∗ ( s ) (cid:12)(cid:12)(cid:12) F τ i (1 − Y ) / ( Ln )= Y ∗ ( τ ) = essinf σ ∈ S τ E [ X ( σ ) | F τ ] . For the reverse inequality fix some some arbitrary stopping time σ with values in [ τ , T ]. Define σ m = σ ∧ ( T − ( T − τ ) /m ) ≥ τ . We observe that, for n ≥ m , u n,m := L − L [ σ m ,σ m +( T − τ ) /n ] { Y =1 − L ( T − τ ) }∩{ τ 0) are subject to three constraints. A local constraint requiresthat u takes values in [0 , L ], a global one imposes that the total volume spent by the investor R T u ( s ) ds is bounded by one, and the third one is the adaptedness condition. In Section 7 of24BD] we relaxed the adaptedness constraint and came up with a continuous time version of aninformation relaxation dual. This kind of dual is well studied for discrete time stochastic controlproblems, see e.g. Brown et al. (2010). We now relax the global constraint and re-inforce it bya more classical Lagrange multiplier approach. It turns out that the Lagrange multiplier can becalculated explicitly in term of the derivative of J . This approach leads to the following result: Theorem 5.1. Suppose LT > , and that X satisfies the standing assumptions and is LCE.Denote by M the set of RCLL martingales on [0 , T ] . Then ¯ J (0 , 0) = inf M ∈ M (cid:18) E (cid:20)Z T L ( X ( t ) − M ( t )) + dt (cid:21) + E [ M (0)] (cid:19) . Moreover, an optimal martingale is given by the (unique up to indistinguishability) RCLL andadapted modification of ¯ M ( t ) = − ∂∂y J (cid:16) t, R t u , ( s ) ds (cid:17) , t < ¯ σY ∗ (¯ σ ) + M ∗ ( t ) − M ∗ (¯ σ ) , t ≥ ¯ σ = ¯ σ U Y ∗ (¯ σ ) + M ∗ ( t ) − M ∗ (¯ σ ) , t ≥ ¯ σ = ¯ σ L , where ¯ σ = ¯ σ U ∧ ¯ σ L , ¯ σ U = inf { t ≥ Z t u , ( s ) ds ≥ } , ¯ σ L = inf { t ≥ Z t u , ( s ) ds ≤ − L ( T − t ) } , and u , is an optimal control for ¯ J (0 , satisfying R T u , ( s ) ds = 1 . Moreoever, M ∗ and M ∗ arethe martingale parts of the Doob-Meyer decompositions of the RCLL supermartingale Y ∗ ( s ) =esssup σ ∈ S s E [ X ( σ ) | F s ] and the RCLL submartingale Y ∗ ( s ) = essinf σ ∈ S s E [ X ( σ ) | F s ] . We prepare the proof of this theorem with the following lemma. Lemma 5.2. Under the assumptions and with the notations of Theorem 5.1 we have: E [ X (¯ σ ) | F t ] = − ∂∂y J (cid:18) t, Z t u , ( s ) ds (cid:19) , t < ¯ σ, (5.1) X (¯ σ ) = ( Y ∗ (¯ σ U ) , ¯ σ U ≤ ¯ σ L Y ∗ (¯ σ L ) , ¯ σ L < ¯ σ U (5.2)25 emark . By Lemma 5.2, the process ¯ M in Theorem 5.1 can be expressed as¯ M ( t ) = E [ X (¯ σ ) | F t ] , t ≤ ¯ σX (¯ σ ) + M ∗ ( t ) − M ∗ (¯ σ ) , t > ¯ σ = ¯ σ U X (¯ σ ) + M ∗ ( t ) − M ∗ (¯ σ ) , t > ¯ σ = ¯ σ L . In particular, this shows that ¯ M is a martingale and, thus, has an adapted RCLL modification,which we still denote by ¯ M . Proof. First note that by Proposition 3.2 in [BD] there is an optimal control u , satisfying R T u , ( s ) ds = 1. Moreover, u , fulfills (2.2) with τ = 0 and Y = 0 by the same proposition.We first prove (5.1). By Theorem 4.7, (i), and Lemma 4.3 we observe that for t < ¯ σ − ∂∂y J (cid:18) t, Z t u , ( s ) ds (cid:19) = E [ X (¯ σ ( t )) | F t ] (5.3)where ¯ σ ( t ) = inf { s ≥ t ; Z st ¯ u , ( r ) dr / ∈ (1 − Z t u , ( r ) dr − L ( T − t ) , − Z t u , ( r ) dr ) } = inf { s ≥ t ; Z s ¯ u , ( r ) dr / ∈ (1 − L ( T − t ) , } Here we used that [ t,T ] u , is optimal for ¯ J ( t, R t u , ( r ) dr ) by the dynamic programming principlein Proposition 3.3 of [BD]. As ¯ σ = ¯ σ (0), we conclude that ¯ σ ( t ) = ¯ σ for t < ¯ σ . Hence, (5.3) implies(5.1).We now turn to the proof of (5.2). Suppose that σ is any predictable stopping time withvalues in (¯ σ L , T ] ∪ { T } and denote an announcing sequence by ( σ n ). We will first show that E [ X (¯ σ ) { ¯ σ L < ¯ σ U } ] ≤ E [ X ( σ ) { ¯ σ L < ¯ σ U } ] . (5.4)To this end we define ˜ σ = ( σ, ¯ σ L < ¯ σ U ¯ σ U , ¯ σ U ≤ ¯ σ L . The stopping time ˜ σ is predictable, since it is announced by the sequence˜ σ n = ( σ n ∨ ¯ σ L , ¯ σ L < ¯ σ U,n ¯ σ U,n , ¯ σ U,n ≤ ¯ σ L , ) ∧ ( T − /n )where the sequence ¯ σ U,n = inf { t ∈ [0 , T ]; Z t u , ( s ) ds ≥ − /n } σ U . Now note that ˜ σ = ¯ σ ∈ B (0 , 0) on ¯ σ U ≤ ¯ σ L by Lemma 4.3. Moreover, (¯ σ L , T ] ⊂ B (0 , 0) on ¯ σ L < ¯ σ U by (2.2). Consequently, ˜ σ belongs to S pB (0 , . Hence, Theorem 4.7, (i), andLemma 4.3 yield E [ X (¯ σ )] ≤ E [ X (˜ σ )] , which in turn implies (5.4) by the definition of ˜ σ . In a next step we fix an optimal stopping time σ ∗ for the optimal stopping problem Y ∗ (¯ σ L ∧ T ), which exists, because X is right-continuous andLCE, see e.g. El Karoui (1981). Then, for every k ∈ N , the stopping time σ ∗ k = ( σ ∗ + 1 /k ) ∧ T is predictable (with announcing sequence (( σ ∗ + 1 /k − /n ) ∧ ( T − /n )) n ≥ k ) and takes values in(¯ σ L , T ] ∪ { T } . Thus, by (5.4), we have E [ X (¯ σ ) { ¯ σ L < ¯ σ U } ] ≤ E [ X ( σ ∗ k ) { ¯ σ L < ¯ σ U } ]Passing to the limit we obtain by right-continuity of X and optimality of σ ∗ , E [ X (¯ σ ) { ¯ σ L < ¯ σ U } ] ≤ E [ X ( σ ∗ ) { ¯ σ L < ¯ σ U } ] = E [ Y ∗ (¯ σ L ) { ¯ σ L < ¯ σ U } ] . As obviously, X (¯ σ L ) ≥ Y ∗ (¯ σ L )on { ¯ σ L < ¯ σ U } , we finally arrive at X (¯ σ L ) = Y ∗ (¯ σ L )on { ¯ σ L < ¯ σ U } . The proof of (5.2) in the case ¯ σ U < ¯ σ L is analogously, while the case ¯ σ L = ¯ σ U istrivial, since this implies ¯ σ U = T . Proof of Theorem 5.1. We introduce the set U consisting of all adapted, [0 , L ]-valued processes.Hence, u ∈ U (0 , u ∈ U and satisfies the global constraint R T u ( s ) ds ≤ 1. Wethen define the Lagrangian L ( u, Λ) = E (cid:20)Z T u ( s ) X ( s ) ds − Λ (cid:18)Z T u ( s ) ds − (cid:19)(cid:21) for u ∈ U and F T -measurable integrable random variables Λ. Put differently, we relax the globalconstraint and try to enforce it by an appropriate choice of the Lagrange multiplier Λ. Apparentlywe have for any Λ L ( u , , Λ) = E (cid:20)Z T u , ( s ) X ( s ) ds (cid:21) = ¯ J (0 , , because u , is optimal and satisfies R T u , ( s ) ds = 1. Thus, we obtain for every F T -measurable27ntegrable random variable Λ¯ J (0 , ≤ sup u ∈ U L ( u, Λ) = sup u ∈ U E (cid:20)Z T u ( s )( X ( s ) − E [Λ | F s ]) ds (cid:21) + E [Λ]= E (cid:20)Z T L ( X ( s ) − E [Λ | F s ]) + ds (cid:21) + E [Λ] . Consequently, we get for every RCLL martingale M with the choice Λ = M ( T ), J (0 , ≤ E (cid:20)Z T L ( X ( s ) − M ( s )) + ds (cid:21) + E [ M (0)] . In order to finish the proof it is, thus, sufficient to show that¯ J (0 , 0) = E (cid:20)Z T L ( X ( t ) − ¯ M ( t )) + dt (cid:21) + E [ ¯ M (0)] , where ¯ M is indeed a martingale thanks to Remark 5.3. To this end recall that the good versionof the value process constructed in Proposition 2.1 is denoted by J . By the characterization ofoptimal controls in Theorem 2.2, (ii), by the martingale property of ¯ M , and by (2.2) we obtain¯ J (0 , 0) = J (0 , 0) = E [ Z T u , ( t ) X ( t ) dt ]= E (cid:20)Z T (cid:18) L ( X ( t ) + D − y J ( t, Z t u , ( s ) ds )) + − D − y J ( t, Z t u , ( s ) ds ) u , ( t ) (cid:19) dt (cid:21) = E (cid:20)Z ¯ σ (cid:0) L ( X ( t ) − ¯ M ( t )) + + E [ ¯ M (¯ σ ) | F t ] u , ( t ) (cid:1) dt (cid:21) + E (cid:20) { ¯ σ U ≤ ¯ σ L } Z T ¯ σ (cid:0) L ( X ( t ) + D − y J ( t, + − D − y J ( t, u , ( t ) (cid:1) dt (cid:21) + E (cid:20) { ¯ σ U > ¯ σ L } Z T ¯ σ (cid:0) L ( X ( t ) + D − y J ( t, − L ( T − t ))) + − D − y J ( t, − L ( T − t )) u , ( t ) (cid:1) dt (cid:21) = ( I ) + ( II ) + ( III ) . Note that E (cid:20)Z ¯ σ E [ ¯ M (¯ σ ) | F t ] u , ( t ) dt (cid:21) = Z T E (cid:2) { ¯ σ>t } ¯ M (¯ σ ) u , ( t ) (cid:3) dt = E (cid:20) ¯ M (¯ σ ) Z ¯ σ u , ( t ) dt (cid:21) . Hence, ( I ) = E (cid:20)Z ¯ σ L ( X ( t ) − ¯ M ( t )) + dt + ¯ M ( σ ) Z ¯ σ u , ( t ) dt (cid:21) . Moreover, by Theorem 4.7, we have,( II ) = E (cid:20) { ¯ σ U ≤ ¯ σ L } Z T ¯ σ U Y ∗ ( t ) u , ( t ) dt (cid:21) = 0 = E (cid:20) { ¯ σ U ≤ ¯ σ L } ¯ M (¯ σ ) Z T ¯ σ U u , ( t ) dt (cid:21) , u , ( t ) = 0 for t > ¯ σ U , and( III ) = E (cid:20) { ¯ σ U > ¯ σ L } Z T ¯ σ L LX ( t ) dt (cid:21) . Hence,¯ J (0 , 0) = E (cid:20)Z T L ( X ( t ) − ¯ M ( t )) + dt (cid:21) + E (cid:20) ¯ M (¯ σ ) Z ¯ σ u , ( t ) dt (cid:21) − E (cid:20) { ¯ σ U ≤ ¯ σ L } Z T ¯ σ L ( X ( t ) − ¯ M ( t )) + dt (cid:21) + E (cid:20) { σ U ≤ σ L } ¯ M (¯ σ ) Z T ¯ σ u , ( t ) dt (cid:21) + E (cid:20) { ¯ σ U > ¯ σ L } Z T ¯ σ (cid:0) LX ( t ) − L ( X ( t ) − ¯ M ( t )) + (cid:1) dt (cid:21) . Now, by the definition of ¯ M and the supermartingale property of Y ∗ we obtain for t > ¯ σ on¯ σ U ≤ ¯ σ L , X ( t ) ≤ Y ∗ ( t ) ≤ Y ∗ (¯ σ ) + M ∗ ( t ) − M ∗ (¯ σ ) = ¯ M ( t )and analogously for t > ¯ σ on ¯ σ L < ¯ σ U , using the submartingale property of Y ∗ X ( t ) ≥ Y ∗ ( t ) ≥ Y ∗ (¯ σ ) + M ∗ ( t ) − M ∗ (¯ σ ) = ¯ M ( t ) . Thus, ¯ J (0 , 0) = E (cid:20)Z T L ( X ( t ) − ¯ M ( t )) + dt (cid:21) + E (cid:20) ¯ M (¯ σ ) Z ¯ σ u , ( t ) dt (cid:21) + E (cid:20) { σ U ≤ σ L } ¯ M (¯ σ ) Z T ¯ σ u , ( t ) dt (cid:21) + E (cid:20) { ¯ σ U > ¯ σ L } Z T ¯ σ L ¯ M ( t ) dt (cid:21) Noting that E (cid:20) { ¯ σ U > ¯ σ L } Z T ¯ σ L ¯ M ( t ) dt (cid:21) = E (cid:20) { ¯ σ U > ¯ σ L } ¯ M (¯ σ ) Z T ¯ σ L Ldt (cid:21) = E (cid:20) { ¯ σ U > ¯ σ L } ¯ M (¯ σ ) Z T ¯ σ u , ( t ) dt (cid:21) by (2.2) and that R T u , ( t ) dt = 1, we finally obtain¯ J (0 , 0) = E (cid:20)Z T L ( X ( t ) − ¯ M ( t )) + dt (cid:21) + E (cid:2) ¯ M (¯ σ ) (cid:3) = E (cid:20)Z T L ( X ( t ) − ¯ M ( t )) + dt (cid:21) + E [ ¯ M (0)] . Acknowledgement The authors gratefully acknowledge financial support by the ATN-DAAD Australia GermanyJoint Research Cooperation Scheme. 29 eferenceseferences