A First-Order BSPDE for Swing Option Pricing
aa r X i v : . [ q -f i n . P R ] M a y A First-Order BSPDE for Swing Option Pricing
Christian Bender , Nikolai Dokuchaev November 10, 2018
Abstract
We study an optimal control problem related to swing option pric-ing in a general non-Markovian setting in continuous time. As a mainresult we show that the value process solves a first-order non-linearbackward stochastic partial differential equation. Based on this resultwe can characterize the set of optimal controls and derive a dual min-imization problem.
Keywords:
Backward SPDE, stochastic optimal control, swing options.
AMS classification:
In a swing option contract, the holder of the option can buy some volume ofa commodity, say electricity, for a fixed strike price during the lifetime of theoption. There are typically local constraints on how much volume can beexercised at a given time, and global constraints on the total volume. Swingoptions are particularly popular in electricity markets, and can be used tohedge against the risk of fluctuating demand, see Carmona and Ludkovski(2010).Mathematically, the pricing problem of such a swing option leads to op-timal control problems, whose formulation varies depending on the way theconstraints are formulated. On the one hand, the constraints can be formu-lated discretely in the following sense: The total volume must be exercised inform of a finite number of packages. Local constraints prescribe how manypackages can at most be exercised at a given time and refraction periods areimposed to enforce a minimal waiting time after one package is exercised.This formulation leads to multiple stopping problems and was studied indiscrete time e.g. by Jaillet et al. (2004), Meinshausen and Hambly (2004), Saarland University, Department of Mathematics, Postfach 151150, D-66041 Saar-br¨ucken, Germany, [email protected] . Department of Mathematics & Statistics, Curtin University, GPO Box U1987, Perth,6845 Western Australia, Australia,
[email protected] X ( t ) denotes the discounted payoff of the option, if oneunit volume is exercised at time t . In the case of swing option pricing wecan set X ( t ) = e − ρt ( S ( t ) − K ) + , where S is the electricity price process, K is the strike price, ρ is the interest rate, and ( x ) + stands for the positivepart of x . Then, we consider the following control problem¯ J ( t, y ) := esssup u E (cid:20) Z Tt u ( s ) X ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , where the supremum is taken over the set of adapted processes with valuesin [0 , L ] which satisfy R Tt u ( s ) ds ≤ − y . Here, a local constraint restrictsthe rate at which the option can be exercised to the interval [0 , L ], while theglobal constraint imposes that the maximal volume which can be exercisedin the remaining time from t to T is 1 − y . Then ¯ J ( t, y ) is a discounted fairprice of the swing option contract, if the expectation is taken with respectto a risk-neutral pricing measure under which all tradable and storable basicsecurities in the market are σ -martingales.As the main result of this paper we will show that a ‘good’ version( J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , 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, y ) = 0 , J ( t,
1) = 0 . (1)Here D − y J denotes the left-hand side derivative of J in y and it can be re-placed by the right-hand side derivative D + y J in the above equation. This re-sult will be obtained under the weak assumptions that X is right-continuous,nonnegative, adapted, and satisfies some integrability condition. We will alsoshow that under these assumptions J is smooth enough to apply a variantof a chain rule, which is sufficient to show that a control u is optimal, if and2nly if 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 . We finally derive a dual minimization problem for ¯ J ( t, y ) in terms of mar-tingales. This type of dual formulations has its origin in the pricing problemof American options, see Rogers (2002) and Haugh and Kogan (2004), andwas later generalized to a pure martingale dual for multiple exercise optionsby Schoenmakers (2012) in discrete time and Bender (2011b) in continuoustime. Our dual representation can be seen as a continuous time version ofgeneral dual formulations for discrete time control problems in Brown et al.(2010), Rogers (2007), and Gyurko et al. (2013).We note that a connection between backward SPDEs and dynamic pro-gramming for a class of non-Markovian control problems was first studiedby Peng (1992). As in most of the existing literature for backward SPDEshe considers parabolic type second order equations such that the matrix ofthe higher order coefficients is positive definite. We also note that someadditional conditions on the coercivity are usually imposed in the literature;see, e.g., condition (0.4) in Rozovskii (1990), Ch. 4. Without these condi-tions, a parabolic type SPDE is regarded as degenerate. For the degeneratebackward SPDEs in the whole space, i.e., without boundaries, regularityresults were obtained in Rozovskii (1990), Ma and Yong (1990), Hu et al.(2002), and more recently by Du et al. (2013) and Du and Zhang (2013).The methods developed in these works cannot be applied in the case of adomain with boundary because of regularity issues that prevent using anapproximation of the differential operator by a non-degenerate one. It turnsout that the theory of degenerate SPDEs in domains is much harder than inthe whole space and was, to the best of our knowledge, not addressed yet inthe existing literature. The present paper consider a problem of this kind.We introduce and prove existence for a first order BSPDE in a domain withboundary. This equation can be interpreted as a limit case of a degeneratesecond order parabolic BSPDE.The paper is organized as follows: In Section 2 we set the problem andderive some basic properties of the control problem, including the existenceof optimal controls and the construction of the good version J ( t, y ). InSection 3 we study the marginal values − D ± y J ( t, y ). It turns out that theleft-hand side derivative D − y J ( t, y ) in general is a submartingale with right-continuous paths, while the right-hand side derivative D + y J ( t, y ) may admitdiscontinuities from the right. For this reason it is more convenient to workwith the left-hand side derivative in most of the proofs. The proof of themain result, namely that J solves the first-order backward stochastic partialdifferential equation (1) is given in Section 4. Finally, the characterization3f optimal strategies and the dual formulation are presented in Sections 5and 6. Uniqueness results for the BSPDE (1) and smoothness of the valueprocess J ( t, y ) will be discussed in a companion paper, which is in prepara-tion. Throughout this paper we assume that (Ω , F , F , P ) is a filtered probabil-ity space satisfying the usual conditions and that ( X ( t ) , ≤ t ≤ T ) is anonnegative, rightcontinuous, F -adapted stochastic process which fulfills E [ sup ≤ t ≤ T X ( t ) p ] < ∞ (2)for some p > X continuously, but she is subjected to the constraint that therate at which she exercises is bounded by a constant L >
0, which is fixedfrom now on. Moreover the maximal total volume of exercise is bounded by1. The investor’s aim is to maximize the expected reward, i.e. she wishes tomaximize E (cid:20)Z T u ( s ) X ( s ) ds (cid:21) over all F -adapted processes with values in [0 , L ] which satisfy R T u ( s ) ds ≤ , T ]-valued stopping time τ and F τ -measurable ( −∞ , Y denote by U ( τ, Y ) the setof all F -adapted processes with values in [0 , L ] such that R Tτ u ( s ) ds ≤ − Y .Hence, the investor enters at time τ and can spend a total volume of 1 − Y .The corresponding 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) As explained in the introduction, the main result of this paper is that a‘good’ version ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , Proposition 2.1.
For every pair ( τ, Y ) , where τ is a stopping time and Y is an ( −∞ , -valued F τ -measurable random variable, there is an optimal ontrol ¯ u ∈ U ( τ, Y ) , i.e. ¯ J ( τ, Y ) = E [ Z Tτ ¯ u ( r ) X ( r ) dr | F τ ] Proof.
We consider U ( τ, Y ) as a subset of L q ( F ⊗ B [0 , T ] , P ⊗ λ [0 ,T ] ) where B [0 , T ] and λ [0 ,T ] denote the Borel σ -field and the Lebesgue measure on[0 , T ], and 1 /q + 1 /p = 1 for p > U ( τ, Y ) isa weakly sequentially compact subset of the reflexive Banach space L q ( F ⊗ B [0 , T ] , P ⊗ λ [0 ,T ] ), because U ( τ, Y ) is bounded and closed in the strongtopology and convex.We now introduce the set M = (cid:26) E [ Z Tτ u ( r ) X ( r ) dr | F τ ] , u ∈ U ( τ, Y ) (cid:27) . It is straighforward to check that M is closed under pathwise maximization,i.e. M , M ∈ M implies that M ∨ M ∈ M . Hence, by Theorem A.3 inKaratzas and Shreve (1998), there is a sequence ( u n ) ⊂ U ( τ, Y ) such that E [ Z Tτ u n ( r ) X ( r ) dr | F τ ] ↑ J ( τ, Y ) , n → ∞ . (3)As U ( τ, Y ) is weakly sequentially compact, we can assume without loss ofgenerality (by passing to a subsequence, if necessary), that u n convergesweakly in L q ( F ⊗ B [0 , T ] , P ⊗ λ [0 ,T ] ) to some ¯ u ∈ U ( τ, Y ). We now showthat ¯ u is indeed optimal. Suppose A ∈ F τ . By weak convergence of ( u n ) to¯ u and considering X A × [ τ,T ] as an element of L p ( F ⊗ B [0 , T ] , P ⊗ λ [0 ,T ] ), weget E [ A E [ Z Tτ u n ( r ) X ( r ) dr | F τ ]] ↑ E [ A E [ Z Tτ ¯ u ( r ) X ( r ) dr | F τ ]] , n → ∞ , which, combined with (3), yields E [ A E [ Z Tτ ¯ u ( r ) X ( r ) dr | F τ ]] = E [ A J ( τ, Y )]As A ∈ F τ was arbitrary, this immediately gives J ( τ, Y ) = E τ [ Z Tτ ¯ u ( r ) X ( r ) dr | F τ ] . At several instances, it will be convenient to switch from the control set U ( τ, Y ) to the subset U ′ ( τ, Y ) of controls u which additionally satisfy u ( r ) = L on { L ( T − r ) ≤ − ( Y + Z rτ u ( s ) ds ) } , (4)5 [0 ,T ] ⊗ P -almost everywhere.We collect some facts on the relation between U ( τ, Y ) and U ′ ( τ, Y ) inthe following proposition. Proposition 2.2. (i) For any control u ∈ U ( τ, Y ) , there is a control ˜ u ∈ U ′ ( τ, Y ) such that ˜ u ( r ) ≥ u ( r ) on [ τ, T ] . In particular, there exists anoptimal strategy ¯ u τ,Y ∈ U ′ ( τ, Y ) for J ( τ, Y ) .(ii) u ∈ U ( τ, Y ) belongs to U ′ ( τ, Y ) , if and only if R Tτ u ( r ) dr = 1 − Y on the set { L ( T − τ ) ≥ − Y } and u ( r ) = L for r ∈ [ τ, T ] on the set { L ( T − τ ) ≤ − Y } .Proof. For u ∈ U ( τ, Y ) define τ L,u = inf { r ≥ τ ; L ( T − r ) ≤ − ( Y + Z rτ u ( s ) ds ) } ∧ T. (i) If u ∈ U ( τ, Y ), then˜ u = u [ τ,τ L,u ) + L [ τ L,u ,T ] ∈ U ′ ( τ, Y ) . (ii) Suppose u ∈ U ′ ( τ, Y ). On the set { L ( T − τ ) ≤ − Y } we have τ L,u = τ .Hence u ( r ) = L on [ τ, T ]. On the set { L ( T − τ ) ≥ − Y } ∩ { τ L,u < T } weget Z Tτ u ( r ) dr = Z τ L,u τ u ( r ) dr + Z Tτ L,u u ( r ) dr = 1 − Y − L ( T − τ L,u ) + L ( T − τ L,u ) = 1 − Y. (5)On the set { L ( T − τ ) ≥ − Y } ∩ { τ L,u = T } , we obtain R Tτ u ( r ) dr ≥ − Y by (4) and the other inequality is trivial.Now suppose that u ∈ U ( τ, Y ) satisfies the two properties stated in theassertion. If τ L,u = τ , then L ( T − τ ) ≤ − Y and hence u ( r ) = L for r ∈ [ τ, T ]. If τ < τ L,u < T , then L ( T − τ ) > − Y , and hence R Tτ u ( r ) dr = 1 − Y .An analogous calculation than in (5) shows Z Tτ L,u u ( r ) dr = L ( T − τ L,u ) , which implies (4).Next, we state the dynamic programming principle for this optimizationproblem. Its simple proof is omitted. Proposition 2.3.
Suppose σ ≤ τ are [0 , T ] -valued stopping times and Y isan F σ -measurable, ( −∞ , -valued random variable. Then, ¯ J ( σ, Y ) = esssup u ∈ U ( σ,Y ) E [ Z τσ u ( r ) X ( r ) dr + ¯ J ( τ, Y + Z τσ u ( r ) dr ) | F σ ]6he next lemma singles out two properties which are related to Lipschitzcontinuity and concavity of J in the y -variable. Lemma 2.4.
Suppose τ is a [0 , T ] -valued stopping time and Y , Y are F τ -measurable ( −∞ , -valued random variables. Then, P -almost surely, | ¯ J ( τ, Y ) − ¯ J ( τ, Y ) | ≤ E [( sup ≤ t ≤ T X ( t )) | F τ ] | Y − Y | . (6)¯ J (cid:18) τ, Y + Y (cid:19) ≥ ¯ J ( τ, Y ) + ¯ J ( t, Y )2 (7) Proof.
We first show (6). Choose an optimal strategy ¯ u τ,Y ∈ U ( τ, Y ) anddefine σ = inf { t ≥ τ ; R tτ ¯ u τ,Y ( s ) ds ≥ − Y } ∧ T . Then, u ( t ) = ¯ u τ,Y [ τ,σ ] ∈ U ( τ, Y ). Consequently, on the set { Y ≤ Y } , we get0 ≤ ¯ J ( τ, Y ) − ¯ J ( τ, Y ) ≤ E [ Z Tτ (¯ u τ,Y ( s ) − u ( s )) X ( s ) ds | F τ ]= E [ Z Tσ ¯ u τ,Y ( s ) X ( s ) ds | F τ ] ≤ E [( sup ≤ r ≤ T X ( r )) Z Tσ ¯ u τ,Y ( s ) ds | F τ ] ≤ E [( sup ≤ r ≤ T X ( r ))( Y − Y ) | F τ ]Changing the roles of Y and Y , we obtain that this inequality also holdson { Y > Y } , which proves (6).For (7) one merely needs to note that for u ∈ U ( τ, Y ) and u ∈ U ( τ, Y ),the control ( u + u ) / U ( τ, ( Y + Y ) / J ( t, y ) as stated in the followingproposition. Proposition 2.5.
There is an adapted random field ( J ( t, y ) , t ∈ [0 , T ] , y ∈ ( −∞ , with the following properties:a) For every pair ( τ, Y ) J ( τ, Y ) = ¯ J ( τ, Y ) P − a.s. In particular, for every y ∈ ( −∞ , , J ( t, y ) is an adapted modification of ¯ J ( t, y ) .b) There is a set ¯Ω ∈ F with P ( ¯Ω) = 1 such that the following propertieshold 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 sat-isfies sup r ∈ [0 ,T ] Z ( r ) < ∞ on ¯Ω . . For every t ∈ [0 , T ] , the mapping y J ( t, y ) is concave. As a preparation we need the following lemma.
Lemma 2.6. (a) For every y ∈ ( −∞ , , the mapping t E [ ¯ J ( t, y )] isrightcontinuous.(b) For every y ∈ ( −∞ , , the process ¯ J ( t, y ) has a modification ˆ J ( t, y ) ,which is a supermartingale whose paths are RCLL with probability one.Proof. We fix some y ∈ ( −∞ , J ( t, y ) is a supermartin-gale on [0 , T ], because U ( t , y ) ⊂ U ( t , y ) for 0 ≤ t ≤ t ≤ T . Hence,by Theorem 1.3.13 in Karatzas and Shreve (1991), (a) implies (b). For(a) we fix some t ∈ [0 , T ) and choose a sequence ( t n ) ⊂ [0 , T ] such that t n ↓ t . By the supermartingale property we have E [ ¯ J ( t, y )] ≥ E [ ¯ J ( t n , y )].So it is sufficient to show that lim inf n →∞ E [ ¯ J ( t n , y )] ≥ E [ ¯ J ( t, y )]. Tothis end we choose an optimal strategy ¯ u t,y ∈ U ( t, y ) for ¯ J ( t, y ). Then, u n = ¯ u t,y [ t n ,T ] ∈ U ( t n , y ). Therefore, by dominated convergence,lim inf n →∞ E [ ¯ J ( t n , y )] ≥ lim inf n →∞ E [ Z Tt n ¯ u t,y ( s ) ds ] = E [ Z Tt ¯ u t,y ( s ) ds ] = ¯ J ( t, y ) . Proof of Proposition 2.5.
Let Q := ([0 , T ] ∩ Q ) ∪ { T } , Q := ( −∞ , ∩ Q We choose a set ¯Ω with P ( ¯Ω) = 1 such that the following properties hold on¯Ω:(i) Z ∗ := sup r ∈ [0 ,T ] Z ( r ) < ∞ .(ii) ˆ J ( t, y ) = ¯ J ( t, y ) for every ( t, y ) ∈ Q × Q (where ˆ J was constructedin the previous lemma).(iii) The mapping t ˆ J ( t, y ) is RCLL for every y ∈ Q .(iv) For every ( t, y , y ) ∈ Q × Q it holds that | ¯ J ( t, y ) − ¯ J ( t, y ) | ≤ Z ∗ | y − y | . (v) For every ( t, y , y ) ∈ Q × Q it holds that¯ J ( t, y + y ≥ ¯ J ( t, y ) + ¯ J ( t, y )2 .
8e briefly check that such a set ¯Ω exists. The martingale E [sup ≤ r ≤ T X ( r ) | F t ]has an RCLL modification which we denote Z ( t ). By Doob’s inequality itsatisfies E [ sup ≤ t ≤ T Z ( t ) p ] ≤ (cid:18) pp − (cid:19) p E [ Z ( T ) p ] = (cid:18) pp − (cid:19) p E [ sup ≤ t ≤ T X ( t ) p ] < ∞ . Hence, the random variable Z ∗ is almost surely finite. Moreover, (ii) and (iii)can be realized by the previous lemma, because Q and Q are countable.The same applies to (iv) and (v) in view of Lemma 2.4.On ¯Ω we wish to define J ( t, y ) in the following way: In a first step wedefine J ( t, y ) = ˆ J ( t, y ) for ( t, y ) ∈ Q × Q . In a second step we let J ( t, y ) = lim Q ∋ ˜ y → y ˆ J ( t, ˜ y )for t ∈ Q , y ∈ ( −∞ , \ Q . Then, J ( t, y ) is defined on Q × ( −∞ , J ( t, y ) = lim Q ∋ ˜ t ↓ t J (˜ t, y )for t ∈ [0 , T ] \ Q and y ∈ [0 , t ∈ Q and y ∈ ( −∞ , \ Q . We choose a sequence (˜ y n ) ⊂ Q such that ˜ y n → y . Then, by (ii) and (iv), | ˆ J ( t, ˜ y n ) − ˆ J ( t, ˜ y m ) | ≤ Z ∗ | ˜ y n − ˜ y m | In view of (i), ( ˆ J ( t, ˜ y n )) is a Cauchy sequence and, as its limit does certainlynot depend on the choice of the sequence, we see that lim Q ∋ ˜ y → y ˆ J ( t, ˜ y ) ex-ists. Hence J ( t, y ) is well-defined on Q × ( −∞ , t ∈ Q and ( y , y ) ∈ ( −∞ , | J ( t, y ) − J ( t, y ) | ≤ Z ∗ | y − y | holds true.Now we fix some t ∈ [0 , T ] \ Q and some y ∈ ( −∞ , t n ) ⊂ Q and ( y k ) ⊂ Q such that t n ↓ t and y k → y . Then, | J ( t n , y ) − J ( t m , y ) |≤ | ˆ J ( t n , y k ) − ˆ J ( t m , y k ) | + | J ( t n , y ) − J ( t n , y k ) | + | J ( t m , y ) − J ( t m , y k ) |≤ | ˆ J ( t n , y k ) − ˆ J ( t m , y k ) | + 2 Z ∗ | y − y k | (8)By (iii) we can conclude that the sequence ( J ( t n , y )) is Cauchy, and, hence,lim Q ∋ ˜ t ↓ t ¯ J (˜ t, y ) exists, because the limit does not depend on the approxi-mating sequence. So J is well-defined.We now prove that J satisfies the properties stated in b) on ¯Ω. The Lip-schitz property b2) can be immediately transferred from t ∈ Q (for which it9as shown above) to general t by the construction of J . Property b1), whichstates that J has RCLL paths in t , can be shown by a similar argument asin (8). It remains to show concavity in y . As J ( t, y ) is continuous in y forfixed t , it is sufficient to show that J ( t, y + y ≥ J ( t, y ) + J ( t, y )2holds for every ( t, y ) ∈ [0 , T ] × ( −∞ , J , it is valid for ( t, y ) ∈ Q × Q . By the continuity properties of ¯ J thisimmediately extends to general ( t, y ).It remains to prove a). Suppose τ is a [0 , T ]-valued stopping time and Y is a F τ measurable, ( −∞ , Y by a nonincreasing sequence ( Y n ) of Q -valued, F τ measurable randomvariables. Moreover, we can choose a sequence ( τ n ) of Q -valued stoppingtimes such that τ n converges nonincreasingly to τ . By (ii) and the continuityproperties b1) and b2) of J , we get on ¯Ω J ( τ, Y ) = lim n →∞ J ( τ n , Y n ) = lim n →∞ ¯ J ( τ n , Y n ) . So it remains to show thatlim n →∞ ¯ J ( τ n , Y n ) = ¯ J ( τ, Y ) , P − a.s. (9)As U ( τ n , Y n ) ⊂ U ( τ, Y ) we observe that E [ ¯ J ( τ n , Y n ) | F τ ] ≤ ¯ J ( τ, Y )Hence, we have that for every A ∈ F τ E [ A ¯ J ( τ n , Y n )] ≤ E [ A ¯ J ( τ, Y )] , which implies lim sup n →∞ ¯ J ( τ n , Y n ) ≤ ¯ J ( τ, Y ) (10)Now choose some optimal strategy ¯ u τ,Y ∈ U ( τ, Y ) for ¯ J ( τ, Y ) and define u n ( t ) = [ τ n ,σ n ] ( t )¯ u τ,Y ( t )where σ n = inf { t ≥ τ n ; Z tτ ¯ u τ,Y ( s ) ds ≥ − Y n } ∧ T. Then u n ∈ U ( τ n , Y n ). As Y n is nonincreasing, the sequence of stopping times σ n is nondecreasing. Denoting its limit by σ we obtain that Z στ ¯ u τ,Y ( s ) ds = 1 − Y σ = T . As ¯ u τ,Y ∈ U ( τ, Y ), we have in any case that Z Tσ ¯ u τ,Y ( s ) ds = 0 . Thus, E [ ¯ J ( τ n , Y n ) | F τ ] ≥ E [ Z Tτ n u n ( s ) X ( s ) ds | F τ ]= ¯ J ( τ, Y ) − E [ Z τ n τ ¯ u τ,Y ( s ) X ( s ) ds | F τ ] − E [ Z σσ n ¯ u τ,Y ( s ) X ( s ) ds | F τ ] . As τ n ↓ τ and σ n ↑ σ , the dominated convergence theorem yields for every A ∈ F τ lim inf n →∞ E [ A ¯ J ( τ n , Y n )] ≥ E [ A ¯ J ( τ, Y )] , which in turn implies lim inf n →∞ ¯ J ( τ n , Y n ) ≥ ¯ J ( τ, Y )and in view of (10) finishes the proof of (9). By Proposition 2.5, there is a set ¯Ω of full measure such that for all ( t, y )the left-hand side derivative D − y J ( t, y ) and the right-hand side derivative D + y J ( t, y ) in y -direction exist on ¯Ω due to concavity. In order to study themarginal values − D − y J ( t, y ) and − D + y J ( t, y ), we first derive some propertiesrelated to the difference process J ( t, y + h ) − J ( t, y ). Proposition 3.1.
Suppose τ is a [0 , T ] -valued stopping time and Y ≥ Y are F τ -measurable, ( −∞ , -valued random variables. Denote by ¯ u τ,Y ∈ U ′ ( τ, Y ) , ¯ u τ,Y ∈ U ′ ( τ, Y ) optimal controls for ¯ J ( τ, Y ) and ¯ J ( τ, Y ) , re-spectively. Then,(i) It holds that ¯ J ( τ, Y ) − ¯ J ( τ, Y ) = esssup u ∈ ˜ U (¯ u τ,Y ,Y − Y ) E [ Z Tt u ( r ) X ( r ) dr | F τ ] , where ˜ U (¯ u τ,Y , Y − Y ) denotes the set of adapted processes u such that R Tτ u ( r ) dr ≤ Y − Y and ≤ u ( r ) ≤ L − ¯ u τ,Y ( r ) for r ∈ [ τ, T ] .(ii) Define ¯ u on [ τ, T ] by ¯ u ( r ) = (¯ u τ,Y ( r ) − ¯ u τ,Y ( r )) + { r ; R rτ (¯ u τ,Y ( s ) − ¯ u τ,Y ( s )) + ds ≤ Y − Y } Then, ¯ u ∈ ˜ U (¯ u τ,Y , Y − Y ) . Moreover, ¯ u + ¯ u τ,Y ∈ U ′ ( τ, Y ) and is anoptimal control for ¯ J ( τ, Y ) . roof. We prove both items at the same time. Let u ∈ ˜ U (¯ u τ,Y , Y − Y ).Then, it is straightforward to check that ¯ u τ,Y + u ∈ U ( τ, Y ). Hence,¯ J ( τ, Y ) ≥ E [ Z Tτ (¯ u τ,Y ( s )+ u ( s )) X ( s ) ds | F τ ] = ¯ J ( τ, Y )+ E [ Z Tτ u ( s ) X ( s ) ds | F τ ] . This implies¯ J ( τ, Y ) − ¯ J ( τ, Y ) ≥ esssup u ∈ ˜ U (¯ u τ,Y ,Y − Y ) E [ Z Tt u ( r ) X ( r ) dr | F τ ] . (11)We will show that ¯ u , defined in (ii), satisfies¯ u ∈ ˜ U (¯ u τ,Y , Y − Y ) (12)¯ u τ,Y − ¯ u ∈ U ( τ, Y ) (13)This proves (i), because¯ J ( τ, Y ) = E [ Z Tτ ¯ u τ,Y ( s ) X ( s ) ds | F τ ]= E [ Z Tτ (¯ u τ,Y ( s ) − ¯ u ( s )) X ( s ) ds | F τ ] + E [ Z Tt ¯ u ( s ) X ( s ) ds | F τ ] ≤ ¯ J ( τ, Y ) + esssup u ∈ ˜ U (¯ u τ,Y ,Y − Y ) E [ Z Tt u ( r ) X ( r ) dr | F τ ]In view of (11), the inequality turns into an identity. Hence we obtain (i)and the optimality of ¯ u for the problem ¯ J ( τ, Y ) − ¯ J ( τ, Y ). This impliesoptimality of ¯ u + ¯ u τ,Y for ¯ J ( τ, Y ), because¯ J ( τ, Y ) = ¯ J ( τ, Y )+ E [ Z Tt ¯ u ( s ) X ( s ) ds | F τ ] = E [ Z Tt (¯ u τ,Y ( s )+¯ u ( s )) X ( s ) ds | F τ ] . We will now verify (12) and (13). Notice first that (12) is rather obvious,because Z Tτ ¯ u ( r ) dr ≤ Y − Y by construction, and0 ≤ ¯ u ( r ) ≤ ¯ u τ,Y ( r ) − ¯ u τ,Y ( r ) ≤ L − ¯ u τ,Y ( r )for r ∈ [ τ, T ].We prove (13) on the sets { L ( T − τ ) ≤ − Y } , { L ( T − τ ) > − Y } ∩{ L ( T − τ ) ≤ − Y } and { L ( T − τ ) > − Y } separately. On the set { L ( T − τ ) ≤ − Y } , we get ¯ u τ,Y ( r ) = ¯ u τ,Y ( r ) = L for r ∈ [ τ, T ] byProposition 2.2. Hence, ¯ u ( r ) = 0 and ¯ u τ,Y − ¯ u = ¯ u τ,Y ∈ U ( τ, Y ). On the12et { L ( T − τ ) > − Y } ∩ { L ( T − τ ) ≤ − Y } , we get ¯ u τ,Y ( r ) = L for r ∈ [ τ, T ] and R Tτ ¯ u τ,Y ( r ) dr = 1 − Y by Proposition 2.2. Hence, we obtainon this set, for every r ∈ [ τ, T ], Z rτ (¯ u τ,Y ( s ) − ¯ u τ,Y ( s )) + ds ≤ L ( T − τ ) − Z Tτ ¯ u τ,Y ( s ) ds ≤ − Y − (1 − Y ) = Y − Y . This again implies ¯ u τ,Y − ¯ u = ¯ u τ,Y ∈ U ( τ, Y ). On the set { L ( T − τ ) > − Y } , we already know, by Proposition 2.2, that Z Tτ ¯ u τ,Y ( s ) ds = 1 − Y , Z Tτ ¯ u τ,Y ( s ) ds = 1 − Y . Hence, Z Tτ (¯ u τ,Y ( s ) − ¯ u τ,Y ( s )) ds = 1 − Y − (1 − Y ) = Y − Y , which yields Z Tτ ¯ u ( s ) ds = Y − Y . Consequently, Z Tτ (¯ u τ,Y ( s ) − ¯ u ( s )) ds = 1 − Y − ( Y − Y ) = 1 − Y Moreover, 0 ≤ min { ¯ u τ,Y ( r ) , ¯ u τ,Y ( r ) } ≤ ¯ u τ,Y ( r ) − ¯ u ( r ) ≤ L for r ∈ [ τ, T ]. So, u τ,Y − ¯ u ∈ U ( τ, Y ) also holds on { L ( T − τ ) > − Y } .By the arguments in the proof of (13) it is easy to see that ¯ u + ¯ u τ,Y ∈ U ′ ( τ, Y ) thanks to by Proposition 2.2. Corollary 3.2.
Suppose σ ≤ τ are [0 , T ] -valued stopping times and Y isan F σ -measurable random variable with values in ( −∞ , . Then there areoptimal controls ¯ u τ,Y for ¯ J ( τ, Y ) and ¯ u σ,Y for ¯ J ( σ, Y ) such that ¯ u τ,Y ( r ) ≥ ¯ u σ,Y ( r ) for r ∈ [ τ, T ] . Moreover, ¯ u τ,Y can be chosen from the set U ′ ( τ, Y ) .Proof. Choose optimal controls ¯ u σ,Y ∈ U ′ ( σ, Y ) for ¯ J ( σ, Y ) and ¯ u τ, ˜ Y ∈ U ′ ( τ, ˜ Y ) for ¯ J ( τ, ˜ Y ), where ˜ Y = Y + R τσ u σ,Y ( r ) dr . By the dynamic pro-gramming principle in Proposition 2.3, we observe that¯ u σ,Y = u σ,Y [ σ,τ ) + u τ, ˜ Y [ τ,T ) is also optimal for ¯ J ( σ, Y ). As ˜ Y ≥ Y , part (ii) of the previous propositionimplies that there is an optimal control ¯ u τ,Y ∈ U ′ ( τ, Y ) for J ( τ, Y ) such that¯ u τ,Y ( r ) ≥ u τ, ˜ Y ( r ) for r ∈ [ τ, T ]. 13he following proposition includes as a special case the statement thatthe difference process ¯ J ( t, y + h ) − ¯ J ( t, y ) is a submartingale for every y ∈ ( −∞ ,
1] and h ∈ [0 , − y ]. Proposition 3.3.
Suppose σ ≤ τ are [0 , T ] -valued stopping times and Y ≤ Y are F σ -measurable, ( −∞ , -valued random variables. Then, E [ ¯ J ( τ, Y ) − ¯ J ( τ, Y ) | F σ ] ≥ ¯ J ( σ, Y ) − ¯ J ( σ, Y ) . Proof.
By the previous corollary, we can choose optimal controls ¯ u τ,Y for¯ J ( τ, Y ) and ¯ u σ,Y for ¯ J ( σ, Y ) such that ¯ u τ,Y ( r ) ≥ ¯ u σ,Y ( r ) for r ∈ [ τ, T ] and¯ u τ,Y ∈ U ′ ( τ, Y ). Moreover, by Proposition 3.1 we can choose ¯ u τ,Y optimalfor J ( τ, Y ) such that ¯ u τ,Y − ¯ u τ,Y ∈ ˜ U (¯ u τ,Y , Y − Y ). Consequently, u ( r ) := ¯ u σ,Y ( r ) + [ τ,T ] ( r )(¯ u τ,Y ( r ) − ¯ u τ,Y ( r ))belongs to U ( σ, Y ). This yields¯ J ( σ, Y ) ≥ E [ Z Tσ u ( s ) X ( s ) ds | F σ ]= E [ Z Tσ ¯ u σ,Y ( s ) X ( s ) ds | F σ ] + E [ Z Tτ ¯ u τ,Y ( s ) X ( s ) ds | F σ ] − E [ Z Tτ ¯ u τ,Y ( s ) X ( s ) ds | F σ ]= ¯ J ( σ, Y ) + E [ ¯ J ( τ, Y ) | F σ ] − E [ ¯ J ( τ, Y ) | F σ ]In view of Proposition 2.5 and 3.3 we immediately obtain the followingresult. It states that the marginal values − D ± y J ( t, y ) are supermartingales,analogously to the situation for discrete time multiple stopping problems inMeinshausen and Hambly (2004) and Bender (2011a). Corollary 3.4. (i) For every y ∈ ( −∞ , , the left-hand side derivative D − y J ( t, y ) is a submartingale.(ii) For every y ∈ ( −∞ , , the right-hand side derivative D + y J ( t, y ) is asubmartingale. We will now study the regularity of the one-sided derivatives D − y J ( t, y )and D + y J ( t, y ) as processes in time. The following example is instructive tosee what kind of results we can expect. Example . Suppose ρ is a stopping time of the filtration F with values in[0 , T ] and consider the RCLL process X ( t ) = [0 ,ρ ) ( t )14hen, certainly it is optimal to exercise as soon as possible, i.e. ¯ u t,y = L [ t,t +(1 − y ) /L ] is an optimal control for J ( t, y ). Therefore, J ( t, y ) = E [min(1 − y, L ( ρ − t )) | F t ] { ρ ≥ t } . Thus, the one sided derivatives of J are D − y J ( t, y ) = − E [ { y> − L ( ρ − t ) } | F t ] D + y J ( t, y ) = − E [ { y ≥ − L ( ρ − t ) } | F t ] . It follows that E [ D + y J ( t, y )] = P ( { ρ < (1 − y ) /L + t } − . If the distribution function of ρ has a jump at (1 − y ) /L + t , then themapping t E [ D + y J ( t, y )] is not rightcontinuous at t . This implies that D + y J ( t, y ) does not admit a rightcontinuous version in t , if the distributionfunction of ρ is discontinuous. Contrarily E [ D − y J ( t, y )] = P ( { ρ ≤ (1 − y ) /L + t } − t for every y . As D − y J ( t, y ) is a submartingale for fixed y , we conclude, that, for every y , D − y J ( t, y ) has an RCLL modification. Proposition 3.6. (i) For every y ∈ ( −∞ , , the submartingale D − y J ( t, y ) has an RCLL modification.(ii) λ [0 ,T ] ⊗ P ( { D − y J ( · , y ) = D + y J ( · , y ) } ) = 0 for λ ( −∞ , -a.e. y .Proof. Notice first, that ¯ J ( t, y ) = E [ L R Tt X ( s ) ds | F t ] for y ≤ − LT . Hence, D ± y J ( t, y ) = 0 for y < − LT . We can hence restrict ourselves to y ∈ [1 − LT,
1] for the rest of the proof.(i) D − y J ( t, y ) is a submartingale by Corollary 3.4 for every y . Hence, itis sufficient to prove that for every y , the mapping t → E [ D − y J ( t, y )] isrightcontinuous. Fix t ∈ [0 , T ) and a sequence ∆ n ↓
0. By the submartin-gale property, E [ D − y J ( t + ∆ n , y )] ≥ E [ D − y J ( t, y )] is nonincreasing. By theconcavity of J ( t, y ) in y , we hence obtain Z − LT | E [ D − y J ( t + ∆ n , y )] − E [ D − y J ( t, y )] | dy = E (cid:20)Z − LT ( D − y J ( t + ∆ n , y ) − D − y J ( t, y )) dy (cid:21) = E [ J ( t, − LT ) − J ( t + ∆ n , − LT )] → n → ∞ by the rightcontinuity of J ( t, − LT ) in t . Thus, for almostevery y , | E [ D − y J ( t + ∆ n , y )] − E [ D − y J ( t, y )] | → , n → ∞ . (14)15ow fix some arbitrary y and choose a sequence y k ↑ y such that (14)holds for every y k . Note that by concavity, E [ D − y J ( t + ∆ n , y )] ≤ E [ D − y J ( t + ∆ n , y k )] . Consequently,0 ≤ E [ D − y J ( t + ∆ n , y )] − E [ D − y J ( t, y )] ≤ E [ D − y J ( t + ∆ n , y k )] − E [ D − y J ( t, y k )] + E [ D − y J ( t, y k )] − E [ D − y J ( t, y )] . By (14) we thus obtainlim sup n →∞ E [ D − y J ( t +∆ n , y )] − E [ D − y J ( t, y )] ≤ E [ D − y J ( t, y k )] − E [ D − y J ( t, y )] . Letting k tend to infinity we observe thatlim n →∞ E [ D − y J ( t + ∆ n , y )] = E [ D − y J ( t, y )] . (ii) We define the measurable set N := { (( t, ω, y ) ∈ [0 , T ] × Ω × [1 − LT, D − y J ( t, y, ω ) = D − y J ( t, y, ω ) } and consider the sections N y = { ( t, ω ) ∈ [0 , T ] × Ω; D − y J ( t, y, ω ) = D − y J ( t, y, ω ) } , y ∈ [1 − LT, N ( t,ω ) = { y ∈ [1 − LT, D − y J ( t, y, ω ) = D − y J ( t, y, ω ) } , ( t, ω ) ∈ [0 , T ] × Ω . It is sufficient to show that Z − LT ( λ [0 ,T ] ⊗ P )( N y ) dy = 0 . By Fubini’s theorem Z − LT ( λ [0 ,T ] ⊗ P )( N y ) dy = Z [0 ,T ] × Ω λ [1 − LT, ( N ( t,ω ) ) d ( λ [0 ,T ] ⊗ P ) . However, λ [1 − LT, ( N ( t,ω ) ) = 0 for every ( t, ω ) ∈ [0 , T ] × ¯Ω, (where ¯Ω isthe set of full measure constructed in Proposition 2.5), by concavity of thefunction y J ( t, ω, y ). In this section we prove that the good version of the value process J ( t, y )indeed solves the BSPDE (1). 16 heorem 4.1. For every y ∈ ( −∞ , and t ∈ [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 holds P -almost surely. Moreover, the left-hand side derivative D − y can bereplaced by the right-hand side derivative D + y .Proof. The boundary condition J ( t,
1) = 0 is obviously satisfied.
Step 1:
We show for every 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) . To this end we first fix some ( t, y ) ∈ [0 , T ] × ( −∞ ,
1) and choose a sequenceof partitions π n = { t n , t n , . . . , t nn } of [ t, T ] such that the mesh size | π n | =max i =1 ,...,n | t ni − t ni − | tends to zero and with t n = t and t nn = T . We denoteby ¯ u t ni ,y an optimal control for ¯ J ( t ni , y ) and define ¯ Y ni := R t ni +1 t ni ¯ u t ni ,y ( r ) dr .Applying the dynamic programming principle (Proposition 2.3) repeatedly,we obtain J ( t, y )= E [ Z t n t n ¯ u t n ,y ( r ) X ( r ) dr + J ( t , y ) + ( J ( t , y + ¯ Y n ) − J ( t , y )) | F t ]= n − X i =0 E [ Z t ni +1 t ni ¯ u t ni ,y ( r ) X ( r ) dr | F t ]+ n − X i =0 E [ J ( t i +1 , y + ¯ Y ni ) − J ( t i +1 , y ) | F t ]= n − X i =0 E [ Z t ni +1 t ni ¯ u t ni ,y ( r )( X ( r ) + D − y J ( r, y )) dr | F t ]+ n − X i =0 E [ Z t ni +1 t ni ¯ u t ni ,y ( r ) (cid:18) J ( t i +1 , y + ¯ Y ni ) − J ( t i +1 , y )¯ Y ni − D − y J ( r, y ) (cid:19) dr | F t ]= ( I ) + ( II )Then, ( I ) ≤ 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) , and it remains to show that the limsup of ( II ) is nonpositive. We denote by [ D − y J ( r, y ) the RCLL modification of D − y J ( r, y ) which exists by Proposition17.6, (i). Moreover, let ¯ π n ( r ) = t ni +1 for r ∈ ( t ni , t ni +1 ]. By concavity we get( II ) ≤ n − X i =0 E [ Z t ni +1 t ni ¯ u t ni ,y ( r ) (cid:0) D − y J ( t ni +1 , y ) − D − y J ( r, y ) (cid:1) dr | F t ] ≤ LE [ Z Tt | [ D − y J (¯ π n ( r ) , y ) − [ D − y J ( r, y ) | dr | F t ] . The right-hand side converges to zero by rightcontinuity and dominatedconvergence.
Step 2:
We show 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) . for every y ∈ A := { η ∈ ( −∞ , λ [0 ,T ] ⊗ P ( { D − y J ( · , η ) = D + y J ( · , η ) } ) = 0 } . We fix a pair ( t, y ) ∈ [0 , T ] × A and choose a sequence of partitions π n = { t n , t n , . . . , t nn } of [ t, T ] such that the mesh size | π n | = max i =1 ,...,n | t ni − t ni − | tends to zero and with t n = t and t nn = T . Now we define the controls, u t ni ,ym ( r ) = L ( t ni ,t ni +1 ] ( r ) { Z m ( t ni ) > } , m ∈ N , where Z m ( r ) = m Z r ( r − /m ) ∧ ( X ( s ) + D − y J ( s, y )) ds, r ∈ [0 , T ] . By Lebesgue’s differentiation theorem and Fubini’s theorem λ [ t,T ] ⊗ P ( { ( r, ω ); lim m →∞ Z m ( r ) = ( X ( r ) + D − y J ( r, y )) } c ) = 0 (15)Note that u t ni ,ym ∈ U ( t ni , y ) for sufficiently large n (independent of m ), whichwe assume from now on. We define Y n,mi := Z t ni +1 t ni u t ni ,ym ( r ) dr, which is F t ni -measurable. Similarly to the first step, but taking the subopti-mality of the controls into account, we get J ( t, y ) ≥ n − X i =0 E [ Z t ni +1 t ni u t ni ,ym ( r )( X ( r ) + D − y J ( r, y )) dr | F t ]+ n − X i =0 E [ Z t ni +1 t ni u t ni ,ym ( r ) (cid:18) J ( t i +1 , y + Y n,mi ) − J ( t i +1 , y ) Y n,mi − D − y J ( r, y ) (cid:19) dr | F t ]= ( I ) + ( II ) 18e first treat the term ( I ). Let π n ( r ) = t ni for r ∈ ( t ni , t ni +1 ]. Then,( I ) = E [ Z Tt L { Z m ( π n ( r )) > } ( X ( r ) + D − y J ( r, y )) dr | F t ] ≥ E [ Z Tt L { Z m ( π n ( r )) > } Z m ( r ) dr | F t ] − LE [ Z Tt | X ( r ) + D − y J ( r, y ) − Z m ( r ) | dr | F t ]Concerning term ( II ) we note that, for r ∈ ( t ni , t ni +1 ], E [ Z t ni +1 t ni u t ni ,ym ( r ) (cid:18) J ( t i +1 , y + Y n,mi ) − J ( t i +1 , y ) Y n,mi − D − y J ( r, y ) (cid:19) dr | F t ]= E [ Z t ni +1 t ni u t ni ,ym ( r ) (cid:18) E [ J ( t i +1 , y + Y n,mi ) − J ( t i +1 , y ) | F r ] Y n,mi − D − y J ( r, y ) dr (cid:19) | F t ] ≥ E [ Z t ni +1 t ni u t ni ,ym ( r ) (cid:18) J ( r, y + Y n,mi ) − J ( r, y ) Y n,mi − D − y J ( r, y ) (cid:19) dr | F t ]= E [ Z t ni +1 t ni u t ni ,ym ( r ) (cid:18) J ( r, y + L ( t ni +1 − t ni ) − J ( r, y ) L ( t ni +1 − t ni ) − D − y J ( r, y ) (cid:19) dr | F t ] ≥ − LE [ Z t ni +1 t ni | J ( r, y + L ( t ni +1 − t ni ) − J ( r, y ) L ( t ni +1 − t ni ) − D − y J ( r, y ) | dr | F t ]Here, we applied the F t ni -measurability of Y n,mi and the submartingale prop-erty in Proposition 3.3. Hence, making use of y ∈ A ,( II ) ≥ − LE [ Z Tt | J ( r, y + L (¯ π n ( r ) − π n ( r ))) − J ( r, y ) L (¯ π n ( r ) − π n ( r )) − D + y J ( r, y ) | dr | F t ] . Gathering the estimates for ( I ) and ( II ) we have J ( t, y ) ≥ E [ Z Tt L { Z m ( π n ( r )) > } Z m ( r ) dr | F t ] − LE [ Z Tt | X ( r ) + D − y J ( r, y ) − Z m ( r ) | dr | F t ] − LE [ Z Tt | J ( r, y + L (¯ π n ( r ) − π n ( r ))) − J ( r, y ) L (¯ π n ( r ) − π n ( r )) − D + y J ( r, y ) | dr | F t ] . As Z m has continuous paths, we get L { Z m ( π n ( r )) > } Z m ( r ) dr → L ( Z m ( r )) + n tends to infinity. Letting n go to infinity, we thus obtain by dominatedconvergence J ( t, y ) ≥ E [ Z Tt L ( Z m ( r )) + dr | F t ] − LE [ Z Tt | X ( r )+ D − y J ( r, y ) − Z m ( r ) | dr | F t ] . In view of (15) the proof of Step 2 can then be completed by letting m tendto infinity. Step 3:
We can now prove the assertion.By step 1 and 2 we have 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) = 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) (16)for y ∈ A . Now fix some y ∈ ( −∞ , − \ A . By Proposition 3.6, (ii), thereare sequences (¯ y k ) and ( y k ) in A such that ¯ y k ↓ y and y k ↑ y . Recallingthat y J ( t, y ) is continuous, y D − y J ( s, y ) is leftcontinuous and y D + y J ( s, y ) is rightcontinuous, we immediately see that the equations in (16)also hold for y .We can slightly reformulate the result that the value process solves theabove BSPDE in the following way. Corollary 4.2.
For every 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) , t ∈ [0 , T ] D − y J ( t, ≤ − X ( t ) , D − y J ( t, − L ( T − t )) = 0 , t ∈ [0 , T ) holds P -almost surely.Proof. In view of the previous theorem, we only need to show that D − y J ( t, ≤ − X ( t ) , D − y J ( t, − L ( T − t )) = 0 , (17)for every t ∈ [0 , T ). The first assertion in (17) in turn implies 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) = 0 = J ( t, t ∈ [0 , T ).Note that the second assertion in (17) is trivial, because J ( t, y ) = E [ Z Tt LX ( s ) ds | F t ]20or y < − L ( T − t ). In order to prove the first assertion we define u t,y ( r ) = L [ t,t +(1 − y ) /L ] ( r ). Then, for y < J ( t, y ) − J ( t, y − − J ( t, y )1 − y ≤ − L − y Z min { t +(1 − y ) /L,T } t X ( s ) ds. By right-continuity of X , the right-hand side converges to − X ( t ), whichconcludes the proof of (17). In this section we characterize optimality of controls. By Corollary 4.2 oneexpects that the following result holds under at most technical conditions:Suppose that u ∈ U ( t, y ). Then u is an optimal control, if and only if 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 (18) λ [ t,T ] ⊗ P -almost surely.To prove such result we require an appropriate version of a chain rule,which is derived in the following lemma. Lemma 5.1.
Suppose V ( t, y ) = E [ Z Tt v ( r, y ) dr | F t ] , t ∈ [0 , T ] , y ∈ ( −∞ , , is an adapted random field which satisfies:1. There is a set ¯Ω of full P measure such that D − y V ( t, ω, y ) exists forevery t ∈ [0 , T ] , y ∈ ( −∞ , and ω ∈ ¯Ω , and such that v ( t, ω, y ) isleftcontinuous in y for every t ∈ [0 , T ] , y ∈ ( −∞ , and ω ∈ ¯Ω .2. v ( t, y ) is ( F t ) -adapted for every y ∈ ( −∞ , and E [ sup ( t,y, ˜ y ) ∈ [0 ,T ] × ( −∞ , , ˜ y = y (cid:18) | v ( t, y ) | + (cid:12)(cid:12)(cid:12)(cid:12) V ( t, y ) − V ( t, ˜ y ) y − ˜ y (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ] < ∞ . Then, for every ( t, y ) ∈ [0 , T ] × ( −∞ , and for every nondecreasing processof the form y ( r ) = y + R rt u ( s ) ds with u ∈ U ( t, y ) , V ( t, y ) = E [ Z Tt (cid:0) v ( r, y ( r )) dr − D − y V ( r, y ( r )) u ( r ) (cid:1) dr | F t ] holds P -almost surely. roof. We first smoothen V in y -direction by setting˜ V ( t, y ) := Z y V ( t, η ) dη, with the usual convention that R y V ( t, η ) dη = − R y V ( t, η ) dη for y < V ( t, y ) = E [ Z Tt ˜ v ( t, y ) dt | F t ]where ˜ v ( t, y ) = Z y v ( t, η ) dη. We now fix a pair ( t, y ) ∈ [0 , T ] × ( −∞ ,
1) and define, for n ∈ N , t ni := t + i ( T − t ) /n and U n ( t, y ) := n u ∈ U ( t, y ); u ( r ) = u ( t ni ) , for every r ∈ [ t ni , t ni +1 ) o Step 1:
For u ∈ U n ( t, y ) and y ( r ) = y + R rt u ( s ) ds ˜ V ( t, y ) = E [ Z Tt ˜ v ( r, y ( r )) dr − V ( r, y ( r )) u ( r ) dr | F t ] . In order to prove Step 1, we fix some n ∈ N and u ∈ U n ( t, y ). Choose asequence of refining partitions ( π N ) N ≥ n of [ t, T ] such that { t n , . . . , t nn } ⊂{ s N , . . . , s NN } = π N for every N ≥ n . We then define π N ( r ) = s Ni , ¯ π N ( r ) = s Ni +1 , r ∈ ( s Ni , s Ni +1 ] . We split ˜ V ( t, y ) = E [ N − X i =0 ˜ V ( s Ni , y ( s Ni )) − ˜ V ( s Ni +1 , y ( s Ni +1 )) | F t ]= E [ N − X i =0 ˜ V ( s Ni , y ( s Ni +1 )) − ˜ V ( s Ni +1 , y ( s Ni +1 )) | F t ]+ E [ N − X i =0 ˜ V ( s Ni , y ( s Ni )) − ˜ V ( s Ni , y ( s Ni +1 )) | F t ]= ( I ) + ( II )Then, ( I ) = E [ Z Tt ˜ v ( r, y (¯ π N ( r ))) dr | F t ] .
22y continuity of ˜ v ( r, · ) and dominated convergence we obtain that( I ) → E [ Z Tt ˜ v ( r, y ( r )) dr | F t ] , N → ∞ . We now observe that( II ) = E [ N − X i =0 Z s Ni +1 s Ni ˜ V ( s Ni , y ( s Ni )) − ˜ V ( s Ni , y ( s Ni +1 )) s Ni +1 − s Ni dr | F t ]= E [ N − X i =0 Z s Ni +1 s Ni − y ( s Ni +1 ) − y ( s Ni ) Z y ( s Ni +1 ) y ( s Ni ) V ( s Ni , η ) dη u ( r ) dr | F t ] . Here, we used that u ( r ) = u ( s Ni ) for r ∈ [ s Ni , s Ni +1 ) and y ( s Ni +1 ) = y ( s Ni ) + u ( s Ni )( s Ni +1 − s Ni ). Then, y ( s Ni +1 ) is F s Ni -measurable and, thus,( II ) = E [ N − X i =0 Z s Ni +1 s Ni − y ( s Ni +1 ) − y ( s Ni ) Z y ( s Ni +1 ) y ( s Ni ) V ( r, η ) dη u ( r ) dr | F t ]+ E [ N − X i =0 Z s Ni +1 s Ni − y ( s Ni +1 ) − y ( s Ni ) × Z y ( s Ni +1 ) y ( s Ni ) E [ V ( s Ni , η ) − V ( r, η ) | F s Ni ] dη u ( r ) dr | F t ]=: ( IIa ) + (
IIb ) . Then,(
IIa ) = E [ Z Tt − y (¯ π N ( r )) − y ( π N ( r )) Z y (¯ π N ( r )) y ( π N ( r )) V ( r, η ) dη u ( r ) dr | F t ]By continuity of V ( r, · ) and dominated convergence we get( IIa ) → − E [ Z Tt V ( r, y ( r )) u ( r ) dr | F t ] , N → ∞ . It thus remains to show that (
IIb ) goes to zero. To see this we note that for r ∈ [ s Ni , s Ni +1 ] | E [ V ( s Ni , η ) − V ( r, η ) | F s Ni ] | ≤ | π N | E [sup ( s,η ) | v ( s, η ) || F s Ni ] , where | π N | denotes the mesh size of the partition π N . Hence, | ( IIb ) | ≤ | π N | E [ Z Tt u ( r ) dr sup ( s,η ) | v ( s, η ) || F t ] → , N → ∞ . tep 2: For u ∈ U ( t, y ) and y ( r ) = y + R rt u ( s ) ds ˜ V ( t, y ) = E [ Z Tt ˜ v ( r, y ( r )) dr − V ( r, y ( r )) u ( r ) dr | F t ]Indeed, given a control u ∈ U ( t, y ), we define u n via u n ( r ) = n R t i t i − u ( s ) dsT − t , r ∈ [ t ni , t ni +1 ) , i = 1 , . . . , n − u n ( r ) = 0 for r ∈ [0 , t n ). Then, u n ∈ U n ( t, y ).Let y ( r ) = y + R rt u ( s ) ds and y n ( r ) = y + R rt u n ( s ) ds . Then it is straight-forward to verify that y ( t ni − ) = y n ( t ni ) , i = 1 , . . . , n. This implies that the sequence ( y n ( r )) converges to y ( r ), as n tends to in-finity, for every r ∈ [ t, T ]. By continuity of ˜ v ( r, · ) and V ( r, · ) and dominatedconvergence we have E [ Z Tt ˜ v ( r, y n ( r )) dr − V ( r, y n ( r )) u ( r ) dr | F t ] → E [ Z Tt ˜ v ( r, y ( r )) dr − V ( r, y ( r )) u ( r ) dr | F t ] . Moreover, y n ( r ) → y ( r ) for every r ∈ [ t, T ], together with the boundednessof the sequence ( u n ) in L ([ t, T ] , λ [ t,T ] ) implies that ( u n ) converges to u weakly in L ([ t, T ] , λ [ t,T ] ). Hence, E [ Z Tt V ( r, y ( r ))( u n ( r ) − u ( r )) dr | F t ] → . This shows that Step 1 implies Step 2.
Step 3:
For u ∈ U ( t, y ) and y ( r ) = y + R rt u ( s ) dsV ( t, y ) = E [ Z Tt v ( r, y ( r )) dr − D − y V ( r, y ( r )) u ( r ) dr | F t ] . Fix some u ∈ U ( t, y ) and note that u also belongs to U ( t, y − ǫ ) for ǫ > V ( t, y ) = E [ Z Tt ˜ v ( r, y ( r )) dr − V ( r, y ( r )) u ( r ) dr | F t ]˜ V ( t, y − ǫ ) = E [ Z Tt ˜ v ( r, y ( r ) − ǫ ) dr − V ( r, y ( r ) − ǫ ) u ( r ) dr | F t ]24here y ( r ) = y + R rt u ( s ) ds . Hence,˜ V ( t, y − ǫ ) − ˜ V ( t, y ) − ǫ = E [ Z Tt ǫ Z y ( r ) y ( r ) − ǫ v ( r, η ) dηdr | F t ] − E [ Z Tt u ( r ) V ( r, y ( r ) − ǫ ) − V ( r, y ( r )) − ǫ dr | F t ]Letting ǫ tend to zero, the right-hand side converges to E [ Z Tt v ( r, y ( r )) − u ( r ) D − y V ( r, y ( r )) dr | F t ]by leftcontinuity of v , and the left-hand side converges to V ( t, y ), because˜ V ( t, y − ǫ ) − ˜ V ( t, y ) − ǫ = 1 ǫ Z yy − ǫ V ( t, η ) dη. By the results established in the previous sections (Corollary 4.2 andProposition 2.5) we, hence, arrive at the following corollary.
Corollary 5.2.
For every ( t, y ) ∈ [0 , T ] × ( −∞ , and for every nondecreas-ing process of the form y ( r ) = y + R rt u ( s ) ds with u ∈ U ( t, y ) , J ( t, y ) = E [ Z Tt L ( X ( r ) + D − y J ( r, y ( r ))) + dr − Z Tt D − y J ( r, y ( r )) u ( r ) dr | F t ] holds P -almost surely. We are now in the position to characterize the set of optimal controls.
Theorem 5.3.
A control u ∈ U ( t, y ) is optimal for J ( t, y ) , if and only if(18) holds.Proof. By Corollary 5.2, J ( t, y )= E [ Z Tt L ( X ( r ) + D − y J ( r, y ( r ))) + dr − Z Tt D − y J ( r, y ( r )) u ( r ) dr | F t ]= E [ Z Tt X ( r ) u ( r ) dr | F t ]+ E [ Z Tt (cid:0) L ( X ( r ) + D − y J ( r, y ( r ))) + − ( X ( r ) + D − y J ( r, y ( r ))) u ( r ) (cid:1) dr | F t ]Hence, u is optimal, if and only if the nonnegative second term on the right-hand side vanishes, which is equivalent to (18).25 A dual formulation
We finally present a dual representation in terms of martingales. This type ofrepresentation was first suggested by Rogers (2002) and Haugh and Kogan(2004) for optimal stopping problems. A corresponding result for generaldiscrete time optimal control problems is due to Brown et al. (2010).The main idea is to relax the adaptedness condition on the set of controlsand to penalize non-adapted controls by a suitable choice of martingales. Wefirst introduce the set U ( t, y ) of deterministic functions u : [ t, T ] → [0 , L ] suchthat R Tt u ( s ) ds ≤ − y . With this notation, U ( t, y ) is the set of adaptedprocesses whose paths take values in U ( t, y ). Definition 6.1.
Suppose ( t, y ) ∈ [0 , T ] × ( −∞ , M : [ t, T ] × Ω × U ( t, y ) → R is called a martingale map , if ( M ( s, u ) , t ≤ s ≤ T ) is a martingale for every u ∈ U ( t, y ). We denote the set of martingale maps by M ( t, y ).We can now represent the value J ( t, y ) as a solution of minimizationproblem over martingale maps. Theorem 6.2.
Suppose ( t, y ) ∈ [0 , T ] × ( −∞ , . Then, J ( t, y ) = essinf M ∈ M ( t,y ) E [esssup u ∈ U ( t,y ) Z Tt u ( r ) X ( r ) dr − ( M ( T, u ) − M ( t, u )) | F t ] . (19) Moreover, ¯ M t,y ( s, u ) = J ( s, y ( s )) + Z st L ( X ( r ) + D − y J ( r, y ( r ))) + − D − y J ( r, y ( r )) u ( r ) dr with y ( r ) = y + R rt u ( l ) dl is an optimal martingale map, which satisfies J ( t, y ) = esssup u ∈ U ( t,y ) Z Tt u ( r ) X ( r ) dr − ( ¯ M t,y ( T, u ) − ¯ M t,y ( t, u )) . (20) Proof.
Suppose u ∈ U ( t, y ) and M is a martingale map. Then, E [ Z Tt u ( r ) X ( r ) dr | F t ] = E [ Z Tt u ( r ) X ( r ) dr − ( M ( T, u ) − M ( t, u )) | F t ] ≤ E [esssup u ∈ U ( t,y ) Z Tt u ( r ) X ( r ) dr − ( M ( T, u ) − M ( t, u )) | F t ] , where the first identity is due to the martingale property of M ( s, u ). Thisshows J ( t, y ) ≤ essinf M ∈ M ( t,y ) E [esssup u ∈ U ( t,y ) Z Tt u ( r ) X ( r ) dr − ( M ( T, u ) − M ( t, u )) | F t ] .
26n order to finish the proof it is now sufficient to show that ¯ M t,y is a mar-tingale map and satisfies J ( t, y ) ≥ esssup u ∈ U ( t,y ) Z Tt u ( r ) X ( r ) dr − ( ¯ M t,y ( T, u ) − ¯ M t,y ( t, u )) . (21)Fix some u ∈ U ( t, y ) and let y ( r ) = y + R rt u ( l ) dl . By Corollary 5.2, we getfor s ≥ tJ ( s, y ( s )) = E [ Z Ts L ( X ( r )+ D − y J ( r, y ( r ))) + dr − Z Ts D − y J ( r, y ( r )) u ( r ) dr | F s ] . This shows that ¯ M t,y is a martingale map. Finally, (21) holds, because esssup u ∈ U ( t,y ) Z Tt u ( r ) X ( r ) dr − ( ¯ M t,y ( T, u ) − ¯ M t,y ( t, u )) ! − J ( t, y )= esssup u ∈ U ( t,y ) Z Tt (cid:0) u ( r )( X ( r ) + D − y J ( r, y ( r ))) − L ( X ( r ) + D − y J ( r, y ( r ))) + (cid:1) dr ≤ . Acknowledgement
The authors gratefully acknowledge financial support by the ATN-DAADAustralia Germany Joint Research Cooperation Scheme.
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