Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank-El Karoui Representation Theorem
aa r X i v : . [ m a t h . O C ] D ec Identifying the Free Boundary of aStochastic, Irreversible Investment Problemvia the Bank-El Karoui Representation Theorem ∗ Maria B. Chiarolla † Giorgio Ferrari ‡ November 10, 2018
Abstract.
We study a stochastic, continuous time model on a finite horizon for a firm thatproduces a single good. We model the production capacity as an Itˆo diffusion controlled by anondecreasing process representing the cumulative investment. The firm aims to maximize itsexpected total net profit by choosing the optimal investment process. That is a singular stochas-tic control problem. We derive some first order conditions for optimality and we characterizethe optimal solution in terms of the base capacity process l ∗ ( t ), i.e. the unique solution of a rep-resentation problem in the spirit of Bank and El Karoui [4]. We show that the base capacity isdeterministic and it is identified with the free boundary ˆ y ( t ) of the associated optimal stoppingproblem, when the coefficients of the controlled diffusion are deterministic functions of time.This is a novelty in the literature on finite horizon singular stochastic control problems. As asubproduct this result allows us to obtain an integral equation for the free boundary, which weexplicitly solve in the infinite horizon case for a Cobb-Douglas production function and constantcoefficients in the controlled capacity process. Key words : irreversible investment, singular stochastic control, optimal stopping, freeboundary, Bank and El Karoui’s Representation Theorem, base capacity.
MSC2010 subsject classification : 91B70, 93E20, 60G40, 60H25.
JEL classification : C02, E22, D92, G31.
We study a continuous time, singular stochastic investment problem on a finite horizon T for afirm that produces a single good. The setting is as in Chiarolla and Haussmann [11] but without ∗ These results extend a portion of the second author Ph.D. dissertation [16] under the supervision of the firstauthor. This paper has been presented by the second author at several conferences thanks to the financial supportby the German Research Foundation (DFG) via grant Ri 1128-4-1. † Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza, Universit`a di Roma ‘La Sapienza’,via del Castro Laurenziano 9, 00161 Roma, Italy; [email protected] ‡ Corresponding author. Center for Mathematical Economics, Bielefeld University, Universitaetsstrasse 25, D-33615 Bielefeld, Germany; [email protected] he Free Boundary via the Representation Theorem ν ( t ) representing the cumulative investment, i.e. dC y,ν ( t ) = C y,ν ( t )[ − µ C ( t ) dt + σ C ( t ) dW ( t )] + f C ( t ) dν ( t ) , t ∈ [0 , T ) ,C y,ν (0) = y > . The optimal investment problem issup ν E (cid:26) Z T e − R t µ F ( s ) ds R ( C y,ν ( t )) dt − Z [0 ,T ) e − R t µ F ( s ) ds dν ( t ) (cid:27) . (1.1)In [11] the Authors proved the existence of the optimal investment process ˆ ν . As expected,the optimal time to invest τ ∗ was the solution of the associated optimal stopping problem. Inparticular, under constant coefficients and a Cobb-Douglas production function, they obtained avariational formulation for the optimal stopping problem, i.e. a free boundary problem. In orderto characterize the moving boundary ˆ y ( t ) through an integral equation, the Authors provedthe left-continuity of ˆ y ( t ) and assumed its right-continuity (cf. [11], Assumption-[Cfb]) sincecontinuity of the free boundary was needed to prove the smooth fit property.In this paper, rather than trying to generalize the variational approach to the case of time-dependent coefficients, we introduce a new approach based on a stochastic generalization of theKuhn-Tucker conditions and we identify the free boundary by exploiting the Bank-El KarouiRepresentation Theorem (cf. [4], Theorems 1 and 3). As a subproduct, we obtain an integralequation for the free boundary which does not require a priori continuity of the free boundaryand the smooth fit property to be derived.The Bank-El Karoui Representation Theorem allows to write an optional process X = { X ( t ) , t ∈ [0 , T ] } such that X ( T ) = 0 as an optional projection of the form X ( t ) = E (cid:26) Z ( t,T ] f (cid:16) s, sup t ≤ v , (2.1)where µ C , σ C and f C are given progressively measurable processes, uniformly bounded in ( ω, t ).Moreover f C is continuous with 0 < k f ≤ f C ( t ) ≤ κ f and µ C ≥
0. Here f C is a conversion factorsince any unit of investment is converted into f C units of production capacity.By setting C ( t ) := C , ( t ) , ν ( t ) := Z [0 ,t ) f C ( s ) C ( s ) dν ( s ) , (2.2)we may write C ( t ) = e − R t µ C ( s ) ds M ( t ) , (2.3)where the exponential martingale M s ( t ) := e − R ts σ C ( u ) du + R ts σ C ( u ) dW ( u ) , t ∈ [ s, T ] , (2.4)is defined for s ∈ [0 , T ]. Without investment, C represents the decay of a unit of initial capitaland we have C y,ν ( t ) = C ( t )[ y + ν ( t )] . (2.5)The production function of the firm is a nonnegative, measurable function R ( C ). We makethe following Assumption 2.1. the mapping C R ( C ) is strictly increasing and strictly concave with con-tinuous derivative R c ( C ) := ∂∂C R ( C ) satisfying the Inada conditions lim C → R c ( C ) = ∞ , lim C →∞ R c ( C ) = 0 . Our Assumption 2.1 is not as general as the Assumption in [11] but it is needed to apply theBank-El Karoui Representation Theorem [4].Each investment plan ν ∈ S o leads to the expected total profit net of investment J ,y ( ν ) = E (cid:26) Z T e − R t µ F ( s ) ds R ( C y,ν ( t )) dt − Z [0 ,T ) e − R t µ F ( s ) ds dν ( t ) (cid:27) (2.6)where S o := { ν : Ω × [0 , T ] R + nondecreasing, left-continuous, adapted s.t. ν (0) = 0 , P -a.s. } is the convex set of irreversible investment processes. Here µ F is the firm’s manager discountfactor; it is a nonnegative, progressively measurable process, uniformly bounded in ( ω, t ).The firm’s problem is then V (0 , y ) := sup ν ∈S o J ,y ( ν ) , (2.7)with V finite thanks to Assumption 2.1 (cf. [11], Proposition 2 . R and the affine nature of C y,ν in ν imply that J ,y ( ν ) is strictly concave on S o . Hence ifa solution ˆ ν of (2.7) exists, it is unique. The existence of the solution has been proved in [11],Theorem 3 .
1. We provide a new characterization of it in Theorem 4.3 below. he Free Boundary via the Representation Theorem As in [2], [5], [25] and [26], among others, we now aim to characterize the optimal solution of(2.7) by some first order conditions for optimality.Let T denote the set of all stopping times with value in [0 , T ], P -a.s. Note that the strictlyconcave functional J ,y ( ν ) admits the supergradient ∇ ν J ,y ( ν )( τ ) := E (cid:26) Z Tτ e − R s µ F ( u ) du C ( s ) f C ( τ ) C ( τ ) R c ( C y,ν ( s )) ds (cid:12)(cid:12)(cid:12) F τ (cid:27) (3.1) − e − R τ µ F ( u ) du { τ The quantity ∇ ν J ,y ( ν )( t ) may be interpreted as the net marginal expected futureprofit resulting from an additional infinitesimal investment at time t . Mathematically, ∇ ν J ,y ( ν ) can be viewed as the Riesz representation of the profit’s gradient at ν . More precisely, we maydefine ∇ ν J ,y ( ν ) as the optional projection of the product-measurable process φ ( t ) := Z Tt e − R s µ F ( u ) du C ( s ) f C ( t ) C ( t ) R c ( C y,ν ( s )) ds − e − R t µ F ( u ) du { t Theorem 3.2. Given problem (2.7), ˆ ν ( t ) is optimal if and only if the following first-orderconditions ∇ ν J ,y (ˆ ν )( τ ) ≤ , ∀ τ ∈ T , P -a.s. , (3.4) E (cid:26) Z [0 ,T ) ∇ ν J ,y (ˆ ν )( t ) d ˆ ν ( t ) (cid:27) = 0 , (3.5) hold true.Proof. We may start by proving the sufficient part. Let ˆ ν satisfy the first-order conditions (3.4)and (3.5) and let ν ∈ S o . Then it follows from (2.5) that C y, ˆ ν ( t ) − C y,ν ( t ) = Z [0 ,t ) C ( t ) f C ( s ) C ( s ) ( d ˆ ν ( s ) − dν ( s )) . Hence concavity of R implies J ,y (ˆ ν ) − J ,y ( ν )= E (cid:26) Z T e − R t µ F ( u ) du h R ( C y, ˆ ν ( t )) − R ( C y,ν ( t )) i dt − Z [0 ,T ) e − R t µ F ( u ) du ( d ˆ ν ( t ) − dν ( t )) (cid:27) he Free Boundary via the Representation Theorem E (cid:26) Z T e − R t µ F ( u ) du h R ( C y, ˆ ν ( t )) − R ( C y,ν ( t )) i dt − Z [0 ,T ) e − R t µ F ( u ) du ( d ˆ ν ( t ) − dν ( t )) (cid:27) ≥ E (cid:26) Z T e − R t µ F ( u ) du R c ( C y, ˆ ν ( t ))( C y, ˆ ν ( t ) − C y,ν ( t )) dt − Z [0 ,T ) e − R t µ F ( u ) du ( d ˆ ν ( t ) − dν ( t )) (cid:27) = E (cid:26) Z T e − R t µ F ( u ) du R c ( C y, ˆ ν ( t )) Z [0 ,t ) C ( t ) f C ( s ) C ( s ) ( d ˆ ν ( s ) − dν ( s )) dt − Z [0 ,T ) e − R t µ F ( u ) du ( d ˆ ν ( t ) − dν ( t )) (cid:27) = E (cid:26) Z [0 ,T ) (cid:20) Z Tt e − R s µ F ( u ) du R c ( C y, ˆ ν ( s )) C ( s ) f C ( t ) C ( t ) ds − e − R t µ F ( u ) du (cid:21) ( d ˆ ν ( t ) − dν ( t )) (cid:27) = E (cid:26) Z [0 ,T ) ∇ ν J ,y (ˆ ν )( t ) ( d ˆ ν ( t ) − dν ( t )) (cid:27) ≥ , where we have used Fubini’s theorem in the third equality, and (3.3), (3.4) and (3.5) in the lastone. It follows that ˆ ν is optimal for problem (2.7).Necessity may be derived from [26], Proposition 3 . 2, with k ( t ) := f − C ( t ) e − R t µ F ( u ) du C ( t )and F ( ω, t, q ) := e − R t µ F ( ω,u ) du R ( qC ( ω, t )), ω ∈ Ω , t ∈ [0 , T ] , q > 0. Although we do not havethat the optional cost process k is a supermartingale, however it is nonnegative and this is allthat is needed in [26], proof of Proposition 3 . 2, to get necessity of (3.4) and (3.5).Theorem (3.2) characterizes the optimal investment plan but it might not be useful if one aimsto find the explicit solution, since the first order conditions are not always binding.In what follows we construct the optimal capacity in terms of the ‘ base capacity ’ { l ∗ ( t ) , t ∈ [0 , T ] } (cf. also [25], Definition 3 . 1) which represents the capacity level that is optimal for a firmstarting at time t with capacity zero. We show that it is optimal for (2.7) to invest up to thebase capacity level if the current capacity level is below it; otherwise no investment is optimal.Mathematically, l ∗ is the solution of the Bank-El Karoui representation problem [4]. The Bank-El Karoui Representation Theorem (cf. [4], Theorem 3 and Remark 2.1) states that,given • an optional process X = { X ( t ) , t ∈ [0 , T ] } of class (D), lower-semicontinuous in expectationwith X ( T ) = 0, • a nonnegative optional random Borel measure µ ( ω, dt ), • f ( ω, t, x ) : Ω × [0 , T ] × R R such that f ( ω, t, · ) : R R is continuous, strictly decreasingfrom + ∞ to −∞ , and the stochastic process f ( · , · , x ) : Ω × [0 , T ] R is progressivelymeasurable and integrable with respect to d P ⊗ µ ( ω, dt ),then there exists an optional process ξ = { ξ ( t ) , t ∈ [0 , T ] } taking values in R ∪ {−∞} such thatfor all τ ∈ T , f ( t, sup τ ≤ u There exists a unique optional, upper right-continuous, positive process l ∗ ( t ) thatsolves E (cid:26) Z Tτ e − R s µ F ( u ) du C ( s ) R c (cid:18) C ( s ) sup τ ≤ u 25, among others). In this paper we follow the point of view of the literature on the Bank-ElKaroui Representation Theorem and its applications (like [3], [4] and [6], among others) and hence we base ourresults on (4.2). he Free Boundary via the Representation Theorem ξ ( t ) := ess inf t ≤ τ ≤ T E (cid:26) Z τt f ( u, ξ ) µ ( du ) + X ( τ ) (cid:12)(cid:12)(cid:12) F t (cid:27) , ξ ∈ R , t ∈ [0 , T ] . (4.6)Recall that Γ ξ of (4.6) may be taken to be right-continuous and it is such that the mapping ξ Γ ξ ( ω, t ) is continuous and nonincreasing for ω ∈ Ω, t ∈ [0 , T ] (cf. [4], Lemma 4 . ξ ∗ ( t ) := sup n ξ ∈ R : Γ ξ ( t ) = X ( t ) o , t ∈ [0 , T ) , (4.7)solves the representation problem e − R τ µ F ( u ) du C ( τ ) f C ( τ ) { τ 12) to writeΓ ζ − ǫ ( τ ) = lim n →∞ Γ ζ − ǫ ( τ n ) = lim n →∞ X ( τ n ) = X ( τ ) = Γ ξ ∗ ( τ ) ( τ ) . (4.12)It thus follows that ζ − ǫ ≤ ξ ∗ ( τ ) for any ǫ > 0, which implies (4.11); i.e., ξ ∗ is upper right-continuous.Finally, to prove that ξ ∗ ( t ) < , T ) define σ := inf { t ∈ [0 , T ) : ξ ∗ ( t ) ≥ } ∧ T, then for ω ∈ { σ < T } , upper right-continuity of ξ ∗ implies ξ ∗ ( σ ) ≥ σ ≤ u For a given positive process l , the capacity process that tracks l is defined as C ( l ) ( t ) := C ( t ) (cid:18) y ∨ sup ≤ u Let l ∗ ( t ) be the unique optional, upper right-continuous, positive solution of (4.3)and let C ( l ∗ ) be the capacity process that tracks l ∗ . Then the investment plan ν ( l ∗ ) that finances C ( l ∗ ) , i.e. dν ( l ∗ ) ( t ) = 1 f C ( t ) C ( l ∗ ) ( t )[ µ C ( t ) dt − σ C ( t ) dW ( t )] + 1 f C ( t ) dC ( l ∗ ) ( t ) , with ν ( l ∗ ) (0) = 0 , is optimal for the firm’s problem (2.7).Proof. In order to prove that C ( l ∗ ) ( t ) is the optimal capacity, we only have to show that C ( l ∗ ) ( t )solves the two first-order conditions of Theorem 3.2. In fact, for all τ ∈ T E (cid:26) Z Tτ e − R s µ F ( u ) du C ( s ) R c (cid:0) C ( l ∗ ) ( s ) (cid:1) ds (cid:12)(cid:12)(cid:12) F τ (cid:27) = E (cid:26) Z Tτ e − R s µ F ( u ) du C ( s ) R c (cid:18) C ( s ) (cid:18) y ∨ sup ≤ u 0. In fact, at such time, wehave C ( l ∗ ) ( t ) = C ( t ) sup τ ≤ u Recall that C y, ˆ ν ( t ) = C ( t )[ y + ν y ( t )] (cf. (2.5)) where ν y ( t ) := R [0 ,t ) f C ( s ) C ( s ) d ˆ ν ( s ) .Hence it follows from (4.14) with l = l ∗ that ν y ( t ) = sup ≤ u 12 and Lemma 4 . 13, guarantee that • the stopping time τ ξ ( t ) := inf (cid:26) s ∈ [ t, T ) : Γ ξ ( s ) = e − R s µ F ( r ) dr C ( s ) 1 f C ( s ) (cid:27) ∧ T (5.2)is optimal for (5.1); • the optional, upper right-continuous process ξ ∗ ( t ) := sup (cid:26) ξ < ξ ( t ) = e − R t µ F ( r ) dr C ( t ) 1 f C ( t ) (cid:27) , t ∈ [0 , T ) , (5.3)uniquely solves the representation problem (4.8). he Free Boundary via the Representation Theorem e P by e P ( A ) := E {M ( T ) A } , for A ∈ F T , with M ( T ) as in (2.4). Thenthe Radon-Nikodym derivative is d e P d P (cid:12)(cid:12)(cid:12) F t = M ( t ) , t ∈ [0 , T ] , (5.4)and the process f W ( t ) := W ( t ) − R t σ C ( u ) du , t ∈ [0 , T ], is a standard Brownian motion under e P .We denote by e E {·} the expectation w.r.t. e P .Hence, under e P , by the continuous time Bayes’ rule (see e.g. [20]) the process e R t µ F ( r ) dr Γ ξ ( t ) C ( t ) may be written as e Γ ξ ( t ) := ess inf t ≤ τ ≤ T e E (cid:26) Z τt e − R ut µ ( r ) dr R c (cid:18) − ξ C ( u ) (cid:19) du + e − R τt µ ( r ) dr f C ( τ ) { τ With e Γ ξ ( t ) as in (5.5), Y t,z ( u ) = z C ( u ) C ( t ) = z e C t ( u ) , u ≥ t , as in (A-2), v ( t, z ) = inf t ≤ τ ≤ T e E (cid:26) Z τt e − R ut µ ( r ) dr R c (cid:16) Y t,z ( u ) (cid:17) du + e − R τt µ ( r ) dr f C ( τ ) { τ The proof borrows arguments from [10], proof of Theorem 4 . 1. For t ∈ [0 , T ) and τ ∈ [ t, T ], notice that e E (cid:26) Z τt e − R ut µ ( r ) dr R c (cid:0) yC ( u ) (cid:1) du + e − R τt µ ( r ) dr f C ( τ ) { τ The base capacity l ∗ ( t ) , unique optional, upper right-continuous, positivesolution of (4.3), admits the representation l ∗ ( t ) = sup (cid:26) yC ( t ) > v ( t, yC ( t )) = 1 f C ( t ) (cid:27) , t ∈ [0 , T ) , (5.10) with v as in (5.7).Proof. For t ∈ [0 , T ) and y > l ∗ ( t ) := − C ( t ) ξ ∗ ( t ) = − C ( t )sup n ξ < e Γ ξ ( t ) = f C ( t ) o = − C ( t )sup n − y < e Γ − y ( t ) = f C ( t ) o = C ( t ) − sup n − y < e Γ − y ( t ) = f C ( t ) o = C ( t )inf n y > e Γ − y ( t ) = f C ( t ) o = C ( t ) sup (cid:26) y > e Γ − y ( t ) = 1 f C ( t ) (cid:27) = sup (cid:26) yC ( t ) > e Γ − y ( t ) = 1 f C ( t ) (cid:27) = sup (cid:26) yC ( t ) > v ( t, yC ( t )) = 1 f C ( t ) (cid:27) , where the last equality follows from Proposition 5.2.Notice that v ( t, y ) ≤ f C ( t ) for all t ∈ [0 , T ) and y > . Continuation Region (or ‘no-action region’) of problem (5.7) D := (cid:26) ( t, y ) ∈ [0 , T ) × (0 , ∞ ) : v ( t, y ) < f C ( t ) (cid:27) . (5.11)Roughly speaking D is the region where it is not profitable to invest, since the shadow value ofinstalled capital is strictly less than the capital’s replacement cost. Similarly its complement isthe Stopping Region (or ‘action region’), i.e. D c := (cid:26) ( t, y ) ∈ [0 , T ) × (0 , ∞ ) : v ( t, y ) = 1 f C ( t ) (cid:27) . (5.12) he Free Boundary via the Representation Theorem y ( t ) of the optimal stopping problem (5.7). Theorem 5.4. The base capacity process l ∗ ( t ) , unique optional, upper right-continuous, positivesolution of (4.3), is deterministic and coincides with the free boundary ˆ y ( t ) associated to theoptimal stopping problem (5.7). Hence l ∗ ( t ) = sup (cid:26) z > v ( t, z ) = 1 f C ( t ) (cid:27) for t ∈ [0 , T ) . (5.13) Proof. Recall (5.10). Fix t ∈ [0 , T ) and set˜ z ( ω, y ) := yC ( ω, t ) . It follows that (cid:26) yC ( ω, t ) > v ( t, yC ( ω, t )) = 1 f C ( t ) (cid:27) = (cid:26) ˜ z ( ω, y ) > v ( t, ˜ z ( ω, y )) = 1 f C ( t ) (cid:27) ⊆ (cid:26) z > v ( t, z ) = 1 f C ( t ) (cid:27) for a.e. ω ∈ Ω and y > 0, hence the inclusion holds a.s. for all y > z > 0, then for each ω ∈ Ω and t ∈ [0 , T ), z may be writtenas z = ˜ z ( ω, y ( ω, z )) , with y ( ω, z ) := zC ( ω,t ) . Therefore (cid:26) z > v ( t, z ) = 1 f C ( t ) (cid:27) = (cid:26) y ( ω, z ) C ( ω, t ) > v ( t, y ( ω, z ) C ( ω, t )) = 1 f C ( t ) (cid:27) ⊆ (cid:26) yC ( ω, t ) > v ( t, yC ( ω, t )) = 1 f C ( t ) (cid:27) . This inclusion holds for a.e. ω ∈ Ω, thus a.s. Hence, it holds e P -a.s. thatsup (cid:26) yC ( ω, t ) > v ( t, yC ( ω, t )) = 1 f C ( t ) (cid:27) = sup (cid:26) z > v ( t, z ) = 1 f C ( t ) (cid:27) (5.14)and l ∗ ( t ) is deterministic (cf. (5.10)). Now the right-hand side of (5.14) (cf. [11], eq. (3 . l ∗ ( t ) with the free boundary ˆ y ( t ) of problem (5.7).Since ˆ y ( t ) coincides with l ∗ ( t ), equation (4.3) provides an integral equation for the freeboundary ˆ y ( t ) which does not require a priori continuity of ˆ y and the smooth fit property (asinstead that in [11]) to be derived. Theorem 5.5. The free boundary ˆ y ( t ) of problem (5.7) is the unique upper right-continuous,positive solution of the integral equation e E (cid:26) Z T − t e − R t + vt µ ( r ) dr R c (cid:18) sup ≤ u ′ Fix t ∈ [0 , T ). Set τ = t and recall that l ∗ ( t ) = ˆ y ( t ). Then write (4.3) under e P and applythe continuous time Bayes’ Rule to obtain e E (cid:26) Z T − t e − R t + vt µ ( r ) dr R c (cid:18) sup ≤ u ′ 0, is independent of F t .As in [11], Section 4, we now make the following Assumption 5.6. R ( C ) = α C α with α ∈ (0 , (i.e. Cobb-Douglas production function);2. µ C ( t ) ≡ µ C , σ C ( t ) ≡ σ C , µ F ( t ) ≡ µ F , f C ( t ) ≡ f C . Remark 5.7. Notice that under the second part of Assumption 5.6, the process C ( t + v ) C ( t + u ′ ) has thesame law as C ( v ) C ( u ′ ) . Hence, the integral equation (5.15) takes the form e E (cid:26) Z T − t e − µv R c (cid:18) sup ≤ u ′ Under Assumption 5.6 the boundary ˆ y ( t ) of the continuation region D satis-fies ˆ y ( t ) ≤ (cid:20) f C (cid:18) − e − ( µ F + αµ C + α (1 − α ) σ C )( T − t ) µ F + αµ C + α (1 − α ) σ C (cid:19)(cid:21) − α =: y ∗ ( t ) , (5.17) for every t ∈ [0 , T ) .Proof. Fix t ∈ [0 , T ). The representation formula (4.3) for τ = t and in the Cobb-Douglas casebecomes e − µ F t f C = E (cid:26) Z Tt e − µ F s C ( s ) C ( t ) (cid:18) sup t ≤ u Notice that the curve y ∗ ( t ) is exactly what in [10] was incorrectly identified as thefree boundary between the ‘action’ and the ‘no-action’ regions. In [11] the authors characterizedthe free boundary ˆ y ( t ) as the unique solution of a nonlinear integral equation (see [11], Theorem4.8). Then, by using a discrete approximation of such integral equation, they showed that ˆ y ( t ) ≤ y ∗ ( t ) , for t ≤ T . That is exactly what we proved here in Proposition 5.8. Remark 5.10. The arguments in the proof of Proposition 5.8 apply even under the more generalconditions of Assumption 5.1. That is, under deterministic, time-dependent coefficients we have ˆ y ( t ) ≤ (cid:20) f C ( t ) Z T − t e − R v + tt ( µ F ( s )+ αµ C ( s )+ α (1 − α ) σ C ( s ) ) ds dv (cid:21) − α , t ∈ [0 , T ) . In this Section we have linked the Bank-El Karoui’s probabilistic approach to the variationalapproach followed by Chiarolla and Haussmann in [10] and [11] for an irreversible investmentproblem similar to (2.7). Under Assumption 5.1 we have proved that the base capacity process l ∗ ( t ) is a deterministic process and it coincides with the free boundary of the optimal stoppingproblem (5.7). We have characterized the free boundary as the unique solution of an integralequation based on the stochastic Representation Theorem of [4]. Even under Assumption 5.6,the integral equation for the free boundary (5.15) cannot be analitically solved when the timehorizon is finite. However it is possible to find a curve bounding the free boundary from above.In Section 6 we shall see that, instead, when T = + ∞ (as in H. Pham [24]) the free boundaryis a constant whose value we find explicitly by applying Proposition 6.1. T = + ∞ In this Section, with T = + ∞ and under Assumption 5.6, we set f C = 1 in order to compareour finding with the results in H. Pham [24]. As one would expect, when the time horizon isinfinite, the free boundary is a point. That is what we show below. Proposition 6.1. The unique solution of the representation problem (4.3) is given by l ∗ ( t ) = h µ F − σ C β − − ασ C (1 + β + ) i − α =: a (6.1) where β ± are, respectively, the positive and negative roots of σ C x + e µ C x − µ F = 0 with e µ C := µ C + σ C .Hence (cf. Definition 4.2 and Theorem 4.3) the optimal capacity is given by C y, ˆ ν ( t ) = C ( a ) ( t ) ≡ C ( t ) (cid:18) y ∨ sup ≤ u ≤ t (cid:18) aC ( u ) (cid:19)(cid:19) . (6.2) Proof. We make the ansatz that l ∗ ( t ) ≡ a for all t ≥ he Free Boundary via the Representation Theorem a α − E (cid:26) Z ∞ τ e − µ F s C ( s ) C ( τ ) (cid:20) sup τ ≤ u ≤ s (cid:18) C ( s ) C ( u ) (cid:19)(cid:21) α − ds (cid:12)(cid:12)(cid:12) F τ (cid:27) = a α − E (cid:26) Z ∞ τ e − µ F s C ( s ) C ( τ ) inf τ ≤ u ≤ s (cid:18)h C ( s ) C ( u ) i α − (cid:19) ds (cid:12)(cid:12)(cid:12) F τ (cid:27) = a α − E (cid:26) Z ∞ τ e − µ F s e σ C ( W ( s ) − W ( τ )) − e µ C ( s − τ ) (6.3) × inf ≤ u ′ ≤ s − τ (cid:20) e σ C ( W ( s ) − W ( u ′ + τ )) − e µ C ( s − u ′ − τ ) (cid:21) ( α − ds (cid:12)(cid:12)(cid:12) F τ (cid:27) = a α − e − µ F τ E (cid:26) Z ∞ e − µ F v e σ C W ( v ) − e µ C v inf ≤ u ′ ≤ v (cid:18) e ( α − σ C ( W ( v ) − W ( u ′ )) − e µ C ( v − u ′ )) (cid:19) dv (cid:27) since the Brownian increments are independent of F τ .If we now define Y ( v ) := e µ C v − σ C W ( v ), Y ( v ) := inf ≤ u ′ ≤ v Y ( u ′ ) and Y ( v ) := sup ≤ u ′ ≤ v Y ( u ′ ),then we have a α − e − µ F τ E (cid:26) Z ∞ e − µ F v e σ C W ( v ) − e µ C v inf ≤ u ′ ≤ v (cid:18) e ( α − σ C ( W ( v ) − W ( u ′ )) − e µ C ( v − u ′ )) (cid:19) dv (cid:27) = a α − e − µ F τ E (cid:26) Z ∞ e − µ F v e − αY ( v ) e ( α − Y ( v ) dv (cid:27) = 1 µ F a α − e − µ F τ E (cid:26) Z ∞ µ F e − µ F v e − α ( Y ( v ) − Y ( v ) ) e − Y ( v ) dv (cid:27) (6.4)= 1 µ F a α − e − µ F τ E n e − α ( Y ( τ ( µ F )) − Y ( τ ( µ F )) ) e − Y ( τ ( µ F )) o , where τ ( µ F ) denotes an independent exponentially distributed random time.Using the Excursion Theory for Levy processes (cf. [8]), Y − Y is independent of Y , and bythe Duality Theorem, Y − Y has the same distribution as Y . Hence from (6.4) we obtain1 µ F a α − e − µ F τ E n e − α ( Y ( τ ( µ F )) − Y ( τ ( µ F )) ) e − Y ( τ ( µ F )) o = 1 µ F a α − e − µ F τ E n e − αY ( τ ( µ F )) o E n e − Y ( τ ( µ F )) o . (6.5)It is well known that for a Brownian motion with drift E n e zY ( τ ( µ F )) o = β + β + − z and E n e zY ( τ ( µ F )) o = β − β − − z , if β + and β − are, respectively, the positive and negative roots of σ C x + e µ C x − µ F = 0, i.e. β ± = − e µ C σ C ± s(cid:18) e µ C σ C (cid:19) + 2 µ F σ C . he Free Boundary via the Representation Theorem e − µ F τ = E (cid:26) Z ∞ τ e − µ F s C ( s ) C ( τ ) (cid:20) C ( s ) sup τ ≤ u ≤ s (cid:18) l ∗ ( u ) C ( u ) (cid:19)(cid:21) α − ds (cid:12)(cid:12)(cid:12) F τ (cid:27) = 1 µ F a α − e − µ F τ E n e − αY ( τ ( µ F )) o E n e − Y ( τ ( µ F )) o (6.6)= 1 µ F a α − e − µ F τ β + β − (1 + β + )( α + β − ) . Then, we solve for a and we obtain a α − = (cid:18) µ F (1 + β + )( α + β − ) β + β − (cid:19) , which may also be written as a = (cid:18) µ F − σ C β − − ασ C (1 + β + ) (cid:19) − α being β + β − = − µ F σ C .Hence (cf. Theorem 4.3) the optimal capacity is C y, ˆ ν ( t ) = C ( a ) ( t ) = C ( t ) (cid:18) y ∨ sup ≤ u ≤ t (cid:18) aC ( u ) (cid:19)(cid:19) . (6.7)From Remark 4.4 we have ν y ( t ) = sup ≤ u ≤ t (cid:18) a − yC ( u ) C ( u ) (cid:19) ∨ , (6.8)and the corresponding control ˆ ν ( t ) (cf. (2.2)) makes the diffusion reflect at the boundary a , itis the local time of C y, ˆ ν ( t ) at a .Notice that the boundary a in (6.1) coincides with the free boundary k b obtained via aviscosity solution approach by H. Pham in [24] for a unit cost of investment p . In fact from [24],Example 1 . . k α − b = 1 − mC ( α − m ) , with C = 1 µ F + α e µ C − α σ C and m = − β + , and it is easy to see that a α − = µ F (1 + β + )( α + β − ) β + β − = 1 − mC ( α − m ) = k α − b , (6.9)hence a = k b . he Free Boundary via the Representation Theorem Remark 6.2. For a general production function R ( · ) satisfying Assumption 2.1, to find the freeboundary a one should solve the analogue of (6.5), i.e. µ F E n e − Y ( τ ( µ F )) R c (cid:16) a e − Y ( τ ( µ F )) (cid:17)o E n e − Y ( τ ( µ F )) o = 1 , or equivalently µ F E n e − Y ( τ ( µ F )) R c (cid:16) a e − Y ( τ ( µ F )) (cid:17)o β + β + = 1 . That is, a is the unique solution of E n e − Y ( τ ( µ F )) R c (cid:16) a e − Y ( τ ( µ F )) (cid:17)o = µ F (1 + β + ) β + . (6.10) Since now − Y ( τ ( µ F )) = ( − Y )( τ ( µ F )) , and ( − Y )( τ ( µ F )) has exponential distribution of pa-rameter γ + := e µ C + √ e µ C +2 µ F σ C σ C > (see, e.g., [8], Chapter VII), then (6.10) may be rewrittenas Z ∞ γ + e x (1 − γ + ) R c ( ae x ) dx = µ F (1 + β + ) β + . A The Variational Approach in the Case of Time-DependentCoefficients In this Appendix we revisit the solution of problem (2.7) obtained in Chiarolla and Haussmann[11] by a variational approach and we generalize some of their results to the case of deterministic,time-dependent coefficients of the controlled diffusion (cf. Assumption 5.1).Denote by C t,y,ν ( s ) the capacity process starting at time t ∈ [0 , T ) from y , controlled by ν ,with dynamics dC t,y,ν ( s ) = C t,y,ν ( s )[ − µ C ( s ) ds + σ C ( s ) dW ( s )] + f C ( s ) dν ( s ) , s ∈ [ t, T ) ,C t,y,ν ( t ) = y > . (A-1)Hence C t,y,ν ( s ) = C ( s ) C ( t ) (cid:26) y + Z [ t,s ) C ( t ) f C ( u ) C ( u ) dν ( u ) (cid:27) with C as defined in (2.3).To simplify notation write e C t ( s ) := C t, , ( s ) = C ( s ) C ( t ) = e − R st ( µ C ( u )+ σ C ( u )) du + R st σ C ( u ) dW ( u ) , Y t,y ( s ) := y e C t ( s ) , (A-2)and note that these processes are e F t,s := σ { W ( u ) − W ( t ) , t ≤ u ≤ s } -measurable.To C t,y,ν we associate the expected total profit, net of investment costs, given by J t,y ( ν ) = E (cid:26) Z Tt e − R st µ F ( u ) du R ( C t,y,ν ( s )) ds − Z [ t,T ) e − R st µ F ( u ) du dν ( s ) (cid:27) , (A-3) he Free Boundary via the Representation Theorem V ( t, y ) := sup ν ∈S t J t,y ( ν ) , (A-4)where S t := { ν : Ω × [ t, T ] R + nondecreasing, left-continuous, adapted s.t. ν ( t ) = 0 , P -a.s. } is the convex set of irreversible investments.We define the opportunity cost of not investing until time s as (compare with [11], Section3) ζ t,y,T ( s ) := Z st e − R ut µ F ( r ) dr e C t ( u ) R c ( y e C t ( u )) du + e − R st µ F ( r ) dr e C t ( s ) 1 f C ( s ) { s Under Assumption 5.1, for every ( t, y ) in [0 , T ) × (0 , ∞ ) the optimal stoppingtime (A-8) may be written as τ ∗ ( t, y ) = inf (cid:26) s ∈ [ t, T ) : v ( s, Y t,y ( s )) = 1 f C ( s ) (cid:27) ∧ T, (A-10) with v as in (A-7) and Y t,y as in (A-2). he Free Boundary via the Representation Theorem Proof. Recall that Y t,y ( s ) = y e C t ( s ) and (A-5). Then, from (A-6) we may write Z t,y,T ( s ) = ess inf s ≤ τ ≤ T E (cid:26) Z st e − R ut µ F ( r ) dr e C t ( u ) R c (cid:0) Y t,y ( u ) (cid:1) du + Z τs e − R ut µ F ( r ) dr e C t ( u ) R c (cid:0) Y t,y ( u ) (cid:1) du + e − R st µ F ( r ) dr e − R τs µ F ( r ) dr e C t ( τ ) 1 f C ( τ ) { τ 5) and Theorem 4 . Acknowledgments. The authors thankfully acknowledge two anonymous referees for theirpertinent and useful comments. References [1] F.M. Baldursson, I. Karatzas, Irreversible Investment and Industry Equilibrium , Financeand Stochastics 1 (1997), pp. 69–89.[2] P. Bank, F. Riedel, Optimal Consumption Choice with Intertemporal Substitution , TheAnnals of Applied Probability 11 (2001), pp. 750–788.[3] P. Bank, H. 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, hence (5.18) and (5.20) imply that e − µ F t f C ≤ e − µ F t [ l ∗ ( t )] α − Z T − t e − ( µ F + αµ C + α (1 − α ) σ C ) v dv (5.21)= e − µ F t [ l ∗ ( t )] α − (cid:18) − e − ( µ F + αµ C + α (1 − α ) σ C ) ( T − t ) µ F + αµ C + α (1 − α ) σ C (cid:19) . Now (5.21) gives[ l ∗ ( t )] − α ≤ f C (cid:18) − e − ( µ F + αµ C + α (1 − α ) σ C ) ( T − t ) µ F + αµ C + α (1 − α ) σ C (cid:19) =: [ y ∗ ( t )] − α , (5.22)and (5.17) follows from the identification of l ∗ ( · ) with ˆ y ( · ) (cf. Theorem 5.4). he Free Boundary via the Representation Theorem Remark 5.9.