LONG-TERM OPTIMAL REAL INVESTMENT STRATEGIES IN THE PRESENCE OF ADJUSTMENT COSTS

Transcription

1 LONG-TERM OPTIMAL REAL INVESTMENT STRATEGIES IN THE PRESENCE OF ADJUSTMENT COSTS ARNE LØKKA AND MIHAIL ZERVOS Abstract. We consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment. In particular, we consider an investment project that yields a payoff at a rate that depends on its installed capacity level and on a random economic indicator such as, for instance, the price of the project s output commodity. We model the economic indicator by a one-dimensional ergodic Itô diffusion. At any time, the capacity level can be increased or decreased at given proportional costs. The aim is to maximise the long-term average profit, which takes the form of an ergodic optimisation criterion. The associated Hamilton-Jacobi-Bellman equation is a two-dimensional differential equation that we solve explicitly. We characterise completely an optimal strategy.. Introduction We consider an investment project that operates within a random evironment and yields a payoff rate that depends on the installed capacity and a stochastic economic indicator such as, for instance, the price of the project s output commodity. The capacity can be increased or decreased dynamically over time at proportional costs. We model the economic indicator as a one-dimensional ergodic Itô diffusion, and address the problem of determining the strategy that maximises the long-term average payoff of the project. In mathematical terms, this formulation takes the form of an optimal control problem with an expected ergodic criterion. We are able to solve the corresponding Hamilton-Jacobi- Bellman equation, and conpletely characterise the corresponding optimal strategy. Irreversible capacity models have attracted considerable interest in the literature. See for instance Davis, Dempster, Sethi and Vermes [9], Davis [8], Kobila K, Øksendal [23], Wang [27], Chiarolla and Haussmann [7], Bank [2] and the references therein. Abel and Eberly [], and Merhi and Zervos [22] consider a model involving both expansion and reduction of a project s capacity level. Guo and Pham [2] consider a partially reversible investment model with entry and exit decisions and a general running payoff function. However, as far as we know, all of the aforementioned references aims at maximising the discounted running payoff of the project. We address the problem of maximising the longterm average payoff rate of the project. This yields a singular stochastic control problem with an ergodic criterion. Jack and Zervos [3], [4] consider an impulse and absolutely continuous control problem with ergodic criterion. Bronstein and Zervos [6] consider a sequential entry and exit decision problem with an ergodic criterion. For the more general Research supported by EPSRC grant no. GR/S22998/.

2 2 LØKKA AND ZERVOS theory of optimal ergodic control, the reader is referred to Kushner [2], Karatzas [5], Gatarek and Stettner [], Borkar and Gosh [5], Bensoussan and Frehse [3], Menaldi, Robin and Taksar [2], Duncan, Maslowski and Pasik-Duncan [], Kurtz and Stockbridge [9], Borkar [4], Kruk [8], Sadowy and Stettner [26], and the references therein. The paper is organised as follows. In Section 2 we formulate the problem and the various technical assumptions. In Section 3 we derive the solution to the Hamilton-Jacobi-Bellman equation. In Section 4 we specify the optimal strategy corresponding to the Hamilton- Jacobi-Bellman equation, and prove that this provides an optimal strategy. Further, we show that the value function corresponding to the problem is constant, and analyse the non-uniquenss of optimal strategies. 2. Problem formulation We consider an investment project which payoff rate depends on the installed capacity and an economic indicator. We model the economic indicator by a one-dimensional Itô diffusion, i.e., the state X t of the indicator at time t is given by 2.) dx t = bx t ) dt σx t ) dw t, X = x >, where b, σ :, ) R are given functions, and W is a one-dimensional Brownian motion. If we think of the investment project as a producer of a single commodity, the state process X can be used to model an economic indicator such as the commodity s demand or the commodity s price. We make the following assumptions regarding the coefficients b and σ. Assumption 2.. The deterministic functions b, σ :, ) R satisfy the following conditions: 2.2) 2.3) σ 2 x) >, for all x, ), for all x, ), there exists ε > such that The scale function p defined by 2.6) satisfies 2.4) Further, 2.5) ε x ε lim px) = and lim px) =. x x lim xbx) x σ 2 x) = c, for some c, ]. bs) σ 2 s) ds <, With reference to the general theory of one-dimensional diffusions e.g., see Section 5.5 in Karatzas and Shreve [6]), the assumptions 2.2) and 2.3) are sufficient for 2.) to define a diffusion that is unique in the sense of probability law. 2.2) and 2.3) also ensure that the scale function p and the speed measure m given by 2.6) p) = and p x) = exp 2 bs) σ 2 s) ds ), for x, ),

3 and 2.7) LONG-TERM OPTIMAL STRATEGIES 3 mdx) = 2 σ 2 x)p x) dx, respectively, are well defined. According to Proposition in Karatzas and Shreve [6], assumption 2.4) is sufficient to ensure that the solution to 2.) is non-explosive and recurrent. Assumption 2.5) is related to conditions formulated by Peskir [24] and bounds on the maximimum of Itô diffusions. We assume that the project s running payoff rate is of the form hx t, Y t ), for some function h, where X denotes the state of the economic indicator and Y denotes the level of installed capacity. We make the following assumptions on h. Assumption 2.2. There exists a measurable function k :, ) [, ) and finite constants C, C 2 > such that 2.8) and 2.9) kx) mdx) <, C y) hx, y) kx) C 2 y), for all x, y), ) 2. Further, hx, y) is concave in y and non-decreasing in x, and there is a unique, strictly increasing function y :, ), ) such that 2.) where H x, y x) ) =, for all x, ), 2.) Hx, y) = h x, y). We assume that there exist finite constants C 3, C 4 >, such that 2.2) y x) C 3 C 4 x, for all x. Since y is strictly increasing, it has a well defined inverse function, denoted by x. We procede to define what we mean by an investment/capacity strategy, and the corresponding set of all such strategies. Definition 2.. Assume given an initial condition x, y), and let Ω, F, F t, X, P x ) be a weak solution to 2.). An investment strategy Y x,y is a F t -adapted cáglád process Y of finite variation with Y = y. Moreover, let Y and Y denote the increasing processes corresponding to the increasing and decreasing parts of Y, respectively, that is Y = y Y Y. Let Y x,y denote the set of all investments strategies with initial condition x, y). We are now ready to formulate the optimisation criterion. This takes the form of an expected ergodic optimisation criterion, which corresponds to the long-term average payoff. Define the performance index 2.3) JY x,y ) = lim sup [ T ] T Ex hx t, Y t ) dt K Y T K Y T,

4 4 LØKKA AND ZERVOS where K is the cost of increasing the capacity one unit and K is the cost/return of reducing the capacity one unit. Assumption 2.3. We assume that K K >. The aim is to determine the strategy that maximise the performance index J given by 2.3). The value function associated with such an optimal control problem is given by 2.4) V x, y) = sup JY x,y ). Y x,y Y x,y It turns out that the value function V is constant, i.e., V does not depend on the initial condition x, y) see the end of Section 4). According to the next remark, the value function takes values in R. Remark 2.. Let x, y), ) 2 be any initial condition, and denote by Y c x,y the strategy for which Y t = y, for all t. From the assumption 2.9) it follows that V x, y) JY c x,y ) C y) >. On the other hand, it follows from the assumptions 2.8) 2.9) and Rogers and Williams [25, p. 3] that [ T ] JY x,y ) lim sup T Ex kx t ) dt = m, ) ) kx) mdx) <, for every strategy Y x,y Y x,y. Hence, V x, y) m, ) ) kx) mdx) <. We conclude that < V x, y) <, for all initial conditions x, y). 3. The Hamilton-Jacobi-Bellman equation In the case of the usual expected discounted optimisation criterion, the standard approach applying the principle of dynamic programming provides a Hamilton-Jacobi- Bellman equation, which determines the value function and the corresponding optimal strategy. Since in our case, the value function is constant, the similar approach is not that straight forward. Regardless of this, it turns out that we can derive the Hamilton- Jacobi-Bellman equation corresponding to the value function, which determines an optimal strategy. The Hamilton-Jacobi-Bellman equation takes the following form: { } 3.) max 2 σ2 w xx bw x h, w y K, w y K =. From the nature of the problem it is natural to conjecture that we are looking for continuous and increasing functions F : [y F, ) [, ) and G : [, y G ) [x G, ) that divides [, ) 2 into three connected sets. Here, y F, y G and x G are positive real numbers. The interpretation is that if the economic indicator X t at time t is less than FY t ), it is optimal to decrease the capacity to a level F X t ). If X t is greater than GY t ) then it is optimal

5 LONG-TERM OPTIMAL STRATEGIES 5 to increase the capacity to a level G X t ). While the indicator X is greater than FY t ) and less than GY t ) it is optimal to take no action until X reaches the boundaries defined by F and G. So, we look for a solution to 3.) of the form: 3.2) 3.3) 3.4) w y x, y) K =, for x, y) D, 2 σ2 x)w xx x, y) bx)w x x, y) hx, y) =, for x, y) C, where D, C and I are given by 3.5) 3.6) 3.7) w y x, y) K =, for x, y) I, D = { x, y) [, ) 2 : y F x) }, C = { x, y) [, ) 2 : G x) y F x) }, I = { x, y) [, ) 2 : y G x) }. Our derivation of a solution to equation 3.) starts with the observation that the ordinary differential equation 3.8) 2 σ2 x)v xx x, y) bx)v x x, y) hx, y) =, has general solutions of the form 3.9) vx, y) = By) Ay)px) p s) for some functions A and B. To see this, first observe that v xx x, y) = Ay)p x) p x) v x = Ay)p x) p x) p x) = 2bx) σ 2 x) p x). hu, y) mdu) ds, hu, y) mdu) 2 hx, y), σ 2 x) hu, y) mdu), Hence, we calculate that 2 σ2 x)v xx x, y) bx)v x x, y) hx, y) = ) 2 σ2 x) Ay)p x) p x) hu, y) mdu) hx, y) bx)ay)p x) bx)p x) hu, y) mdu) hx, y)

6 6 LØKKA AND ZERVOS = Ay)bx)p x) bx)p x) =, bx)ay)p x) bx)p x) hu, y) mdu) hu, y) mdu) which proves that functions v of the form 3.9) satisfy equation 3.8). Having established the general solution to equation 3.8), our next task is to determine the unknown functions A and B, as well as the functions F and G. We start by defining candidates for y F and y G. Definition 3.. Let β : [, ) [, ) be given by { b } βy) = inf b [, ) : Hu, y) mdu) >, if and βy) =, if Hu, y) mdu) <. Hu, y) mdu), Observe that x y) < βy). Moreover, β satisfies βy) Hu, y) mdu) =, if βy) <. From Assumption 2.2, it follows that 3.) β y) = σ2 βy) ) p βy) ) 2H βy), y ) if βy) <. Now, let ȳ be given by Then β is strictly increasing for y < ȳ. Definition 3.2. Let y F be given by { 3.) y F = inf y : βy) ȳ = inf { y : βy) = }. βy) H u, y) mdu) >, pu)hu, y) mdu) = K K }. In view of Assumption 2.2 and the equation for β y) given by 3.), we calculate that βy) ) βy) pu)hu, y) mdu) = pu) p βy) )) H u, y) mdu) >, for every y < ȳ. Definition 3.3. Let α : [, ) [, ) be given by { } αy) = sup a, ) : Hu, y) mdu) <, if and, if Hu, y) mdu). a Hu, y) mdu) <,

7 LONG-TERM OPTIMAL STRATEGIES 7 Note that αy) < x y) < βy), and that αy) =, for y ȳ. Moreover, αy) satisfies Hu, y) mdu) =, if αy) >. Assumption 2.2 and the inequality H αy), y) <, αy) implies that α y) = σ2 αy) ) p αy) ) 3.2) 2H αy), y ) H u, y) mdu) >, for y > ȳ. αy) We conclude that α is strictly increasing for y > ȳ. Definition 3.4. Let y G be given by { 3.3) y G = inf y y F : αy) pu)hu, y) mdu) = K K }, if the set is non empty, and y G =, if the set over which the infimum is taken is empty. Assumption 2.2 and the expression for α y) given by 3.2) imply that ) 3.4) pu)hu, y) mdu) = pu) p αy) )) H u, y) mdu) <, αy) αy) for y > ȳ. We conclude that y G < if and only if lim sup y αy) pu)hu, y) mdu) < K K. Our next step is to construct the solution to equation 3.), for y y F. The case y y F. According to the ansatz that the solution to 3.) takes the form 3.2) 3.4), w satisfies equation 3.8) for x, y) in C. Hence, for x, y) C, the function w is given by 3.5) wx, y) = By) Ay)px) p s) hu, y) mdu) ds, for some functions A and B. We require that lim x wx, y) exists for every y [, y F ], which implies that 3.6) Ay) = hu, y) mdu), for y [, y F ]. In addition we look for a solution w to 3.) that is continuous. This implies that 3.7) wx, y) = wx, G x)) K G x) y ), for x, y) I. With reference to the principle of smooth fit, we further require w to satisfy 3.8) 3.9) Equation 3.8) implies that 3.2) B y) = K w y Gy), y) = w y Gy), y), w xy Gy), y) = w xy Gy), y). Hu, y) mdu)pgy)) p s) Hu, y) mdu) ds,

8 8 LØKKA AND ZERVOS and equation 3.9) implies that p Gy)) Hu, y) mdu) =, fory [, y F ]. From the latter equation it follows that G should satisfy 3.2) Hu, y) mdu) =, for y [, y F ]. We remark that Assumption 2.2 ensures that equation 3.2) uniquely determines G. Moreover, we observe that Gy) = βy), from which it follows that G is differentiable and strictly increasing, for every y [, y F ]. Equation 3.2) determines B up to a constant. We may choose this constant such that 3.22) for y y F. By) = K y y y Gx) Hu, x), mdu)p Gx) ) dx p s) Hu, s) mdu) ds dx, The case y F y y G. We require w to be continuous at the points, y) and Gy), y), which imply that w must have the form 3.23) and 3.24) wx, y) = wx, G x)) K G x) y ), wx, y) = wx, F x)) K y F x) ), for x Gy), for x. Appealing to the so called principle of smooth fit, we further require w to satisfy 3.25) 3.26) 3.27) 3.28) Conditions 3.27) and 3.28) imply that 3.29) 3.3) from which it follows that 3.3) A y) = w y, y) = w y, y), w y Gy), y) = w y Gy), y), w xy, y) = w xy, y), w xy Gy), y) = w xy Gy), y). A y)p ) p ) A y)p Gy)) p Gy)) Hu, y) mdu) =, Hu, y) mdu) =, Hu, y) mdu), or equivalently A y) = Hu, y) mdu),

9 and 3.32) LONG-TERM OPTIMAL STRATEGIES 9 Hu, y) mdu) =. Further, the smooth fit conditions 3.25) and 3.26) imply that 3.33) 3.34) B y) A y)p) B y) A y)pgy)) p s) p s) Hu, y) mdu) ds = K, Hu, y) mdu) ds = K. In view of equations 3.3), 3.33) and 3.34), we calculate that = K K A y) [ pgy)) p) ] p s) Hu, y) mdu) ds = K K A y) [ pgy)) p) ] [ pgy)) p) ] Hu, y) mdu) = K K from which it follows that 3.35) p s) p s) Hu, y) mdu) ds, p s) Hu, y) mdu) ds = K K ). Hu, y) mdu) ds p s) Hu, y) mdu) ds By 3.32) and Fubini s theorem, we calculate that the left-hand side of the latter equation is given by p s) Hu, y) mdu) ds = = = pgy)) p s)hu, y) {u s} mdu) ds ) p s) {u s} ds Hu, y) mdu) Hu, y) mdu) pu)hu, y) mdu)

10 LØKKA AND ZERVOS 3.36) = pu)hu, y) mdu). From equations 3.35) and 3.36), we conclude that, in addition to 3.32), the functions F and G must satisfy 3.37) pu)hu, y) mdu) = K K, The next result states that 3.32) and 3.37) has a unique solution. Lemma 3.. The system of equations 3.32) and 3.37) have a unique solution and Gy), for every y F y y G. Moreover, the functions F and G are both continuously differentiable and strictly increasing. Proof. Define the function L y : αy), x y)] [x y), βy)) by 3.38) Lya) a Hu, y) mdu) =. We remark that Assumption 2.2 ensures that equation 3.38) has a unique solution L y a), for every y F y y G and a αy), x y)]. Moreover, L satisfies lim a αy) L y a) = βy) and lim a x y) L y a) = x y). By differentiating L ya) Hu, y) mdu) with respect to a we obtain a L y a) = Ha, y)σ2 L y a))p L y a)) 3.39). HL y a), y)σ 2 a)p a) From the expression for L y a) obtained in 3.39), we calculate that [ d Lya) ] ) 2Ha, y) pu)hu, y) mdu) = pl da σ 2 a)p y a)) pa) a) a since p is strictly increasing and a < L y a) and Ha, y) <. It follows from the definition of y G that 3.32) and 3.35) has a unique solution for every y F y y G, Differentiating equation 3.38) with respect to y shows that 3.4) L y a) = L y a))p L y a)) σ2 2HL y a), y) Lya) a H u, y) mdu), which is strictly positive since H u, y) is strictly negative for all u, ). Denote by ãy) the unique function of y which satisfies 3.4) Lyeay)) eay) pu)hu, y) mdu) = K K. By construction of ã and L, we have in particular that Lyeay)) eay) Hu, y) mdu) =. <,

11 LONG-TERM OPTIMAL STRATEGIES Differentiating this identity with respect to y, and inserting the expressions for L y and Ly given by 3.39) and 3.4), we obtain 3.42) ã y) = σ2 ãy) ) p ãy) ) Lyeay)) eay) [ p Ly ãy)) ) p u )] Hu, y) mdu) 2H ãy), y )[ p L y ãy)) ) p ãy) )], which is strictly positive since H is concave in y, H ãy), y ) is negative and p is strictly increasing. Since ã y) > and Ly a) >, it follows that the functions F and G both are continuously differentiable and strictly increasing. Our next task is to verify that the solutions for y y F and y F y y G are consistent with each other, and to piece them together in a smooth way. Since F and G satisfy equations 3.32) and 3.37), for y y F, it follows that Fy F ) = and Gy F ) = βy F ) = Gy F ). Based on the expressions 3.39) 3.42), we calculate that G y) = σ2 Gy) ) p Gy) ) 2H Gy), y ) p Gy) ) pu) p Gy) ) p ) ) H u, y) mdu), from which it follows that G y F ) = σ2 Gy F ) ) p Gy F ) ) GyF ) 2H H ) Gy F ), y F u, y F) mdu) = G y F ). We conclude that G is continuously differentiable for every y [, y G ]. Further, equations 3.2) and 3.34) imply that B y F ) = B y F ), and equations 3.6) and 3.3) imply that A y F ) = A y F ). Hence, by choosing 3.43) and 3.44) Ay) = hu, y F ) mdu) By) = By F ) K y y F ) y Gx) y F p s) y Fs) y F y y F Hu, s) mdu) ds, Hu, x), mdu)p Gx) ) dx Hu, s) mdu) ds dx, for y > y F, we have that Ay F ) = Ay F ) and By F ) = By F ). The case y y G. We start by remarking that if y G =, then there is nothing to prove. Therefore, assume that y G <. For x, the solution w to equation 3.) is of the form 3.9), for some functions A and B. We require w to be continuous at, y), which implies that 3.45) wx, y) = wx, F x)) K y F x) ), We postulate that F should solve 3.46) Hu, y) mdu) =, for y y G. for x.

12 2 LØKKA AND ZERVOS Assumption 2.2 ensures that equation 3.46) has a unique solution F. Further, the smooth fit conditions w y, y) = w y, y) and w xy, y) = w xy, y), imply that 3.47) and 3.48) A y) = B y) A y)p) Hu, y) mdu), p s) Hu, y) mdu) ds = K. From the definition of y G and equations 3.32) and 3.37), it follows that Gy G ) = and that Fy G ) satisfies Fy G ) Hu, y G ) mdu) =. We conclude that Fy G ) = Fy G ). Comparing 3.3) and 3.33) with 3.47) and 3.48) verifies that A y G ) = A y G ) and B y G ) = B y G ). Moreover, from equation 3.42) it follows that 3.49) F y G ) = σ2 Fy G ) ) p Fy G ) ) 2H Fy G ), y G ) Fy G ) H Hu, y G) mdu). Differentiating equation 3.46) with respect to y shows that F y G ) coincides with the right hand side of 3.49), which verifies that F is continuously differentiable. Finally, by choosing 3.5) and 3.5) Ay) = Ay G ) y y G By) = By G ) K y y G ) y Fx) y G p s) Hu, x) mdu) dx, y y G Hu, x) mdu) ds dx, Hu, x) mdu)p Fx) ) dx we see that A and B are continuous at y G. We have the following characterisation of the solution to the differential equation 3.), which we later show plays a similar role as a Hamilton-Jacobi-Bellman equation. Proposition 3.2. Let y F and y G be given by 3.) and 3.3), respectively. Let F and G be the unique solution to 3.2), for y < y F, be the unique solution to 3.32) and 3.37), for y F y < y G, and be the unique solution to 3.46), for y y G. Further, let v be given by 3.9) where A and B be given by 3.3) and 3.22), for y < y F, by 3.43)

13 LONG-TERM OPTIMAL STRATEGIES 3 and 3.44), for y F y < y G, and by 3.5) and 3.5), for y y G. Then F and G are strictly increasing and continuously differentiable, and vx, G x)) K G x) y ), for x Gy), 3.52) wx, y) = vx, y), for x Gy), vx, F x)) K y F x) ), for x, is up to a constant the unique solution to 3.) of class C 2, [, ) 2 ). Proof. First note that the existence and claimed properties for F and G follows from the previous analysis. This also proves the existence and uniqueness of A and the existence of B, and that B is unique up to a constant. Further, by construction and the function v given by 3.9) satisfies equation 3.8). Assume that x, y) I. Using the fact that w y Gy), y) = w y Gy), y) = K, we calculate that 3.53) w x x, y) = v x x, G x)) v y x, G x)) dg x) KdG dx dx x) ) dg = v x x, G x)) v y x, G x)) K dx x) = v x x, G x)) = w x x, G x)), Further, since w yx Gy), y) = w xy Gy), y) =, we obtain that 3.54) w xx x, y) = v xx x, G x)) v yx x, G x)) dg dx x) = w xxx, G x)). A similar calculation shows that w x x, y) = w x x, F x) ) and w xx x, y) = w xx x, F x) ), for x, y) in D. We conclude that w belongs to C 2, [, ) 2 ). Moreover, up to a constant, w is the unique solution to the smooth pasting conditions, hence up to a constant the unique solution to 3.) of class C 2, [, ) 2 ). In view of equations 3.53) and 3.54), we calculate that 2 σ2 x)w xx x, y) bx)v x x, y) hx, y) = 2 σ2 x)w xx x, G x) ) bx)v x x, G x) ) hx, y) = y G x) Hx, s) ds, for all x, y) I, since Hx, y), for x, y) I. A similar argument shows that 2 σ2 x)w xx x, y) bx)v x x, y) hx, y), for all x, y) D. Assume that x, y) C. With reference to 2.), 3.22), 3.43) and 3.44), we calculate w y x, y) K = p s) Hu, y) mdu) ds p Gy) ) Hu, y) mdu)

14 4 LØKKA AND ZERVOS 3.55) Define a function R I y by and observe that R I yx) = px) px) Hu, y), mdu) = p Gy) ) Hu, y) mdu) px) Hu, y) mdu) Hu, y) mdu) Ry) I x) = p x) Hx, y) mdu). p s) Hu, y) mdu) ds pu)hu, y) mdu) pu)hu, y) mdu). pu)hu, y) mdu), From the definition of the functions F and G, and Assumption 2.2, it follows that Ry I) x), for every x Gy). This observation and equation 3.55) imply that w y x, y) K, for every x, y) C. For completeness we remark that for x, y) D, we have that w y x, y) = K K. Furthermore, for x, y) I, we have w y x, y) = K K, since K K >. Next, observe that the construction of y F, y G, F and G imply that Consequently, pu)hu, y) mdu) K K w y x, y) K Define a function R D y by We calculate that R D y x) = and Hu, y) mdu) =. pu)hu, y) mdu) px) Hu, y) mdu). pu)hu, y) mdu) px) Hu, y) mdu). Ry D ) x) = p x) Hu, y) mdu). This implies that = R D y ) R D y x). We conclude that w y x, y) K. 4. The optimal investment strategy The aim of this section is to formulate the strategy corresponding to the solution to the Hamilton-Jacobi-Bellman equation, and prove that this strategy is optimal. However, we will start by establishing a couple of technical results.

15 Definition 4.. Let Φ :, ), ) be given by LONG-TERM OPTIMAL STRATEGIES 5 Φx) = m[, u])pu)du, and let Φ denote its inverse, i.e. Φ Φ y) = y, and Φ Φx) = x. The following result provides a relationship between conditions formulated by Peskir [24] and condition 2.5). Lemma 4.. Assumption 2.5) implies that { Φy) 4.) y sup y> y } du <, Φu) and Φ x) 4.2) lim =. x x Proof. Assume that 2.5) holds. Then there exists an ɛ > and a constant C such that bx) σ 2 x) ɛ, for x C x. This implies that p x) C 2 x 2ɛ, for some positive constant C 2, and that px) C C 2 x 2ɛ, for some positive constants C and C 2. Observe that there exist constants C, C 2 > such that m[, x]) C, for x C 2. By L Hopital s rule and the previous estimate for the scale function p, 4.3) Next, we claim that 4.4) Φx) lim x x 2 xφ x) lim x Φx) =. In order to verify this, assume that 4.4) does not hold. Then for every ɛ > there exists a C > such that xφ x) ɛ, for x C Φx). This implies that Φx) C 2 x ɛ, for some constant C 2 >, which contradicts 4.3). Further, it follows from 4.3) that x dy Φy) where C > is a constant. This shows that x L Hopital s rule and 4.4), 4.5) lim x Φx) x Together with the observation 4.6) this implies that 4.) holds. x x >. dy <, for all x, C y2 ) dy x = lim Φy) x dy Φy) dy Φy) x Φx) is well defined for all x. By = lim Φx) ) dy lim =, x x x Φy) x xφ x) Φx) <.

16 6 LØKKA AND ZERVOS Since Φx), for all x, it follows that Φ x), for all x. Therefore, if lim x Φ x) <, then we can conclude that 4.2) holds. Now, assume that lim x Φ x) =. Then by L Hopital s rule, we calculate that lim x Φ x) x for some ɛ >. This verifies that 4.2) holds. = lim ) Φ x) = lim x x Φ x) lim x x 2ɛ =, The next result shows that strategies with a certain limit behaviour as time tends to infinity can be ruled out as optimal strategies. Lemma 4.2. Let x, y) be any initial condition and Y x,y Y x,y be any strategy for which either of the following holds. Then V x, y) > JY x,y ) =. ) lim inf 2) lim inf E x [Y T ] >. T E x [Y T ] =. T Proof. First, observe that assumption 2.9) implies that [ T JY x,y ) = lim sup T Ex lim sup T Ex [ h ) X t, Y t dt K Y T K Y T [ ] kx t ) dt C lim inf T Ex max{k, K K E x [Y T 4.7) } lim inf ]. T By Rogers and Williams [25, p. 3] and assumption 2.8), it follows that 4.8) lim sup T Ex [ ] kx t ) dt = m, ) ) ] ] Y t dt kx) mdx) <. Now, assume that Y x,y satisfies ). Then there exists a κ > and a finite positive constant C such that E [ ] Y T > κt, for every T greater C. Hence, [ T ] T C 4.9) lim inf T Ex Y t dt κ lim inf t dt =. T It follows from 4.7) 4.9) that JY x,y ) =, from which the claimed result, when Y x,y satisfies ), follows from Remark 2.. Assume that Y x,y satisfies 2). Then it follows from 4.7) 4.8) that and the result follows from Remark 2.. JY x,y ) =, C

17 LONG-TERM OPTIMAL STRATEGIES 7 We are now ready to formulate the strategy corresponding to the solution to the Hamilton- Jacobi-Bellman equation that we derived the the previous section. Definition 4.2. Let C be given by 3.6, and F, G, and w be as in Proposition 3.2. Denote by Y x,y the strategy consisting of immediately raising or decreasing the capacity to the closest boundary point of C if x, y) / C, and then take minimal action so as to reflect X, Y ) on the boundary of C. That is, at time t =, Y has a positive jump of size G x) t), if G x) > y, and Y has a negative jump of size y F x), if y > F x). For t >, the strategy Y is a continuous process given by 4.) dy t = {Yt=G X t)} {Yt=F X t)} ) dy t. According to the next result, the strategy Y provides an optimal strategy corresponding to the optimisation problem with value function V given by 2.4). Theorem 4.3. Fix any initial condition x, y), the strategy Y x,y provides an optimal strategy. That is V x, y) = JY x,y ). given by Definition 4.2 Proof. We need to prove that if Y x,y is any strategy in Y x,y, then JY x,y ) JY x,y). Observe that since Y t >, for all t, and Y is increasing, it follows from Lemma 4.2 that it is sufficient to prove the inequality for strategies Y x,y which satisfy 4.) E[Y T ] lim = and lim T E[Y T ] T <. So, let Y x,y be any strategy which satisfy the conditions in 4.). We claim that 4.2) and 4.3) hx t, Y t ) dt K Y T K Y T wx, y) wx T, Y T ) σx t )w x X t, Y t ) dw t, hx t, Y t ) dt K Y T ) K Y T ) = wx, y) wx T, Y for all T <. To prove this, observe that by Itô s formula = wx, y) wx T, Y T ) σx t )w x X t, Y t ) dw t T ) σx t )w x X t, Y t ) dw t, 2 σ2 X t )w xx X t, Y t ) bx t )w x X t, Y t ) dt w y X t, Y t ) dy t ) c w y X t, Y t ) dy t ) c

18 8 LØKKA AND ZERVOS 4.4) wx t, Y t Y t ) wx t, Y t ). t T By re-arranging equation 4.4), we get 4.5) hx t, Y t ) dt K Y T K Y T = wx, y) wx T, Y T ) t T 2 σ2 X t )w xx X t, Y t ) bx t )w x X t, Y t ) hx t, Y t ) dt σx t )w x X t, Y t ) dw t w y X t, Y t ) K dy t ) c w y X t, Y t ) K dyt ) c wx t, Y t Y t ) wx t, Y t ) K Y t wx, y) wx T, Y T ) σx t )w x X t, Y t ) dw t wx t, Y t Y t ) wx t, Y t ) K Y t t T K Y t K Y t, since w is a solution to 3.). The same calculations holds for Y x,y, except that in this case the inequality in 4.5) holds with equality. Further, we can make the following calculations regarding the last term in 4.5), ) wx t, Y t Y t ) wx t, Y t ) K Y t K Yt 4.6) t T = t T =, ) wx t, Y t Y t ) wx t, Y t ) K Y t t T Y t t T ) wx t, Y t Yt ) wx t, Y t ) K Yt w y X t, Y t u) K du t T Y t w y X t, Y t u) K du since w is a solution to 3.). The same calculations hold for the case Y x,y, with the exception that the inequality in 4.6) holds with equality. The claimed inequalities 4.2) and 4.3) then follows from 4.5), 4.6) and the comments regarding the case Y x,y proceeding these inequalities.

19 It follows from assumptions 2.9) and 4.) that h ) X t, Y t ) dt K Y T K Y T C LONG-TERM OPTIMAL STRATEGIES 9 Y t dt max { K, K K } Y T, which belongs to L P x ), for all T <. Here ) denotes the absolute value of the negative part. Moreover, from assumptions 2.8) 2.9) it follows that h ) X t, Yt ) dt K YT ) K YT ) kx t ) dt, which belongs to L P x ), for all T <. Assumption 2.2) implies that G x) y x) C C 2 x, for some finite constants C and C 2. Therefore, YT y sup G X t ) y C C 2 sup X t ), t T t T for some finite constants C and C 2. By the assumption 4.2), Lemma 4. and [24, Theorem 2.5], [ ] lim T E[ ] YT lim T E Φ T) 4.7) C C 2 sup X t C 2 lim =. t T T Since w satisfies the equation 3.), we calculate that wxt, Y T ) wx T, YT ) Y T = w y X T, y) dy max{ K, K } YT YT. Y T By condition 4.) and equation 4.7), it follows that WX T, Y T ) WX T, Y T ) L P x ), for all T <, and that 4.8) lim WXT, Y T ) WX T, YT T ) =. According to the previous results and the inequalities 4.2) and 4.3), the negative part of σx t ) { w x X t, Y t ) w x X t, Y t )} dw t belongs to L P x ), for all T <. Hence, by Fatou s lemma and 4.8), [ T lim sup T E x hx t, Y t ) hx t, Yt ) dt K Y T YT )) K Y T Y T ) )] T ) wx T, Y T ) ] lim sup lim sup T Ex[ wx T, Y [ T T Ex σx t ) { w x X t, Y t ) w x X t, Y t )} dw t ]

20 2 LØKKA AND ZERVOS 4.9) lim sup lim inf n [ T τn T Ex σx t ) { w x X t, Y t ) w x X t, Yt )} dw t ]), where {τ n } n= given by τ n = inf { t : X t, Y t ) / [, n) 2} is a sequence of stopping times converging to infinity, P x -a.s., since X t, Y t ), ) 2, P x -a.s., for all t <. The calculation τn σx t ) { w x X t, Y t ) w x X t, Yt ) }) 2 dt implies that 4.2) T max σ 4 x) w x x, y) w x x, z) )2 <, x,y) [,n] 2,z [,y n)] E x [ τn σx t ) { w x X t, Y t ) w x X t, Y t ) } dw t ] =, for all n and T <. From 4.9) and 4.2) we conclude that 4.2) lim sup [ T T E x hx t, Y t ) hx t, Yt ) dt K Y T YT )) K Y T Y T ) )]. Finally, in view of 4.2), we calculate [ T JY x,y ) = lim sup T Ex lim inf which completes the proof. T Ex lim sup T Ex = JY x,y), [ [ hx t, Y t ) dt K Y T K Y T hx t, Y t ) dt K Y T ) K Y T ) ] hx t, Y t ) dt K Y T ) K Y T ) ] The next result shows that the strategy Y is not the only optimal strategy. Essentially this is due to the fact that certain actions taken before any finite stopping time does not affect the performance index. Lemma 4.4. Let τ be a stopping time satisfying τ < a.s., and let Ȳx,y be given by Y t = y {t τ} Yt {t>τ}, i.e. the strategy that consists of taking no action up to time τ and then proceed according to the optimal strategy described in Theorem 4.3. Then JȲx,y) = JY x,y ) = vx, y), for every x, y), )2. ]

21 LONG-TERM OPTIMAL STRATEGIES 2 Proof. We calculate that 4.22) JȲx,y) JY x,y) = lim sup T Ex [ τ [ τ lim sup E x lim inf [ τ Ex ] hx t, y) dt hx t, Y t ) dt K Y T τ ) K Y T τ ) ] hx t, y) hx t, Y t ) dt K Y T τ ) K Y T τ ) ]. From assumptions 2.8) 2.9), it follows that the absolute value of the negative part of τ hx t, y) hx t, Y t ) dt K Y T τ ) K Y T τ ) is less than or equal to C y)t τ) τ kx t ) dt, which belongs to L P x ), for all deterministic finite T. Hence, by Fatou s lemma, 4.23) lim inf T Ex E x [lim inf =. [ τ τ T hx t, y) hx t, Y t ) dt K Y T τ) K Y T τ) ] hx t, y) hx t, Y t ) dt K Y T τ ) K Y T τ ) )] From 4.22) and 4.23) we conclude that JȲx,y) JY x,y ). The result then follows from Theorem 4.3. Our assumptions on b and σ assure that X is recurrent. Hence, for any x, ) and initial value x, ) the hitting time of x, given by τ x = inf{t : X t = x}, is finite almost surely see, e.g., Karatzas and Shreve [6, Section 5.5]). By Lemma 4.4, it follows that 4.24) V x, y) = V x, y), for all x, x, y, ). Moreover, given any initial condition x, y), choose x such that K y < x <. Then 4.25) V x, y) V x K y, ) = V x, ) V x K y, y) = V x, y), where we have used that immediately increasing and decreasing the capacity level is suboptimal, and that the value function does not depend on the initial level of the economic indicator. From 4.24) and 4.25) we conclude that the value function V given by 2.4) is a constant that does not depend on the initial value x, y).

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23 LONG-TERM OPTIMAL STRATEGIES 23 [25] L. C. G. Rogers and D. Williams 2), Diffusions, Markov Processes and Martingales, volume 2, Cambridge University Press. [26] R. Sadowy and L. Stettner 22), Om risk-sensitive ergodic impulsive control of Markov processes, Applied Mathematics and Optimization, vol. 45, pp [27] H. Wang 23), Capacity expansion with exponential jump diffusion processes, Stochastics and Stochastics Reports, vol.75, pp Arne Løkka) Department of Mathematics King s College London Strand, London WC2R 2LS United Kingdom address: arne.lokka@kcl.ac.uk Mihail Zervos) Department of Mathematics King s College London Strand, London WC2R 2LS United Kingdom address: mihail.zervos@kcl.ac.uk