Liquidity Management with Decreasing-returns-to-scale and Secured Credit Line.

Transcription

1 Liquidity Management with Deceasing-etuns-to-scale and Secued Cedit Line. Ewan Piee Stéphane Villeneuve Xavie Wain axiv: v2 [q-fin.pm 4 Nov 215 Abstact: This pape examines the dividend and investment policies of a cash constained fim, assuming a deceasing-etuns-to-scale technology and adjustment costs. We extend the liteatue by allowing the fim to daw on a secued cedit line both to hedge against cash-flow shotfalls and to invest/disinvest in poductive assets. We fomulate this poblem as a bi-dimensional singula contol poblem and use both a viscosity solution appoach and a veification technique to get qualitative popeties of the value function. We futhe solve quasi-explicitly the contol poblem in two special cases. Keywods: Investment, dividend policy, singula contol, viscosity solution JEL Classification numbes: C61; G35. MSC Classification numbes: 6G4; 91G5; 91G8. 1 Intoduction In a wold of pefect capital maket, fims could finance thei opeating costs and investments by issuing shaes at no cost. As long as the net pesent value of a poject is positive, it will find investos eady to supply funds. This is the cental assumption of the Modigliani and Mille theoem [22. On the othe hand, when fims face extenal financing costs, these costs geneate a pecautionay demand fo holding liquid assets and etaining eanings. This depatue fom the Modigliani-Mille famewok has eceived a lot of attention in ecent yeas and has given bith to a seie of papes explaining why fims hold liquid assets. Pioneeing papes ae Jeanblanc and Shiyaev [17, Radne and Shepp [26 while moe ecent studies include Bolton, Chen and Wang [4, Décamps, Maiotti, Rochet and Villeneuve [7 and Hugonnie, Malamud and Moellec [16. In all of these papes, it is assumed that fims ae all equity financed. Should it uns out of liquidity, the fim eithe liquidates o aises new funds in ode to continue opeations by issuing equity. This binay decision only depends on the seveity of issuance costs. The pimay objective of ou pape is to study a setup whee a cash-constained fim has a EDF R&D OSIRIS. ewan.piee@edf.com Toulouse School of Economics (CRM-IDEI, Manufactue des Tabacs, 21, Allée de Bienne, 31 Toulouse, Fance. stephane.villeneuve@tse-f.eu. This authos gatefully acknowledges the financial suppot of the eseach initiative IDEI-SCOR Risk Maket and Ceation Value unde the aegis of the isk foundation. EDF R&D & FiME, Laboatoie de Finance des Machés de l Enegie ( 1

2 mixed capital stuctue. To do this, we build on the pape by Bolton, Chen and Wang [4 chapte V to allow the fim to access a secued cedit line. While [4 assumed a constantetuns-to-scale and homogeneous adjustment costs which allows them to wok with the fim s cash-capital atio and thus to educe the dimension of thei poblem, we athe conside a deceasing-etuns-to-scale technology with linea adjustments costs. Bank cedit lines ae a majo souce of liquidity povision in much the same way as holding cash does. Kashyap, Rajan and Stein [18 found that 7% of bank boowing by US small fims is though cedit line. Howeve, access to cedit line is contingent to the solvency of the boowe which makes the use on cedit line costly though the inteest ate and thus makes it an impefect substitute fo cash (Sufi [28. Fom a theoetical viewpoint, the use of cedit lines can be justified by moal hazad poblems (Holmstom-Tiole [15 o fom the fact that banks can commit to povide liquidity to fims when capital maket cannot because banks have bette sceening and monitoing skills (Diamond [9 In this pape, we model cedit line as a full commitment lending elationship between a fim and a bank. The lending contact specifies that the fim can daw on a line of cedit as long as its outstanding debt, measued as the size of the fim s line of cedit, is below the value of total assets (cedit limit. The liability side of the balance sheet of the fim consists in two diffeent types of ownes: shaeholdes and bankes. Should the fim be liquidated, bankes have senioity ove shaeholdes on the total assets. We assume that the secued line of cedit continuously chages a vaiable spead 1 ove the isk-fee ate indexed on the fim s outstanding debt, the highe the size of fim s line of cedit, the highe the spead is. With this assumption, the secued line of cedit is somehow simila to the pefomance-sensitive debt studied in [21 except that the shaeholdes ae hee foced to go bankupt when they ae no moe able to secue the cedit line with thei assets. Many models initiated by Black and Cox [3 and Leland [19 that conside the taditional tadeoff between tax and bankuptcy costs as an explanation fo debt issuance study fims liabilities as contingent claims on its undelying assets, and bankuptcy as an endogenous decision of the fim management. On the othe hand, these models assume costless equity issuance and thus put aside liquidity poblems. As a consequence, the fim s decision to boow on the cedit maket is independent fom liquidity needs and investment decisions. A notable exception is a ecent pape by Della Seta, Moellec, Zucchi [8 which studies the effects of debt stuctue and liquid eseves on banks insolvency isk. Ou model belongs to the class of models that conside endogenous bankuptcy of a fim with mixed capital stuctue eplacing taxes with liquidity constaints. Fom a mathematical point of view, poblems of cash management have been fomulated as singula stochastic optimal contol poblems. As efeences fo the theoy of singula stochastic contol, we may mention the pioneeing woks of Haussman and Suo [12 and [13 and fo application to cash management poblems Højgaad and Taksa [14, Asmussen, Højgaad and Taksa [1, Choulli, Taksa and Zhou [5, Paulsen [24 among othes. To mege copoate liquidity, investment and financing in a tactable model is challenging because it involves a athe difficult thee-dimensional singula contol poblem with stopping whee the state vaiables ae the book value of equity, the size of poductive asset and the size of the 1 The spead may be justified by the cost of equity capital fo the bank. Indeed, the full commitment to supply liquidity up to the fim s cedit limit pevents bank s shaeholdes to allocate pat of thei equity capital to moe valuable investment oppotunities. 2

3 fim cedit line while the stopping time is the decision to default. The liteatue on multidimensional contol poblems elies mainly on the study of leading examples. A seminal example is the so-called finite-fuel poblem intoduced by Benes, Shepp and Witsenhausen [2. This pape povides a ae example of a bi-dimensional optimization poblem that combines singula contol and stopping that can be solved explicitly by analytical means. Moe ecently, Fedeico and Pham [1 have solved a degeneate bi-dimensional singula contol poblem to study a evesible investment poblem whee a social planne aims to contol its capacity poduction in ode to fit optimally the andom demand of a good. Ou pape complements the pape by Fedeico and Pham [1 by intoducing fims that ae cashconstained 2. To ou knowledge, this is the fist time that such a combined appoach is used. This makes the poblem much moe complicated and we do not petend solving it with full geneality, but athe, we pave the way fo futue developments of these multidimensional singula contol models. In paticula, we lose the global convexity popety of the value function that leads to the necessay smooth-fit popety in [1 (see Lemma 8. Instead, we will give popeties of the value function (see Poposition 6 and chaacteize it by means of viscosity solution (see Theoem 2. Futhemoe, we will solve explicitly by a standad veification agument the peculia case of costless evesible investment. A last new esult is ou chaacteization of the endogenous bankuptcy in tems of the pofitability of the fim and the spead function. The emainde of the pape is oganized as follows. Section 2 intoduces the model with a poductive asset of fixed size, fomalizes the notion of secued line of cedit and defines the shaeholdes value function. Section 3 contains ou fist main esult, it descibes the optimal cedit line policy and gives the analytical chaacteization of the value function in tems of a fee bounday poblem fo a fixed size of poductive assets. Section 4 is a technical section that builds the value function by solving explicitly the fee bounday poblem. Section 5 extends the analysis to the case of evesible investment on poductive assets and paves the way to a complete chaacteization of the dividend and investment policies. 2 The No-investment Model We conside a fim owned by isk-neutal shaeholdes, with a poductive asset of fixed size K, whose pice is nomalized to unity, that has an ageement with a bank fo a secued line of cedit. The cedit line is a souce of funds available at any time up to a cedit limit defined as the total value of assets. The fim has been able to secue the cedit line by posting its poductive assets as collateal. Nevetheless, in ode to make the cedit line attactive fo bank s shaeholdes that have dedicated pat of thei equity to this ageement, we will assume that the fim will pay a vaiable spead ove the isk-fee ate depending on the size of the used pat of the cedit line. In this pape, the cedit line contact is given and thus the spead is exogenous, see Assumption 1. Finally, building on Diamond s esult [9 we assume that the costs of equity issuance ae so high that the fim is unwilling to incease its cash eseves by aising funds in the equity capital maket and pefes dawing on the cedit line. The fim is chaacteized at each date t by the following balance sheet: 2 Ly Vath, Pham and Villeneuve [2 have also studied a evesible investment poblem in two altenative technologies fo a cash-constained fim that has no access to extenal funding 3

4 K X t M t L t K epesents the fim s poductive assets, assumed to be constant 3 and nomalized to one. M t epesents the amount of cash eseves o liquid assets. L t epesents the size of the cedit line, i.e. the amount of cash that has been dawn on the line of cedit. Finally, X t epesents the book value of equity. The poductive asset continuously geneates cash-flows ove time. The cumulative cashflows pocess R = (R t t is modeled as an aithmetic Bownian motion with dift µ and volatility σ which is defined ove a complete pobability space (Ω, F, P equipped with a filtation (F t t. Specifically, the cumulative cash-flows evolve as dr t = µ dt + σ db t whee (B t t is a standad one-dimensionnal Bownian motion with espect to the filtation (F t t. Cedit line equies the fim to make an inteest payment that is inceasing in the size of the used pat of the cedit line. We assume that the inteest payment is defined by a function α(. whee Assumption 1 α is a stictly inceasing, continuously diffeentiable convex function such that x, α (x and α( =. (1 The cedit line spead α(. is thus stictly positive and inceasing. The liquid assets ean a ate of inteest δ whee δ (, epesents a cay cost of liquidity 4. Thus, in this famewok, the cash eseves evolve as dm t = ( δm t dt + (µ α(l t dt + σdb t dz t + dl t (2 whee (Z t t is an inceasing ight-continuous (F t t adapted pocess epesenting the cumulative dividend payment up to time t and (L t t is a positive ight-continuous (F t t adapted pocess epesenting the size of the cedit line (outstanding debt at time t. Using the accounting elation 1 + M t = X t + L t, we deduce the dynamics fo the book value of equity dx t = ( δx t dt + (µ ( δ + ( δl t α(l t dt + σdb t dz t. (3 Finally, we assume the fim is cash-constained in the following sense: 3 The extension to the case of vaiable size will be studied in Section 4 4 This assumption is standad in models with cash. It captues in a simple way the agency costs, see [7, [16 fo moe details 4

5 Assumption 2 The cash eseves must be non negative and the fim management is foced to liquidate when the book value of equity hits zeo. Using the accounting elation, this is equivalent to assume bankes get back all the poductive assets afte bankuptcy. The goal of the management is to maximize shaeholdes value which is defined as the expected discounted value of all futue dividend payouts. Because shaeholdes ae assumed to be isk-neutal, futue cash-flows ae discounted at the isk-fee ate. The fim can stop its activity at any time by distibuting all of its assets to stakeholdes. Thus, the objective is to maximize ove the admissible contol π = (L, Z the functional ( τ V (x, l; π = E x,l e t dz t whee τ = inf{t, X π t } accoding to Assumption 2. Hee x (esp. l is the initial value of equity capital (esp. debt. We denote by Π the set of admissible contol vaiables and define the shaeholdes value function by V (x, l = sup V (x, l; π. (4 π Π Remak 1 We suppose that the cash eseves must be non negative (Assumption 2 so to be admissible, a contol π = (L, Z must satisfy at any time t dz t X t. 3 No-investment Model solution This section deives the shaeholdes value and the optimal dividend and cedit line policies. It elies on a standad HJB chaacteization of the contol poblem and a veification pocedue. 3.1 Optimal cedit line issuance The shaeholdes optimization poblem (4 involves two state vaiables, the value of equity capital X t and the size of the cedit line L t, making its esolution difficult. Fotunately, the next poposition will enable us to educe the dimension and make it tactable the computation of V. Poposition 1 shows that cedit line issuance is only optimal when the cash eseves ae depleted. Poposition 1 A necessay and sufficient condition to daw on the cedit line is that the cash eseves ae depleted, that is t R +, L t M t = o equivalently L t = (1 X t +. 5

6 Poof: Fist, by Assumption 2, it is clea that the fim management must daw on the cedit line when cash eseves ae nonpositive. Convesely, assume that the level of cash eseves m is stictly positive. We will show that it is always bette off to educe the level of outstanding debt by using the cash eseves. We will assume that the initial size of the cedit line is L = l > and denote π t = (L t, Z t any admissible stategy. Let us define by φ the cost of the cedit line on the vaiation of the book value of equity, that is φ(l = α(l(δl such that the book value of equity dynamics is dx t = ( δx t dt + (µ ( δ φ(l t dt + σdb t dz t. (5 Note that φ is stictly inceasing. We fist assume that the fim does not daw on the cedit line at time, L = l. Because m >, we will built a stategy fom π as follows: { L ɛ = l ɛ fo < ɛ < min(m, l and L ɛ t L t, Z ɛ t = Z t + t (φ(l s φ(l ɛ s ds Note that the cedit line issuance stategy L ɛ consists in always having less debt that unde the cedit line issuance stategy L and because φ is inceasing, the dividend stategy Z ɛ t pays moe than the dividend stategy Z t. Futhemoe, denoting by π ɛ = (L ɛ t, Z ɛ t, equation (5 shows that the bankuptcy time unde π ɛ stating fom (x, l ɛ and the bankuptcy time unde π stating fom (x, l have the same distibution. Theefoe, ( τ π ɛ V (x, l; π ɛ = E (x,lɛ ( τ π ɛ > E (x,lɛ = E (x,l ( τ π = V (x, l; π, e s dz ɛ s e s dz s e s dz s which shows that it is bette off to follow π ɛ than π. So if m > l, it is optimal to set l = by using m l units of cash eseves while if m < l, it is optimal to educe the debt to l m. In any case, at any time L t = (1 X t +. Now, if we assume that the fim daw on the cedit line at time, i.e. L, two cases have to be consideed. L = which is possible only if m > l. In that case, we set L t = L t and Z t = Z t fo t >. L >. In that case, we take the same stategy π ɛ with < ɛ < min(m, l + L. Accoding to Poposition 1, we define the value function as v (x = V (x, (1 x +. The est of the section is concened with the deivation of v. 6

7 3.2 Analytical Chaacteization of the fim value Because the level of capital is assumed to be constant, Poposition 1 makes ou contol poblem one-dimensional. Thus, we will follow a standad veification pocedue to chaacteize the value function in tems of a fee bounday poblem. In ode to focus on the impact of cedit line on the liquidity management, we will assume heeafte that δ =. This assumption is without loss of geneality but allow us to be moe explicit in the analytical deivation of the HJB fee bounday poblem. We denote by L the diffeential opeato: LΦ = (µ α((1 x + Φ (x + σ2 2 Φ (x Φ. (6 We stat by poviding the following standad esult which establishes that a smooth solution to a fee bounday poblem coincides with the value function v. Poposition 2 Assume thee exists a C 1 and piecewise twice diffeentiable function w on (, + togethe with a pai of constants (a, b R + R + such that, x a, Lw and w(x = x a x b, Lw = and w (x 1 x > b, Lw and w (x = 1. (7 then w = v. with w (b = (8 Poof: Fix a policy π = (Z Π. Let : dx t = (µ α((1 X t + dt + σdb t dz t, X( = x be the dynamic of the book value of equity unde the policy π. Let us decompose Z t = Zt c + Z t fo all t whee Zt c is the continuous pat of Z. Let τ the fist time when X t =. Using the genealized Itô s fomula, we have : e (t τ w(x t τ = w(x + t τ t τ e s Lw(X s ds + t τ e s w (X s dzs c + e s [w(x s w(x s. s t τ σe s w (X s db s Because w is bounded, the thid tem is a squae integable matingale. Taking expectation, we obtain [ t τ w(x = E x [e (t τ w(x t τ E x e s Lw(X s ds [ t τ + E x e s w (X s dzs c [ E x e s [w(x s w(x s. s t τ 7

8 Because w 1, we have w(x s w(x s X s = Z s theefoe the thid and the fouth tems ae bounded below by ( t τ E x e s w (X s dz s. Futhemoe w is positive because w is inceasing with w( = and Lw thus the fist two tems ae positive. Finally, ( t τ ( t τ w(x E x e s w (X s dz s E x e s dz s. Letting t + and we obtain w(x v (x. To show the evese inequality, we will pove that thee exists an admissible stategy π such that w(x = v(x, π. Let (X t, Z t be the solution of whee, with X t = t Z t = (x 1 {x a} + (x b + 1 {t= } + (µ α((1 X s + ds + σb t Z t (9 t τ a τ a = inf{t, X t a} 1 {X s =b}dz s + a1 {t τa} (1 whose existence is guaanteed by standad esults on the Skookhod poblem (see fo example Revuz and Yo [27. The stategy π = (Zt is admissible. Note also that Xt is continuous on [, τa. It is obvious that v(x, π = x = w(x fo x a. Now suppose x > a. Along the policy π, the liquidation time τ coincides with τ a because Xτ a =. Poceeding analogously as in the fist pat of the poof, we obtain [ w(x = E x e (t τ w(xt τ [ t τ + E x [ = E x e (t τ w(xt τ [ t τ + E x e s w (X s dz s e s w (bdz s [ = E x e (t τ w(xt τ [ t τ + E x e s dzs, + E x [ 1t>τ e τ (w(x τ w(x τ + E x [ 1t>τ e τ a whee the last two equalities uses, w(a = a w (b = 1 and ( Z τ = a. Now, because w( =, [ E x e (t τ w(xt τ [ = E x e t w(xt 1 t τ. Futhemoe, because w has at most linea gowth and π is admissible, we have lim E [ x e t w(xt 1 t τ =. t 8

9 Theefoe, we have by letting t tend to +, [ τ w(x = E x e s dzs = v(x, π which concludes the poof. Remak 2 We notice that the poof emains valid when a = and w ( is infinite by a standad localisation agument which will be the case in section Optimal Policies The veification theoem allows us to chaacteize the value function. The following theoem summaizes ou findings. Theoem 1 Unde Assumption 1 and 2, the following holds: If µ, it is optimal to liquidate the fim, v (x = x. If µ α(1, the value of the fim is an inceasing and concave function of the book value of equity. Any excess of cash above the theshold b = inf{x >, (v (x = 1} is paid out to shaeholdes.(see Figue 1. If µ < α(1, the value of the fim is an inceasing convex-concave function of the book value of equity. When the book value of equity is below the theshold a = sup{x >, v (x = x}, it is optimal to liquidate. Any excess of cash above the theshold b a = inf{x > a, (v (x = 1} is paid out to shaeholdes.(see Figue 2 It is inteesting to compae ou esults with those obtained in the case of all equity financing. Fist, because the use of cedit line is costly, it is optimal to wait that the cash eseves ae depleted to daw on it. Moeove, thee exists a taget cash level above which it is optimal to pay out dividends. These two fist findings ae simila to the case of all equity financing. On the othe hand, the maginal value of cash may not be monotonic in ou case. Indeed, when the cost of the cedit line is high, it becomes optimal fo shaeholdes to teminate the lending elationship. This embedded option value makes the shaeholde value locally convex in the neighbohood of the liquidation theshold a. The highe is the cost, measued by λ in ou simulation, the soone is the stategic default o equivalently, the value function deceases, while the embedded exit option inceases, with the cost of the cedit line. The stategic default comes fom the fact that the instantaneous fims pofitability µ α(x becomes negative fo low value of equity capital. This is a key featue of ou model that neve happens when the fim is all-equity whee the maginal value of cash at zeo is the only statistic eithe to tigge the equity issuance o to liquidate. Figue 1 plots some value functions, when µ α(1, using a linea function fo α, α(x = λx with diffeent values of λ. 9

10 Figue 1: Compaing shaeholdes value functions with µ =.25, =.2, σ =.3 and µ α(1 fo diffeent values of λ whee α(x = λx. Figue 2 plots some value functions, when α(1 > µ, using a linea function fo α, α(x = λx fo diffeent values of λ. 1

11 Figue 2: Compaing shaeholdes value functions with µ =.25, =.2, σ =.3 and α(1 > µ fo diffeent values of λ whee α(x = λx. Next section is devoted to the poof of Theoem 1. The poof is based on an explicit constuction of a smooth solution of the fee bounday poblem and necessitates a seies of technical lemmas. 4 Solving the fee bounday poblem The fist statement of Theoem 1 comes fom the fact that the function w(x = x satisfies Poposition 2 when µ. To see this, we have to show that L w(x is nonpositive fo any x. A staightfowad computation gives fo x > 1, L w(x = µ x < µ, fo x 1, L w(x = µ α(1 x x. Using Equation (1 of Assumption 1, we obseve that L w(x is nondeceasing fo x 1 and nonpositive at x = 1 when µ. Heeafte, we will assume that µ > and focus on the existence of a function w and a pai of constants (a, b satisfying Poposition 2. We will poceed in two steps. Fist we ae going to establish some popeties of the solutions of the diffeential equation Lw =. Second, we will conside two diffeent cases- one whee the poductivity of the fim is always highe than the maximal inteest payment α(1 µ, the othe whee the inteest payment of the loan may exceed the poductivity of the fim α(1 > µ. 11

12 Standad existence and uniqueness esults fo linea second-ode diffeential equations imply that, fo each b, the Cauchy poblem : w(x = (µ α((1 x + w (x + σ2 2 w (x w (b = 1 (11 w (b = has a unique solution w b ove [, b. By constuction, this solution satifies w b (b = µα((1b+. Extending w b linealy to [b, [ as w b (x = x b + µα((1b+, fo x b yields a twice continuously diffeentiable function ove [, [, which is still denoted by w b. 4.1 Popeties of the solution to the Cauchy Poblem We will establish a seie of peliminay esults of the smooth solution w b of (11. Lemma 1 Assume b > 1. If w b ( = then w b is inceasing and thus positive. Poof: Because w b ( =, w b (b = µ and Lw b =, the maximum pinciple implies w b > on (, +. Let us define c = inf{x >, w b(x = } If c = then w b ( = w b ( = w b ( =. By unicity of the Cauchy poblem, this would imply w b = which contadicts w b (b = µ. Thus, c >. If c < b, we would have w b(c >, w b (c = and w b (c and thus Lw b(c < which is a contadiction. Theefoe w b is always positive. Lemma 2 Assume b > 1. We have w b > 1 and w b < on [1, b[. Poof: Because w b is smooth on 1, b, we diffeentiate Equation (11 to obtain, w b (b = 2 σ 2 >. As w b (b = and w b (b = 1, it follows that w b <, and thus w b > 1 ove some inteval b ɛ, b[, whee ɛ >. Now suppose by way of contadiction that w b (x 1 fo some x [1, b ɛ and let x = sup{x [1, b ɛ, w b (x 1}. Then w b ( x = 1 and w b (x > 1 fo x x, b[, so that w b (b w b (x > b x fo all x x, b[. Because w b (b = µ, this implies that fo all x x, b[, w b (x = 2 σ [w b(x µw b(x < 2 2 σ [(x b + w b(b µ = 2 (x b < 2 σ2 12

13 which contadicts w b (b = w b ( x = 1. Theefoe w b > 1 ove [1, b[. Futhemoe, using Lemma 1, w b (x = 2 σ 2 [w b(x µw b(x < 2 σ 2 [w b(x µ < 2 σ 2 [w b(b µ =. The next esult gives a sufficient condition on b to ensue the concavity of w b on (, b. Coollay 1 Assume b α(1 and µ α(1, we have w b > 1 and w b < ove, b[. Poof: Poceeding analogously as in the poof of Lemma 2, we define x = sup{x [, b ɛ, w b (x 1} such that w b ( x = 1 and w b (x > 1 fo x x, b[, so that w b(b w b (x > b x fo all x x, b[. Because b > α(1 > 1, w b (b = µ, we have Denote by g the function w b (x = 2 σ 2 [w b(x (µ α((1 x + w b(x < 2 σ 2 [(x b + w b(b (µ α((1 x + < 2 σ 2 [(x b + α((1 x+. g(x = 2 [(x b + α(1 x, x [, 1[. σ2 We have g (x = 2 [ α (1 x < by Assumption 1. Because g( = 2 [b + α(1 σ 2 σ 2 if b α(1, we have w b (x < fo x, 1 which contadicts w b ( x = 1 and w b (1 > 1 by Lemma 2. Theefoe w b > 1 ove [, 1[, fom which it follows w b < and w b is concave on, 1[. Because Lemma 2 gives the concavity of w b on [1, b[, we conclude. The next poposition establishes some esults about the egulaity of the function b w b (y fo a fixed y [, 1[. Lemma 3 Fo any y [, 1[, b w b (y is an inceasing function of b ove [y, 1 and stictly deceasing ove 1, + [. Poof: Conside the solutions H y and H y 1 to the linea second-ode diffeential equation LH = ove [y, [ chaacteized by the initial conditions H y (y = 1, (H y (y =, H y 1 (y =, (H y 1 (y = 1. We fist show that (H y and (H y 1 ae stictly positive on y, [. Because H y (y = 1 and (H y (y =, one has (H y (y = 2 >, such that σ 2 13

14 (H y (x > ove some inteval y, y + ɛ[ whee ɛ >. Now suppose by way of contadiction that x = inf{x y + ɛ, (H y (x } <. Then (H y ( x = and (H y ( x. Because LH y =, it follows that H y ( x, which is impossible because H y (y = 1 and H y is stictly inceasing ove [y, x. Thus (H y > ove y; [, as claimed. The poof fo H y 1 is simila, and is theefoe omitted. Next, let W H y,hy 1 (y = 1 and W H y,hy 1 = H y (H y 1 H y 1 (H y be the Wonskian of H y and H y 1. One has x y, W H y,hy 1 (x =H y (x(h y 1 (x H y 1 (x(h y (x = 2 σ 2 [Hy (x(h y 1 (x (µ α((1 x + (H y 1 (x H y 1 (x(h y (x (µ α((1 x + (H y (x = 2[µ α((1 x+ W σ 2 H y (x.,hy 1 Because α is integable, the Abel s identity follows by integation: [ ( 2 x x y, W H y (x = exp µ(x y + α((1 u + du.,hy 1 σ 2 Because W H y,hy 1 >, Hy and H y 1 ae linealy independent. As a esult of this, (H y, H y 1 is a basis of the two-dimensional space of solutions to the equation LH =. It follows in paticula that fo each b >, on can epesent w b as : y x [y, b, w b (x = w b (yh y (x + w b(yh y 1 (x. Using the bounday conditions w b (b = µα((1b+ and w b (b = 1, on can solve fo w b(y as follows: w b (y = (Hy 1 (b µα((1b+ H y 1 (b W H y (b.,hy 1 Using the deivative of the Wonskian along with the fact that H y 1 is solution to LH =, it is easy to veify that: ( (H y 1 (b α (1b + 1 b [y, 1[, dw b(y db b 1, [, dw b(y db = W H y (b,hy 1 = (Hy 1 (b W H y (b.,hy 1 So w b (y is an inceasing function of b ove [y, 1 and stictly deceasing ove 1, [. Coollay 2 If b 2 > b 1 > 1, then w b2 < w b1. Poof: Let us define W = w b1 w b2. Clealy, W > on [b 2, + [. Moeove, we have LW = on [, b 1 and W ( > by Lemma 3. Moeove, w b1 (b 1 = w b2 (b 2 and 14

15 w b2 (b 2 > w b2 (b 1 by Lemma 2. Theefoe, the maximum pinciple implies w b2 < w b1 on [, b 1. Finally, w b2 is concave and w b 2 (b 2 = 1 theefoe fo b 1 x b 2, w b2 (x w b2 (b 2 + x b 2 = < µ + x b 2 + x b 1 = w b1 (x. 4.2 Existence of a solution to the fee bounday poblem We ae now in a position to chaacteize the value function and detemine the optimal dividend policy. Two cases have to be consideed: when the pofitability of the fim is always highe than the maximal inteest payment (µ α(1 and when the inteest payment exceeds the pofitability of the fim (µ < α( Case: µ α(1 The next lemma establishes the existence of a solution w b to the Cauchy poblem (11 such that w b ( =. Lemma 4 Thee exists b 1, µ [ such that the solution to (11 satisfies w b ( =. Poof: Because µ α(1, we know fom Coollay 1 that w µ is a concave function on [, µ. Moeove, because µ >, w µ ( µ = µ. Because w µ is stictly concave ove, µ [ with w µ ( µ = µ and w µ = 1, w µ (x x fo all x, µ [. In paticula, w µ ( <. Moeove, we have : w ( = µ α(1. Theefoe, Lemma 3 implies w 1 ( >. Finally by continuity thee is some b 1, µ [ such that w b ( = which concludes the poof. The next lemma establishes the concavity of w b. Lemma 5 The function w b is concave on [, b Poof: Because b > 1, Lemma 2 implies that w b is concave on [1, b thus w b (1. Fo x < 1, we diffeentiate the diffeential equation satisfied by w b to get, σ 2 2 w b (x + (µ α(1 xw b (x + (α (1 x w b (x =. (12 15

16 Because w b ( = we have w b ( = 2 σ 2 (µ α(1w b (. Now, suppose by a way of contadiction that w b is continuous and nonpositive at the boundaies of [, 1, thee is some c such that w b w b > on some subinteval of [, 1. Because (c = and w (c >. But, this implies b w b α(1 cw b (c = (µ (c α (1 c < which is a contadiction with Lemma 1. Poposition 3 If µ α(1, w b is the solution of the contol poblem (9. Poof: Because w b is concave on [, b and w (b = 1, w 1 on [, b. Theefoe we have a twice continuously diffeentiable concave function w b and a pai of constants (a, b = (, b satisfying the assumptions of Poposition 2 and thus w b = v. When the maximal inteest payment is lowe than the fim pofitability, the value function is concave. This illustates the shaeholdes fea to liquidate a pofitable fim. In paticula, the shaeholdes value is a deceasing function of the volatility Case: µ < α(1 We fist show that, fo all y [, 1[, thee exists b y such that w by Cauchy Poblem (11 with w by (y = y. is the solution of the Lemma 6 Fo all y [, 1[, we have w 1 (y > y. Poof: Because ( α is continuous with α( = and µ >, thee exists ɛ such that w 1ɛ (1 ɛ = µα(ɛ > 1. Diffeentiating Equation (11, we obseve w 1ɛ(1 ɛ = 2 σ 2 ( α (ɛ < using Equation (1. Theefoe w 1ɛ is convex in a left neighbohood of 1. If w 1ɛ is convex on (, 1 then w 1ɛ (x x (1 ɛ + µα(ɛ > x fo small enough and the esult is poved. If w 1ɛ is not convex on (, 1 then it will exist some x < 1 such that w 1ɛ( x =, w 1ɛ( x > and w 1ɛ convex on x, 1ɛ. Diffeentiating Equation (11 at x gives w 1ɛ( x <. Theefoe w 1ɛ is noninceasing in a neighbohood of x. Assume by a way of contadiction that w 1ɛ is inceasing at some point ˆx [, x[. This would imply the existence of x < x such that w 1ɛ( x =, w 1ɛ( x < and w 1ɛ ( x > which contadicts Equation (11. Theefoe w 1ɛ is deceasing on (, x and convex on ( x, 1 which implies that w 1ɛ (x > x fo all x 1. To conclude, fo any y < 1, we can find small enough to have w 1ɛ (y > y which can be extended to w 1 (y > y by Lemma 3. 16

17 Coollay 3 Fo all y [, 1[, thee is an unique b y 1, 1 + µ [ such that w b y (y = y. Poof: By Lemma 1, w 1+ µ is concave on 1, 1 + µ [, thus w 1+ µ (1 < µ + (1 (1 + µ =. Suppose that thee exists c in [, 1[ such that w 1+ µ (c >, then thee exists x c, 1[ such that w 1+ µ ( x <, w 1+ ( x =, w µ 1+ ( x > yielding to the standad contadiction with µ the maximum pinciple. We thus have w 1+ µ (y < y fo all y [, 1 + µ. Using Lemma 6 and the continuity of the function b w b (y, it exists fo all y < 1 a theshold b y 1, 1 + µ [ such that w by (y = y. The uniqueness of b y comes fom Coollay 2. We will now study the behavio of the fist deivative of w by. Lemma 7 Thee exists ɛ > such that w b 1 (1 1 and b 1ɛ < µ. Poof: Because α( = and µ >, it exists η > such that x [1 η, 1, α(1 x + x µ <. (13 Moeove w µ is stictly concave on [1, µ [ by Lemma 2 and thus w µ (1 w µ (µ + (1 µ w µ ( µ = 1. Because by Lemma 2, we have w µ > 1 on [1, µ [, thee exists ν > such that x [1 ν, 1, w µ (x < x. Let = min(η, ν. By Coollay 3, it exists b 1ɛ 1, 1 + µ [ such that w b1ɛ (1 ɛ = 1 ɛ. We have w b1ɛ (1 ɛ > w µ (1 ɛ and then b 1ɛ < µ by Coollay 2. Let us conside the function W (x = w b1ɛ (x x, we have W (1 ɛ =, W (b 1ɛ = µ b 1ɛ >. Moeove, W is solution (µ α((1 x + W (x + σ2 2 W (x W (x = α((1 x + + x µ. (14 On [1 ɛ, 1, the second membe of Equation (14 is negative due to Equation (13. On [1, b 1ɛ, it is equal to x µ which is negative because b 1ɛ < µ. Assume by a way of contadiction that thee is some x [1 ɛ, b 1ɛ such that W (x <, then it would exist x [1 ɛ, b 1ɛ such that W ( x <, W ( x = and W ( x > which is in contadiction with Equation (14. Hence, W is a positive function on [1 ɛ, b 1ɛ with W (1 ɛ = which implies w b 1ɛ (1 ɛ 1. Lemma 8 When µ < α(1, w b is a convex-concave function. Poof: Accoding to Coollay 3, thee exists b 1, 1 + µ [ such that w b ( = and by Lemma 1, w b > on (, b. Using Equation (11, we thus have w b ( > implying that w b is stictly convex on a ight neighbohood of. Because b > 1, Lemma 2 implies w b (x < on [1, b [. If thee is moe than one change in the concavity of w b, it will exist x [, 1[ such that w b ( x >, w b ( x = and w b ( x yielding the standad contadiction. 17

18 Poposition 4 If µ < α(1 and w b ( 1, w b is the shaeholdes value function (4 Poof: It is staightfowad to see that the function w b satisfies Poposition 2 when w b ( 1. Now, we will conside the case w b ( < 1. Lemma 9 If w b ( < 1, it exists a, 1[ such that w ba (a = a and w b a (a = 1. Poof: Let φ(x = w b x (x. By assumption, we have φ( < 1 and by Lemma 7, φ(1 > 1. By continuity of φ, thee exists a, 1[ such that w b a (a = 1. By definition, the function w ba satisfies w ba (a = a. Lemma 1 w ba is a convex-concave function on [a, b a. Poof: Fist, we show that w ba is inceasing on [a, b a. Because w b a (a = 1, we can define x = min{x > a, w b a (x }. If x b a, we will have w b a ( x =, w ba ( x > and w b a ( x yielding the standad contadiction. Accoding to Lemma 1, we have w b a (x < ove [1, b a [ because b a > 1. Poceeding analogously as in the poof of Lemma 8, we pove that w ba is a convex-concave function because it cannot change of concavity twice. Lemma 11 We have w ba > 1 on (a, b a with b a < µ. Poof: Accoding to Lemma 1, w ba is convex-concave with w b a (a = 1 and w b a (b a = 1, theefoe x a, b a [, w b a (x > 1. As a consequence, w ba (x > x on a, b a and in paticula w ba (1 > 1. Remembeing that w µ (1 < 1 and using Coollay 2, we have b a < µ. Poposition 5 If w b ( < 1, the function x w(x = w ba (x x b a + µ fo x a fo a x b a fo x b a is the shaeholdes value function (4. Poof: it is staightfowad to check that w satisfies Poposition 2. 18

19 5 The Investment Model In this section, we enich the model to allow vaiable investment in the poductive assets. We will assume a deceasing-etuns-to-scale technology by intoducing an inceasing concave function β with lim x β(x = β that impacts the dynamic of the book value of equity as follows: { dxt = β(k t (µdt + σdw t α((k t X t + dt γ di t dz t dk t = di t = di t + dit (15 whee I t + (esp. It is the cumulative capital invested (esp. disinvested in the poductive assets up to time t, γ > is an exogenous popotional cost of investment. Assumption (2 thus foces liquidation when the level of outstanding debt eaches the sum of the liquidation value of the poductive assets and the liquid assets, (1 γk t + M t. The goal of the management is to maximize ove the admissible stategies π = (Z t, I t t the isk-neutal shaeholdes value ( τ V (x, k = sup E x,k e t dz t (16 π whee τ = inf{t, L t (1 γk t + M t } = inf{t, X t γk t }. By definition, we have k, V (γk, k =. ( Dynamic pogamming and fee bounday poblem In ode to deive a classical analytic chaacteization of V in tems of a fee bounday poblem, we ely on the dynamic pogamming pinciple as follows Dynamic Pogamming Pinciple: Fo any (x, k S whee S = {(x, k R 2 +, x γk}, we have ( θ V (x, k = sup E e t dz t + e θ V (X θ, K θ (18 π whee θ is any stopping time. Take the suboptimal contol π which consists in investing only at time t = a cetain amount h. Then, accoding to the dynamic pogamming pinciple, we have with θ = +, So, Dividing by h, we have V (x, k V (X +, K + = V (x γh, k + h. V (x, k V (x γh, k + V (x γh, k V (x γh, k + h. γ V (x, k V (x γh, k γh V (x γh, k + h V (x γh, k h If V wee smooth enough, we can let h tend to to obtain. γ V x V k. 19

20 Likewise, we can pove that and γ V x + V k V x 1 L k V whee L k is the second ode diffeential opeato L k w = ( β(kµ α((k x + w x + σ2 β(k 2 2 w w. (19 2 x2 The aim of this section is to chaacteize via the dynamic pogamming pinciple the shaeholdes value as the unique continuous viscosity solution to the fee bounday poblem in ode to use a numeical pocedue to descibe the optimal policies. F (x, k, V, DV, D 2 V = (2 whee ( F (x, k, w, Dw, D 2 w = min L k w, w w 1, γ x x w k, γ w x + w. k We will fist establish the continuity of the shaeholdes value function which elies on some peliminay well-known esults about hitting times we pove below fo sake of completeness. Lemma 12 Let a < b and (x n n a sequence of eal numbes such that lim n + x n = b and min n x n > a. Let (X n t n the solution of the stochastic diffeential equation { dx n t = µ n (X n t dt + σ n dw t X n = x n whee µ n and σ n satisfy the standad global Lipschitz and linea gowth conditions. Moeove, (σ n n ae stictly positive eal numbes conveging to σ > and (µ n n is a sequence of bounded functions conveging unifomly to µ. Let us define T n = inf{t, Xt n = a} and θ n = inf{t, Xt n = b}. We have lim P(θ n < T n = 1. n + Poof: (a, b as Let us define the functions U n, F n : I R, on some bounded inteval I containing U n (y = y µ n (z + x n dz, F n (y = y e 2Un(z σn 2 dz. 2

21 Because (µ n n conveges unifomly to µ, we note that (F n, U n n conveges unifomly to (F, U whee and F (y = U(y = y y e 2U(z σ 2 dz µ(z + bdz. Let Yt n = Xt n x n, Mt n = F n (Yt n and τ n = inf{t, Yt n / a n, b n [} with a n = a x n and b n = b x n. We fist show that τ n is integable. Because F n is the scale function of the pocess Yt n, Mt n is a local matingale with quadatic vaiation Because < M n > t = t σne 2 4Un(Y σn 2 n s ds. ( E(< M n > t τn σnt 2 exp 4 min σ U n(y < + n 2 y [a n,b n the pocesses (M n t τ n t and ((M n t τ n 2 < M n > t τn t ae both matingales. By Optional sampling theoem E[(M n t τ n 2 < M n > t τn = which implies and [ t E ( σn 2 exp 4 σn 2 1 [,τn(sσne 2 4Un(Y σn 2 thus thee is a constant K n > such that n s ds = E[Fn(Y 2 t τ n n max U n(y E[t τ n y [a n,b n t, E[t τ n K n. max y [a n,b n F 2 n(y We conclude by dominated convegence that τ n is integable. The matingale popety implies E[F n (Yt τ n n = which yields by dominated convegence because This is equivalent to E[F n (Y n τ n =, t, F n (Yt τ n n max F n(y. y [a n,b n F n (a n (1 p(a n, b n + F n (b n p(a n, b n = 21

22 with p(a n, b n = P(Y n τ n = b n. Hence, Moeove, p(a n, b n = F n (a n F n (b n F n (a n. P(θ n < T n = P(X n τ n = b Using the unifom convegence of F n, we have = P(Y n τ n = b x n = p(a n, b n. lim P(θ n < T n = lim p(a n, b n n + n + F (a b = F ( F (a b =1. Lemma 13 Let a < b and (x n n a sequence of eal numbes such that lim n + x n = b and min n x n > a. Let (X n t n the solution to { dx n t = µ n (X n t dt + σ n dw t X n = x n with the same assumptions as in Lemma 12. Thee exist constants A n and B n such that ( exp b x n ( ( A σ 2 n 2 n + 2σn 2 A n E[e θn exp b x n ( B σ 2 n 2 n + 2σn 2 B n. (21 Poof: Because µ n ae bounded functions, thee ae two constants A n and B n such that A n µ n (x B n fo all a < x < b. We define X t n = x n + A n t + σ n W t. By compaison, we have X t n Xt n and θ n θ n, with θ n = inf{t, X t n = b}. But the Laplace tansfom of θ n is explicit and given by ( E[e θ n = exp b x n ( A σ 2 n 2 n + 2σn 2 A n which gives the left inequality of (21. The poof is simila fo the ight inequality intoducing X n t = x n + B n t + σ n W t. Poposition 6 The shaeholdes value function is jointly continuous. 22

23 Poof: Let (x, k S and let us conside (x n, k n a sequence in S conveging to (x, k. Theefoe, {(x n γ k k n, k, (x γ k k n, k n } S 2 fo n lage enough. We conside the following two stategies that ae admissible fo n lage enough: Stategy πn: 1 stat fom (x, k, invest if k n k > (o disinvest if k n k < and do nothing up to the minimum between the liquidation time and the hitting time of (x n, k n. Denote (X π1 n t, K π1 n t t the contol pocess associated to stategy πn. 1 Stategy πn: 2 stat fom (x n, k n, invest if k n k < (o disinvest if k n k > and do nothing up to the minimum between the liquidation time and the hitting time of (x, k. Denote (X π2 n t, K π2 n t t the contol pocess associated to stategy πn. 2 To fix the idea, assume k n > k. The stategy π 1 makes the pocess (X, K jump fom (x, k to (x γ(k n k, k n. Define and θn 1 = inf{t, (X π1 n t, K π1 n t = (x n, k n }, θ 2 n = inf{t, (X π2 n t T 1 n = inf{t, X π1 n,x t T 2 n = inf{t, X π2 n,x n t, K π2 n t = (x, k}, γk π1 n,k t } γk π2 n,k n t }. Dynamic pogamming pinciple and V (X T 1 n, K T 1 n = on Tn 1 θn 1 yield [ θ 1 n T V n 1 (x, k E e t dz π1 n t + e (θ1 n T n 1 1 {θ 1 n <Tn}V 1 (X θ 1 n, K θ 1 n [ E e θ1 n 1{θ 1 n <Tn 1} V (x n, k n ( E ( ( e n θ1 E e θn 1 1{θ 1 n Tn} 1 V (x n, k n ( E ( ( e n θ1 P θ 1 n Tn 1 V (x n, k n. (22 23

24 On the othe hand, using V (X T 2 n, K T 2 n = on Tn 2 θn 2 [ θ 2 n T V n 2 (x n, k n E e t dz π2 n t + e (θ2 n T n 2 1 {θ 2 n <Tn 2} V (X θ 2 n, K θ 2 n [ E e θ2 n 1{θ 2 n <Tn}V 2 (x, k ( E ( ( e n θ2 E e θn 2 1{θ 2 n T n 2} V (x, k ( E ( ( e n θ2 P θ 2 n Tn 2 V (x, k. (23 The convegence of (x n, k n implies fom which we deduce using Lemma 12 that and lim (x n γ k k n, k = (x, k n + lim n + P(θ1 n Tn 1 = (24 lim n + P(θ2 n T 2 n =. (25 Let µ n (X n t = β(k n µ α((k n X n t + and σ n = β(k n σ. The function µ n is bounded by thus, accoding to Lemma 13 exp ( κn ( A σ 2 n 2 n + 2σn 2 A n with κ n = x x n + γ k n k. Letting n tend to + and using we obtain Finally, we have fom (22 and (23, A n = β(k n µ α(k n B n = β(k n µ lim n + A n lim n + B n lim n + σ n E[e θ1n exp ( κn = β(kµ α(k = β(kµ = β(kσ ( B σ 2 n 2 n + 2σn 2 B n lim n + E(eθ1 n = lim n + E(eθ2 n = 1. (26 V (x, k lim sup V (x n, k n lim inf V (x n, k n V (x, k, n n which poves the continuity of V. We ae now in a position to chaacteize the shaeholdes value in tems of viscosity solution of the fee bounday poblem (2. 24

25 Theoem 2 The shaeholdes value V is the unique continuous viscosity solution to (2 on S with linea gowth. Poof: The poof is postponed to the Appendix The main inteest of Theoem 2 is to guaantee that the standad numeical pocedue to solve HJB fee bounday poblems poposed in [11 will convege to the shaeholdes value function. We obtain the following desciption of the contol egions (Figue 3. Ou numeical analysis demonstates that unlike [4, thee exists an optimal level of poductive assets (top of the yellow egion and thus an objective measue of manageial oveinvestment in ou context. This is clealy due to the deceasing-etuns-to-scale assumption. constained fims with low cash eseves, that is when equity capital is close to poductive asset size, and low equity capital will athe disinvest to offset cash-flows shotfalls. constained fims with low cash eseves and high equity capital will fist daw on the cedit line to offset cash-flows shotfalls. the cedit line is neve used to invest. 25

26 Figue 3: Optimal contol with µ =.25, =.2, σ =.3, λ =.8, β max = 2, β = 1 and an investment cost γ = 5e 4. While the numeical esults give the above insights about the optimal policies, we have not been able to pove igoously the shape of the optimal contol egions. Nonetheless, making the stong assumption that thee is no tansaction cost γ = allows us to fully descibe the contol egions and gives us easons to believe in Figue 3. This is the object of ou last subsection. 5.2 Absence of Investment cost Using a veification pocedue analogous to section 3, we chaacteize the value function and the optimal policies in tems of a fee bounday poblem. The following poposition poved in the Appendix summaizes ou findings. Poposition 7 When thee is no cost of investment/disinvestment, γ =, the following holds: If µβ ( then it is optimal to liquidate the fim thus v (x = x. If α ( > µβ ( > and σ 2 β ( µ, the shaeholdes value is an inceasing and (1δ concave function of the book value of equity. Any excess of cash above the theshold 26

27 b = inf{x >, (v (x = 1} is paid out to shaeholdes (see Figue 4. The optimal size of the poductive asset is chaacteized by a deteministic function of equity capital (see Figue 5 given by [ x a, k(x = β 1 µx σ 2 (1 δ x a, k(x = x. whee a is the unique nonzeo solution of the equation σ 2 (1 δβ(a = µa. (27 with δ = 2σ2 µ 2 + 2σ 2 (28 If α ( > µβ ( > and σ 2 β ( < µ, the shaeholdes value is an inceasing and (1δ concave function of the book value of equity (see Figue 6. Any excess of cash above the theshold b = inf{x >, (v (x = 1} is paid out to shaeholdes. Moeove, all the cash eseves ae invested in the poductive assets. The above poposition has two inteesting implications. When the volatility of eanings is low σ 2 β ( < µ, it is optimal to invest all the (1δ cash eseves in the poductive assets and use it as a complementay substitute fo cash which is bette off than using a costly cedit line. Nonetheless, when the volatility of eanings is high, poductive assets ae not a pefect substitute of cash because it implies a high isk of bankupcy when the book value of equity is low. Figue 4 plots the shaeholdes value functions with α ( > µβ ( and σ 2 β ( µ (1δ fo diffeent values of β ( using : a linea function fo α, α(x = λx. an exponential function fo β, β(x = β max ( 1 e β ( βmax. x 27

28 Figue 4: Compaing shaeholdes value functions with µ =.25, =.2, σ =.6, λ =.8, β max = 5, fo diffeent values of β ( (case σ 2 β ( µ (1δ. Figue 5 plots the optimal level of poductive assets fo diffeent values of σ. It shows that, fo a given level of the book value of equity, the investment level in poductive assets is a deceasing function of the volatility. 28

29 Figue 5: Compaing optimal level of poductive assets with µ =.25, =.2, λ =.8, β max = 5, β ( = 2 fo diffeent values of σ. Figue 6 plots the shaeholdes value functions when α ( > µβ ( and σ 2 β ( µ (1δ fo diffeent values of β ( using : a linea function fo α, α(x = λx. an exponential function fo β, β(x = β max ( 1 e β ( βmax. x 29

30 Figue 6: Compaing shaeholdes value functions with µ =.25, =.2, σ =.6, λ =.8, β max = 5, fo diffeent values of β ( (case σ 2 β ( µ (1δ. 6 Appendix 6.1 Poof of Theoem 2 Supesolution popety. Let ( x, k S and ϕ C 2 (R 2 + s.t. ( x, k is a minimum of V ϕ in a neighbohood B ( x, k of ( x, k with small enough to ensue B S and V ( x, k = ϕ( x, k. Fist, let us conside the admissible contol ˆπ = (Ẑ, Î whee the shaeholdes decide to neve invest o disinvest, while the dividend policy is defined by Ẑt = η fo t, with η. Define the exit time τ = inf{t, (Xt x, K k t / B ( x, k}. We notice that τ < τ fo small enough. Fom the dynamic pogamming pinciple, we have [ τ h ϕ( x, k = V ( x, k E e t dẑt + e (τ h V (Xτ x h, K k τ h [ τ h E e t dẑt + e (τ h ϕ(xτ x h, K k τ h. (29 3

31 Applying Itô s fomula to the pocess e t ϕ(xt x, K k t between and τ h, and taking the expectation, we obtain [ [ τ h E e (τ h ϕ(xτ x h, K k τ h = ϕ( x, k + E e t Lϕ(Xt x, K k t dt [ + E e t [ϕ(xt x, K k t ϕ(x x t, K k t. (3 t τ h Combining elations (29 and (3, we have [ τ h [ τ h E e t (Lϕ(Xt x, K k t dt E e t dẑt [ E e t [ϕ(xt x, K k t ϕ(x x t, K k t t τ h. (31 Take fist η =. We then obseve that X is continuous on [, τ h and only the fist tem of the elation (31 is non zeo. By dividing the above inequality by h with h, we conclude that Lϕ( x, k. Take now η > in (31. We see that Ẑ jumps only at t = with size η, so that [ τ h E e t (Lϕ(Xt x, K k t dt η (ϕ( x η, k ϕ( x, k. By sending h, and then dividing by η and letting η, we obtain ϕ x ( x, k 1. Second, let us conside the admissible contol π = ( Z, Ī whee the shaeholdes decide to neve payout dividends, while the investment/disinvestment policy is defined by Īt = η R fo t, with < η. Define again the exit time τ = inf{t, (Xt x, K k t / B ( x, k}. Poceeding analogously as in the fist pat and obseving that Ī jumps only at t =, thus [ τ h E e t (Lϕ(Xt x, K k t dt (ϕ( x γ η, k + η ϕ( x, k. Assuming fist η >, by sending h, and then dividing by η and letting η, we obtain γ ϕ x ( x, k ϕ k ( x, k. When η <, we get in the same manne γ ϕ x ( x, k + ϕ k ( x, k. This poves the equied supesolution popety. 31

32 Subsolution Popety: We pove the subsolution popety by contadiction. Suppose that the claim is not tue. Then, thee exists ( x, k S and a neighbouhood B ( x, k of x, k, included in S fo small enough, a C 2 function ϕ with (ϕ V ( x, k = and ϕ V on B ( x, k, and η >, s.t. fo all (x, k B ( x, k we have Lϕ(x, k > η, (32 ϕ (x, k 1 x > η, (33 (γ ϕ x ϕ (x, k k > η. (34 (γ ϕ x + ϕ (x, k k > η. (35 Fo any admissible contol π, conside the exit time τ = inf{t, (X x t, K k t / B ( x, k} and notice again that τ < τ. Applying Itô s fomula to the pocess e t ϕ(x x t, K k t between and τ, we have E[e τ ϕ(x τ, K τ = ϕ( x, k E + E + E E + E [ τ [ τ [ τ [ [ τ e u Lϕdu e u (γ ϕ x + ϕ k dic,+ u e u (γ ϕ x ϕ u ϕ e x dzc u k dic, u <s<τ e s [ϕ(x s, K s ϕ(x s, K s (36 (37 (38 (39 (4 Using elations (32,(33,(34,(35, we obtain 32

33 V ( x, k = ϕ( x, [ k τ ηe e u du + E[e τ ϕ(x τ, K τ (41 + ηe + ηe [ τ [ τ + (1 + ηe E [ e u di c,+ u e u di c, u [ τ e u dz c u <s<τ e s [ϕ(x s, K s ϕ(x s, K s Note that X s = Z s γ( I + s + I s, K s = I + s I s and by the Mean Value Theoem, thee is some θ, 1[ such that, ϕ(x s, K s ϕ(x s, K s = ϕ x (X s + θ X s, K s + θ K s X s + ϕ k (X s + θ X s, K s + θ K s K s = ϕ x (X s + θ X s, K s + θ K s ( Z s γ( I s + + Is + ϕ k (X s + θ X s, K s + θ K s ( I s + Is = ϕ x (X s + θ X s, K s + θ K s Z s ( + γ ϕ x (X s + θ X s, K s + θ K s + ϕ k (X s + θ X s, K s + θ K s ( + γ ϕ x (X s θ X s, K s + θ K s + ϕ k (X s + θ X s, K s + θ K s (42 (43 (44 (45 I + s I s Because (X s + θ X s, K s + θ K s B ( x, k, we use the elations (33,(34,(35 again (ϕ(x s, K s ϕ(x s, K s (1 + η Z s + η I + s + η I s Theefoe, [ τ V ( x, k E[e τ ϕ(x τ, K τ + E e u dz u ( [ [ τ [ τ τ + η E e u du + E e u di u + + E 33 [ τ e u diu + E e u dz u

34 Notice that while (X τ, K τ B ( x, k, (X τ, K τ is eithe on the bounday B ( x, k o out of B ( x, k. Howeve, thee is some andom vaiable α valued in [, 1 such that: (X (α, K (α = (X τ, K τ + α( X τ, K τ = (X τ, K τ + α( Z τ γ I + τ γ I τ, I + τ I τ B ( x, k. Poceeding analogously as above, we show that Obseve that ϕ(x (α, K (α ϕ(x τ, K τ α[(1 + η Z τ + η I + τ + η I τ. (X (α, K (α = (X τ, K τ + (1 α( Z τ + γ I + τ + γ I τ, I + τ + I τ. Stating fom (X (α, K (α, the stategy that consists in investing (1α I + τ o disinvesting (1 α I τ depending on the sign of K (α K τ and payout (1 α Z τ as dividends leads to (X τ, K τ and theefoe, V (X (α, K (α V (X τ, K τ (1 α Z τ. Using ϕ(x (α, K (α V (X (α, K (α, we deduce ϕ(x τ, K τ V (X τ, K τ (1 + αη Z τ + αη( I + τ + I τ. Hence, [ V ( x, k ( τ η E e u du + E [ τ e u di + u + E[e τ α( Z τ + γ I τ + + γ Iτ + E[e τ V (X τ, K τ + E [ τ e u dz u + E [ τ [ τ e u diu + E e u dz u (46 We now claim thee is c > such that fo any admissible stategy [ τ τ τ c E e u du + e u di u + + e u diu + + E [ e τ α( Z τ + γ I + τ + γ I τ Let us conside the C 2 function, φ(x, k = c [1 (x x2 with, 2 { < c min 2, 2γ, 1, 2 σn 2 β, 2 2d max } τ e u dz u (47 whee { } β(kµ α((k x + d max = sup, (x, k B ( x, k >, 34