An Optimal Dividend Problem with Capital Injections over a Finite Horizon
aa r X i v : . [ q -f i n . M F ] M a y AN OPTIMAL DIVIDEND PROBLEM WITH CAPITAL INJECTIONSOVER A FINITE HORIZON
GIORGIO FERRARI, PATRICK SCHUHMANN
Abstract.
In this paper we propose and solve an optimal dividend problem with capi-tal injections over a finite time horizon. The surplus dynamics obeys a linearly controlleddrifted Brownian motion that is reflected at the origin, dividends give rise to time-dependentinstantaneous marginal profits, whereas capital injections are subject to time-dependent in-stantaneous marginal costs. The aim is to maximize the sum of a liquidation value at terminaltime and of the total expected profits from dividends, net of the total expected costs for cap-ital injections. Inspired by the study of El Karoui and Karatzas [14] on reflected followerproblems, we relate the optimal dividend problem with capital injections to an optimal stop-ping problem for a drifted Brownian motion that is absorbed at the origin. We show thatwhenever the optimal stopping rule is triggered by a time-dependent boundary, the valuefunction of the optimal stopping problem gives the derivative of the value function of theoptimal dividend problem. Moreover, the optimal dividend strategy is also triggered by themoving boundary of the associated stopping problem. The properties of this boundary arethen investigated in a case study in which instantaneous marginal profits and costs fromdividends and capital injections are constants discounted at a constant rate.
Keywords : optimal dividend problem; capital injections; singular stochastic control; op-timal stopping; free boundary.
MSC2010 subject classification : 93E20, 60G40, 62P05, 91G10, 60J651.
Introduction
The literature on optimal dividend problems started in 1957 with the work of de Finetti[11], where, for the first time, it was proposed to measure an insurance portfolio by thediscounted value of its future dividends’ payments. Since then, the literature in Mathematicsand Actuarial Mathematics experienced many scientific contributions on the optimal dividendproblem, which has been typically modeled as a stochastic control problem subject to differentspecifications of the control processes and of the surplus dynamics (see, among many others,the early work by Jeanblanc-Piqu´e and Shiryaev [17], the more recent works by Akyildirim etal. [1], De Angelis and Ekstr¨om [10] and Jiang and Pistorius [19], the review by Avanzi [2],and the book by Schmidli [32]).Starting from the observation that ruin occurs almost surely when the fund’s managerpays dividends by following the optimal strategy of de Finetti’s problem, Dickson and Watersproposed in [12] several modifications to the original formulation of the optimal dividendproblem. In particular, in [12] a model has been suggested in which the shareholders areobliged to inject capital in order to avoid bankruptcy. This is the so-called optimal dividendproblem with capital injections .The literature on the optimal dividend problem with capital injections is not as rich asthat on the classical de Finetti’s problem. In Kulenko and Schmidli [24], the authors studyan optimal dividend problem with capital injections in which the surplus process is reflectedat the origin, and on (0 , ∞ ) evolves according to a classical Cram´er-Lundberg risk model. InSchmidli [33], an optimal dividend problem with capital injections and taxes in a diffusive Date : May 22, 2019. setting is formulated and solved. In Lokka and Zervos [26] the shareholders can choose thecapital injections’ policy and, in absence of any interventions, the surplus process follows aBrownian motion with drift. Other works in which the surplus process evolves as a generalone-dimensional diffusion are the ones by Ferrari [16], Zhu and Yang [35], and Shreve et al.[34]. Optimal dividends and capital injections in a jump-diffusion setting are determined byAvanzi et al. in [3]. In all those papers the optimal dividend problem with capital injectionsis formulated as a singular stochastic control problem over an infinite time horizon. Giventhe stationarity of the setting, in those works it is shown that (apart a possible initial lumpsum payment) it is usually optimal to pay just enough dividends in order to keep the surplusprocess in the interval [0 , b ], for some constant b > T ∈ (0 , ∞ ). This horizon might beseen as a pre-specified future date at which the fund is liquidated.As is common in the literature (see [1], [10], and [26], among many others), in absence ofany interventions, the surplus process evolves as a Brownian motion with drift µ and volatil-ity σ . This dynamics for the fund’s value can be obtained as a suitable (weak) limit of aclassical dynamics `a la Cram´er-Lundberg (see Appendix D.3 in [32] for details). We also as-sume that, after time-dependent transaction costs/taxes have been paid, shareholders receivea time-dependent instantaneous net proportion of leakages f from the surplus. Moreover,shareholders are forced to inject capital whenever the surplus attempts to become negative,and injecting capital they incur a time-dependent marginal administration cost m . Finally,a surplus-dependent liquidation reward g is obtained at liquidation time T . Notice, that,under suitable requirements on f , m and g (see Remark 2.4), injecting capital at the originturns out to be optimal within the class of dividends/capital injections that keep the surplusnonnegative for any time with probability one (see also [24], [31], and [33]).Within this setting, the fund’s manager takes the point of view of the shareholders andthus aims at solving(1.1) V ( t, x ) := sup D E (cid:20) Z T − t f ( t + s ) dD s − Z T − t m ( t + s ) dI Ds + g ( T, X DT − t ( x )) (cid:21) , for any initial time t ∈ [0 , T ] and any initial value of the fund x ∈ R + . In (1.1) the fund’svalue evolves as X Ds ( x ) = x + µs + σW s − D s + I Ds , s ≥ , and the optimization is performed over a suitable class of nondecreasing processes D . Infact, the quantity D s represents the cumulative amount of dividends paid to shareholders upto time s , whereas I Ds is the cumulative amount of capital injected by the shareholders upto time s . We take I D as the minimal nondecreasing process which ensures that X D staysnonnegative, and it is flat off { t ≥ X Dt = 0 } .If we attempt to tackle problem (1.1) via a dynamic programming approach, we will findthat the dynamic programming equation for V takes the form of a parabolic partial differentialequation (PDE) with gradient constraint (i.e. a variational inequality), and with a Neumannboundary condition at x = 0 (the latter is due to the fact that the state process X is reflectedat the origin through the capital injections process). Proving that a solution to this PDEproblem has enough regularity to characterize an optimal control is far from being trivial.Starting from the observation that the optimal dividend problem with capital injections(1.1) is actually a reflected follower problem (see, e.g., Baldursson [4], El Karoui and Karatzas[13], and Karatzas and Shreve [21]) with costly reflection at the origin, and inspired by theresults of El Karoui and Karatzas in [14], here we solve (1.1) without relying on PDE methods,but relating (1.1) to a (still complex but) more tractable optimization problem; i.e., to anoptimal stopping problem with absorption at the origin and with value function u (cf. (3.2) PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 3 below). In this auxiliary optimal stopping problem, the functions f , m , and g x give thepayoff of immediate stopping, the payoff from absorption at the origin, and the final reward,respectively.Then, if the optimal stopping time for that problem is given in terms of a continuous andstrictly positive time-dependent boundary b ( · ) (cf. the structural Assumption 3.1 below),one has that V x = u , and the optimal dividends’ payments strategy D ⋆ is triggered by b (see Theorem 3.2 below). In fact, if the optimization starts at time t ∈ [0 , T ], the couple( D ⋆ , I D ⋆ ) keeps at any instant in time s ∈ [0 , T − t ] the optimally controlled fund’s value X D ⋆ s nonnegative and below the time-dependent critical level b ( s + t ).This result is obtained via an almost exclusively probabilistic study in which we suitablyintegrate in the space variable two different representations of the value function u of theauxiliary optimal stopping problem. It is worth noticing that although we borrow argumentsfrom the study in [14] on the connection between reflected follower problems and questionsof optimal stopping (see also [21]), differently to [14], in our performance criterion (1.1) wealso have a cost of reflection and this requires a careful and not immediate adaptation of theideas and results of [14].We then show that the structural Assumption 3.1, needed to establish the relation between(1.1) and the optimal stopping problem, does indeed hold in a canonical formulation of theoptimal dividend problem with capital injections in which marginal benefits and costs areconstants discounted at a constant rate, and the liquidation value at time T is proportional tothe terminal value of the fund. In particular, we show that the optimal dividend strategy isgiven in terms of an optimal boundary b that is decreasing, continuous, bounded, and null atterminal time. To the best of our knowledge, also this result appears here for the first time.The rest of the paper is organized as follows. In Section 2 we set up the problem, andin Section 3 we state the connection between (1.1) and the optimal stopping problem withabsorption. Its proof is then performed in Section 4. In Section 5 we consider the case studywith (discounted) constant marginal benefits and costs, whereas in the Appendices we collectthe proofs of some results needed in the paper.2. Problem Formulation
In this section we introduce the optimal dividend problem that is the object of our study.Let (Ω , F , P ) be a complete probability space with a filtration F := ( F s ) s ≥ which satisfiesthe usual conditions. We assume that the fund’s value is described by the one-dimensionalprocess(2.1) X Ds ( x ) = x + µs + σW s − D s + I Ds , s ≥ , where x ≥ µ ∈ R , σ >
0, and W is an F -standard Brownianmotion. For any s ≥ D s represents the cumulative amount of dividends paid to shareholdersup to time s , whereas I Ds is the cumulative amount of capital injected by the shareholders upto time s in order to avoid bankruptcy of the fund.Define the (nonempty) set A = (cid:26) ν : Ω × R + → R + , F − adapted s.t. s ν s ( ω ) is a.s.nondecreasing and left-continuous, and ν = 0 a.s. (cid:27) . For fixed x ≥
0, we assume that the fund’s manager can pick a dividends’ distribution strategyamong the processes D ∈ A and such that a.s.(2.2) D s + − D s ≤ X Ds ( x ) for all s ≥ FERRARI, SCHUHMANN that is, bankruptcy cannot be obtained with a single lump sum dividend’s payment. For anysuch dividend policy D , the capital injections process I D is given as the minimal cumulativeamount of capital needed to ensure that X D ( x ) stays nonnegative, and which is flat off { t ≥ X Dt ( x ) = 0 } . In particular, for x ≥
0, we take the couple ( X D ( x ) , I D ) as the uniquesolution to the (discontinuous) Skorokhod reflection problem (see, e.g., Chaleyat-Maurel etal. [8] and Ma [27]):(2.3) Find ( X D ( x ) , I D ) s.t. I D ∈ A , X Ds ( x ) = x + µs + σW s − D s + I Ds , s ≥ ,X Ds ( x ) ≥ s ≥ , Z ∞ X Ds ( x ) d ( I Ds ) c = 0 a.s. , ∆ I Ds := I Ds + − I Ds = 2 X Ds + ( x ) ∀ s ∈ { s ≥ I Ds > } . Here, ( I D ) c denotes the continuous part of I D . Notice that, given (2.2), the process(2.4) I Dt := 0 ∨ sup ≤ s ≤ t ( D s − ( x + µs + σW s )) , t ≥ , I D = 0 , uniquely solves (2.3) and t I Dt is continuous (see, e.g., Propositions 2 and 3 in [8], orTheorem 3.1 and Corollary 3.2 in [21]). As a consequence, the last condition in (2.3) is notbinding, since ∆ I Dt = 0 a.s. for all t ≥ T ∈ (0 , ∞ ) representing, e.g., a finite liquidation time, the fund’smanager takes the point of view of the shareholders, and is faced with the problem of choosinga dividends’ distribution strategy D maximizing the performance criterion(2.5) J ( D ; t, x ) = E (cid:20)Z T − t f ( t + s ) dD s − Z T − t m ( t + s ) dI Ds + g ( T, X DT − t ( x )) (cid:21) , for ( t, x ) ∈ [0 , T ] × R + given and fixed. That is, the fund’s manager aims at solving(2.6) V ( t, x ) := sup D ∈D ( t,x ) J ( D ; t, x ) , ( t, x ) ∈ [0 , T ] × R + . Here, for any ( t, x ) ∈ [0 , T ] × R + , D ( t, x ) denotes the class of dividend payments belonging to A and satisfying (2.2), when the surplus process X D starts from level x and the optimizationruns up to time T − t . In the following, any D ∈ D ( t, x ) will be called admissible for( t, x ) ∈ [0 , T ] × R + .In the reward functional (2.5) the term E [ R T − t f ( t + s ) dD s ] is the total expected cash-flow from dividends. The function f might be seen as a time-dependent instantaneous netproportion of leakages from the surplus received by the shareholders after time-dependenttransaction costs/taxes have been paid. The term E [ R T − t m ( t + s ) dI Ds ] gives the total ex-pected costs of capital injections, and m is a time-dependent marginal administration cost forcapital injections. Finally, E (cid:2) g ( T, X DT − t ( x )) (cid:3) is a liquidation value.The functions f , m , and g satisfy the following conditions. Assumption 2.1. f : [0 , T ] → R + , m : [0 , T ] → R + , g : [0 , T ] × R + → R + are continuous, f and m are continuously differentiable with respect to t , and g is continuously differentiablewith respect to x . Moreover, (i) g x ( T, x ) ≥ f ( T ) for any x ∈ (0 , ∞ ) , (ii) m ( t ) > f ( t ) for any t ∈ [0 , T ] . Remark 2.2.
Requirement ( i ) ensures that the marginal liquidation value is at least as highas the marginal profits from dividends. This will ensure that the value function of the optimalstopping problem considered below is not discontinuous at terminal time. PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 5
Condition ( ii ) means that the marginal costs for capital injections are bigger than themarginal profits from dividends. Notice that in the case in which m < f the value functionmight be infinite, as it shown in the next example. Take f ( s ) = η, m ( s ) = κ for all s ∈ [0 , T ],and η > κ . For arbitrary β > b D s := βs , and notice that b I Ds = sup ≤ u ≤ s ( − x − µu − σB u + βu ) ∨
0. Then b I Ds ≤ βs + Y s , with Y s := sup ≤ u ≤ s ( − x − µu − σB u ) ∨
0, and using that g ≥ b DV ( t, x ) ≥ βη ( T − t ) − βκ ( T − t ) − κ E [ Y T − t ]= β ( T − t )( η − κ ) − κ E [ Y T − t ] . However, the latter expression can be made arbitrarily large by increasing β if η > κ .On the other hand, by taking m ( t ) = f ( t ) = e − rt , is has been recently shown in Ferrari[16] for a problem with T = + ∞ (see Theorem 3.8 therein) that an optimal control may notexist, but only an ε -optimal control does exist.In order to avoid pathological situations as the ones described above, here we assumeAssumption 2.1-(ii). Remark 2.3.
Notice that our formulation is general enough to accommodate also a problemin which profits and costs are discounted at a deterministic time-dependent discount rate( r s ) s ≥ . Indeed, if we consider the optimal dividend problem with capital injections b V ( t, x ) := sup D ∈D ( t,x ) E (cid:20) Z T − t e − R t + st r α dα b f ( t + s ) dD s − Z T − t e − R t + st r α dα b m ( t + s ) dI Ds + e − R Tt r α dα b g ( T, X DT − t ( x )) (cid:21) , then, for any ( t, x ) ∈ [0 , T ] × R + we can set f ( t ) := e − R t r α dα b f ( t ) , m ( t ) := e − R t r α dα b m ( t ) , g ( t, x ) := e − R t r α dα b g ( t, x ) , and V ( t, x ) := e − R t r α dα b V ( t, x ) is of the form (2.6).In Section 5 we will consider a problem with constant marginal profits and costs discountedat a constant rate r > Remark 2.4.
Notice that in our model shareholders are forced to inject capital wheneverthe surplus process attempts to become negative; that is, the capital injection process is nota control variable of their, and shareholders do not choose when and how to invest in thecompany.Injecting capital at the origin, under the condition that bankruptcy is not allowed, can beshown to be optimal in the canonical formulation of the optimal dividend problem of Section5 in which marginal costs and profits are constants discounted at a constant interest rate.Indeed, in such a case, due to discounting, shareholders will inject capital as late as possiblein order to minimize the total costs of capital injections. See also Kulenko and Schmidli [24]and Schmidli [33] for a similar result in stationary problems. More in general, the policy“inject capital at the origin” is optimal when m is decreasing and min t ∈ [0 ,T ] m ( t ) > g x ( T, x )for all x ∈ R + . Under these conditions, shareholders postpone injection of capital, and injectonly as much capital as necessary since any additional capital injection cannot be compensatedby the reward at terminal time.The dynamic programming equation for V takes the form of a parabolic partial differentialequation (PDE) with gradient constraint, and with a Neumann boundary condition at x = 0(the latter is due to the fact that the state process X is reflected at the origin through the FERRARI, SCHUHMANN capital injections process). Indeed, it readsmax n ∂ t V + 12 σ ∂ xx V + µ∂ x V, f − ∂ x V o = 0 , on [0 , T ) × (0 , ∞ ) , with boundary conditions ∂ x V (0 , t ) = m ( t ) for all t ∈ [0 , T ], and V ( T, x ) = g ( T, x ) for any x ∈ (0 , ∞ ). Proving that such a PDE problem admits a solution that has enough regularityto characterize an optimal control is far from being trivial.In order to solve the optimal dividend problem (2.6) we then follow a different approach,and we relate (2.6) to an optimal stopping problem with absorbing condition at x = 0. Thisis obtained by borrowing arguments from the study of El Karoui and Karatzas in [14] on theconnection between reflected follower problems and questions of optimal stopping (see alsoBaldursson [4] and Karatzas and Shreve [21]). However, differently to [14], in our performancecriterion (2.5) we also have a cost of reflection which requires a careful and not immediateadaptation of the ideas and results of [14].In particular, introducing a problem of optimal stopping with absorption at the origin, weshow that a proper integration of the value function of the latter leads to the value function ofthe optimal control problem (2.6). This result is stated in the next section, and then provedin Section 4. 3. The Main Result
Let S ( x ) := inf { s ≥ x + µs + σW s = 0 } , x ≥
0, and for any s ≥
0, introduce theabsorbed drifted Brownian motion(3.1) A s ( x ) := ( x + µs + σW s , s < S ( x ) , ∆ , s ≥ S ( x ) , where ∆ is a cemetery state isolated from R + (i.e. ∆ < g x ( T, ∆) := 0, for ( t, x ) ∈ [0 , T ] × R + , consider the optimalstopping problem(3.2) u ( t, x ) := sup τ ∈ [0 ,T − t ] E h f ( t + τ ) { τ< ( T − t ) ∧ S ( x ) } + m ( t + S ( x )) { τ ≥ S ( x ) } + g x (cid:0) T, x + µ ( T − t ) + σW T − t (cid:1) { τ = T − t Assume that the continuation region of the stopping problem (3.2) is givenby (3.3) C := { ( t, x ) ∈ [0 , T ) × (0 , ∞ ) : u ( t, x ) > f ( t ) } = { ( t, x ) ∈ [0 , T ) × (0 , ∞ ) : x < b ( t ) } , and that its stopping region by S := { ( t, x ) ∈ [0 , T ) × (0 , ∞ ) : u ( t, x ) ≤ f ( t ) } ∪ (cid:0) { T } × (0 , ∞ ) (cid:1) = { ( t, x ) ∈ [0 , T ) × (0 , ∞ ) : x ≥ b ( t ) } ∪ (cid:0) { T } × (0 , ∞ ) (cid:1) , (3.4) PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 7 for a continuous function b : [0 , T ) → (0 , ∞ ) . We refer to the function b as to the optimalstopping boundary of problem (3.2) . Further, assume that the stopping time (3.5) τ ⋆ ( t, x ) := inf { s ∈ [0 , T − t ) : A s ( x ) ≥ b ( t + s ) } ∧ ( T − t ) (with the usual convention inf ∅ = + ∞ ) is optimal; that is, u ( t, x ) = E h f ( t + τ ⋆ ( t, x )) { τ ⋆ ( t,x ) < ( T − t ) ∧ S ( x ) } + m ( t + S ( x )) { τ ⋆ ( t,x ) ≥ S ( x ) } + g x ( T, x + µ ( T − t ) + σW T − t ) { τ ⋆ ( t,x )= T − t Let Assumption 3.1 hold. Then, the process D ⋆ defined through (3.7) providesthe optimal dividends’ distribution policy, and the value function V of (2.6) is such that (3.8) V ( t, x ) = V ( t, b ( t )) − Z b ( t ) x u ( t, y ) dy, ( t, x ) ∈ [0 , T ] × R + . Assume further that lim t ↑ T b ( t ) =: b ( T ) < ∞ . Then V ( t, b ( t )) = − µ Z T − t f ′ ( t + s ) s ds + Z T − t f ′ ( t + s ) b ( t + s ) ds + g ( T, b ( T )) + f ( T ) µ ( T − t ) + f ( t ) b ( t ) − f ( T ) b ( T ) . (3.9)Consistently with the result of El Karoui and Karatzas in [14] (see also Karatzas and Shreve[21]), we find that also in our problem with costly reflection at the origin the value of an optimalstopping problem (namely, problem (3.2)) gives the marginal value of the value function (2.6).The optimal stopping boundary b thus triggers the timing at which it is optimal to pay anadditional unit of dividends. Moreover, once the optimal stopping value function u and itscorresponding free boundary b are known, (3.8) and (3.9) provide a complete characterizationof the optimal dividend problem’s value function V . Notice that the condition b ( T ) < ∞ issatisfied in the case study of Section 5, where we actually prove that b ( T ) = 0. The proof ofTheorem 3.2 is quite lengthy and technical, and it is relegated to Section 4.4. On the Proof of Theorem 3.2 This section is entirely devoted to the proof of Theorem 3.2. This is done through aseries of intermediate results which are proved by employing mostly probabilistic arguments.Assumption 3.1 will be standing throughout this section. FERRARI, SCHUHMANN On a Representation of the Optimal Stopping Value Function. Here we derivean alternative representation for the value function of the optimal stopping problem (3.2),by borrowing ideas from El Karoui and Karatzas [14], Section 3. In the following we set g x ( T, ∆) = 0.The idea that we adopt here is to rewrite the optimal stopping problem (3.2) in terms ofthe function b of Assumption 3.1. To accomplish that, for given ( t, x ) ∈ [0 , T ] × R + , definethe payoff associated to the admissible stopping rule “never stop” as(4.1) G ( t, x ) := E (cid:2) m ( t + S ( x )) { S ( x ) ≤ T − t } + g x ( T, A T − t ( x )) (cid:3) , where we have used that g x ( T, A T − t ( x )) { T − t Theorem 4.1. The dual predictable projection Θ( t, x ) of e C ( t, x ) exists, is nondecreasing andit is given by Θ α ( t, x ) = Z α − f ′ ( t + θ ) { A θ ( x ) >b ( t + θ ) } dθ + h f ( T ∧ ( t + S ( x ))) − ˜ g ( T − t, S ( x ) , A T − t ( x ); t ) i { A T − t ( x ) >b ( T ) } { It holds that (i) (cid:2) f ( T ∧ ( t + S ( x ))) − ˜ g ( T − t, S ( x ) , A T − t ( x ); t ) (cid:3) { A T − t ( x ) >b ( T ) } = 0 a.s. (ii) { t ∈ [0 , T ) : f ′ ( t ) ≤ } ⊇ S ;Proof. (i) On the set { A T − t ( x ) > b ( T ) } we obtain by the definition of ˜ g (see (4.2)) that f ( T ∧ ( t + S ( x ))) − ˜ g ( T − t, S ( x ) , A T − t ( x ); t ) = f ( T ) − g x ( T, A T − t ( x )) . (4.9)Since Θ · ( t, x ) is nondecreasing, the last term in (4.9) has to be positive, thus implying f ( T ) − g x ( T, A T − t ( x )) ≥ { A T − t ( x ) > b ( T ) } . However, by Assumption 2.1-(i) one has f ( T ) ≤ g x ( T, x ) for all x ∈ (0 , ∞ ). Hence the claim follows.(ii) Since α Θ α ( t, x ) is a.s. nondecreasing, it follows from (i) above and (4.8) that f ′ ( t + θ ) { A θ ( x ) >b ( t + θ ) } ≤ θ ∈ [0 , T − t ]. But f ′ ( · ), A · ( x ) and b ( t + · ) arecontinuous up to ( T − t ) ∧ S ( x ), and therefore the latter actually holds a.s. for all θ ∈ [0 , T − t ].Hence, { t ∈ [0 , T ) : f ′ ( t ) ≤ } ⊇ S . (cid:3) Remark 4.3. As a byproduct of Corollary 4.2-(i) (see in particular (4.9)), Assumption 2.1-(i),and of the fact that A T − t ( x ) has a transition probability that is absolutely continuous withrespect to the Lebesgue measure on R + (cf. (A.5)), one has (cid:0) f ( T ) − g x ( T, y ) (cid:1) { y>b ( T ) } = 0for y ≥ u of problem (3.2). Theorem 4.4. For any ( t, x ) ∈ [0 , T ] × R + one has u ( t, x ) = E (cid:20) Z ( T − t ) ∧ S ( x )0 − f ′ ( t + θ ) { x + µθ + σW θ ≥ b ( t + θ ) } dθ + m ( t + S ( x )) { S ( x ) ≤ T − t } + g x ( T, A T − t ( x )) (cid:21) . (4.10) Proof. Since by Theorem 4.1 Θ( t, x ) is the dual predictable projection of e C ( t, x ), from (4.6)we can write for any ( t, x ) ∈ [0 , T ] × R + (4.11) v ( t, x ) = E h e C T − t ( t, x ) i = E [Θ T − t ( t, x )] . Due to (4.8) and Corollary 4.2-(i), (4.11) gives v ( t, x ) = E "Z ( T − t ) ∧ S ( x )0 − f ′ ( t + θ ) { x + µθ + σW θ ≥ b ( t + θ ) } dθ . (4.12)Here we have also used that the joint law of S ( x ) and of the drifted Brownian motion isabsolutely continuous with respect to the Lebesgue measure in R (cf. (A.2)) to replace { x + µθ + σW θ >b ( t + θ ) } with { x + µθ + σW θ ≥ b ( t + θ ) } inside the expectation in (4.8).However, since by definition v = u − G , we obtain from (4.12) and (4.1) the alternativerepresentation u ( t, x ) = v ( t, x ) + G ( t, x ) = E (cid:20) Z ( T − t ) ∧ S ( x )0 − f ′ ( t + θ ) { x + µθ + σW θ ≥ b ( t + θ ) } dθ + m ( t + S ( x )) { S ( x ) ≤ T − t } + g x ( T, A T − t ( x )) (cid:21) . (cid:3) Remark 4.5. Notice that representation (4.10) coincides with that one might obtain by anapplication of Itˆo’s formula if u were C , ([0 , T ) × (0 , ∞ )) ∩ C ([0 , T ] × R + ), and satisfies (asit is customary in optimal stopping problems) the free-boundary problem(4.13) ∂ t u + σ ∂ xx u + µ∂ x u = 0 , < x < b ( t ) , t ∈ [0 , T ) u = f, x ≥ b ( t ) , t ∈ [0 , T ) u ( T, x ) = g x ( T, x ) , x > u ( t, 0) = m ( t ) , t ∈ [0 , T ] . Indeed, in such a case an application of Dynkin’s formula gives E (cid:2) u ( t + ( T − t ) ∧ S ( x ) , Z ( T − t ) ∧ S ( x ) ( x )) (cid:3) = u ( t, x )+ E "Z ( T − t ) ∧ S ( x )0 f ′ ( t + θ ) { Z θ ( x ) ≥ b ( t + θ ) } dθ , where we have set Z s ( x ) := x + µs + σW s , s ≥ 0, to simplify exposition. Hence, using (4.13)we have from the latter u ( t, x ) = E (cid:20) m ( t + S ( x )) { S ( x ) ≤ T − t } + g x ( T, x + µ ( T − t ) + σW T − t ) { S ( x ) >T − t } − Z ( T − t ) ∧ S ( x )0 f ′ ( t + θ ) { Z θ ( x ) ≥ b ( t + θ ) } dθ (cid:21) = E (cid:20) m ( t + S ( x )) { S ( x ) ≤ T − t } + g x ( T, A T − t ( x )) { S ( x ) >T − t } − Z ( T − t ) ∧ S ( x )0 f ′ ( t + θ ) { Z θ ( x ) ≥ b ( t + θ ) } dθ (cid:21) = E (cid:20) m ( t + S ( x )) { S ( x ) ≤ T − t } + g x ( T, A T − t ( x )) − Z ( T − t ) ∧ S ( x )0 f ′ ( t + θ ) { Z θ ( x ) ≥ b ( t + θ ) } dθ (cid:21) , where in the last step we have used that g x ( T, A T − t ( x )) { S ( x ) >T − t } = g x ( T, A T − t ( x )) becauseof (3.1) and the fact that g x ( T, ∆) = 0. Remark 4.6. Notice that the representation (4.10) immediately gives an integral equationfor the optimal stopping boundary b . Indeed, since (4.10) holds for any ( t, x ) ∈ [0 , T ] × R + , PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 11 by taking x = b ( t ), t ≤ T , on both sides of (4.10), and by recalling that u ( t, b ( t )) = f ( t ), wefind that b solves f ( t ) = E (cid:20) Z ( T − t ) ∧ S ( b ( t ))0 − f ′ ( t + θ ) { b ( t )+ µθ + σW θ ≥ b ( t + θ ) } dθ + m ( t + S ( b ( t ))) { S ( b ( t )) ≤ T − t } + g x ( T, A T − t ( b ( t ))) (cid:21) . (4.14)By adapting arguments as those in Section 25 of Peskir and Shiryaev [28], based on thesuperharmonic characterization of u , one might then prove that b is the unique solution to(4.14) among a suitable class of continuous and positive functions.The next result follows from (4.10) by expressing the expected value as an integral withrespect to the probability densities of the involved processes and random variables. Its proofcan be found in the Appendix for the sake of completeness. Corollary 4.7. The function u ( t, · ) is continuously differentiable on (0 , ∞ ) for all t ∈ [0 , T ) . In the next section we will suitably integrate the two alternative representations of u (3.6)and (4.10) with respect to the space variable, and we will show that such integrations givethe value function (2.6) of the optimal dividend problem. As a byproduct, we will also obtainthe optimal dividend strategy D ⋆ .4.2. Integrating the Optimal Stopping Value Function. In the next two propositionswe integrate with respect to the space variable the two representations of u given by (3.6) and(4.10). The proofs will employ pathwise arguments. However, in order to simplify exposition,we will not stress the ω -dependence of the involved random variables and processes. Proposition 4.8. Let b the optimal stopping boundary of problem (3.2) , recall I s ( x ) = max ≤ θ ≤ s {− x − µθ − σW θ } ∨ , s ≥ , and define R s ( x ) := x + µs + σW s + I s ( x ) , s ≥ . (4.15) Then for any ( t, x ) ∈ [0 , T ] × R + one has (4.16) Z b ( t ) x u ( t, y ) dy = N ( t, b ( t )) − N ( t, x ) , where N ( t, x ) := E (cid:20) − Z T − t (cid:0) R s ( x ) − b ( t + s ) (cid:1) + f ′ ( t + s ) ds − Z T − t m ( t + s ) dI s ( x )+ g ( T, R T − t ( x )) (cid:21) . (4.17) Proof. To prove (4.16) we use representation (4.10) of the value function of the optimal stop-ping problem (3.2). Using Fubini-Tonelli’s Theorem we obtain Z b ( t ) x u ( t, y ) dy = Z b ( t ) x E (cid:20) Z ( T − t ) ∧ S ( y )0 − f ′ ( t + s ) { y + µs + σW s ≥ b ( t + s ) } ds + m ( t + S ( y )) { S ( y ) ≤ T − t } + g x ( T, A T − t ( y )) (cid:21) dy = E (cid:20) − Z ( T − t )0 f ′ ( t + s ) (cid:18) Z b ( t ) x { y + µs + σW s ≥ b ( t + s ) } { s ≤ S ( y ) } dy (cid:19) ds (4.18) + Z b ( t ) x m ( t + S ( y )) { S ( y ) ≤ T − t } dy + Z b ( t ) x g x ( T, A T − t ( y )) dy (cid:21) . In the following we investigate separately the three summands of the last term on the right-hand side of (4.18).Recalling S ( x ) = inf { u ≥ x + µu + σW u = 0 } it is clear that(4.19) S ( y ) ≥ s ⇔ M s ≤ y for any ( s, y ) ∈ R + × (0 , ∞ ), where we have defined(4.20) M s := max ≤ θ ≤ s ( − µθ − σW θ ) , s ≥ . We can then rewrite (4.15) in terms of (4.20) and obtain(4.21) R s ( x ) = ( x ∨ M s ) + µs + σW s , s ≥ . By using (4.19) we find Z b ( t ) x { y + µs + σW s ≥ b ( t + s ) } { S ( y ) ≥ s } dy = Z b ( t ) ∨ (cid:2) b ( t + s ) − µs − σW s (cid:3) x ∨ (cid:2) b ( t + s ) − µs − σW s (cid:3) { S ( y ) ≥ s } dy = Z b ( t ) ∨ (cid:2) b ( t + s ) − µs − σW s (cid:3) x ∨ (cid:2) b ( t + s ) − µs − σW s (cid:3) { M s ≤ y } dy = (cid:2) ( b ( t ) ∨ ( b ( t + s ) − µs − σW s ) ∨ M s ) − ( x ∨ ( b ( t + s ) − µs − σW s ) ∨ M s ) (cid:3) = (cid:2) ( b ( t ) ∨ M s ) ∨ ( b ( t + s ) − µs − σW s ) − ( x ∨ M s ) ∨ ( b ( t + s ) − µs − σW s ) (cid:3) (4.22) = (cid:2)(cid:0) [( b ( t ) ∨ M s ) + µs + σW s ] ∨ b ( t + s ) (cid:1) − (cid:0) [( x ∨ M s ) + µs + σW s ] ∨ b ( t + s ) (cid:1)(cid:3) = (cid:2)(cid:0) R s ( b ( t )) ∨ b ( t + s ) (cid:1) − (cid:0) R s ( x ) ∨ b ( t + s ) (cid:1)(cid:3) = (cid:2)(cid:0) R s ( b ( t )) − b ( t + s ) (cid:1) + − (cid:0) R s ( x ) − b ( t + s ) (cid:1) + (cid:3) . For the third summand of the last term of the right-hand side of (4.18) we have, due tothe fact that g x ( T, ∆) = 0, Z b ( t ) x g x ( T, A T − t ( y )) dy = Z b ( t ) x g x ( T, y + µ ( T − t ) + σW T − t ) { S ( y ) >T − t } dy = Z b ( t ) x g x ( T, y + µ ( T − t ) + σW T − t ) { M T − t Here we take x ∈ { y ∈ R + : S ( y ) ≥ T − t } ; that is, the initial point x > R s ( x ) in (4.15) equals x + µs + σW s and so I s ( x ) = 0 for all s ∈ [0 , T − t ]. Hence, we canwrite Z b ( t ) x m ( t + S ( y )) { S ( y ) ≤ T − t } dy = 0 = Z T − t m ( t + s ) dI s ( x ) − Z T − t m ( t + s ) dI s ( b ( t )) , (4.25)where we have used that S ( y ) > S ( x ) ≥ T − t for any y > x and { x } has zero Lebesguemeasure to obtain the first equality, and the fact that 0 = I s ( x ) ≥ I s ( b ( t )) ≥ x < b ( t ). Case 2. Here we take x ∈ { y ∈ R + : S ( y ) < T − t } ; i.e., the drifted Brownian motionreaches 0 before the time horizon. Define(4.26) z := inf { y ∈ R + : S ( y ) ≥ T − t } , with the usual convention inf ∅ = + ∞ . In the sequel we assume that z < + ∞ , since otherwisethere is no need for the following analysis to be performed. Note that, by continuity in timeand in the initial datum of the paths of the drifted Brownian motion, we have S ( z ) ≤ T − t .Furthermore, it holds for all y ∈ [ x, z ] that (cf. (4.20))(4.27) y + I s ( y ) = M s , ∀ s ≥ S ( y ) , (4.28) I s ( y ) = 0 , ∀ s < S ( y ) . Using (4.27), (4.28), (4.19), and the change of variable formula in Section 4 of Chapter 0of the book by Revuz and Yor [29] (see also equation (4.7) in Baldursson and Karatzas [5])we obtain Z z ∧ b ( t ) x m ( t + S ( y )) { S ( y ) ≤ T − t } dy = Z z ∧ b ( t ) x m ( t + S ( y )) dy = Z S ( z ∧ b ( t )) S ( x ) m ( t + s ) dM s = Z S ( z ∧ b ( t )) S ( x ) m ( t + s ) (cid:0) dI s ( x ) − dI s ( z ∧ b ( t )) (cid:1) )(4.29) = Z T − t m ( t + s ) (cid:0) dI s ( x ) − dI s ( z ∧ b ( t )) (cid:1) = Z T − t m ( t + s ) dI s ( x ) − Z T − t m ( t + s ) dI s ( z ∧ b ( t )) . For the integral R b ( t ) z ∧ b ( t ) m ( t + S ( y )) { S ( y ) ≤ T − t } dy we can use the result of Case 1 due to thedefinition of z (4.26). Then, combining (4.25) and (4.29) leads to (4.24).By (4.22), (4.23) and (4.24), and recalling (4.17) and (4.18) we obtain (4.16). (cid:3) Proposition 4.9. Let ( D ⋆ , I ⋆ ) be the solution to system (3.7) . Then, for any ( t, x ) ∈ [0 , T ] × R + one has (4.30) Z b ( t ) x u ( t, y ) dy = M ( t, b ( t )) − M ( t, x ) , where b is the optimal stopping boundary of problem (3.2) and (4.31) M ( t, x ) := E (cid:20)Z T − t f ( t + s ) dD ⋆s ( t, x ) − Z T − t m ( t + s ) dI ⋆s ( t, x ) + g ( T, X D ⋆ T − t ( x )) (cid:21) . Proof. For this proof we use instead the representation of u (cf. (3.6)) u ( t, x ) = E h f ( t + τ ⋆ ( t, x )) { τ ⋆ ( t,x ) Here we take x ∈ { y ∈ R + : τ ⋆ (0 , y ) < S ( y ) } ; that is, the initial point x > L s ) s ≥ such that(4.33) L s := max ≤ θ ≤ s { µθ + σW θ − b ( θ ) } , ≤ s ≤ T. Then we have that for all y ∈ [ x, b (0)](4.34) { τ ⋆ (0 , y ) ≤ s } = { L s ≥ − y } , (4.35) { τ ⋆ (0 , y ) = T } = { L T ≤ − y } , (4.36) D ⋆s (0 , y ) = ( , ≤ s ≤ τ ⋆ (0 , y ) ,y + L s , τ ⋆ (0 , y ) ≤ s ≤ S ( y ) , and(4.37) X D ⋆ s ( y ) = ( y + µs + σW s , ≤ s ≤ τ ⋆ (0 , y ) ,µs + σW s − L s , τ ⋆ (0 , y ) ≤ s ≤ S ( y ) , and in particular (cf. (3.7)) I ⋆s (0 , y ) = I ⋆s (0 , b (0)) = 0 for any s ∈ [0 , τ ⋆ (0 , y )].Moreover, it follows by definition of τ ⋆ (0 , x ), S ( x ) and X D ⋆ ( x ) that for all y ∈ [ x, b (0)] wehave(4.38) 0 = τ ⋆ (0 , b (0)) ≤ τ ⋆ (0 , y ) ≤ τ ⋆ (0 , x ) , (4.39) τ ⋆ (0 , y ) < τ ⋆ (0 , x ) < S ( x ) ≤ S ( y ) , and(4.40) on { τ ⋆ (0 , x ) < T } : X D ⋆ s ( y ) = X D ⋆ s ( x ) , ∀ s > τ ⋆ (0 , x ) . PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 15 With these results at hand, we now show that for all x ∈ [0 , b (0)] such that τ ⋆ (0 , x ) < S ( x ) itholds that(4.41) Z b (0) x f ( τ ⋆ (0 , y ))1 { τ ⋆ (0 ,y ) Here we take x ∈ { y ∈ R + : τ ⋆ (0 , y ) > S ( y ) , τ ⋆ (0 , q ) < S ( q ) ∀ q ∈ ( y, b (0)) } . Fora realization like that, such an x is such that the drifted Brownian motion touches the originbefore hitting the boundary, but it does not cross the origin. This in particular implies that I ⋆s (0 , x ) = 0 for all s ≤ τ ⋆ (0 , x ). Hence the same arguments employed in Step 1 hold true,and (4.41) – (4.43) follow. Step 3. Here we take x ∈ { y ∈ R + : τ ⋆ (0 , y ) > S ( y ) } ; that is, the drifted Brownian motionhits the origin before reaching the boundary.Define(4.47) z := inf { y ∈ [0 , b (0)] : τ ⋆ (0 , y ) < S ( y ) } which exists finite since y τ ⋆ (0 , y ) − S ( y ) is decreasing and τ ⋆ (0 , b (0)) = 0 and S (0) = 0a.s. We want to prove that(4.48) Z zx m ( S ( y )) { τ ⋆ (0 ,y ) ≥ S ( y ) } dy = Z T m ( s ) dI ⋆s (0 , x ) − Z T m ( s ) dI ⋆s (0 , z ) , (4.49) Z zx f ( τ ⋆ (0 , y ))1 { τ ⋆ (0 ,y ) On the other hand, for the left-hand side of (4.48), we use the change of variable formulaof Section 4 in Chapter 0 of Revuz and Yor [29]. This leads to(4.57) Z zx m ( S ( y )) { τ ⋆ (0 ,y ) ≥ S ( y ) } dy = Z zx m ( S ( y )) dy = Z S ( z ) S ( x ) m ( s ) dM s , where we use (4.55), the fact that { z } is a Lebesgue zero set, and that M is the right-continuousinverse of S (see (4.19)). Combining (4.56) and (4.57) proves (4.48).Equation (4.49) follows by observing that (4.53)–(4.54) imply that the processes D ⋆ (0 , z )and D ⋆ (0 , x ) coincide, and the left-hand side equals 0 by definition. Notice that for such anargument particular care has to be put when considering z of (4.47) as a starting point forthe drifted Brownian motion. In particular, if the realization of the Brownian motion is suchthat τ ⋆ (0 , z ) < S ( z ), then by definition of z , the drifted Brownian motion only touches theboundary at time τ ⋆ (0 , z ), but does not cross it. Hence, we still have D ⋆s (0 , z ) = 0 for all s ≤ S ( z ), which implies (4.53) and therefore still D ⋆s (0 , z ) = D ⋆s (0 , x ). In turn, this givesagain that (4.49) holds also for such a particular realization of the Brownian motion.Finally, to prove equation (4.50) remember that x ∈ { y ∈ R + : τ ⋆ (0 , y ) > S ( y ) } . By defi-nition of z we obtain τ ⋆ (0 , y ) ≥ S ( y ) for all y ∈ [ x, z ) and the left-hand side of (4.50) equalszero. By (4.53) the processes X D ⋆ s ( z ) = X D ⋆ s ( x ) coincides for all s ≥ S ( z ), and S ( z ) ≤ T a.s.by Lemma A.1 in the Appendix. Therefore, the right-hand side of (4.50) equals zero as well. Step 4. For x ∈ { y ∈ R + : τ ⋆ (0 , y ) < S ( y ) } , (4.30) follows by the results of Step 1. If,instead, x ∈ { y ∈ R + : τ ⋆ (0 , y ) > S ( y ) } , then we can integrate u separately in the intervals[ x, z ] and [ z, b (0)]. When integrating u in the interval [ x, z ] we use the results of Step 3. Onthe other hand, integrating u over [ z, b (0)] we have to distinguish two cases. Now, if z belongsto { y ∈ R + : τ ⋆ (0 , y ) < S ( y ) } , then we can still apply the results of Step 1 to conclude.If z belongs to { y ∈ R + : τ ⋆ (0 , y ) > S ( y ) , τ ⋆ (0 , q ) < S ( q ) ∀ q ∈ ( y, b (0)) } , we can employ theresults of Step 2 to obtain the claim. Thus, in any case, (4.30) holds. (cid:3) We now prove that the two functions N and M of (4.17) and (4.31), respectively, are suchthat N = M . To accomplish that we preliminary notice that by their definitions and strongMarkov property, one has that the processes(4.58) N ( t + s ∧ τ ⋆ ( t, x ) , R s ∧ τ ⋆ ( t,x ) ( x )) − Z s ∧ τ ⋆ ( t,x )0 m ( t + θ ) dI θ ( x ) , ≤ s ≤ T − t, and(4.59) M ( t + s ∧ τ ⋆ ( t, x ) , R s ∧ τ ⋆ ( t,x ) ( x )) − Z s ∧ τ ⋆ ( t,x )0 m ( t + θ ) dI ⋆θ ( t, x ) , ≤ s ≤ T − t, are F -martingales for any ( t, x ) ∈ [0 , T ] × R + . Moreover, by (4.16) one has N ( t, x ) = N ( t, b ( t )) − R b ( t ) x u ( t, y ) dy and, due to (4.30), M ( t, x ) = M ( t, b ( t )) − R b ( t ) x u ( t, y ) dy . Hence,(4.60) Ψ( t ) := M ( t, x ) − N ( t, x ) , t ∈ [0 , T ] , is independent of the x variable. We now prove that one actually has Ψ = 0 and therefore N = M . Theorem 4.10. It holds Ψ( t ) = 0 for all t ∈ [0 , T ] . Therefore, N = M on [0 , T ] × R + .Proof. Since ( N − M ) is independent of x , it suffices to show that ( N − M )( t, x ) = 0 at some x for any t ≤ T . To accomplish that we show Ψ ′ ( t ) = 0 for any t < T , since by (4.16) and(4.30) we already know thatΨ( T ) = N ( T, x ) − M ( T, x ) = g ( T, x ) − g ( T, x ) = 0 . Then take 0 < x < x , t ∈ [0 , T ) and ε > t + ε < T given and fixed, considerthe rectangular domain R := ( t − ε, t + ε ) × ( x , x ) such that cl ( R ) ⊂ C (where C hasbeen defined in (3.3)). Also, denote by ∂ R := ∂ R\ ( { t − ε } × ( x , x )). Then consider theproblem ( P ) ( h t ( t, x ) = L h ( t, x ) , ( t, x ) ∈ R ,h ( t, x ) = ( N − M )( t, x ) , ( t, x ) ∈ ∂ R , where L is the second-order differential operator that acting on ϕ ∈ C , ([0 , T ] × R ) gives( L ϕ )( t, x ) = µ ∂ϕ∂x ( t, x ) + 12 σ ∂ ϕ∂x ( t, x ) , ( t, x ) ∈ [0 , T ] × R . By reversing time, t T − t , Problem (P) corresponds to a classical initial value problemwith uniformly elliptic operator (notice that σ > 0) and parabolic boundary ∂ R . Since N − M is continuous, and all the coefficients in the first equation of ( P ) are smooth (actuallyconstant), by classical theory of partial differential equations of parabolic type (see, e.g.,Chapter V in the book by Lieberman [25]) problem ( P ) admits a unique solution h that iscontinuous, with continuous derivatives h t , h x , h xx . Moreover, by the Feynman-Kac’s formula,such a solution admits the representation h ( t, x ) = E [( N − M )( t + b τ ( t, x ) , Z b τ ( t,x ) ( x ))] , where b τ ( t, x ) := inf { s ∈ [0 , T − t ) : ( t + s, Z s ( x )) ∈ ∂ R} ∧ ( T − t ) , and Z s ( x ) = x + µs + σW s , s ≥ 0. Notice that we have b τ ( t, x ) ≤ τ ⋆ ( t, x ) a.s., since cl ( R ) ⊂ C .Also, the integral terms in (4.58) and (4.59) are equal since dI θ ( x ) = dI ⋆θ ( t, x ) = 0 for any θ ≤ b τ ( t, x ) ≤ τ ⋆ ( t, x ). Hence by the martingale property of (4.58) and (4.59) we have(4.61) h ( t, x ) = ( N − M )( t, x ) in R , and, by arbitrariness of R , Ψ( t ) = ( N − M )( t, x ) = h ( t, x ) in C . Therefore, since Ψ = N − M is independent of x , continuous in t and solves the first equationof ( P ) in C , we obtain Ψ ′ ( t ) = 0 for any t < T . Hence Ψ( t ) = 0 for any t ≤ T since Ψ( T ) = 0,and thus N ( t, x ) − M ( t, x ) = 0 for any t ≤ T and for any x ∈ (0 , ∞ ). (cid:3) In the following we show that the function N is an upper bound for the value function V of (2.6). We first prove the following result. Theorem 4.11. For any ( t, x ) ∈ R + × [0 , T ] the process (4.62) e N s := N ( t + s, R s ( x )) − Z s m ( t + u ) dI u ( x ) , ≤ s ≤ T − t, is an F -supermartingale.Proof. It is enough to show that E (cid:2) e N θ (cid:3) ≤ E [ e N τ ] for all bounded F -stopping times θ, τ suchthat θ ≥ τ (see Karatzas and Shreve [22], Chapter 1, Problem 3.26). PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 19 By the strong Markov property and the definition of N (4.17), we get that for any bounded F -stopping time ρ one has E [ e N ρ ] = E (cid:20) N ( t + ρ, R ρ ( x )) − Z ρ m ( t + s ) dI s ( x ) (cid:21) = E (cid:20) − Z T − tρ f ′ ( t + s )[ R s ( x ) − b ( t + s )] + ds − Z T − t m ( t + s ) dI s ( x ) + g ( R T − t ( x )) (cid:21) = N ( t, x ) + E (cid:20)Z ρ f ′ ( t + s ) (cid:0) R s ( x ) − b ( t + s ) (cid:1) + ds (cid:21) =: N ( t, x ) + ∆ ρ , for any ( t, x ) ∈ [0 , T ] × R + . Hence, taking θ, τ such that T − t ≥ θ ≥ τ we get from the latterthat E [ e N θ ] = N ( t, x )+∆ θ ≤ N ( t, x )+∆ τ = E [ e N τ ], where the inequality is due to the fact that f ′ ≤ S (cf. Corollary 4.2-(ii)). This proves the claimed supermartingale property. (cid:3) To proceed further, we need the following properties of the function N of (4.17). Its proofis relegated to the Appendix. Lemma 4.12. The function N ∈ C , ([0 , T ) × (0 , ∞ )) ∩ C ([0 , T ] × R + ) . Thanks to Lemma 4.12, an application of Itˆo’s formula allows us to obtain the following(unique) Doob-Meyer decomposition of the F -supermartingale e N (cf. (4.62)). Corollary 4.13. The F -supermartingale e N of (4.62) is such that for all ( t, x ) ∈ [0 , T ] × R + and s ∈ [0 , T − t ](4.63) N ( t + s, R s ( x )) − Z s m ( t + θ ) dI θ ( x ) = N ( t, x ) + σ Z s u ( t + θ, R θ ( x )) dW θ + Π s ( t, x ) , where Π · ( t, x ) is a continuous, nonincreasing and F -adapted process.Proof. By the Doob-Meyer decomposition, the F -supermartingale in (4.62) can be (uniquely)written as the sum of an F -martingale and a continuous, F -adapted nonincreasing process(Π s ) s ≥ . Applying the martingale representation theorem to the martingale part of e N , yieldsthe decomposition(4.64) e N s = N ( t, x ) + Z s φ θ dW θ + Π s ( t, x ) , for some φ ∈ L (Ω × [0 , T ] , P ⊗ dt ). Finally, an application of Itˆo’s lemma shows that φ θ = σu ( t + θ, R θ ( x )) a.s. (cid:3) Theorem 4.14. For any process D ∈ D ( t, x ) and any ( t, x ) ∈ [0 , T ] × R + , the process (4.65) Q s ( D ; t, x ) := Z [0 ,s ] f ( t + θ ) dD θ − Z s m ( t + θ ) dI Dθ + N ( t + s, X Ds ( x )) ,s ∈ [0 , T − t ] , is such that (4.66) E [ Q s ( D ; t, x )] ≤ N ( t, x ) , for any s ∈ [0 , T − t ] . Proof. The proof is organized in 3 steps. Step 1. For D ≡ 0, the proof is given by Theorem 4.11. Step 2. Let D s := R s z u du , s ≥ 0, where z is a bounded, nonnegative, F -progressivelymeasurable process. To show (4.66) we use Girsanov’s Theorem and we rewrite the stateprocess X Ds ( x ) = x + µs + σW s + D s − I Ds as a new drifted Brownian motion reflected at theorigin. We therefore introduce the exponential martingale Z s = exp (cid:18)Z s z u σ dW u − σ Z s z u du (cid:19) , s ≥ , and we obtain that under the measure b P = Z T P , the process c W s := W s − σ Z s z u du, s ≥ , is an F - Brownian motion.We can now rewrite the process Q of (4.65) under b P as(4.67) Q s ( D ; t, x ) = Z [0 ,s ] f ( t + θ ) dD θ − Z s m ( t + θ ) d b I Dθ + N ( t + s, b R s ( x )) , for any s ∈ [0 , T − t ], where under b P b X Ds ( x ) = x + µs + σ c W s + b I Ds =: b R s ( x ) . Here b I D · is flat off { s ≥ b R s ( x ) = 0 } and reflects the drifted Brownian motion at the origin.By employing (4.63), equation (4.67) reads as Q s ( D ; t, x ) = N ( t, x ) + σ Z s u ( t + u, b R u ( x )) d c W u + b Π s ( t, x ) , s ∈ [0 , T − t ] , (4.68)where we have set(4.69) b Π s ( t, x ) := Π s ( t, x ) + Z s (cid:18) f ( t + θ ) − u ( t + θ, R θ ( x )) (cid:19) z θ dθ, s ∈ [0 , T − t ] . Since b Π is nonincreasing due to the fact that u ≥ f and Π · ( t, x ) is nonincreasing, we can takeexpectations in (4.68) so to obtain E [ Q s ( D ; t, x )] ≤ N ( t, x ) , ∀ s ∈ [0 , T − t ] . Step 3. Since any arbitrary D ∈ D ( t, x ) can be approximated by an increasing sequence( D n ) n ∈ N of absolutely continuous processes as the ones considered in Step 2 (see El Karouiand Karatzas [13], Lemmata 5.4, 5.5 and Proposition 5.6), we have for all n ∈ NE [ Q s ( D n ; t, x )] ≤ N ( t, x ) . Applying monotone and dominated convergence theorem, this property holds for Q ( D ; t, x )as well, for any D ∈ D ( t, x ). (cid:3) By Theorem 4.14 and the definition of Q as in (4.65) we immediately obtain(4.70) V ( t, x ) = sup D ∈D ( t,x ) J ( D ; t, x ) = sup D ∈D ( t,x ) E [ Q T − t ( D ; t, x )] ≤ N ( t, x ) . Moreover, by definition (4.31) one has(4.71) M ( t, x ) = J ( D ⋆ ( t, x ); t, x ) ≤ V ( t, x ) . With all these results at hand, we can now finally prove Theorem 3.2. Proof of Theorem 3.2 . By combining (4.70), (4.71), and Theorem 4.10 we obtain the seriesof inequalities N ( t, x ) ≥ V ( t, x ) ≥ M ( t, x ) = N ( t, x )which proves the claim that V = M , and the optimality of D ⋆ . It just remains to prove (3.9).To accomplish that we adapt and expand arguments as those used by El Karoui and Karatzasin the proof of Corollary 4.2 in [14].Observe that optimality of D ⋆ implies that for all x > b ( t )(4.72) V ( t, b ( t )) + f ( t )( x − b ( t )) = V ( t, x ) . PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 21 Using (4.17) and the fact that V = N as proved above, we then find from (4.72) V ( t, b ( t )) = V ( t, x ) − f ( t )( x − b ( t ))= E (cid:20) − Z T − t f ′ ( t + s )( R s ( x ) − b ( t + s )) + ds − Z T − t m ( t + s ) dI s ( x )+ g ( T, R T − t ( x )) − f ( t )( x − b ( t )) (cid:21) = E (cid:20) − Z T − t f ′ ( t + s ) h ( R s ( x ) − b ( t + s )) + − ( x − b ( t )) i ds − Z T − t m ( t + s ) dI s ( x )+ g ( T, R T − t ( x )) − f ( T )( x − b ( t )) (cid:21) . Recall (4.15), and observe that under the condition b ( T ) < ∞ we can write E h g ( T, R T − t ( x )) i = g ( T, b ( T )) + E (cid:20)(cid:18) Z R T − t ( x ) b ( T ) g x ( T, y ) dy (cid:19) { R T − t ( x ) >b ( T ) } − (cid:18) Z b ( T ) R T − t ( x ) g x ( T, y ) dy (cid:19) { R T − t ( x ) ≤ b ( T ) } (cid:21) = g ( T, b ( T ))+ E (cid:20) f ( T ) (cid:0) R T − t ( x ) − b ( T ) (cid:1) { R T − t ( x ) >b ( T ) } − (cid:18) Z b ( T ) R T − t ( x ) g x ( T, y ) dy (cid:19) { R T − t ( x ) ≤ b ( T ) } (cid:21) , where the last equality follows from Remark 4.3. Therefore, we obtain that V ( t, b ( t )) = E (cid:20) − Z T − t f ′ ( t + s ) h ( R s ( x ) − b ( t + s )) + − ( x − b ( t )) i ds − Z T − t m ( t + s ) dI s ( x )+ g ( T, b ( T )) + f ( T ) (cid:0) R T − t ( x ) − b ( T ) (cid:1) { R T − t ( x ) >b ( T ) } − f ( T ) (cid:0) x − b ( t ) (cid:1) − (cid:18) Z b ( T ) R T − t ( x ) g x ( T, y ) dy (cid:19) { R T − t ( x ) ≤ b ( T ) } (cid:21) . Notice now that I s ( x ) → R s ( x ) → ∞ , and ( R s ( x ) − b ( t + s )) + − ( x − b ( t )) → µs + σW s − b ( t + s ) + b ( t ) a.s. for any s ≥ x ↑ ∞ (cf. (4.15)). Then, letting x → ∞ in the lastexpression for V ( t, b ( t )), and invoking the monotone and dominated convergence theorems,we find (after evaluating the expectations and rearranging terms) V ( t, b ( t )) = E (cid:20) − Z T − t f ′ ( t + s ) (cid:16) µs + σW s − b ( t + s ) + b ( t ) (cid:17) ds + g ( T, b ( T )) + f ( T ) ( µ ( T − t ) + σW T − t − b ( T ) + b ( t )) (cid:21) = − µ Z T − t f ′ ( t + s ) s ds + Z T − t f ′ ( t + s ) b ( t + s ) ds + g ( T, b ( T )) + f ( T ) µ ( T − t ) + f ( t ) b ( t ) − f ( T ) b ( T ) . (cid:3) Remark 4.15. As a byproduct of the fact that V = N and of Lemma 4.12, we have that V ∈ C , ([0 , T ) × (0 , ∞ )) ∩ C ([0 , T ] × R + ). Moreover, from (3.8) and (3.2) we have that V satisfies the Neumann boundary condition V x ( t, 0) = m ( t ) for all t ∈ [0 , T ]. Remark 4.16. The pathwise approach followed in this section seems to suggest that someof the intermediate results needed to prove Theorem 3.2 remain valid also in a more general setting in which profits and costs in (2.6) are discounted at a stochastic rate. We leave theanalysis of this interesting problem for future work.5. Verifying Assumption 3.1:a Case Study with Discounted Constant Marginal Profits and Costs In this section we consider the optimal dividend problem with capital injections b V ( t, x ) := sup D ∈D ( t,x ) E (cid:20)Z T − t ηe − rs dD s − Z T − t κe − rs dI Ds + ηe − r ( T − t ) X DT − t ( x ) (cid:21) (5.1) = e rt V ( t, x ) , where we have defined(5.2) V ( t, x ) := sup D ∈D ( t,x ) E (cid:20)Z T − t ηe − r ( t + s ) dD s − Z T − t κe − r ( t + s ) dI Ds + ηe − rT X DT − t ( x ) (cid:21) . It is clear from (5.2) and (2.5) that such a problem can be accommodated in our generalsetting (2.6) by taking (cf. Assumption 2.1)(5.3) f ( t ) = ηe − rt , m ( t ) = κe − rt , g ( t, x ) = ηe − rt x, for some κ > η (see also Remark 2.3).In b V of (5.1) the coefficient κ can be seen as a constant proportional administration costfor capital injections. On the other hand, if we immagine that transaction costs or taxes haveto be paid on dividends, the coefficient η measures a constant net proportion of leakages fromthe surplus received by the shareholders. Remark 5.1. Problem (5.1) is perhaps the most common formulation of the optimal dividendproblem with capital injections (see, e.g., Kulenko and Schmidli [24], Lokka and Zervos [26],Zhu and Yang [35] and references therein). However, to the best of our knowledge, no previouswork has considered such a problem in the case of a finite time horizon, whereas problem (5.1)has been extensively studied when T = + ∞ (see, e.g., Ferrari [16] and references therein).In particular, it has been shown, e.g., in [16] that in the case T = + ∞ the optimal dividendstrategy is triggered by a boundary b ∞ > F -stopping times).Thanks to Theorem 3.2 we know that, whenever Assumption 3.1 is satisfied, the optimalcontrol D ⋆ for problem (5.2) is triggered by the optimal stopping boundary b of the optimalstopping problem u ( t, x ) = sup τ ∈ Λ( T − t ) E h e − rτ η { τ Also, notice that we have u ( t, x ) ≤ κ for ( t, x ) ∈ [0 , T ] × R + since η < κ .Since the reward process φ t := e − rt η { t 0, the process(5.6) e − r ( s ∧ τ ⋆ ( t,x ) ∧ S ( x )) u ( t + ( s ∧ τ ⋆ ( t, x ) ∧ S ( x )) , Z ( s ∧ τ ⋆ ( t,x ) ∧ S ( x )) ( x )) , s ∈ [0 , T − t ] , is an F -martingale (cf. Proposition 1.6 and Remark 1.7 in Kobylanski and Quenez [23]).The next proposition proves some preliminary properties of u . Proposition 5.2. The value function u of (5.4) satisfies the following: (i) u ( T, x ) = η for any x > and u ( t, 0) = κ for any t ∈ [0 , T ] ; (ii) t u ( t, x ) is nonincreasing for any x > ; (iii) x u ( t, x ) is nonincreasing for any t ∈ [0 , T ] .Proof. We prove each item separately.(i) The first property easily follows from definition (5.4).(ii) The second property is due to the fact that Λ( T − · ) shrinks and the expected value onthe right-hand side of (5.4) is independent of t ∈ [0 , T ].(iii) Fix t ∈ [0 , T ], x > x ≥ S ( x ) > S ( x ). Then, from (5.4) we can write u ( t, x ) − u ( t, x ) ≤ sup τ ∈ Λ( T − t ) E (cid:20) e − rτ η { τ The free boundary b is such that (i) t b ( t ) is nonincreasing; (ii) One has b ( t ) > for all t ∈ [0 , T ) . Moreover, there exists b ∞ > such that b ( t ) ≤ b ∞ for any t ∈ [0 , T ] . Proof. We prove each item separately.(i) The claimed monotonicity of b immediately follows from (ii) of Proposition 5.2.(ii) To show that b ( t ) > t ∈ [0 , T ) it is enough to observe that u ( t, 0) = κ > η forall t ∈ [0 , T ).To prove b ( t ) < ∞ notice that u ( t, x ) ≤ u ∞ ( x ) for all ( t, x ) ∈ [0 , T ] × R + , where u ∞ ( x ) := sup τ ≥ E h ηe − rτ { τ Proposition 5.4. The function ( t, x ) u ( t, x ) is lower semicontinuous on [0 , T ) × (0 , ∞ ) . The lower semicontinuity of u implies that the martingale of (5.6) has right-continuoussample paths, and that the stopping region is closed. The latter fact in turn plays an importantrole when proving continuity of the free boundary, as it is shown in the next proposition. Proposition 5.5. The free boundary b is such that t b ( t ) is continuous on [0 , T ) . Moreover, b ( T ) := lim t ↑ T b ( t ) = 0 .Proof. We prove the two properties separately.Here we show that b is continuous, and this proof is divided in two parts. We start with theright-continuity. Note that, by lower semicontinuity of u (cf. Proposition 5.4), the stoppingregion S is closed. Then fix an arbitrary point t ∈ [0 , T ), take any sequence ( t n ) n ≥ suchthat t n ↓ t , and notice that ( t n , b ( t n )) ∈ S , by definition. Setting b ( t +) := lim t n ↓ t b ( t n )(which exists due to Proposition 5.3-(i)), we have ( t n , b ( t n )) → ( t, b ( t +)), and since S isclosed ( t, b ( t +)) ∈ S . Therefore, it holds b ( t +) ≥ b ( t ) by definition (5.7) of b . However, b ( · )is nonincreasing, and therefore b ( t ) = b ( t +).Next we show left-continuity for all t ∈ (0 , T ) and for this we adapt to our setting ideasas those in the proof of Proposition 4.2 in De Angelis and Ekstr¨om [10]. Suppose that b makes a jump at some t ∈ (0 , T ). By Proposition 5.3-(i) we have lim t n ↑ t b ( t n ) := b ( t − ) ≥ b ( t ).We employ a contradiction scheme to show b ( t − ) = b ( t ), and we assume b ( t − ) > b ( t ). Let x := b ( t − )+ b ( t )2 , recall Z s ( x ) = x + µs + σW s , s ≥ 0, and define τ ε := inf { s ≥ Z s ( x ) / ∈ ( b ( t − ) , b ( t )) } ∧ ε for ε ∈ (0 , t ). Then noticing that τ ε < τ ⋆ ( t − ε, x ) ∧ S ( x ), by the martingale property of (5.6)we can write u ( t − ε, x ) = E (cid:2) e − rτ ε u ( t − ε + τ ε , Z τ ε ( x )) (cid:3) = E (cid:2) e − rε u ( t, Z ε ( x )) { τ ε = ε } + e − rτ ε u ( t − ε + τ ε , Z τ ε ( x )) { τ ε <ε } (cid:3) ≤ E (cid:2) e − rε η { τ ε = ε } + e − rτ ε κ { τ ε <ε } (cid:3) ≤ e − rε η + κ P ( τ ε < ε ) , where the last step follows from the fact that u ≤ κ , and that Z τ ε ( x ) ≥ b ( t ) on the set { τ ε = ε } . Since e − rε η + κ P ( τ ε < ε ) = η (1 − rε ) + κo ( ε ) as ε ↓ 0, we have found a contradictionto u ( t, x ) ≥ η . Therefore, b ( t − ) = b ( t ) and b is continuous on [0 , T ).To prove the claimed limit, notice that if b ( T ) := lim t ↑ T b ( t ) > 0, then any point ( T, x )with x ∈ (0 , b ( T )) belongs to C . However, we know that ( T, x ) ∈ S for all x > 0, and we thusreach a contradiction. (cid:3) PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 25 Thanks to the previous results all the requirements of Assumption 3.1 are satisfied forproblem (5.4). Hence Theorem 3.2 holds, and one has that V of (5.2) and u of (5.4) are suchthat V x = u on [0 , T ] × R + . In particular, by (5.1) and Theorem 3.2 we can write b V ( t, x ) = b V ( t, b ( t )) − e rt Z b ( t ) x u ( t, y ) dy, where by (3.9), (5.3), and the fact that b ( T ) = 0 we have b V ( t, b ( t )) = ηb ( t ) + µηr (cid:0) − e − r ( T − t ) (cid:1) − rη Z Tt e − r ( u − t ) b ( u ) du. Moreover, the optimal dividend distributions’ policy D ⋆ given through (3.7) is triggered bythe free boundary b whose properties have been derived in Theorem 5.5.5.1. A Comparative Statics Analysis. We conclude by providing the monotonicity of thefree boundary with respect to some of the problem’s parameters. In the following, for anygiven and fixed t ∈ [0 , T ], we write b ( t ; · ) in order to stress the dependence of the free boundarypoint b ( t ) with respect to a given parameter. Similarly, we write u ( t, x ; · ) when we need toconsider the dependence of u ( t, x ), ( t, x ) ∈ [0 , T ] × R + , with respect to a given problem’sparameter. Proposition 5.6. Let t ∈ [0 , T ] be given and fixed. It holds that (i) κ b ( t ; κ ) is nondecreasing; (ii) η b ( t ; η ) is nonincreasing; (iii) r b ( t ; r ) is nonincreasing; (iv) µ b ( t ; µ ) is nonincreasing.Proof. Recalling that u ( t, x ) = sup τ ∈ Λ( T − t ) E h e − rτ η { τ Financial support by the German Research Foundation (DFG) through the CollaborativeResearch Centre 1283 “Taming uncertainty and profiting from randomness and low regularityin analysis, stochastics and their applications” is gratefully acknowledged.Both the authors thank Miryana Grigorova and Hanspeter Schmidli for fruitful discussionsand comments. Part of this work has been finalized while the first author was visiting theDepartment of Mathematics of the University of Padova thanks to the program “Visiting Sci-entists 2018”. Giorgio Ferrari acknowledges the Department of Mathematics of the Universityof Padova for the hospitality. We also wish to thank three anonymous referees for their carefulreading and inspiring comments. PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 27 Appendix A. Appendix A.1. Proof of Corollary 4.7. Notice that from (4.10) we can write for any x > t ∈ [0 , T ] u ( t, x ) = E (cid:20) Z T − t − f ′ ( t + θ ) { x + µθ + σW θ ≥ b ( t + θ ) } { θ 0, as S ( x ) = inf { s ≥ x + µs + σW s = 0 } = inf { s ≥ µσ s + W s = − xσ } L = inf { s ≥ − µσ s + c W s = xσ } . (A.3)where c W is a standard Brownian motion. Hence equation (3 . . 3) in Jeanblanc et al. [18]applies and allows us to write the probability density of S ( x ) as(A.4) ρ S ( x ) ( u ) := d P ( S ( x ) ∈ du ) du = xσ √ πu e − ( xσ + µσ u )22 u , u ≥ . For the third summand we notice that the absorbed process A T − t ( x ) of (3.1) is the driftedBrownian motion started in x and killed at the origin. Denote by ρ A ( t, x, y ) its transitiondensity of moving from x to y in t units of time. Then, by employing the result of Borodinand Salminen [7], Section 15 in Appendix 1 (suitably adjusted to our case with σ = 1), we obtain ρ A ( T − t, x, y ) := d P ( A T − t ( x ) ∈ dy ) dy = 1 p π ( T − t ) σ exp (cid:18) − (cid:18) µ ( x − y ) σ (cid:19) − µ σ ( T − t ) (cid:19) × (cid:18) exp (cid:18) − ( x − y ) σ ( T − t ) (cid:19) − exp (cid:18) − ( x + y ) σ ( T − t ) (cid:19)(cid:19) . (A.5)Feeding (A.2), (A.4) and (A.5) back into (A.1) we obtain u ( t, x ) = Z T − t − f ′ ( t + θ ) (cid:20) N (cid:18) x − b ( t + θ ) σ + µσ θ √ θ (cid:19) − e − µxσ N (cid:18) − b ( t + θ )+ xσ + µσ θ √ θ (cid:19)(cid:21) dθ + Z T − t m ( t + u ) ρ S ( x ) ( u ) du + Z ∞ g x ( T, y ) ρ A ( T − t, x, y ) dy, (A.6)and it is easy to see by the dominated convergence theorem that x u ( t, x ) is continuouslydifferentiable on (0 , ∞ ) for any t < T .A.2. Proof of Lemma 4.12. By (4.16) and Corollary 4.7 the function N of (4.17) is twice-continuously differentiable with respect to x on (0 , ∞ ). To show that N is also continuouslydifferentiable with respect to t on [0 , T ) we express the expected value on the right-hand sideof (4.17) as an integral with respect to the probability densities of the involved processes.We thus start computing the transition density of the reflected Brownian motion R of (4.21),which we call ρ R . By Appendix 1, Chapter 14, in Borodin and Salminen [7] (easily adaptedto our case with σ = 1) we have ρ R ( u, x, y ) := d P ( R u ( x ) ∈ dy ) dy = 1 √ πuσ exp (cid:18) − µσ (cid:18) x − yσ (cid:19) − µ σ u (cid:19) × (cid:18) exp (cid:18) − ( x − y ) σ u (cid:19) − exp (cid:18) − ( x + y ) σ u (cid:19)(cid:19) − µ σ Erfc (cid:18) x + y + µu √ σ u (cid:19) , (A.7)where Erfc( x ) := R x −∞ √ π e − y dy for x ∈ R . Hence, by using Fubini’s Theorem, (4.17) readsas N ( t, x ) = E (cid:20) − Z T − t ( R s ( x ) − b ( t + s )) + f ′ ( t + s ) ds − Z T − t m ( t + s ) dI s ( x )+ g ( T, R T − t ( x )) (cid:21) = − Z Tt E h ( R u − t ( x ) − b ( u )) + i f ′ ( u ) du − E (cid:20) Z T − t m ( t + s ) dI s ( x ) (cid:21) + E h g ( T, R T − t ( x )) i = − Z Tt (cid:18) Z ∞ ( y − b ( u )) + ρ R ( u − t, x, y ) dy (cid:19) f ′ ( u ) du − E (cid:20) Z Tt m ( u ) dI u − t ( x ) (cid:21) (A.8) + Z ∞ g ( T, y ) ρ R ( T − t, x, y ) dy. PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 29 Recalling that m is continuously differentiable by Assumption 2.1 and using an integrationby parts, we can write E (cid:20) Z Tt m ( u ) dI u − t ( x ) (cid:21) = E (cid:20) m ( T ) I T − t ( x ) − Z Tt I u − t ( x ) m ′ ( u ) du (cid:21) = m ( T ) E (cid:2) I T − t ( x ) (cid:3) − Z Tt E (cid:2) I u − t ( x ) (cid:3) m ′ ( u ) du = m ( T ) E (cid:2) ∨ ( σξ T − t − x ) (cid:3) − Z Tt E (cid:2) ∨ ( σξ u − t − x ) (cid:3) m ′ ( u ) du, where we have used that I s ( x ) = 0 ∨ ( σξ s − x ) with ξ s := sup θ ≤ s ( − µσ θ − W θ ). Since (cf.Chapter 3.2.2 in Jeanblanc et al. [18])(A.9) P ( ξ s ≤ z ) = N (cid:18) z − µσ s √ s (cid:19) − exp (cid:16) µσ z (cid:17) N (cid:18) − z − µσ s √ s (cid:19) , we get E (cid:20) ∨ ( σξ u − t − x ) (cid:21) = Z ∞ xσ ( σz − x ) ρ ξ ( u − t, z ) dz, (A.10)where we have defined ρ ξ ( s, z ) := d P ( ξ s ≤ z ) dz . Because ρ ξ ( · , z ) and ρ R ( · , x, y ) are continuouslydifferentiable on (0 , T ], it follows that N ( t, x ) as in (A.8) is continuously differentiable withrespect to t , for any t < T . The continuity of N on [0 , T ] × R + also follows from the previousequations.A.3. Proof of Proposition 5.4. Let ( t, x ) ∈ [0 , T ) × (0 , ∞ ) be given and fixed, and takeany sequence ( t n , x n ) ⊂ [0 , T ) × (0 , ∞ ) such that ( t n , x n ) → ( t, x ). Then, let τ ⋆ := τ ⋆ ( t, x ) bethe optimal stopping time for u ( t, x ) of (5.9). From (5.4) and the fact that τ ⋆ ≤ T − t a.s. wethen find u ( t, x ) − u ( t n , x n ) ≤ E h ηe − rτ ⋆ { τ ⋆ Recall that (cf. (4.47) ) z = inf { y ∈ [0 , b (0)] : τ ⋆ (0 , y ) < S ( y ) } . Then it holds that (A.12) S ( z ) ≤ T a.s. Proof. In order to simplify exposition, in the following we shall stress the dependence on ω only when strictly necessary. Suppose that there exists a set Ω ⊂ Ω s.t. P (Ω ) > 0, and thatfor any ω ∈ Ω we have S ( z ) > T . Then take ω ∈ Ω , recall that Z s ( x ) = x + µs + σW s for PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 31 any x > s ≥ 0, and notice that min ≤ s ≤ T Z s ( z ; ω ) = ℓ := ℓ ( ω ) > 0. Then, defining b z ( ω o ) := b z = z − ℓ , one hasmin ≤ s ≤ T Z s ( b z ; ω ) = min ≤ s ≤ T (cid:18) z + µs + σW s ( ω ) − ℓ (cid:19) = ℓ − ℓ ℓ > . Hence, S ( b z ) > T ≥ τ ⋆ (0 , b z ), but this contradicts the definition of z since b z < z . Therefore weconclude that S ( z ) ≤ T a.s. (cid:3) References [1] Akyildirim, E., Guney, I.E., Rochet, J.C., and Soner, H.M. (2014). Optimal dividend policy withrandom interest rates . J. Math. Econ. , pp. 93–101.[2] Avanzi, B. (2009). Strategies for dividend distribution: a review . N. Am. Actuar. J. , pp. 217–251.[3] Avanzi, B., Gerber, H.U., Shiu, E.S.W. (2007). Optimal dividends in the dual model . Insur. Math.Econ. , pp. 111–123.[4] Baldursson, F.M. (1987). Singular stochastic control and optimal stopping . Stochastics , pp. 1–40.[5] Baldursson, F.M., Karatzas, I. (1996). Irreversible investment and industry equilibrium . FinanceStoch. , pp. 69–89.[6] Blumenthal, R.M., Getoor, R.K. (1968). Markov processes and potential theory . Academic Press, NewYork.[7] Borodin, W.H., Salminen, P. (2002). Handbook of Brownian motion-Facts and Formulae . 2nd Edition.Birkh¨auser.[8] Chaleyat-Maurel, M., El Karoui, N., Marchal, B. (1980). R´eflexion discontinue et syst´emesstochastiques . Ann. Probab. , pp. 1049–1067.[9] Chaleyat-Maurel, M. (1981). R´eflexion discontinue et syst`emes stochastiques . Annales scientifiques del’Universit´e de Clermont-Ferrand 2, s´erie Math´ematiques , pp. 115–124.[10] De Angelis, T., Ekstr¨om, E. (2017). The dividend problem with a finite horizon .Ann. Appl. Probab. , pp. 3525–3546.[11] de Finetti, B. (1957). Su un’impostazione alternativa della teoria collettiva del rischio . Transactions ofthe XVth International Congress of Actuaries Vol. 2 No. 1, pp. 433–443.[12] Dickson, D.C.M., Waters, H.R. (2004). Some optimal dividend problems . ASTIN Bull. , pp. 49–74.[13] El Karoui, N., Karatzas, I. (1988). Probabilistic aspects of finite-fuel, reflected follower problems , ActaAppl. Math. , pp. 223–258.[14] El Karoui, N., Karatzas, I. (1989). Integration of the optimal risk in a stopping problem with absorp-tion , S´eminaire de Probabilit´es, tome 23, pp. 405–420.[15] El Karoui, N., Karatzas, I. (1991). A new approach to the Skorohod problem and its applications ,Stoch. Stoch. Rep. , pp. 57–82.[16] Ferrari, G. (2019). On a class of singular stochastic control problems for reflected diffusions , J. Math.Anal. Appl. , pp. 952–979.[17] Jeanblanc-Piqu´e, M. Shiryaev, A. (1995). Optimization of the flow of dividends . Russian Math. Sur-veys , pp. 257–277.[18] Jeanblanc, M., Yor, M., Chesney, M. (2009). Mathematical methods for financial markets . Springer.[19] Jiang, Z., Pistorius, M. (2012). Optimal dividend distribution under Markov regime switching . FinanceStoch. , pp. 449–476.[20] Karatzas, I. (1983). A class of singular stochastic control problems . Adv. Appl. Probab. , pp. 225–254.[21] Karatzas, I., Shreve, S.E. (1985). Connections between optimal stopping and singular stochastic controlII. Reflected follower problems . SIAM J. Control Optim. , pp. 433–451.[22] Karatzas, I., Shreve, S.E. (1991). Brownian motion and stochastic calculus (Second Edition). GraduateTexts in Mathematics 113, Springer-Verlag, New York.[23] Kobylanski , M., Quenez , M. (2012). Optimal stopping in a general framework. . Electron. J. Probab. , pp. 1–28.[24] Kulenko, N., Schmidli, H. (2008). Optimal dividend strategies in a Cram´er-Lundberg model with capitalinjections . Insur. Math. Econ. , pp. 270–278.[25] Lieberman, G.M. (1996). Second order parabolic differential equations . World Scientific.[26] Lokka, A., Zervos, M. (2008). Optimal dividend and issuance of equity policies in the presence ofproportional costs . Insur. Math. Econ. , pp. 954–961.[27] Ma, J. (1993). Discontinuous reflection, and a class of singular stochastic control problems for diffusions .Stoch. Stoch. Rep. , pp. 225–252. [28] Peskir, G., Shiryaev, A. (2006). Optimal stopping and free-boundary problems . Springer, Berlin.[29] Revuz, D., Yor, M. (1999). Continuous martingales and Brownian motion . Springer, Berlin.[30] Rogers, L., Williams, D. (2000). Diffusions, Markov processes and martingales . Cambridge Mathemat-ical Library, Cambridge.[31] Scheer, N., Schmidli, H. (2011). Optimal dividend strategies in a Cramer-Lundberg model with capitalinjections and administration costs. Europ. Actuar. J. , pp. 57–92.[32] Schmidli, H. (2008). Stochastic control in insurance . Springer-Verlag, Berlin.[33] Schmidli, H. (2016). On capital injections and dividends with tax in a diffusion approximation . Scand.Actuar. J. , pp. 751–760.[34] Shreve, S.E., Lehoczky, J.P., Gaver, D.P. (1984). Optimal consumption for general diffusions withabsorbing and reflecting barriers . SIAM J. Control Optim. , pp. 55–75.[35] Zhu, J., Yang, H. (2016). Optimal capital injection and distribution for growth restricted diffusion modelswith bankruptcy . Insur. Math. Econ. , pp. 259–271. G. Ferrari: Center for Mathematical Economics (IMW), Bielefeld University, Universit¨atsstrasse25, 33615, Bielefeld, Germany E-mail address : [email protected] P. Schuhmann: Center for Mathematical Economics (IMW), Bielefeld University, Univer-sit¨atsstrasse 25, 33615, Bielefeld, Germany E-mail address :: } { τ
S ( y ) , With these observations at hand we can now show (4.48)-(4.50).By (4.53) we have that dI ⋆s (0 , x ) = dI ⋆s (0 , z ) for all s ≥ S ( z ). Further, we have that I ⋆s (0 , z ) = 0 for all s ≤ S ( z ). Therefore, by (4.54) I ⋆s (0 , z ) = I ⋆s (0 , x ) = 0 for s ≤ S ( x ), andthe right-hand side of (4.48) rewrites as Z T m ( s ) dI ⋆s (0 , x ) − Z T m ( s ) dI ⋆s (0 , z ) = Z S ( z ) S ( x ) m ( s ) [ dI ⋆s (0 , x ) − dI ⋆s (0 , z )]= Z S ( z ) S ( x ) m ( s ) dI ⋆s (0 , x ) = Z S ( z ) S ( x ) m ( s ) dM s . (4.56)Here we have used (4.51) with y = x . PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 17 } + e − rS ( x ) κ { A τ ( x ) ≤ } i . (5.4)In the following we study optimal stopping problem (5.4) and verify the requirements ofAssumption 3.1.Moreover, by taking the sub-optimal stopping time τ = 0 in (5.4) clearly gives u ( t, x ) ≥ η for ( t, x ) ∈ [0 , T ] × (0 , ∞ ). Therefore, we can define the continuation and the stopping regionof problem (5.4) as C := { ( t, x ) ∈ [0 , T ) × (0 , ∞ ) : u ( t, x ) > η } , S := { ( t, x ) ∈ [0 , T ] × (0 , ∞ ) : u ( t, x ) = η } . PTIMAL DIVIDENDS WITH CAPITAL INJECTIONS 23
u ( t, x ) ≤ η } , t ∈ [0 , T ] , it is clear that(5.8) C = { ( t, x ) ∈ [0 , T ) × [0 , ∞ ) : 0 < x < b ( t ) } , S = { ( t, x ) ∈ [0 , T ] × [0 , ∞ ) : x ≥ b ( t ) } . Moreover, the optimal stopping time of (5.5) reads(5.9) τ ⋆ ( t, x ) := inf { s ∈ [0 , T − t ) : A s ( x ) ≥ b ( t + s ) } ∧ ( T − t ) . In the following we will refer to b as to the free boundary . The next theorem provespreliminary properties of b . Proposition 5.3. u ∞ ( x ) = η } (which exists finite, e.g., by Proposition 3.2 inFerrari [16]; see also Remark 5.1 above), we have b ( t ) ≤ b ∞ for all t ∈ [0 , T ]. (cid:3) The proof of the next proposition is quite lenghty, and it is therefore postponed in theAppendix in order to simplify the exposition. µ and denote by S ( x ; µ ) (resp. S ( x ; µ )) the hitting time of the originof the drifted Brownian Motion with drift µ (resp. µ ). Since S ( x ; µ ) ≥ S ( x ; µ ) a.s. weobtain u ( t, x ; µ ) − u ( t, x ; µ ) ≤ sup τ ∈ Λ( T − t ) E h e − rτ η (cid:0) { ττ ≥ S ( x,µ ) } + κ { τ ≥ S ( x ; µ ) } (cid:16) e − rS ( x ; µ ) − e − rS ( x ; µ ) (cid:17) i = sup τ ∈ Λ( T − t ) E h { S ( x,µ ) ≤ τ κ and using (1) and (5.7) we have b ( t ; κ ) := inf { x > u ( t, x ; κ ) ≤ η } ≥ inf { x > u ( t, x ; κ ) ≤ η } = b ( t ; κ ) . (ii) Taking η > η and using (2) and (5.7) we have b ( t ; η ) := inf { x > u ( t, x ; η ) − η ≤ } ≤ inf { x > u ( t, x ; η ) − η ≤ } = b ( t ; η ) . (iii) Taking r > r and using (3) and (5.7) we have b ( t ; r ) := inf { x > u ( t, x ; r ) ≤ η } ≤ inf { x > u ( t, x ; r ) ≤ η } = b ( t ; r ) . (iv) Taking µ > µ and that u ( t, x ; µ ) − u ( t, x ; µ ) ≤ b ( t ; µ ) := inf { x > u ( t, x ; µ ) ≤ η } ≤ inf { x > u ( t, x ; µ ) ≤ η } = b ( t ; µ ) . (cid:3) The last proposition allows us to draw some economic implications. Increasing the pa-rameters η , r , and µ , leads, at each time t , to an earlier dividends’ distribution. This resultis quite intuitive since an higher interest rate r lowers future profits due to discounting, anhigher η increases the marginal value of dividends, and an higher µ increases the surplus’trend and lowers the probability of bankruptcy, hence of capital injections. On the otherhand, an increase of κ postpones the dividends’ distribution since capital injections becomemore expensive, and the fund’s manager thus acts in a more cautious way.Proving the monotonicity of the free boundary with respect to the surplus’ volatility σ seems not to be feasible by following the arguments of the proof of Proposition 5.6. One shouldthen rely on a careful numerical analysis of the dynamic programming equation associatedto the optimal dividend problem, and we believe that such a study falls outside the scopesof this work. However, we conjecture that an increase of σ should postpone the dividends’distribution. Indeed, the larger σ is, the higher becomes the risk of the need of costly capitalinjections. As a consequence, the fund’s manager wants to wait longer before distributing anadditional unit of dividends. Such a monotonicity of the free boundary with respect to σ hasbeen recently proved by Ferrari in Proposition 4.1 of [16] in the case of a stationary optimaldividend problem with capital injections. Acknowledgments θ (cid:1) dθ (A.1) + E (cid:2) m ( t + S ( x )) { S ( x ) ≤ T − t } (cid:3) + E (cid:2) g x ( T, A T − t ( x )) (cid:3) , where Fubini’s theorem and the fact that f ′ is deterministic has been used for the integralterm above.We now investigate the three summands separately. By using Proposition 3.2.1.1 in Jean-blanc et al. [18], and recalling that the stopping boundary b is strictly positive by Assumption3.1, we have P (cid:18) x + µθ + σW θ ≥ b ( t + θ ) , S ( x ) > θ (cid:19) = P (cid:18) x + µθ + σW θ ≥ b ( t + θ ) , inf s ≤ θ ( x + µs + σW s ) > (cid:19) = P (cid:18) µσ θ + W θ ≥ b ( t + θ ) − xσ , inf s ≤ θ (cid:16) µσ s + W s (cid:17) > − xσ (cid:19) (A.2) = N (cid:18) x − b ( t + θ ) σ + µσ θ √ θ (cid:19) − e − µxσ N (cid:18) − b ( t + θ )+ xσ + µσ θ √ θ (cid:19) . Here N ( · ) denotes the cumulative distribution function of a standard Gaussian random vari-able. Note that the last term in (A.2) is continuously differentiable with respect to x for any θ > S ( x ), for x ≥T − t n } (cid:26) ηe − rτ ⋆ { τ ⋆ τ ⋆ ≥ S ( x ) } (cid:17) (cid:27)(cid:21) + E (cid:20) { τ ⋆ >T − t n } (cid:26) ηe − r ( T − t n ) (cid:0) { T − t n S ( x ) } (cid:16) e − rS ( x ) { τ ⋆ ≥ S ( x ) } − e − rS ( x n ) { T − t n ≥ S ( x n ) } (cid:17) + κ { T − t = S ( x ) } (cid:16) e − rS ( x ) { τ ⋆ ≥ S ( x ) } − e − rS ( x n ) { T − t n ≥ S ( x n ) } (cid:17) + κ { T − tτ ⋆ ≥ S ( x ) } (cid:17)i + E (cid:20) { τ ⋆ >T − t n } (cid:26) ηe − r ( T − t n ) { S ( x n ) ≤ T − t n S ( x ) } (cid:16) e − rS ( x ) { T − t ≥ S ( x ) } − e − rS ( x n ) { T − t n ≥ S ( x n ) } (cid:17) + κ { T − t = S ( x ) } + κ { T − tτ ⋆ ≥ S ( x ) } (cid:17) (cid:21) − lim n →∞ E (cid:20) { τ ⋆ >T − t n } (cid:26) ηe − r ( T − t n ) { S ( x n ) ≤ T − t n S ( x ) } (cid:16) e − rS ( x ) { T − t ≥ S ( x ) } − e − rS ( x n ) { T − t n ≥ S ( x n ) } (cid:17) + κ { T − t = S ( x ) } + κ { S ( x ) ≤ τ ⋆ ≤ T − t