On the optimality of joint periodic and extraordinary dividend strategies
OOn the optimality of joint periodic and extraordinary dividend strategies
Benjamin Avanzi a , Hayden Lau b, ∗ , Bernard Wong b a Centre for Actuarial Studies, Department of Economics, University of Melbourne VIC 3010, Australia b School of Risk and Actuarial Studies, UNSW Australia Business School, UNSW Sydney NSW 2052, Australia
Abstract
In this paper, we model the cash surplus (or equity) of a risky business with a Brownian motion.Owners can take cash out of the surplus in the form of “dividends”, subject to transaction costs.However, if the surplus hits 0 then ruin occurs and the business cannot operate any more.We consider two types of dividend distributions: (i) periodic, regular ones (that is, dividendscan be paid only at countable many points in time, according to a specific arrival process); and(ii) extraordinary dividend payments that can be made immediately at any time (that is, thedividend decision time space is continuous and matches that of the surplus process). Both types ofdividends attract proportional transaction costs, and extraordinary distributions also attracts fixedtransaction costs, a realistic feature. A dividend strategy that involves both types of distributions(periodic and extraordinary) is qualified as “hybrid”.We determine which strategies (either periodic, immediate, or hybrid) are optimal, that is, weshow which are the strategies that maximise the expected present value of dividends paid until ruin,net of transaction costs. Sometimes, a liquidation strategy (which pays out all monies and stopsthe process) is optimal. Which strategy is optimal depends on the profitability of the business,and the level of (proportional and fixed) transaction costs. Results are illustrated.
Keywords:
Risk analysis, Dividend decision processes, Control, Affine transaction costsMSC classes: 93E20, 91G70, 62P05, 91B30
1. Introduction
The literature on risk processes and their optimal control is rich (see, e.g. Albrecher and Thon-hauser, 2009; Øksendal and Sulem, 2010, for reviews). Such processes consider the surplus (orequity) of a risky business. A risky but profitable business will see cash accumulate (on average).They typically would not let their surplus grow to infinity, but to guard against the downsiderisks, they would retain some cash earnings in order to prevent bankruptcy or financial distress.In this paper, we model such surplus of cash with a stochastic process, and money distributed toshareholders will be interpreted as ‘dividends’; see also Avanzi et al. (2016a) for a discussion ofsuch surplus models from a corporate finance perspective. The question then is to determine whatthe optimal way of distributing surplus is, that is, what the optimal (so-called) ‘dividend’ strategy ∗ Corresponding author.
Email addresses: [email protected] (Benjamin Avanzi), [email protected] (Hayden Lau), [email protected] (Bernard Wong)
June 2, 2020 a r X i v : . [ q -f i n . R M ] J un s. Note that this problem is equivalent to that of determining what the optimal level of retainedcash earnings is (noting that inflows come from the business dynamics, and all outflows are labelledas ‘dividends’), but is formulated in function of what owners can control (the ‘dividends’).The natural and usual objective of this optimisation problem is to maximise the expectedpresent value of dividends paid until ruin (which occurs as soon as the surplus becomes negative).Additional historical notes and discussion of dividends in that context can be found in Avanzi(2009). This objective is also a good criterion of “stability” for the company, as it balancesprofitability (more dividends but earlier ruin) with safety (less dividends but delayed or evenabsence of ruin); see, e.g. B¨uhlmann (1970). Nevertheless, quantities such as the finite time ruinprobability have also been considered (e.g. Diasparra and Romera, 2010; Dimitrova et al., 2014),and the maximisation can be achieved on more sophisticated objectives, such as involving utilityfunctions (e.g. Bayraktar and Egami, 2010; B¨auerle and Ja´skiewicz, 2015).The recent decade has focused a lot on more realistic formulations for the dividends (see Avanziet al., 2016a, for a detailed discussion of what we mean by ‘realistic’). One of the axes of develop-ment recognises that whilst surplus models are continuous, in real life often delays occur (see, e.g.,Cheung and Wong, 2017, who consider dividend payments with implementation delays), and alsodividend decisions are usually made at periodic intervals (see, e.g., Albrecher et al., 2011b).In this spirit, we consider two types of dividend distributions: (i) periodic, regular ones (thatis, dividends can be paid only at countable many points in time, according to a specific arrivalprocess); and (ii) immediate dividend payments that can be made at any time (that is, the divi-dend decision time space is continuous and matches that of the surplus process). This matches thebehaviour of companies in real life, as most established firms would pay dividends regularly. If theyfeel the need to distribute more, then they would clearly label those extra payments as ‘extraor-dinary’ (and sometimes also do it in a different way, such as with share buy-backs, which is notin contradiction with our framework). One can find real life examples (e.g., Woodside Petroleum,2013; Wesfarmers, 2014), and was further explained by, for instance, Morningstar (2014): “Fromtime to time, companies pay out special dividends when they have had an extraordinarily goodperiod of profitability. These dividends fall outside the scope of the “normal” half-year or full-yearresult.” This possibly is to avoid signalling the fact that those extra payments should be expectedto continue in the future. Furthermore, it does make sense that those extra distributions carryheavier costs than the regular ones (actual costs, but also undesirable signalling costs such as wejust explained). We will hence penalise them with heavier fixed transaction costs.Literature on “periodic” dividends is relatively new, but attracted a lot of attention. Albrecheret al. (2011a) first proposed to use an erlangisation technique (Asmussen et al., 2002) to approx-imate the time between dividend decision times. The idea of the erlangisation technique is to setparameters such that the time between decisions is Erlang( n/γ, n ) distributed (hereafter denoted“Erlang( n )”) such that the time between decisions becomes deterministic with mean 1 /γ as n goesto infinity. This convergence was illustrated in the dual model setting (with surplus as a spectrallypositive compound Poisson process) in Avanzi et al. (2013). Avanzi et al. (2014) confirmed that aperiodic barrier strategy is optimal in dual model when the inter-dividend decision time is a sim-pler Erlang(1) variable. P´erez and Yamazaki (2017) extended those results by considering generalspectrally positive L´evy processes as the underlying surplus model. Avanzi, Tu, and Wong (2018)studied the optimal problem when the inter-dividend times is a Erlang( n ) random variable. Theauthors provided a verifying method for Brownian setting and demonstrated the optimality of aperiodic barrier strategy when n = 2. In all those cases, the type of the optimal periodic dividend2trategy is that of a barrier strategy, mirroring the analogous result for dividend decisions that canoccur at any time (see Bayraktar et al., 2014). Optimal strategies with spectrally negative L´evyprocesses are considered in Noba et al. (2018). Of closest relevance to this paper are considerationof optimal periodic (only) dividend strategies with fixed transaction costs, developed in Avanziet al. (2020b,a) for spectrally positive and negative L´evy processes, respectively.Avanzi et al. (2016b), in a dual model framework and with both types of dividends beingadmissible, show that when transaction costs are moderately cheaper for periodic dividends, thenboth types of dividends can be optimal, leading to an optimal hybrid dividend strategy. Theseresults are extended to spectrally positive L´evy processes by P´erez and Yamazaki (2018). However,those papers consider proportional transaction costs only, and in reality fixed costs are likely to bethe ones that truly differentiate the cost of “periodic” versus “immediate” dividends.In this paper we extend results on “hybrid” dividend strategies by introducing fixed transactioncosts on periodic dividends, which results in a comprehensive, more realistic treatment of optimalhybrid strategies. Furthermore, results are materially different, richer and more realistic as ex-plained below. Furthermore, the cash flow of the company is modeled by a diffusion process, whichleads to transparent and many explicit results, and is sufficient to get insights about the optimalstrategies.When the company is profitable, an optimal strategy is a hybrid ( a p , a c , b ) strategy which (1)pays non-regular dividends only when the surplus is too high (2) pays regular (periodic) dividendswhen the surplus is moderate. This strategy has some desired properties. Namely, the regulardividends are either zero or bounded. When the regular dividend is zero, either the company isat risk of bankruptcy or a recent special dividend has been paid. In either case, such behaviour isreasonable. When the company is non-profitable, the model has a different (and no less interesting)interpretation. The main results of the paper are summarised in Section 2.5, after our notation isintroduced.This paper is organised as follows. Section 2 introduces our mathematical framework. Section3 proposes a set of sufficient conditions for a strategy to be optimal, regardless of whether thebusiness is profitable. From there until Section 7, it is assumed that when the business is profitable.As an application of the results developed in Section 3, an optimal strategy is formulated whenthe proportional cost is higher than a certain threshold. Section 5 introduces the class of hybrid( a p , a c , b ) strategy and calculates the value function of a general hybrid ( a p , a c , b ) strategy. Section5 shows constructively that our candidate strategy exists among the class of hybrid ( a p , a c , b )strategy, when the proportional cost is low (lower than a certain threshold). Section 7 provesthat our candidate strategy is optimal, when the proportional cost is low. Section 8 studies theremaining case when the business is strictly non-profitable. Section 9 discusses how the differentoptimal strategies “connect” (i.e., across the Table in Section 2.5). Finally, 10 presents numericalillustrations, and Section 11 concludes.
2. The model
We define the surplus process X = { X ( t ); t ≥ } under the family of laws ( P x ; x ∈ R ) to be adiffusion process that starts at x ≥
0, i.e. X ( t ) = x + µt + σW ( t ) , (2.1)3here W = { W ( t ); t ≥ } is a standard Brownian motion. This surplus process is to be interpretedas the excess, discretionary equity available to the company to pay dividends. It is assumed thatit is sufficiently liquid to pay dividends immediately when it is so decided.We denote the expected profit per unit of time of the business as E [ X ( t + 1) − X ( t )] := µ .Unless stated otherwise, we assume that µ ≥ , (2.2)which means that the business is profitable. The opposite case will be studied in Section 8, andthe connection of the optimal strategies between the cases µ greater than, equal to, and small than0 is conducted in Section 9.1 (continuity of the barriers). In this paper, a dividend strategy is comprised of two components. Dividends can be paidat any time, but there are periodic opportunities to pay dividends at lower transaction costs.A dividend strategy must hence determine how much periodic dividends to pay and how much“immediate” (extraordinary) dividends to pay and when. For a dividend strategy π , we denote theaccumulated periodic “regular” dividend process as D πp = { D πp ( t ); t ≥ } and the accumulated non-periodic “immediate” dividend process as D πc = { D πc ( t ); t ≥ } . The strategy π is then specifiedthrough ( D πp , D πc ), and the accumulated total dividend process under strategy π is denoted as D π = { D π ( t ); t ≥ } . This means D π ( t ) = D πp ( t ) + D πc ( t ) , t ≥ . (2.3)Note that the subscripts p and c refer to the timing of the dividend decision process, be it ‘periodic’or ‘continuous’, in line with previous literature.We need to clarify mathematically how the “regular”, or periodic payment times are defined.Define N γ = { N γ ( t ); t ≥ } as a Poisson process (independent of W ) with rate E [ N γ (1)] = γ > dividend decision times . In other words, periodic dividends can onlybe paid when N γ has increments. Such times are denoted as T = { T i ; i ∈ N } with T i = inf { t ≥ N γ ( t ) = i } . (2.4)This implies that T and T i +1 − T i , i ∈ N , are i.i.d. exponential random variable with mean 1 /γ ,for all i ∈ N .A Markovian stationary strategy is a strategy where the control at time t is a deterministicfunction of X π ( t − ) known at time 0 − which maps the surplus and its characteristics into a dividendpayment, i.e. (∆ D πp ( t ) , ∆ D πc ( t )) = ( f p ( X π ( t − ))1 { t ∈ T } , f c ( X π ( t − ))1 { t/ ∈ T } ) for a given function f =( f p , f c ). For such a strategy π , if D πc ( t ) ≡
0, we call it a (pure) periodic strategy (with regularpayments only). If D πp ( t ) ≡
0, we call it a (pure) continuous strategy (with immediate paymentsonly). Otherwise, we refer it as a hybrid strategy, as there is a non-zero probability that bothcomponents are present.
Remark 2.1.
Note that if dividends can only be paid after every n increment then the time be-tween dividend decision times is Erlang distributed with shape parameter n and rate parameter nγ (the sum of n independent exponential( nγ ) random variables). This random variable can havearbitrarily small variance for appropriate choice of parameters. This is what led to the so-called“Erlangisation” technique as discussed in Asmussen et al. (2002); Albrecher et al. (2011a). Indeed, etting the parameter n increase to infinity leads to the variance of the Erlang( n ) random variableto vanish, which means arbitrary large n can approximate deterministic quantities.This motivates model setups with ‘simple’ Poissonian distribution strategies (whereby inter-dividend decision times are exponentially distributed), such as in this paper. These are an importantfirst step to solving the more general Erlang with n ≥ case. Showing optimality for n ≥ issurprisingly difficult, but not impossible; see Avanzi et al. (2018). Now, the surplus process after the dividend payments is X π = { X π ( t ); t ≥ } with X π ( t ) = X ( t ) − D π ( t ) . (2.5)We define τ π to be the ruin time of the process X π , i.e. τ π = inf { t ≥ X π ( t ) < } , (2.6)that is, the company must stop operations as soon as its surplus hits zero, and no further dividendswill be paid.By defining the filtration generated by the process ( X, N γ ) by F = { F t : t ≥ } , we say a(hybrid) dividend strategy π := { ( D πp ( t ) , D πc ( t )); t ≥ } is admissible if both D πp and D πc are non-decreasing, right continuous and F -adapted process where the sample path of the process D πc is anincreasing step function in time (as a fixed cost will be incurred at each payment), and where thecumulative amount of periodic dividends D πp admits the form D πp ( t ) = (cid:90) [0 ,t ] ν π ( s ) dN γ ( s ) , t ≥ . (2.7)Furthermore, note by definition the sample paths of X are continuous ( X ( t ) = X ( t − )) and hencewe require ∆ D π ( t ) ≤ X π ( t − ) , t ≤ τ π (2.8)that is, the dividend paid at T i —denoted ξ πi := ν π ( T i )—cannot exceed the current value of thesurplus. Denote this set of admissible strategies Π. To measure the performance of the strategies, we will focus on the expected present value ofdividends until ruin V − β,χ ( x ; π ) = V ( x ; π ) := E x (cid:90) τ π e − δt (cid:16) dD πp ( t ) + ( βdD πc ( t ) − χ )1 { ∆ D πc ( t ) > } (cid:17) , (2.9)where E x [ · ] := E [ ·| X (0) = x ] is the mathematical expectation under the law P x (for each x ∈ R ),and where δ > ξ > − β ) ξ + χ . In otherwords, there is a proportional transaction rate of 1 − β , and fixed transaction costs of χ .We seek to maximise the expected present value of dividends, which means that we will lookfor an optimal strategy π ∗ ∈ Π such that V ( x ; π ∗ ) = sup π ∈ Π V ( x ; π ) := v ( x ) = v − β,χ ( x ) , x ≥ . (2.10)5ecause the process is ruined immediately when it reaches 0, we have V (0; π ) = 0 for π ∈ Π . (2.11)Note that we will also write P and E for P and E respectively. Remark 2.2.
An optimal strategy should demonstrate the following 2 rational behaviours: The non-periodic dividend payment at time t , ∆ D π ( t ) , is either or strictly greater than χ/β . This is because any strategy that pays a non-periodic dividend less that χ/β does notcontribute positively to the value function and therefore has at most the same value functionas the same strategy without negative contributions. At periodic dividend times t = T i for some i ∈ N , we do not pay non-periodic dividends.Otherwise, a higher transaction cost is paid, yielding at most the same value function. Remark 2.3.
Note that in (2.9) periodic dividends do not attract any transaction costs. Withrespect to proportional transaction costs this is without of loss of generality, as long as proportionaltransaction costs on periodic dividends (say, − β p ) are smaller than that on immediate dividends(say, − β c ), which is what you would expect in practice as discussed earlier. In this case, (2.9) would become V ( x ; π ) = E x (cid:90) τ π e − δt (cid:16) β p dD πp ( t ) + ( β c dD πc ( t ) − χ )1 { ∆ D πc ( t ) > } (cid:17) (2.12) ≡ β p E x (cid:90) τ π e − δt (cid:16) dD πp ( t ) + ( βdD πc ( t ) − χ/β p )1 { ∆ D πc ( t ) > } (cid:17) with β = β c β p ≤ . (2.13) That is, the objective is simply scaled by a constant ( β p ), which will not affect the generality ofour set-up. In other words, an optimal strategy in problem (2.13) is optimal in problem (2.12) .However, note that the fixed transaction cost amount χ needs to be appropriately scaled if one wantsto obtain accurate numerical valued for one problem from the other.On the other hand, introduction of fixed transaction costs χ p on periodic dividends would likelyalter the form of the optimal dividend strategy fundamentally. We expect that the optimal periodicbarrier would be split into a higher trigger barrier, and lower dividend payment barrier, as is oftenthe case in classical impulse cases (because of the reason explained under item 1 in Remark 2.2).Furthermore, we postulate that ascertaining which type of dividends attracts higher transactioncosts on average or in an expected sense would be critical in determining the optimal dividendstrategy. We believe the optimal dividend strategy would depend on some sort of ‘expected’ overalltransaction costs for each type, which is not trivial to determine as the number and timing ofdividends are random in both cases, and do not match. That being said, if one assume that bothproportional and fixed transaction costs are lower on regular dividends (as opposed to immediatedividends), then extension of the current paper should be relatively straightforward.2.4. Definition of relevant dividend strategies In this section, we define all dividend strategies that we will refer to in this paper. Note thatthey are all Markovian stationary strategies as defined above just after (2.4).
Definition 2.1.
A periodic b strategy, denoted π b , is a periodic dividend strategy which pays adividend ∆ D πp ( t ) = ( X π b ( T i − ) − b )1 ( X π ( T i − ) ≥ b ) , ∆ D πc ( t ) ≡ . t time T i , as long as ruin has not occurred yet, that is, for all T i ≤ τ π b , i ∈ N . We now define the class of strategies that we prove optimal in some cases later in the paper.
Definition 2.2.
A hybrid ( a p , a c , b ) strategy with ≤ a p ≤ a c ≤ b , denoted as π a p ,a c ,b , is a strategywhich pays (before ruin) periodic dividend that brings the surplus down to a p whenever the (con-trolled) surplus X π ap,ac,b is above or equal to a p right before the dividend payment times, pays (before ruin) an immediate dividend that brings the surplus down to a c whenever thesurplus X π ap,ac,b is above or equal to b outside the periodic dividend times.In mathematical notation, it means (cid:40) ∆ D πp ( T i ) = ( X π ( T i − ) − a p )1 ( X π ( T i − ) ≥ b ) { T i ≤ τ π } ∆ D πc ( t ) = ( X π ( t − ) − a c )1 ( X π ( t − ) ≥ b ) ( t (cid:54) = T i ) { t ≤ τ π } , (2.14) with π = π a p ,a c ,b . Figure 1a illustrates the strategy described in Definition 2.2. It charts a typical sample pathwhen a hybrid ( a p , a c , b ) is applied, where a p = 1, a c = 2, and b = 4. The dotted vertical linesindicate periodic dividend decision times T i ’s. From the graph, we see that there is an immediatepayment just before T , of amount b − a c = 2 (before transaction costs). On the other hand, all T i ’s in the graph trigger periodic payments. Note that Definition 2.2 indicates that a p ≤ a c ≤ b ,but in fact Remark 2.2 implies that only the cases b > a c + χ/β (and hence b ≥ a p + χ/β since a c > a p ) make sense in order to avoid negative contribution to the value function. Note also thatthe solid vertical lines are not part of the processes. They are displayed “artificially” to illustratethe “jump” of the processes. a p a c b (a) An illustration of a hybrid ( a p , a c , b ) strategy b b T T ˜ (b) An illustration of a liquidation ( b , b ) strategy Figure 1: Illustrations of the main optimal strategies
Definition 2.3.
A liquidation ( b , b ) strategy, characterised by 2 parameters < b < b ≤ ∞ ,denoted as π b ,b , is the strategy that pays (before ruin) non-periodic dividend X ( θ ) (surplus just before ruin caused by this finaldividend), where θ = inf { t ≥ X ( t ) ∈ ( b , b ) } , the first time the surplus is within the openinterval ( b , b ) ; pays (before ruin) periodic dividend of size X π ( T − ) (surplus just before ruin caused by thisdividend) when X π ( T − ) ≤ b or X π ( T − ) ≥ b , where π = π b ,b .In mathematical notation, it means (cid:40) ∆ D πc ( t ) = X ( θ )1 ( t = θ 0, and we distinguish two cases here for notation purposes:1. Liquidation ( b , b ) strategy, with b < ∞ .2. Liquidation ( b, ∞ ) strategy, denoted as π b, ∞ , which is the liquidation ( b, b ) strategy with b = ∞ .Figure 1b illustrates the strategy of Definition 2.3. It shows different sample paths from when X (0) = 2 . X (0) = 0 . b = 1 and b = 2. When X (0) = 2 . (cid:101) T in the figure) comesbefore the surplus reaches b . As a result, the company is liquidating at (cid:101) T at that surplus level.With the other scenario (represented by the path in black), the first dividend decision time (notshown in the figure) comes after the surplus hits b and therefore it immediately liquidates at thattime, and a surplus of b = 2 (before transaction costs) is distributed. Similar behaviour can beseen when X (0) = 0 . 5. When to liquidate depends on whether the surplus touches b first (blackpath) or T comes first (grey path), conditioning on survival. Otherwise, the company is ruined(the lowest black path). Remark 2.4. The strategies mentioned above are related: The periodic strategy, denoted as π (see Definition 2.1) will sometimes also be optimalwhen µ < ; see Section 2.5. This strategy pays X π ( T − ) when T ≤ τ π , and can be seenas the limit of a liquidation ( b , b ) strategy when b − b ↓ , or simply b ↑ ∞ . It is alsodenoted as π a,a , or π ∞ , ∞ . The liquidation ( b, ∞ ) strategy can be seen as a hybrid (0 , , b ) strategy, i.e. π b, ∞ = π , ,b .Further convergence results are developed and illustrated in Section 9.2.5. Main results The nature of the optimal strategy will depend on the value of some key parameters, as isshown in this paper. Our main results are summarised in Table 1, which can also be used as aroad map for reading the paper. In addition, the transition between cells is “continuous”, exceptfor the cells in the second row for µ < 0, as they are disjoint unless β = γ/ ( γ + δ ); see Section 9for details and proofs of that statement. 8 ≥ µ < χ ≥ χ/β ≥ − µγ + δ χ/β < − µγ + δ ≤ β ≤ β Periodic barrier π b (Thm 4.1) Periodic barrier π (Thm 8.2) β < β ≤ γγ + δ Periodic barrier π b (Thm 4.1) Periodic barrier π (Thm 8.2) Liquidation π b ,b (Thm 8.2) γγ + δ < β ≤ π a p ,a c ,b (Thm 7.1) Liquidation π b, ∞ (Thm 8.2) Table 1: Map of the dividend strategies proven as optimal in the different cases considered in the paper The results can be interpreted as follows.Recall that γ/ ( γ + δ ) is the expected present value at time 0 of a payment 1 paid after an γ -exponentially distributed random amount of time, discounted with a continuous force of interest δ .At a very high level, this explains why this ratio is involved in most thresholds in the table: at anypoint in time, the model balances the choice between (i) a dollar of dividend paid immediately, withnet value involving β and χ , and (ii) a dollar paid at the next periodic time (without transactioncosts), with expected present value γ/ ( γ + δ ).Let us first focus first on the threshold γ/ ( γ + δ ) for β . Ruin is unlikely in the next instant,and if µ ≥ ξ is to be paid, the decision between paying now (as immediate dividend) orpaying later (as a periodic dividend) should depend on whether βξ − χ > γγ + δ ξ, (2.16)in an expected sense. This would be the case if and only if ξ > χβ − γγ + δ and β > γγ + δ . (2.17)This condition will indeed re-appear later when we construct our candidate strategy (which will beproved optimal); for instance in Proposition 5.7 where we use α to denote β − γ/ ( γ + δ ). It playsa significant role in determining the minimum distance between barriers a c and b .Now, when µ < 0, we must liquidate as soon as possible so only ξ > χ/β is required forimmediate dividend, because this is the amount of fixed transaction costs that need to be paid, andthe optimisation won’t require extra to compensate for future possible gains (since the business isnot profitable)—we only need the dividend to be admissible. This explains the distinction betweenthe two right columns. This can also be interpreted as follows. The quantity − µγ + δ = − µγ γγ + δ (2.18)is in fact the expected present value of the expected loss − µ/γ (in absolute terms) that will beaccumulated until the next periodic payment. Whether β times this quantity is more or less thanthe fixed transaction cost χ impacts the optimal strategy, which makes intuitive sense. This isespecially the case when χ is large, that is, when immediate dividends are very expensive. In thiscase, even for sufficiently high β < β ≤ γ/ ( γ + δ ), it will be optimal to liquidate with a periodicpayment at first opportunity ( π ), but not immediately. The threshold β will be defined in Section8. 9he expected present value of a Periodic barrier π b can be found in Avanzi et al. (2016b)or P´erez and Yamazaki (2018), that of a hybrid strategy ( a p , a c , b ) in Section 5 (with optimalparameters in Section 5), an that of a Liquidation ( b , ∞ ) strategy in Section 8. Optimality ofthose strategies is established thanks to the Verification lemma in Section 3 through the referencedTheorems in Sections 4, 7, and 8, respectively. Further illustrations are provided in Section 10. 3. A verification lemma In this section, we provide a set of sufficient conditions for a strategy π ∈ Π to be optimal,in the sense of (2.10). Recall that for a real-valued function F , the extended generator for thestochastic process X on a real-valued function F is defined to be A F ( x ) := σ F (cid:48)(cid:48) ( x ) + µF (cid:48) ( x )for x ∈ R such that the above makes sense. Throughout this paper, we will repeatedly use thefollowing lemma to prove the optimality of different dividend strategies in different cases. Lemma 3.1. For a strategy π ∗ ∈ Π , denote its value function H ( x ) := V ( x ; π ∗ ) . Suppose there isa finite set E ⊆ R + such that H satisfies H ≥ , H ∈ C ( R + ) ∩ C ( R + \ E ) , H (cid:48) is bounded on sets [1 /n, n ] for all n ∈ N , On R + \ E , H satisfies ( A − δ ) H ( x ) + γ sup ξ ∈ [0 ,x ] (cid:16) ξ + H ( x − ξ ) − H ( x ) (cid:17) ≤ , (3.1)5. On R + , H satisfies sup ξ ∈ [0 ,x ] (cid:16) ( βξ − χ )1 { ξ> } + H ( x − ξ ) − H ( x ) (cid:17) = 0 , (3.2) then π ∗ is optimal, i.e. V ( x ; π ∗ ) = v ( x ) for all x ≥ .Proof. The proof which is provided in Appendix A is standard; see for instance P´erez and Yamazaki(2017). However, it requires careful treatment of (1) the different types of strategies (2) immediatedividend at time 0 (3) the approximation for Itˆo’s lemma at the points when H is not smooth. 4. Optimality of a periodic barrier strategy when proportional transaction costs 1 − β are high In this section, we show that a periodic b strategy (see Definition 2.1) is optimal when β ≤ γ/ ( γ + δ ) and µ ≥ 0. This case corresponds to the top left cell of Table 1.From P´erez and Yamazaki (2017) it follows that there exists a optimal barrier b ∗ ≥ b ∗ strategy, π b ∗ , is optimal when dividends are only allowed to paid at the dividendpayment times. Note that this strategy is also admissible in our setting and our definition ofvalue functions for π b ∗ are the same, as there are no dividends to be paid outside the (Poissonian)10ividend payment times and periodic dividends do not attract transaction costs. Therefore, we canborrow the results from P´erez and Yamazaki (2017) regarding the behaviour of the value function V ( · ; π b ∗ ), which is summarised as follows:1. The first 4 conditions in Lemma 3.1 hold, with the finite set E = { b ∗ } .2. When b ∗ > 0, the function is concave. In particular, we have V (cid:48) ( x ; π b ∗ ) > , x ∈ (0 , b ∗ )= 1 , x = b ∗ ∈ ( γγ + δ , , x ∈ ( b ∗ , ∞ ) . 3. When b ∗ = 0, we have(a) If µ > 0, then 1 ≥ V (cid:48) (0+; π b ∗ ) > V (cid:48) ( x ; π b ∗ ) > γ/ ( γ + δ ) ≥ β > 0, for x > µ = 0, then 1 > V (cid:48) ( x ; π b ∗ ) = γ/ ( γ + δ ) ≥ β > 0, for x > π b ∗ is optimal, it suffices to show (3.2), i.e.sup ξ ∈ [0 ,x ] (cid:16) ( βξ − χ )1 { ξ> } + V ( x − ξ ; π b ∗ ) − V ( x ; π b ∗ ) (cid:17) ≤ , x > . Denote H ( ξ ) = βξ − χ + V ( x − ξ ; π b ∗ ) − V ( x ; π b ∗ ) , x > , and by taking derivative w.r.t. ξ , we get H (cid:48) ( ξ ) = β − V (cid:48) ( x − ξ ; π b ∗ )which is always non-positive when µ ≥ 0. Hence, the supremum of H on [0 , x ] is attained at ξ = 0with value H (0) = − χ < 0. This shows that the left hand side of (3.2) is max(0 , − χ ) = 0.The above result is restated as the following theorem. Theorem 4.1. When µ ≥ and ≤ β ≤ γ/ ( γ + δ ) , the periodic b ∗ strategy is optimal, where b ∗ is specified in the third item in Proposition 9.2. 5. The hybrid ( a p , a c , b ) strategy In this section, we calculate the expected present value of dividends of a general hybrid ( a p , a c , b )strategy and then pick a candidate strategy from the class using a “maximisation principle”. Wewill first use scale functions to derive some general results then use the classical PDE method tospecialise to the case of diffusions. We make use of the fluctuation theory for L´evy processes (see,e.g. P´erez and Yamazaki, 2018, Section 6, and references therein). Denote Ψ( θ ) = 1 /t log E ( e θX ( t ) ), θ ∈ R , as the Laplace exponent of a spectrally negative L´evyprocess X . Then for q ≥ 0, the q -scale function W q is the mapping from R to [0 , ∞ ) that takesthe value zero on the negative half-line, while on the positive half-line, it is a strictly increasingfunction that is defined by its Laplace transform: (cid:90) ∞ e − θx W q ( x ) dx = 1Ψ( θ ) − q , θ > φ ( q ) , where φ ( q ) = sup { λ ≥ λ ) = q } . 11n particular, when X is a diffusion process (defined by (2.1)) and q > 0, we have W q ( x ) = e r ( q ) x − e s ( q ) xσ ( r ( q ) − s ( q ) ) , (5.1)where r ( q ) > s ( q ) < θ ) − q = 0 ⇐⇒ σ θ + µθ − q = 0 . In addition, we also define W r,q,a ( x ) := W q ( x ) + r (cid:90) x − a W q + r ( x − y − a ) W q ( y + a ) dy, x ≥ a,W q ( x ) := (cid:90) x W q ( y ) dy, x ≥ ,W q ( x ) := (cid:90) x W q ( y ) dy, x ≥ . In the following, we slightly abuse the notation and assume that the barriers ( a p , a c , b ) are givenas ( a, a c , b ) and therefore denote the value function V ( x ) := V ( x ; π a,a c ,b ). If the dependence on thestrategy π or costs 1 − β and χ needs to be stressed, we will write the full version V ( x ; π a,a c ,b ), or V − β,κ ( · ), respectively. The value function V is given by the following lemma. Lemma 5.1. For a given hybrid ( a p , a c , b ) strategy (with barrier levels < a p = a ≤ a c < b ), itsvalue function is continuous and is given by V ( x ) = V ( a ) W δ ( a ) W δ ( x ) x ∈ ( −∞ , a ) , V ( a ) W δ ( a ) G ( a, x ) − γW γ + δ ( x − a ) x ∈ ( a, b ) ,β ( x − a c − κ ) + V ( a c ) x ∈ ( b, ∞ ) , (5.2) with G ( a, x ) := W δ ( x ) + (cid:90) x − a W γ + δ ( x − a − y ) (cid:16) W δ ( a + y ) − W δ ( a ) (cid:17) dy (5.3) where V ( a ) , V ( a c ) and V ( b ) can be found by solving three linear equations in them.For barrier levels a p = a ≤ a c < b , we use V ( a ) W δ ( a ) := β ( y − κ ) + γ ( W γ + δ ( y + l ) − W γ + δ ( l )) G ( a, y + l ) − G ( a, l ) , (5.4) which also holds for the above case when a p = a > .Proof. Using the notations introduced above, with a minor modification of the proofs in Sections5.1, 6.1 and 6.2 in P´erez and Yamazaki (2018), we can deduce that for a > V ( x ) = W δ ( x ) W δ ( a p ) V ( a p ) , x ∈ ( −∞ , a p ]12nd V ( x ) = V ( a p )( W γ,δ,a p ( x ) W δ ( a p ) − γW γ + δ ( x − a p )) − γW γ + δ ( x − a p ) , x ∈ [ a p , b ] , which is the same as (5.2) after some rearrangement. The case for a = 0 can be carried over by alimit argument as in Noba et al. (2018).From (5.2) we see that V ( x ) = β ( x − κ ) + (cid:16) V ( a c ) − βa c (cid:17) , x > b,V ( x ) = W δ ( x ) (cid:16) V ( a ) W δ ( a ) (cid:17) , x ≤ a. Hence it is reasonable to attempt maximisation of V ( a c ) − βa c or V ( a ) /W δ ( a ) w.r.t. the parameters( a, a c , b ) (and we will see both approaches are equivalent). We now proceed to pick a candidate strategy from the class of hybrid ( a p , a c , b ) strategies. A“nice” hybrid ( a p , a c , b ) strategy is characterised by the derivatives of its value function at theboundaries, see e.g. Avanzi et al. (2020b, Remark 9.2) for an intuitive explanation. We postulate(and later show) that those “nice” properties will lead to the optimal set of strategies, and hencerefer to those as candidates. Definition 5.2. A strategy is said to be a nice hybrid ( a p , a c , b ) strategy if the following are satisfied: It is a hybrid ( a p , a c , b ) strategy (see Definition 2.2); b ≥ a c + χ/β and V (cid:48) ( b ) = β ; Either a c = a p and V (cid:48) (0) ≤ β , or V (cid:48) ( a c ) = β ; Either a p = 0 and V (cid:48) (0) ≤ , or V (cid:48) ( a p ) = 1 . In the following, we will re-parametrise ( a, a c , b ) using ( a, l, y ) with l := a c − a and y := b − a c .The support of ( a, l, y ) is [0 , ¯ a ] × [0 , ∞ ) × [ κ, ∞ ), where ¯ a is the unique solution for W (cid:48)(cid:48) δ ( x ) = 0 if itexists, otherwise ¯ a = 0. We chose to maximise V ( a c ) − βa c . Regarding the auxiliary function G ,it is easy to see ∂∂a G ( a, d ) = ∂∂d G ( a, d ) − γW (cid:48) δ ( a ) W γ + δ ( d ) and ∂∂d G ( a, d ) > . (5.5)For the derivatives of the value function at a p = a , a c and b , we have V (cid:48) ( a ) = V ( a ) W δ ( a ) W (cid:48) δ ( a ) , (5.6) V (cid:48) ( a c ) = V ( a ) W δ ( a ) ∂∂l G ( a, l ) − γW γ + δ ( l ) , (5.7) V (cid:48) ( b ) = V ( a ) W δ ( a ) ∂∂ ( y + l ) G ( a, y + l ) − γW γ + δ ( y + l ) . (5.8)13e will first show that the derivative conditions are satisfied, provided a maximiser exists forour objective function V ( a c ) − βa c = V ( a ) W δ ( a ) (cid:16) W δ ( x )+ (cid:90) l W γ + δ ( l − y ) (cid:16) W δ ( a + y ) − W δ ( a ) (cid:17) dy (cid:17) − γW γ + δ ( l ) − β ( a + l ) . (5.9)We will then show the existence of the maximiser. The following proposition illustrate the proper-ties of a set of optimal parameters ( a, l, y ), assuming its existence. For the moment, we make thefollowing assumption which will be lifted by Proposition 5.6. Assumption 5.3. For any a ≥ , we have ∂∂x G ( a, x ) γW γ + δ ( x ) < , x ≥ . (5.10) Proposition 5.4. Denote ( a ∗ , l ∗ , y ∗ ) a maximiser of the objective function V ( a c ) − βa c and werecall the support is a subset of a ∈ [0 , ¯ a ] . Under Assumption 5.3, if ( a ∗ , l ∗ , y ∗ ) lie in the interiorof the support, then we can conclude that with such choice of parameters, we have V (cid:48) ( a ) = 1 , V (cid:48) ( a c ) = β, V (cid:48) ( b ) = β. (5.11) Otherwise, if a ∗ = 0 , then V (cid:48) (0) ≤ ; if l ∗ = 0 , we have a ∗ = l ∗ = 0 and V (cid:48) (0) ≤ . These are theonly boundary cases.Proof. Since ( a ∗ , l ∗ , y ∗ ) is a maximiser and the objective function is differentiable in the arguments,all the partial derivatives are zero (except in the boundary which requires extra care). In summary,the proof requires a direct checking in the argument y , then l then a , assisted with the help ofequations (5.6)-(5.8); see Appendix B for details.Although the proof of Proposition 5.4 is simple and is similar to existing proofs in the literature(e.g., Loeffen, 2008a), it presents the main ingredients in showing the existence of a candidatestrategy characterised by three non-zero parameters. This is generally a difficult problem sinceexplicit calculation is often impossible. To our best knowledge, this is the first time such problemhas been solved. Remark 5.1. From the proof in Appendix B, we see that maximising V ( a c ) − βa c is the same asmaximising V ( a ) /W δ ( a ) . From the formula of V ( a ) /W δ ( a ) in (5.4) , we see that the a -argumentof maximiser of V ( a ) /W δ ( a ) cannot live outside [0 , ¯ a ] , which justifies our choice of a narrowersupport a ∈ [0 , ¯ a ] . We now proceed to show the existence of a local maximiser ( a ∗ , l ∗ , y ∗ ). Due to the complexityof the calculation using scale functions (generally one assumes completely monotonic L´evy densityand proceed with complicated calculations, see e.g. Noba et al., 2018), we specialise our calculations14sing the classical PDE methods. Denote the following functions: ψ ( θ ) := σ θ + µθ (5.12) f ( x ) := e r x − e s x , (5.13) g ( x ) := e r x − e s x , (5.14) J ( x ) := − s g ( x ) + ( r − s )( e s x − , (5.15) J (cid:48) ( x ) = − r s g ( x ) , (5.16)where ( r , s ) and ( r , s ) are the positive and negative roots of equations ψ ( θ ) − δ = 0 and ψ ( θ ) − γ − δ = 0 respectively, with | s i | > | r i | , i = 0 , µ > f , g and J areproportional to the scale functions W δ , W γ + δ and W γ + δ , respectively, see equation (5.1).Before stating the value function in terms of f , g and J , we shall discuss the smoothness of thevalue function for the PDEs to be solved. From the proof of Lemma 5.1, we can conclude that thevalue function V is continuous, continuously differentiable except at { } , and twice differentiableexcept at 0 and at b , i.e. V ∈ C ( R ) ∩ C ( R \{ } ) ∩ C ( R \{ , b } ).The following proposition provides an alternative characterisation for the value function of ahybrid ( a p , a c , b ) strategy. Proposition 5.5. For given ( a, a c , b ) with b > a c + χ/β and a c ≥ a ≥ , the value function of thehybrid ( a, a c , b ) strategy is given by V ( x ; π a,a c ,b ) = , x ∈ ( −∞ , C ( e r x − e s x ) , x ∈ [0 , a ) A ( e r ( x − a ) − e s ( x − a ) ) + Be s ( x − a ) + γγ + δ ( x − a + µγ + δ + V ( a )) , x ∈ [ a, b ) β ( x − a c ) − χ + V ( a c ) , x ∈ [ b, ∞ ) , (5.17) with l = a c − a , d = b − a , g ( d, l ) = g ( d ) − g ( l ) , J ( d, l ) = J ( d ) − J ( l ) , C = ( r − s ) (cid:0) ( β − γγ + δ )( d − l ) − χ (cid:1) + γγ + δ g ( d, l ) + γµ ( γ + δ ) J ( d, l ) δγ + δ f ( a ) J ( d, l ) + f (cid:48) ( a ) g ( d, l ) , (5.18) B = δγ + δ Cf ( a ) − γγ + δ µγ + δ , (5.19) A = 1 r − s (cid:16) Cf (cid:48) ( a ) − Bs − γγ + δ (cid:17) (5.20) and V ( a ) = Cf ( a ) (5.21) V ( a c ) = Ag ( l ) + Be s l + γγ + δ ( l + µγ + δ + Cf ( a )) . (5.22) We also adopt the (unusual) convention that [0 , 0) = ∅ in (5.17) .Proof. Formulas can be derived by either directly substitute the formula of (5.1) in Lemma 5.1, orby a classical PDE approach. 15o far, Proposition 5.4 holds for general spectrally negative L´evy processes (where Assump-tion 5.3 may or may not hold). We now specialise our results in the diffusion setting, and showin Proposition 5.6 that Assumption 5.3 always holds for diffusion processes, so we can use theconclusion of Proposition 5.4 freely. Proposition 5.6. Assumption 5.3 holds for diffusion processes.Proof. The result stems directly from the explicit formula given by Proposition 5.5; see AppendixC for details.Thanks to Proposition 5.5, we have an explicit formula for the value function. We are nowready to show the following proposition. Proposition 5.7. There exists a triplet ( a ∗ , l ∗ , y ∗ ) ∈ B such that the “derivative conditions” (5.11) hold.Proof. Thanks to Propositions 5.6 and 5.4, it remains to construct a large enough box to containthe maximum of the objective V ( a c ) − βa c . See Appendix D for details. We are now able to derive some sufficient conditions for liquidation strategies to be optimal.Denote the functions Q ( a ) := 1 − f ( a ) /f (cid:48) ( a ) µ/δ (5.23)and I ( x, q ) := β − γγ + δ + (cid:16) γγ + δ − − s γγ + δ − s + q ( r + s ) (cid:17) e s x g (cid:48) ( x ) + g ( x )( − r s ) µγ + δ (1 − q ) , (5.24)where Q maps the periodic lower barrier a p ∈ [0 , ¯ a ] to a number q ∈ [0 , 1] in an decreasing manner,whereas I ( · , q ) is a function decreasing to 0 at infinity after it achieves its maximum.For a hybrid ( a p , a c , b ) strategy, if V (cid:48) ( b ) = β , we have (using the formula for C given by (5.18)) V (cid:48) ( a p ) − − s γγ + δ − s + Q ( a p )( r + s ) = ( r − s ) I ( y + l, q ) < ( r − s ) βg (cid:48) ( g ) =: ε k , with g is a constant representing the minimum of the denominator of (5.24) w.r.t. ( x, q ).Hence, if − s γγ + δ r ≤ − ε k , we will choose a = 0 ( Q ( a ) = 1). If we consider ε k = 0, this condition is equivalent to the conditionin Remark 4.1(i) in Noba et al. (2018), i.e. γ ≤ ( σ / r .Likewise, if − s γγ + δ r ≤ β − ε k , 16e will choose a = l = 0 ( a p = a c = 0). In fact, for any q such that − s γγ + δ r + q ( r + s ) ≥ β, we have I (0 , q ) ≤ q there are always ( l, y + l ) with l > I ( l, q ) = I ( y + l, q ).This implies that when 1 − ε k ≥ − s γγ + δ r ≥ β, we will choose a = 0 but l > a p = 0 < a c ).In practice, ε k is usually negligible, compared to the size of β . Likewise, we can see that once wehave the smoothness condition V (cid:48) ( b ) = β , the right hand side is (almost) negligible and thereforethe derivative at a is (almost) independent of both y and l .In terms of the parameters in the model, we have − s r = 1 + 1 + √ νν , with ν := (cid:16) σµ (cid:17) ( γ + δ )and therefore the above two sufficient conditions can be rewritten as γγ + δ √ νν + ε k ≤ δγ + δ and γγ + δ √ νν + ε k ≥ β − γγ + δ . Note further (1 + √ x ) /x is decreasing in x , hence there are thresholds ν ( γ, δ ) and ν β ( γ, δ )(with ν β > ν unless β ≥ − ε k ) such that ν β ( γ, δ ) ≥ (cid:16) σµ (cid:17) ( γ + δ ) ≥ ν ( γ, δ ) = ⇒ a = 0 , l > (cid:16) σµ (cid:17) ( γ + δ ) ≥ ν β ( γ, δ ) = ⇒ a = 0 , l = 0 . Note the sufficient conditions for liquidation at first opportunity ( a = 0) do not depend onthe transaction costs 1 − β and χ as one can always ignore the opportunities to pay immediatedividends. However, it does depend on the time parameters for the frequency periodic paymentsand discounting ( γ and δ ), as well as the “coefficient of variation” σ/µ , a measurement of theriskiness of the business. 6. The derivative of the value function of our candidate strategy From the previous section, we see that there exists a nice hybrid ( a p , a c , b ) strategy (see Defini-tion 5.2 and Propositions 5.4, 5.7). In other words, there are ( a p , a c , b ) such that a p ≤ ¯ a , V (cid:48) ( a p ) = 1(or a p = 0 and V (cid:48) (0) ≤ V (cid:48) ( a c ) = β (or a p = a c = 0 and V (cid:48) (0) ≤ β ) and V (cid:48) ( b ) = β . We will pickthis strategy and refer it as our candidate strategy, and use V to denote its value function.In this section, we first establish some results regarding the derivative of the value function ofour candidate strategy. They will then be used to verify the optimality of our strategy in Section7. As explained before (e.g., Sections 2.5 and 4) we must have γγ + δ < β ≤ , (6.1)which becomes apparent in some areas of the proof.17ur goal in this section is to establish Lemma 6.2. In order to do that, we need to firstestablish Lemma 6.1 below, which concerns the behaviour of the derivative of the value functionof our candidate strategy. Lemma 6.1. Regarding the derivative of the value function, we have the following: Suppose a p > , then V (cid:48) ( x ) > , x ∈ [0 , a p )= 1 , x = a p ∈ ( β, , x ∈ [ a p , a c )= β, x = a c ∈ (0 , β ) , x ∈ ( a c , b )= β, x ∈ [ b, ∞ ) . (6.2)2. Suppose a p = 0 and a c > , then V (cid:48) ( x ) ∈ ( β, , x = 0 ∈ ( β, , x ∈ (0 , a c )= β, x = a c ∈ (0 , β ) , x ∈ ( a c , b )= β, x ∈ [ b, ∞ ) . (6.3)3. Suppose a p = a c = 0 , then V (cid:48) ( x ) ∈ (0 , β ] , x = 0 ∈ (0 , β ) , x ∈ (0 , b )= β, x ∈ [ b, ∞ ) . (6.4) In any case, we have V (cid:48) > on [0 , ∞ ) .Proof. The proof requires analysing the functional form of the value function with the derivativeconditions imposed for the candidate strategy; see Appendix E for details.The next Lemma shows that our candidate strategy satisfies the last 2 conditions in Lemma3.1. Lemma 6.2. The value function of a nice hybrid ( a p , a c , b ) strategy, V , satisfies ( A − δ ) V ( x ) + γ sup ξ ∈ [0 ,x ] (cid:16) ξ + V ( x − ξ ) − V ( x ) (cid:17) ≤ , x ∈ R + \{ b } (6.5) and sup ξ ∈ [0 ,x ] (cid:16) ( βξ − χ )1 { ξ> } + V ( x − ξ ) − V ( x ) (cid:17) = 0 , x ∈ R + . (6.6) Proof. The result comes as a straightforward consequence of Lemma 6.1; see Appendix F fordetails. 18 . Optimality in case of profitable business ( µ ≥ In this section, we show the optimality of a nice hybrid ( a p , a c , b ) strategy, which is the followingtheorem. Theorem 7.1. Suppose µ ≥ . Denote V the value function of a nice hybrid ( a p , a c , b ) strategy.Suppose π ∈ Π , then V ( x ) ≥ V ( x ; π ) for all x ≥ . In other words, any nice hybrid ( a p , a c , b ) strategy is optimal.Proof. Thanks to Lemma 3.1, we only need to check that V satisfies all conditions proposed, whichis essentially Lemma 6.1 and 6.2, with the finite set E = { b } .We now present a corollary regarding the uniqueness of nice hybrid ( a p , a c , b ) strategies. Corollary 7.2. Suppose µ ≥ . There is one and only one nice hybrid ( a p , a c , b ) strategy. Denoteits parameters ( a ∗ p , a ∗ c , b ∗ ) . Hence, the hybrid ( a ∗ p , a ∗ c , b ∗ ) strategy is optimal.Proof. Lemma 6.1 characterised the derivative of the value function, which together with theoptimality implies uniqueness. A similar proof can be found in Avanzi et al. (2020b, Lemma9.3). 8. Optimality in case of unprofitable business ( µ < In this section, we discuss the optimal strategy when the business is strictly unprofitable, i.e. µ < 0. As such, solely in this section, we make the following assumption. Assumption 8.1. We assume µ < . Recall that µ < χ > χ = 0 can be seen as thecase when χ ↓ b , b ) strategy, characterised by 2 parameters0 < b < b ≤ ∞ , denoted as π b ,b (see Definition 2.1) and the periodic 0 strategy, denoted as π (see Definition 2.3). The latter pays X π ( T − ) when T ≤ τ π , and it can also be seen as the limitof a liquidation ( b , b ) strategy when b − b ↓ 0, or simply b ↑ ∞ . Therefore, it is also denotedas π a,a , or π ∞ , ∞ .For β > γ/ ( γ + δ ), it should be intuitively clear that the form of π b, ∞ is optimal if we can choosethe lower barrier b nicely. On the other hand, for β < γ/ ( γ + δ ), we proceed the following. It isknown that (e.g. from P´erez and Yamazaki, 2017) that V (cid:48) ( x ; π ) is increasing in x to γ/ ( γ + δ ).Therefore, for V (cid:48) (0; π ) < β < γ/ ( γ + δ ), there is a unique a β > V (cid:48) ( a β ; π ) = β . Ifmoreover V ( a β ; π ) < βa β − χ, (8.1)then there is a unique c β,χ such that 0 < c β,χ < a β with V ( c β,χ ; π ) = βc β,χ − χ. Note that (8.1)is equivalent to χβ < a β − V ( a β ; π ) β . (8.2)Denote the right hand side as a function of β , i.e.Λ( β ) = a β − V ( a β ; π ) β , (8.3)19hen we have thatΛ is increasing from 0 to the limit − µγ + δ when β increases on the interval [ V (cid:48) (0; π ) , γγ + δ );(8.4)a proof of which is provided in Appendix G. Hence, (8.1) is only possible when χ < β ( − µ/ ( γ + δ )).To further explain what it means, note when the surplus is x and we can choose either (1) liquidatenow, or (2) liquidate in the next Poissonian time, we need to consider the trade-off. If we liquidatenow, the fixed cost is χ . If we wait, assuming ruin is not an issue, the expected (discounted) lossin surplus is then E ( − µT e − δT ) = − γµ ( γ + δ ) , where we recall that T is an exponential random variable with mean 1 /γ . Hence, when χ ≥ β ( − µ/ ( γ + δ )) and β ≤ γ/ ( γ + δ ), we shall never liquidate immediately. On the other hand, if χ < β ( − µ/ ( γ + δ )), then in view of (8.2) and (8.4), there is a β ∈ ( V (cid:48) (0; π ) , γ/ ( γ + δ )) definedby Λ( β ) = χ/β such that (8.1) does not hold whenever β ∈ ( V (cid:48) (0) , β ) and (8.1) holds whenever β ∈ ( β , γ/ ( γ + δ )).In light of the above analysis, we should not be surprised with the results in this section. Theyare summarised by the following theorem. Theorem 8.2. For µ < , we have the following:Case 1: χ ≥ β − µγ + δ . We have (a) For β ∈ (0 , γ/ ( γ + δ )] , the periodic strategy is optimal. (b) For β ∈ ( γ/ ( γ + δ ) , , a liquidation ( b, ∞ ) strategy is optimal with b > characterisedby V (cid:48) ( b − ; π b, ∞ ) = β = V (cid:48) ( b +; π b, ∞ ) . Case 2: χ < β − µγ + δ . Denote β := Λ − ( χβ ) , we have (a) For β ∈ (0 , β ] , the periodic strategy is optimal. (b) For β ∈ ( β , γ/ ( γ + δ )) , a liquidation ( b , b ) strategy is optimal, with ( b , b ) such that < c β,χ < b < a β < b < ∞ and V (cid:48) ( b − ; π b ,b ) = V (cid:48) ( b +; π b ,b ) = β = V (cid:48) ( b − ; π b ,b ) = V (cid:48) ( b +; π b ,b ) . (c) For β ∈ [ γ/ ( γ + δ ) , , a liquidation ( b, ∞ ) strategy is optimal, with b > characterisedby V (cid:48) ( b − ; π b, ∞ ) = β = V (cid:48) ( b +; π b, ∞ ) . In order to prove Theorem 8.2, we will need the following lemmas. Our first lemma calculatesthe value function for each strategy. Lemma 8.3. The value function of a periodic strategy is given by V ( x ; π ) = − γµ ( γ + δ ) e s x + γγ + δ ( x + µγ + δ ) , x ≥ . (8.5)20 he value function of a liquidation ( b , b ) strategy (with < b l < b < ∞ ) is given by V ( x ; π b ,b ) = Ag ( x ) − γµ ( γ + δ ) e s x + γγ + δ ( x + µγ + δ ) , x ∈ [0 , b ) βx − χ, x ∈ [ b , b ) Be s x + γγ + δ ( x + µγ + δ ) , x ∈ [ b , ∞ ) , (8.6) with A = A ( b ) := ( β − γγ + δ ) b − χ − γµ ( γ + δ ) (1 − e s b ) g ( b ) (8.7) and B can be determined using V ( b − ) = V ( b ) , in the case where b < ∞ .The value function of a liquidation ( b, ∞ ) strategy, with b > is simply (8.6) without the x ∈ [ b , ∞ ) branch, where the formula for A is still the same and we do not need to compute B . We omit the proof for Lemma 8.3 as it can be obtained easily by solving PDE with the continuityof the value functions on the boundaries being the boundary conditions. Remark 8.1. The following results will be repeatedly used in what follows, V ( x ; π b ,b ) = V ( x ; π ) + Ag ( x ) , and V (cid:48)(cid:48) ( x ; π b ,b ) = Ag (cid:48)(cid:48) ( x ) − γµ ( γ + δ ) s e s x , x ≤ b , where b can possibly be infinity. Therefore, we have V (cid:48)(cid:48) ( x ) > if A ≥ . The next lemma is also an argument which will be used over time. Lemma 8.4. For a liquidation ( b, b ) strategy, with < b < b ≤ ∞ , we have ∂∂b A ( b ) > if and only if V (cid:48) ( b − ; π b,b ) < β. We can replace the inequalities by equalities simultaneously.Proof. This can be derived through direct computation using (8.7).The next 2 lemmas establish the existence of the candidate strategy described in Theorem 8.2(for every possible case). Lemma 8.5. If V (cid:48) (0; π ) < β < γγ + δ and V ( a β ; π ) < βa β − χ , then there are ( b , b ) with < c β < b < a β < b < ∞ such that V (cid:48) ( b − ; π b ,b ) = V (cid:48) ( b +; π b ,b ) = β = V (cid:48) ( b − ; π b ,b ) = V (cid:48) ( b +; π b ,b ) . Proof. The proof is based on continuity arguments. For example, for b , if one denotes the objectivefunction b (cid:55)→ (cid:101) O ( b ) := V (cid:48) ( b +) − β , it suffices to show that (cid:101) O ( c β ) (cid:101) O ( a β ) < 0. Details are providedin Appendix H. Lemma 8.6. If γ/ ( γ + δ ) < β ≤ , then there is a b > χ/β such that V (cid:48) ( b − ; π b, ∞ ) = β .Proof. The proof is similar to that of Lemma 8.5; see Appendix ILemmas 8.5-8.6 shows that our candidate strategy (as described in Theorem 8.2) exists. Ourlast lemma below shows that the derivative of its value functions is increasing, which essentiallycompletes the proof of Theorem 8.2. 21 emma 8.7. In each case considered in Theorem 8.2, the derivative of the value function isincreasing for our candidate strategy.Proof. We only need to prove the case when the liquidation ( b , b ) strategy (with b ≤ ∞ ) isoptimal.From the proof of Lemma 8.5, if we choose b to be the smallest one, then we have that A ( b ) > 0, as A is increasing from 0 at c β to b . This implies that V (cid:48)(cid:48) ( x ; π b ,b ) > V (cid:48) ( x ; π b b ) is an increasing function on [0 , b ]. Now, from V (cid:48) ( b − ; π b ,b ) = Bs e s b + γγ + δ = β, we can deduce that B > V (cid:48) ( x ; π b ,b ) is also increasing on [ b , ∞ ). Thiscompletes the proof for the case when V (cid:48) (0; π ) < β < γ/ ( γ + δ ) and V ( a β ; π ) < βa β − χ .From the proof of Lemma 8.6, if we choose b to be the smallest one, using the same argument,we have that A ( b ) > 0, which shows that V (cid:48) ( x ; π b, ∞ ) is increasing on [0 , b ] and hence on [0 , ∞ ).Note the value function of the candidate strategy in each case is continuously differentiable andtwice differentiable except at the boundary points. Therefore the first 3 conditions of Lemma 3.1are automatically satisfied. What is yet to be shown are the last 2 conditions of Lemma 3.1, whichcan be done in a similar way to what has been done in Sections 3 and 6. Hence, Theorem 8.2 holds. 9. On the convergence of strategies across solution thresholds In this section, we discuss the convergence of strategies when µ ↑ β ↓ γ/ ( γ + δ ).These correspond to major thresholds in Table 2.5 on page 8. By convergence in strategy, wemean point-wise converge of the value function at each x ≥ 0. Since barriers determine the valuefunction, convergence in the barriers implies convergence in strategy. µ ↑ µ goes from negative to positive.We start by some numerical exploration. Our baseline parameters are ( σ, χ ) = (0 . , . γ, δ ) = (1 , . 15) (time parameters). Note that the crucial constant γ/ ( γ + δ ) ≈ . 87, and the form of optimal strategies (rather, their connection here) will potentiallydiffer on either side of that constant. Hence, we will illustrate the cases β = 0 . β = 0 . − β ), we can see that around the neigh-bourhood of 0, the periodic 0 strategy is optimal. On the other hand, when the proportional cost islow, we can see from Figure 2b that a liquidation ( b, ∞ ), or equivalently a hybrid (0 , , b ) strategyis optimal around the neighbourhood of 0. This observation is true in general thanks to Lemma9.1 below.Further analysis of Figure 2a is interesting. The vertical grey dashed line in Figure 2a cor-responds to the threshold between the second-last and last columns of Table 2.5 (in the secondrow), so that obviously β < . b becomes strictly positive)corresponds to the first column. 22 - - b * b * b * (a) β = 0 . - - b * b * a c * a p * (b) β = 0 . Figure 2: Continuity of optimal barriers at µ = 0 Lemma 9.1. When β > γ/ ( γ + δ ) , a hybrid (0 , , b ) strategy is optimal for µ = 0 . Moreover,when µ ↑ , the lower barrier of the optimal ( b , ∞ ) strategy, denoted as b = b ( µ ) converges to b . When β ≤ γ/ ( γ + δ ) , the periodic barrier strategy with barrier level , π , is optimal on theneighbourhood of µ = 0 .Proof. For each case, one needs to check the corresponding barrier converges to the barrier at µ = 0when µ ↑ 0; see details in Appendix J. β ↓ γ/ ( γ + δ )The convergence of the barriers when β ↓ γ/ ( γ + δ ) is described in the following proposition. Proposition 9.2. Recall the function Q in (5.23) . When β ↓ γγ + δ , we have the following: If − s r γγ + δ > , a p = Q − (cid:16) s δγ + δ r + s (cid:17) , a c = ∞ , b = ∞ , where “ = ” is in limit sense. If − s r γγ + δ ≤ , a p = 0 , a c = ∞ , b = ∞ , where “ = ” is in limit sense. Q − (cid:16) s δγ + δ r + s (cid:17) = b ∗ and lim a c →∞ ,b →∞ V ( x ; π a p ,a c ,b ) = V ( x ; π b ∗ ) for all x ≥ . That is, both thebarrier and the value function exhibit continuity behaviour at β = γγ + δ .Proof. The proof requires functions Q and I ( x, q ) introduced in Section 5.3. The first two partsrequires an investigation of the condition I ( l, Q ( a )) = I ( l + y, Q ( a )), whereas the last part calculatesrequire a verification of the condition V (cid:48)(cid:48)(cid:48) ( b ∗ +) = V (cid:48)(cid:48)(cid:48) ( b ∗ − ) with the given formula for b ∗ . Detailsare provided in Appendix K. Note when µ < 0, similar continuity results hold, as shown in Appendix M sequentially for the4 different cases enumerated in Theorem 8.2. 23 0. Numerical illustrations In this section, We illustrate numerically some results from previous sections. The first andthe second sections are devoted to the case when µ > µ < µ = 0) has been discussed in Section 9.1. µ > ) Our baseline setting includes: scale parameters ( µ, σ, χ ) = (1 , . , . γ, δ ) = (1 , . 15) and proportional transaction cost parameter β = 0 . 0. In particular, we have β > γ/ ( γ + δ ) which guarantees that the hybrid ( a ∗ p , a ∗ c , b ∗ ) strategy is optimal. In the following,we will study the impact of the parameters on the optimal barriers. χ a p * b * a c * (a) Fixed cost χ - b * a p * b * a c * (b) Proportional cost 1 − β - b * a p * a c * (c) 1 − β (zoomed in) Figure 3: Impact of transaction costs. Solid vertical line: β = γ/ ( γ + δ ). Dotted line: approximation line (seeAppendix L) Figure 3a plots the barriers ( a ∗ p , a ∗ c , b ∗ ) when the fixed cost χ increases from 0 . 001 to 0 . 1. As wecan see, the increase in χ is compensated primarily by the increase in b ∗ , with almost insignificantdrops in both a ∗ p and a ∗ c . This makes sense because with an increased difficulty in paying dividendsoutside the periodic times, one would simply choose to pay more often at the periodic times.Although not obvious in the figure, one should expect that a ∗ p and b ∗ coincide when χ = 0. Thiscorresponds to the special case described in Avanzi et al. (2016b, without fixed transaction costs).Figure 3b plots the barriers ( a ∗ p , a ∗ c , b ∗ ) when the proportional cost rate 1 − β increases from 0to 0 . 15, i.e. across and beyond the threshold δ/ ( γ + δ ). As 1 − β increases from 0, the two barriers a ∗ p and a ∗ c split. In addition, from Figure 3c, we can see that while a ∗ p decreases with a convergingbehaviour (to b ∗ ), a ∗ c is increasing with a diverging behaviour, as predicted and due to Lemma 9.2.As another illustration of Lemma 9.2, there is a continuity behaviour between a ∗ p and the optimalperiodic barrier b ∗ at β = γ/ ( γ + δ ). Figure 4a plots the barriers ( a ∗ p , a ∗ c , b ∗ ) when the volatility parameter σ increases from 0 . 01 to20. When the volatility is small, the business is virtually riskless and excess capital is not neededas a buffer. Therefore, both a ∗ p and a ∗ c are close to zero. When σ increases, the business becomesmore risky and hence all 3 barriers increase. However, as σ further increases beyond a certainlevel, the business is deemed too risky and early exit would be a better choice. This is reflected by24 σ * a p * b * a c * (a) Small σ σ a p * b * a c * (b) Large σ Figure 4: Sensitivities to the volatility parameter σ . the decrease in the lower barrier a ∗ p . Furthermore, we can see from Figure 4b that a ∗ c is also goingdown eventually but the behaviour of b ∗ is unclear. Heuristically we expect that b ∗ ↑ ∞ so thatthe optimal strategy converges to a “liquidation at first opportunity” strategy. The idea is that σ ↑ ∞ is equivalent to µ, κ ↓ , , b ) strategy is optimal (Lemma9.1). Converting to the original scale, we have b ∗ ↑ ∞ . γ b * a p * b * a c * (a) γ δ b * a p * b * a c * (b) δ Figure 5: Sensitivities to the time parameters Solid line: β = γ/ ( γ + δ ). Dotted line: approximation line (seeAppendix L) Figure 5a plots the barriers ( a ∗ p , a ∗ c , b ∗ ) when the dividend frequency parameter γ increases from0 . 01 to beyond 1 . 5. It is clear that all three barriers are increasing with γ . This is consistent withthe intuition that with more frequent chances to pay periodic dividends (which attract no fixedcosts), one does not have the urgency to pay more which puts the company at risk. When γ increases to the point that γ/ ( γ + δ ) approaches β , both a ∗ c and b ∗ should increase to infinity while a ∗ p increases to b ∗ , which resemble a periodic b ∗ strategy. This behaviour is similar to the changein 1 − β studied in Figure 3b.Finally, Figure 5b plots the barriers ( a ∗ p , a ∗ c , b ∗ ) when the time preference parameter δ rangesfrom 0 . 01 to 3. Unsurprisingly, the effect is qualitatively the reverse of that of γ in Figure 5a, with25lso a smooth connection with the periodic b ∗ strategy. µ < ) Our baseline setting includes: scale parameters ( µ, σ, χ ) = ( − , . , . γ, δ ) = (1 , . 15) and proportional transaction cost parameter β = 0 . 7. Section 10.2.1 explores theimpact of the 2 types of costs to the optimal barriers. Following that, Section 10.2.2 illustrates thesensitivities of other parameters to the optimal barriers. We start by discussing the impact of the two types of transaction costs (proportional and fixed)on the optimal barriers. Remember that β ≤ − β is the net level of proportionaltransaction costs and high levels further penalise the immediate dividends as compared to theperiodic ones. Recall as well that immediate liquidation occurs as soon as the surplus level its thearea between b and b , and at first opportunity otherwise. - b * b * b * b * γ + (a) Small fixed transaction costs χ Solid line: χ = 0 . 15; Dashed line: χ = 0 . - b * b * γ + (b) Large fixed transaction costs χ Solid line: χ = 0 . 9; Dashed line: χ = 1 Figure 6: Interplay between proportional and fixed transaction costs. An empty region means π is optimal. Figure 6 illustrates the change in the optimal barriers ( b ∗ , b ∗ ) with increasing proportionaltransaction cost 1 − β and different fixed transaction costs χ . First, when χ is relatively large, theperiodic 0 strategy is optimal for 1 − β > δ/ ( γ + δ ) (high proportional transaction cost), which isevident in Figure 6b, compared to Figure 6a. If we imagine the periodic 0 strategy as a liquidation( b , b ) strategy with both barriers being infinity, then the two graphs are consistent. Hence, wecan focus on Figure 6a.From Figure 6a, we can see that when the proportional transaction cost 1 − β decreases,immediate dividends become optimal, and the associated two barriers appear and diverge. Theupper barriers increase to infinity when 1 − β approaches δ/ ( γ + δ ) from above, and the lowerbarrier stabilises to a certain level. On the other hand, we can see that the two barriers degenerateto one level which corresponds to the periodic 0 strategy when the proportional transaction cost1 − β is large. This continuity feature is quite surprising and remarkable, especially the continuityof the lower barrier b ∗ at β = γ/ ( γ + δ ).The collapse of the the area between the two barriers is quite intuitive as an increase in thecost 1 − β makes the decision to liquidate the company immediately very expensive compared to26aiting for the next dividend decision time and liquidate at that first opportunity. Further, when1 − β is too large, we totally ignore the option to liquidate the company immediate and choose towait. Similarly, when the fixed cost χ increases, the option to liquidate the company now becomesless favourable. This is indicated by the smaller area covered by the 2 dotted lines compared tothe solid lines, when χ increases from 0 . 15 to 0 . 3. Obviously, when χ increases, we should expectan increase in b ∗ , as displayed in Figure 6a. γ b * b * γγ+ δ (a) γ δ b * b * γγ + δ (b) δ σ b * b * (c) σ Figure 7: Sensitivities to parameters. Empty region means π is optimal. Figure 7 displays the sensitivities of the barriers to the parameters γ , δ and σ . Note that the(baseline) fixed cost χ is chosen to be “small” to showcase the presence of two barriers. When γ increases, the chance of being able to liquidate the company at low cost improves. This favours theoption to wait instead of liquidating the company now and is clearly indicated in Figure 7a wherethe area between the two barriers is shrinking. The opposite effect is present for the impatienceparameter δ . Note if we chose a larger base value for γ , we will see both barriers meet just as inFigure 7a. This is because δ and γ have somewhat inverse roles, and are both functions of howtime is defined.Because the company is non-profitable (negative µ ), waiting is speculative because there isnothing left if the company gets ruined before it is liquidated. Figure 7c shows that increasedvolatility makes such a speculation increasingly worthwhile. When σ is low, the lower barrier tobe very close to χ/β , which means we will liquidate the company as long as the outcome gives usa positive value, since there is no chance of recovering: indeed the negative drift µ < σ is too low.On the other hand a very high σ means it is worth trying one’s luck and wait. Note that Figure 7cuses β ≤ γ/γ + δ ; the case when β > γ/γ + δ is similar, except we do not have b ∗ as it is infinity. 11. Conclusion In this paper, we considered a diffusion model for the retained cash earnings of a risk business,and studied comprehensively its optimal control via dividends (cash payments) of two differenttypes as observed in real life. Under realistic transaction cost assumptions, we were able to replicatedividend payment behaviour actually observed as optimal. In particular, for realistic ranges ofparameters a hybrid dividend strategy is optimal, whereby periodic dividends are paid regularly,and extraordinary dividends are paid when the surplus becomes too high. All results summarisedin Section 2.5 and Table 1 are rigorously shown in the paper and its online supplements.27 cknowledgments This paper was presented at the 23rd International Congress on Insurance: Mathematics andEconomics (IME) in July 2019 (Munich, Germany) and at the 54th Actuarial Research Confer-ence (ARC) in August 2019 (Purdue University, USA). The authors are grateful for constructivecomments received from colleagues who attended those events.This research was supported under Australian Research Council’s Linkage (LP130100723) andDiscovery (DP200101859) Projects funding schemes. Hayden Lau acknowledges financial supportfrom an Australian Postgraduate Award and supplementary scholarships provided by the UNSWAustralia Business School. The views expressed herein are those of the authors and are not neces-sarily those of the supporting organisations. References Albrecher, H., Cheung, E. C. K., Thonhauser, S., 2011a. Randomized observation periods for the compound poissonrisk model: dividends. ASTIN Bulletin 41 (2), 645–672.Albrecher, H., Gerber, H. U., Shiu, E. S. W., 2011b. The optimal dividend barrier in the gamma-omega model.European Actuarial Journal 1 (1), 43–55.Albrecher, H., Thonhauser, S., 2009. Optimality results for dividend problems in insurance. RACSAM Revista de laReal Academia de Ciencias; Serie A, Mathem´aticas 100 (2), 295–320.Asmussen, S., Avram, F., Usabel, M., 2002. Erlangian approximations for finite-horizon ruin probabilities. ASTINBulletin 32 (2), 267–281.Avanzi, B., 2009. Strategies for dividend distribution: A review. North American Actuarial Journal 13 (2), 217–251.Avanzi, B., Cheung, E. C. K., Wong, B., Woo, J.-K., 2013. On a periodic dividend barrier strategy in the dual modelwith continuous monitoring of solvency. Insurance: Mathematics and Economics 52 (1), 98–113.Avanzi, B., Lau, H., Wong, B., 2020a. Optimal periodic dividend strategies for spectrally negative l´evy processeswith fixed transaction costs. arXiv math.OC 2004.01838.Avanzi, B., Lau, H., Wong, B., 2020b. Optimal periodic dividend strategies for spectrally positive l´evy risk processeswith fixed transaction costs. Insurance: Mathematics and Economics in press.Avanzi, B., Tu, V., Wong, B., 2014. On optimal periodic dividend strategies in the dual model with diffusion.Insurance: Mathematics and Economics 55, 210–224.Avanzi, B., Tu, V. W., Wong, B., 2016a. A note on realistic dividends in actuarial surplus models. Risks 4 (4), 37.Avanzi, B., Tu, V. W., Wong, B., 2016b. On the interface between optimal periodic and continuous dividend strategiesin the presence of transaction costs. ASTIN Bulletin 46 (3), 709–746.Avanzi, B., Tu, V. W., Wong, B., 2018. Optimal dividends under Erlang(2) inter-dividend decision times. Insurance:Mathematics and Economics 79, 225–242.B¨auerle, N., Ja´skiewicz, A., 2015. Risk-sensitive dividend problems. European Journal of Operational Research242 (1), 161 – 171.Bayraktar, E., Egami, M., 2010. A unified treatment of dividend payment problems under fixed cost and implemen-tation delays. Mathematical Methods of Operations Research 71 (2), 325–351.Bayraktar, E., Kyprianou, A. E., Yamazaki, K., 2014. Optimal dividends in the dual model under transaction costs.Insurance: Mathematics and Economics 54, 133–143.B¨uhlmann, H., 1970. Mathematical Methods in Risk Theory. Grundlehren der mathematischen Wissenschaften.Springer-Verlag, Berlin, Heidelberg, New York.Cheung, E., Wong, J., 2017. On the dual risk model with parisian implementation delays in dividend payments.European Journal of Operational Research 257, 159–173.Diasparra, M., Romera, R., 2010. Inequalities for the ruin probability in a controlled discrete-time risk process.European Journal of Operational Research 204 (3), 496 – 504.Dimitrova, D., Kaishev, V., Zhao, S., 2014. On finite-time ruin probabilities in a generalized dual risk model withdependence. European Journal of Operational Research 242, 134–148.Loeffen, R., 2008a. An optimal dividends problem with transaction costs for spectrally negative L´evy processes.Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences.Loeffen, R. L., 2008b. On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negativeL´evy processes. Annals of Applied Probability 18 (5), 1669–1680. orningstar, 30 June 2014. 5 traps in using the dividend yield by Karl Siegling (last accessed on 18 December 2015on ).Noba, K., P´erez, J.-L., Yamazaki, K., Yano, K., 2018. On optimal periodic dividend strategies for L´evy processes.Insurance: Mathematics and Economics 80, 29–44.Øksendal, B., Sulem, A., 2010. Applied stochastic control of jump diffusions, 2nd Edition. Springer, Berlin.P´erez, J., Yamazaki, K., 2018. Mixed periodic-classical barrier strategies for l´evy risk processes. Risks 6 (2).P´erez, J.-L., Yamazaki, K., 2017. On the optimality of periodic barrier strategies for a spectrally positive L´evyprocess. Insurance: Mathematics and Economics 77, 1–13.P´erez, J.-L., Yamazaki, K., 2018. Optimality of hybrid continuous and periodic barrier strategies in the dual model.Applied Mathematics & Optimization.Peskir, G., 2005. A change-of-variable formula with local time on curves. J. Theoret. Probab. 18 (3), 499–535.Protter, P., 2005. Stochastic Integration and Differential Equations, 2nd Edition. Springer-Verlag, Berlin-Heidelberg.Wesfarmers, 20 August 2014. 2014 Capital Management Initiative — A quick guide (last accessed on 18 December2015 on ).Woodside Petroleum, 23 April 2013. Special dividends and dividend payout announcement (last accessed on 18 De-cember 2015 on ). . Proof of Lemma 3.1 By the definition of v , it suffices to show that under the hypothesis, we have H ( x ) ≥ V ( x ; π )for all π ∈ Π.We first prove the case when D π (0) = 0, i.e. there is no dividend at time 0.Since H ∈ C ( R + ) ∩ C ( R + \ E ), we need to use It¯o-Meyer (e.g. Thm IV.70 in Protter, 2005),where Peskir (2005) shows that H ∈ C is enough to kill the local time at E . As a result, we canstill apply the It¯o Lemma in its standard form.There is nothing to prove when x = 0, see (2.11). Hence, we assume x > 0. For each n ∈ N , wedefine a family of increasing stopping time ( T n , n ∈ N ) with T n := inf { t > X π ( t ) > n or X π ( t ) < n } . By applying the It¯o Lemma to the semi-martingale { e − δ ( t ∧ T n ) H ( X π ( t ∧ T n )); t ≥ } (with a ∧ b = min( a, b ) for a, b ∈ R ), conditioning on the event { X (0) = x } , we have e − δ ( t ∧ T n ) H ( X π ( t ∧ T n )) − H ( x )= (cid:90) t ∧ T n − δe − δs H ( X π ( s − )) ds + (cid:90) t ∧ T n e − δs H (cid:48) ( X π ( s − )) dX π ( s )+ 12 (cid:90) t ∧ T n e − δs H (cid:48)(cid:48) ( X π ( s − ))1 { X π ( s − ) / ∈ E } d [ X π , X π ] c ( s )+ (cid:88) 0, we denote (cid:101) π the same strategy for t > 0, i.e. D (cid:101) π ( t ) = { D πp ( t ) , D πc ( t ) − D π (0) } .Then we have V ( x ; π ) = E x ( βD π (0) − χ + V ( x − D π (0); (cid:101) π )) ≤ E x ( sup ξ ∈ (0 ,x ] (cid:16) βξ − χ + H ( x − ξ ) (cid:17) ) ≤ H ( x )by an application of the previous result for (cid:101) π and Condition 5. B. Proof of Proposition 5.4 Note Existence is established in Proposition 5.7 and here we assume ( a ∗ , l ∗ , y ∗ ) exists.It is clear from (5.9) that the objective function is differentiable w.r.t. ( a, l, y ). Therefore, beingoptimal implies the partial derivatives are zero (except at the boundary). It is straight-forward toshow ∂∂y V ( a ) W δ ( a ) = 0 ⇐⇒ V (cid:48) ( b ) = β. From this, we see that at ( a ∗ , l ∗ , y ∗ )0 = ∂∂y (cid:16) V ( a c ) − βa c (cid:17) = ∂∂y (cid:16) V ( a ) W δ ( a ) G ( a, l ) (cid:17) = (cid:16) ∂∂y V ( a ) W δ ( a ) (cid:17) G ( a, l )= ⇒ ∂∂y V ( a ) W δ ( a ) = 0 ⇐⇒ V (cid:48) ( b ) = β. A further calculation (to appear later in (D.9)) shows that it is never optimal to have y = κ (i.e.at the boundary) so the equality above always hold.Now, using V ( a c ) + β ( y − κ ) = V ( b ) = V ( a ) W δ ( a ) G ( a, y + l ) − γW γ + δ ( y + l ) , 32e get ∂∂l (cid:16) V ( a c ) − βa c (cid:17) = ∂∂l (cid:16) V ( a ) W δ ( a ) G ( a, y + l ) − γW γ + δ ( y + l ) (cid:17) − β = (cid:16) ∂∂l V ( a ) W δ ( a ) (cid:17) G ( a, y + l ) + V (cid:48) ( b ) − β = (cid:16) ∂∂l V ( a ) W δ ( a ) (cid:17) G ( a, y + l )so we have ∂∂l (cid:16) V ( a c ) − βa c (cid:17) = ∂∂l V ( a c ) − β = ∂∂l (cid:16) V ( a ) W δ ( a ) G ( a, l ) − γW γ + δ ( l ) (cid:17) − β = (cid:16) ∂∂l V ( a ) W δ ( a ) (cid:17) G ( a, l ) + V ( a ) W δ ( a ) ∂∂l G ( a, l ) − γW γ + δ ( l ) − β = (cid:16) ∂∂l V ( a ) W δ ( a ) (cid:17) G ( a, l ) + V (cid:48) ( a c ) − β = ⇒ (cid:16) ∂∂l V ( a ) W δ ( a ) (cid:17)(cid:16) G ( a, y + l ) − G ( a, l ) (cid:17) = V (cid:48) ( a c ) − β so we have V (cid:48) ( a c ) = β. if l ∗ > 0. Otherwise, if l ∗ = 0 (i.e. at the boundary), we see that l (cid:55)→ V ( a c ) − βa c is decreasing in l near zero, i.e. 0 ≥ ∂∂l (cid:16) V ( a c ) − βa c (cid:17) = (cid:16) ∂∂l V ( a ) W δ ( a ) (cid:17) G ( a, y + l )and therefore by noting G ( a, y + l ) > G ( a, l ) we get V (cid:48) ( a c ) ≤ β. Similarly, we have ∂∂a (cid:16) V ( a c ) − βa c (cid:17) = (cid:16) V (cid:48) ( b ) − β (cid:17) + G ( a, y + l ) ∂∂a V ( a ) W δ ( a ) + γW γ + δ ( y + l ) (cid:16) − V (cid:48) ( a ) (cid:17) , (B.1) ∂∂a (cid:16) V ( a c ) − βa c (cid:17) = (cid:16) V (cid:48) ( a c ) − β (cid:17) + G ( a, l ) ∂∂a V ( a ) W δ ( a ) + γW γ + δ ( l ) (cid:16) − V (cid:48) ( a ) (cid:17) . (B.2)Now, Assumption 5.3 implies∆ := G ( a, y + l ) γW γ + δ ( l ) − G ( a, l ) γW γ + δ ( y + l ) < . (B.3)Hence, we can eliminate the term with ∂∂a ( V ( a ) /W δ ( a )) to get (cid:16) G ( a, y + l ) − G ( a, l ) (cid:17) ∂∂a (cid:16) V ( a c ) − βa c (cid:17) = G ( a, y + l ) (cid:16) V (cid:48) ( a c ) − β (cid:17) + | ∆ | ( V (cid:48) ( a ) − . (B.4)33ow, suppose a ∗ = ¯ a , then we have V (¯ a ) W δ (¯ a ) ≤ W δ (¯ a ) W (cid:48) δ (¯ a ) W δ (¯ a ) = 1 W (cid:48) δ (¯ a ) = ⇒ V (cid:48) ( a ) = V (cid:48) (¯ a ) = V (¯ a ) W δ (¯ a ) W δ (¯ a ) ≤ , because the value function (of our strategy) is smaller in the current setting than the setting whenthere is no transaction costs (e.g. in Loeffen (2008b)) and the optimal value function at ¯ a is givenabove. Hence the right hand side of the above equation is negative and so as the left hand side.This means it is impossible for a ∗ = ¯ a to be a maximiser for V ( a c ) − βa c . On the other hand, itis possible for a ∗ = 0. In that case, we have ∂∂a ( V ( a c ) − βa c ) ≤ 0. If furthermore l ∗ > 0, we have V (cid:48) ( a c ) = β and therefore we can conclude V (cid:48) ( a ) ≤ a ∗ > 0, i.e. ∂∂a ( V ( a c ) − βa c ) = 0. If l ∗ = 0, we have V (cid:48) ( a ) = V (cid:48) ( a c ) ≤ β which isa contradiction in view of (B.4). Therefore, we must have l ∗ > V (cid:48) ( a ) = 1.This completes the proof. C. Proof of Proposition 5.6 Note γW γ + δ ( x ) = kH ( x ) for some positive constant k , and recall from equation (D.8) and thedefinition of J that G ( a, x ) = g ( x ) f (cid:48) ( a ) − s δγ + δ f ( a ) r − s + e s x δγ + δ f ( a ) + γγ + δ f ( a ) , (C.1) J ( x ) = g ( x )( − s ) + e s ( r − s ) + ( − ( r − s )) , (C.2)we want to show that (for any a ≥ J ( x ) ∂∂x G ( a, x ) − G ( a, x ) ∂∂x J ( x ) < , x > . (C.3)By direct computation, we see that J ( x ) ∂∂x G ( a, x ) − G ( a, x ) ∂∂x J ( x )= − f (cid:48) ( a ) (cid:16) g (cid:48) ( x ) − ( r − s ) e ( r + s ) x (cid:17) (C.4)+ s f ( a ) r g ( x ) . Denote the function F with F ( x ) := g (cid:48) ( x ) − ( r − s ) e ( r + s ) x . It is easy to see F (0) = 0 and F (cid:48) ( x ) > x > F ( x ) > x > J ( x ) ∂∂x G ( a, x ) − G ( a, x ) ∂∂x J ( x ) = − f (cid:48) ( a ) F ( x ) + s f ( a ) r g ( x ) , 34e can conclude that J ( x ) ∂∂x G ( a, x ) − G ( a, x ) ∂∂x J ( x ) < . This completes the proof. D. Proof of Proposition 5.7 The statement is a directly consequence of Propositions 5.6 and 5.4. Therefore, we are left toshow the hypothesis in Proposition 5.4.Using the formulas in Proposition 5.5, we have( r − s ) A = α ( r − s )( y − χα )( f (cid:48) ( a ) − s δγ + δ f ( a )) + γγ + δ ( J ( d, l ) + s g ( d, l ))) δγ + δ f ( a ) J ( d, l ) + f (cid:48) ( a ) g ( d, l ) , (D.1) ∂∂l A = − A δγ + δ f ( a ) J (cid:48) ( d, l ) + f (cid:48) ( a ) g (cid:48) ( d, l ) δγ + δ f ( a ) J ( d, l ) + f (cid:48) ( a ) g ( d, l ) , (D.2)(with J (cid:48) ( d, l ) = J (cid:48) ( d ) − J (cid:48) ( l ) , g (cid:48) ( d, l ) = g (cid:48) ( d ) − g (cid:48) ( l )) (D.3)lim l →∞ C = γµ ( γ + δ ) − γγ + δ s δγ + δ f ( a ) − f (cid:48) ( a ) s < ∞ , (D.4)lim l →∞ (cid:16) g ( l )(1 + s g ( d, l ) J ( d, l ) ) (cid:17) = lim l →∞ ( r − s ) g ( l )( e s d − e s l ) J ( d, l ) = 0 , (D.5)lim l →∞ Ag ( l ) = αe r y − (cid:0) y − χα (cid:1) < ∞ , (D.6)lim l →∞ g ( d, l ) J ( d, l ) = − s , lim l →∞ g (cid:48) ( d, l ) J ( d, l ) = − r s , lim l →∞ J (cid:48) ( d, l ) J ( d, l ) = r , lim l →∞ ∂∂l C = 0 . (D.7)This implieslim l →∞ ∂∂l ( Ag ( l )) = lim l →∞ (cid:16) Ag (cid:48) ( l ) + g ( l ) ∂∂l A (cid:17) = lim l →∞ Ag ( l ) (cid:16) lim l →∞ g (cid:48) ( l ) g ( l ) − lim l →∞ δγ + δ f ( a ) J (cid:48) ( d, l ) + f (cid:48) ( a ) g (cid:48) ( d, l ) δγ + δ f ( a ) J ( d, l ) + f (cid:48) ( a ) g ( d, l ) (cid:17) = lim l →∞ Ag ( l )( r − r ) = 0 . Therefore, we have lim l →∞ ∂∂l ( V ( a c ) − γγ + δ l ) = lim l →∞ ∂∂l ( Ag ( l )) = 0and hence lim l →∞ ∂∂l ( V ( a c ) − βl ) = γγ + δ − β < . V ( a c ) − βa c is decreasing for large enough l (independent of ( a, y )), say ¯ l ,i.e. l (cid:55)→ ( V ( a c ) − βa c ) cannot attain its local maximum for l > ¯ l . Since we have already chosen a ∈ [0 , ¯ a ], we do not worry about the a dimension.On the other hand, we have V ( a c ) = C (cid:32) f ( a ) (cid:16) γγ + δ + δγ + δ ( e s l − s g ( l ) r − s ) (cid:17) + f (cid:48) ( a ) g ( l ) r − s (cid:33) + g ( l ) r − s ( s γµ ( γ + δ ) − γγ + δ ) − e s l γµ ( γ + δ ) + γγ + δ ( l + µγ + δ ) . (D.8) Remark D.1. It is easy to see that C = V ( a ) /W δ ( a ) , the terms inside the bracket after C is G ( a, l ) , and the terms in the second line correspond to γW γ + δ ( l ) . We are left to work with the y dimension. From (D.8), it is clear that in terms of y , theobjective function V ( a c ) − βa c solely depends on C , as we have also discovered before using scalefunctions. There is no shortcut but to compute the derivative w.r.t. y . From C = ( r − s ) (cid:0) ( β − γγ + δ )( d − l ) − χ (cid:1) + γγ + δ g ( d, l ) + γµ ( γ + δ ) J ( d, l ) δγ + δ f ( a ) J ( d, l ) + f (cid:48) ( a ) g ( d, l ) , we get (after some tedious algebric operations) ∂∂y C × ( δγ + δ f ( a ) J ( d, l ) + f (cid:48) ( a ) g ( d, l )) = α ( r − s ) (cid:16) δγ + δ f ( a ) J ( d, l ) + f (cid:48) ( a ) g ( d, l ) − ( y − χα )( δγ + δ f ( a ) J (cid:48) ( d ) + f (cid:48) ( a ) g (cid:48) ( d )) (cid:17) + γγ + δ f (cid:48) ( a ) (cid:16) µγ + δ − δγ + δ f ( a ) f (cid:48) ( a ) (cid:17)(cid:16) g ( d, l ) J (cid:48) ( d ) − J ( d, l ) g (cid:48) ( d ) (cid:17) , where µγ + δ − δγ + δ f ( a ) f (cid:48) ( a ) ≥ a ∈ [0 , ¯ a ] and it can be checked by taking derivative w.r.t. y that g ( d, l ) J (cid:48) ( d ) − J ( d, l ) g (cid:48) ( d ) = g (cid:48) ( d ) e s l − s g ( l ) e s d − ( r − s ) e ( r + s ) d > y ≥ χ/α . This implies ∂∂y V ( a ) W δ ( a ) (cid:12)(cid:12)(cid:12) y = χ/α = ∂∂C (cid:12)(cid:12)(cid:12) y = χ/α > , ( a, l ) ∈ [0 , ¯ a ] × [0 , ¯ l ] . (D.9)36ext, we take the limit y → ∞ , and see (after some algebraic operations) that ∂∂y C × ( δγ + δ f ( a ) J ( d, l ) + f (cid:48) ( a ) g ( d, l )) = f (cid:48) ( a ) α ( r − s ) (cid:16) g ( d, l ) − ( y − χα ) g (cid:48) ( d ) + 1 α ( r − s ) γµ ( γ + δ ) (cid:0) g (cid:48) ( d ) e s l − s g ( l ) e s d − ( r − s ) e ( r + s ) d (cid:1)(cid:17) + δγ + δ f ( a ) α ( r − s ) (cid:16) J ( d, l ) − ( y − χα ) J (cid:48) ( d ) + γγ + δ (cid:0) g (cid:48) ( d ) e s l − s g ( l ) e s d − ( r − s ) e ( r + s ) d (cid:1)(cid:17) ≤ f (cid:48) ( a ) α ( r − s ) (cid:16) g ( d ) − ( y − χα − α ( r − s ) γµ ( γ + δ ) ) g (cid:48) ( d ) (cid:17) + δγ + δ f ( a ) α ( r − s ) (cid:16) J ( d ) − ( y − χα ) J (cid:48) ( d ) + γγ + δ g (cid:48) ( d ) (cid:17) which drifts to −∞ when y → ∞ . Hence, we can choose d such that d > d implies ∂∂y C is decreasingfor all ( a, l ) ∈ [0 , ¯ a ] × [0 , ¯ l ]. In particular, we can choose ¯ y = (¯ l + d ) ∨ χ/α such that the sameholds for y ≥ ¯ y .To conclude, we have find a box for B := [0 , ¯ a ] × [0 , ¯ l ] × [ χ/α, ¯ y ] for ( a, l, y ) such that1. The objective function V ( a c ) − βa c attains its maximum inside B ,2. Its maximum ( a ∗ , l ∗ , y ∗ ) either occurs in the interior of B , or we have a ∗ = 0 or l ∗ = 0 orboth, but not other cases.This concludes the hypothesis in Proposition 5.4 and hence completes the proof. E. Proof of Lemma 6.1 Denote (cid:101) A = A and (cid:101) B = B − A so that the derivative of the value function on [ a p , b ] is V (cid:48) ( a p + x ) = (cid:101) Ar e r x + (cid:101) Bs e s x + γγ + δ , x ∈ [0 , d ] . (E.1)From V (cid:48) ( b ) = β , we have V (cid:48) ( b ) = (cid:101) Ar e r d + (cid:101) Bs e s d + γγ + δ = β, or (cid:101) Ar e r d + (cid:101) Bs e s d = α. (E.2)Moreover, we have V (cid:48)(cid:48) ( a p + x ) = (cid:101) Ar e r x + (cid:101) Bs e s x (E.3)and V (cid:48)(cid:48)(cid:48) ( a p + x ) = (cid:101) Ar e r x + (cid:101) Bs e s x . (E.4)We first show that (cid:101) A > (cid:101) A ≤ (cid:101) B ≥ 0, then the L.H.S. of (E.2)is negative, which is impossible. On the other hand, if we assume (cid:101) A ≤ (cid:101) B < 0, then we havefrom (E.3) V (cid:48)(cid:48) < 0, which implies that V (cid:48) is decreasing on [ a p , b ]. However, from V ∈ C ( R + ), wehave (cid:90) b − a c ( β − V (cid:48) ( x )) dx = χ > , V (cid:48) ≥ β on [ a c , b ], which is also impossible.Now we have established (cid:101) A > 0. If we further assume (cid:101) B ≥ 0, then we have from (E.3) V (cid:48)(cid:48) ≥ 0, which implies that V (cid:48) is increasing on [ a p , b ]. Note that this would not be possible unless V (cid:48) (0) < β = ⇒ a c = a p = 0 because otherwise we have V (cid:48) ( a c ) = β = V (cid:48) ( b ). Regardless, as V (cid:48) increases to V (cid:48) ( b ) = β , we have V (cid:48) < β on [0 , b ) and V (cid:48) ≡ β on [ b, ∞ ), which also holds if b = 0. Furthermore, the fact that V is positive (by the definition of the value function) impliesthat V (cid:48) (0) > 0, which in turn implies that V (cid:48) > , ∞ ).For the last case, (cid:101) A > (cid:101) B < 0, we can deduce from (E.4) that V (cid:48)(cid:48)(cid:48) ≥ a p , b ], orequivalently, V (cid:48) is convex on [ a p , b ]. This together with V (cid:48) ( a c ) ≤ β = V (cid:48) ( b ) gives V (cid:48) ≤ β on[ a c , b ]. This fact combining with V (cid:48) ( a p ) ≤ V (cid:48) is decreasing from a p to a c , then furtherdecreasing and finally increasing to β at b , as V (cid:48) is convex on [ a p , b ], or simply increasing to β if a p = a c = 0 and V (cid:48) (0) < β . Furthermore, in view of (E.1), we have that V (cid:48) > , ∞ ). Remark E.1. Note in any cases, we have V (cid:48)(cid:48) ( b − ) > as V (cid:48) is increasing at b − ε for all smallenough ε > . On the other hand, we have V (cid:48)(cid:48) ( b +) = 0 . F. Proof of Lemma 6.2 For x > b , we have( A − δ ) V ( x ) + γ (cid:16) x − a p + V ( a p ) − V ( x ) (cid:17) = ( A − δ ) V ( b +) + γ (cid:16) b − a p + V ( a p ) − V ( b ) (cid:17) − ( γ + δ )( V ( x ) − V ( b )) + γ ( x − b )= ( A − δ ) V ( b +) + γ (cid:16) b − a p + V ( a p ) − V ( b ) (cid:17) − ( γ + δ ) (cid:16) β ( x − b ) − γγ + δ ( x − b ) (cid:17) ≤ ( A − δ ) V ( b +) + γ (cid:16) b − a p + V ( a p ) − V ( b ) (cid:17) as β > γ/ ( γ + δ ) by the assumption in (6.1).Together with Remark E.1 and the fact that V ∈ C ( R + ), we have0 = ( A − δ ) V ( b − ) + γ (cid:16) x − a p + V ( a p ) − V ( b ) (cid:17) > ( A − δ ) V ( b +) + γ (cid:16) x − a p + V ( a p ) − V ( b ) (cid:17) ≥ ( A − δ ) V ( x ) + γ (cid:16) x − a p + V ( a p ) − V ( x ) (cid:17) (F.1)for x > b . Now, denote H ( ξ ) := ξ + V ( x − ξ ) − V ( x ) (F.2)and by taking derivative with respect to ξ , we have H (cid:48) ( ξ ) = 1 − V (cid:48) ( x − ξ ) . In view of Lemma 6.1, V (cid:48) ( x − ξ ) < x − ξ > a p , or equivalently ξ < x − a p .Therefore, we can deduce that H is increasing on [0 , a p ] if a p > a p , ∞ ),38hich implies that in any case it attains its maximum at ξ = x − a p . Therefore, we have( A − δ ) V ( x ) + γ sup ξ ∈ [0 ,x ] (cid:16) ξ + V ( x − ξ ) − V ( x ) (cid:17) = ( A − δ ) V ( x ) + γ (cid:16) x − a p + V ( a p ) − V ( x ) (cid:17) , x > b ( A − δ ) V ( x ) + γ (cid:16) x − a p + V ( a p ) − V ( x ) (cid:17) , a p ≤ x < b ( A − δ ) V ( x ) , x < a p ≤ x ≤ χ/β , since V is increasing. For x > χ/β , we denote H ( ξ ) := βξ − χ + V ( x − ξ ) − V ( x ) , ξ ≥ . Clearly, we have H (0) = − χ < 0. Taking derivative w.r.t. ξ , we get H (cid:48) ( ξ ) = β − V (cid:48) ( x − ξ ) . In view of Lemma 6.1, V (cid:48) ( x − ξ ) < β is equivalent to x − ξ ∈ ( a c , b ), or equivalently x − b < ξ < x − a c .Therefore, we can deduce that H is decreasing on [0 , x − b ] if x − b ≥ x − b, , x − a c ] if x − a c ≥ 0, then decreasing on ( a c , ∞ ). Hence, on [0 , x ], its maximum isattained at either 0, or x − a c if x − a c > 0. Suppose ξ = x − a c > 0, we have H ( x − a c ) = β ( x − a c ) − χ + V ( a c ) − V ( x ) , which as a function of x , is increasing on [ a c + χ/β, b ]. This implies that H ( x − a c ) ≤ H ( b − a c ) = β ( b − a c ) − χ + V ( a c ) − V ( b ) = 0 > H (0) . Therefore, sup ξ ∈ [0 ,x ] H ( ξ ) ≤ 0. This completes the proof as H and the term inside the bracket inthe second component differs only at ξ = 0, where both of them have value less than or equal to 0. G. Proof of Equation (8.4)First, we have Λ( V (cid:48) (0; π )) = 0 as β = V (cid:48) (0; π ) implies that a β = 0. In addition, Λ is anincreasing function because ∂∂β Λ( β ) = ∂∂β a β − V (cid:48) ( a β ; π ) ∂∂β a β β + V ( a β ; π ) β = (cid:16) ∂∂β a β (cid:17)(cid:16) − V (cid:48) ( a β ; π ) β (cid:17) + V ( a β ; π ) β > . β ↑ γγ + δ , we have a β ↑ ∞ and thereforelim β ↑ γγ + δ Λ( β ) = lim x ↑∞ ( x − V ( x ; π )( γγ + δ ) )= γ + δγ lim x ↑∞ ( γγ + δ x − γµ ( γ + δ ) (1 − e s x ) − γγ + δ x )= − µγ + δ , where we have used Lemma 8.3 to compute V ( x ; π ). H. Proof of Lemma 8.5 Using the fact that the value function is continuous at b , we get Be s b = βb − χ − γγ + δ b − γγ + δ µγ + δ , (H.1)which implies that V (cid:48) ( b +; π b ,b ) = γγ + δ + Bs e s b = ( βb − χ ) s − γγ + δ b s − γγ + δ µγ + δ s + γγ + δ Therefore V (cid:48) ( b +; π b ,b ) = β is equivalent to s ( β − γγ + δ ) b − (cid:16) χs + s γµ ( γ + δ ) − ( γγ + δ − β ) (cid:17) = 0 . (H.2)Similarly, by replacing the equality of (H.1) by “ ≤ ” and B by (cid:101) B = − γµ ( γ + δ ) , we have β = V (cid:48) ( a β ; π ) ≥ γγ + δ + (cid:101) Bs e s a β = ( βa β − χ ) s − γγ + δ a β s − γγ + δ µγ + δ s + γγ + δ , which shows that when b = a β , the left hand side of (H.2) is negative. As the left hand sideof (H.2) is linear in b with positive coefficient, we establish the existence (and uniqueness) of b ∈ ( a β , ∞ ).For b , we note (from definitions of c β,χ , π c β,χ ,b and Remark 8.1) that A ( c β,χ ) g ( c β,χ ) + βc β,χ − χ = V ( c β,χ ; π c β,χ ,b ) = βc β,χ − χ, which shows that A ( c β,χ ) = 0. Therefore, we have V (cid:48) ( c β,χ ; π c β,χ ,b ) = A ( c β,χ ) g (cid:48) ( c β,χ ) + V (cid:48) ( c β,χ ; π ) = V (cid:48) ( c β,χ ; π ) < V (cid:48) ( a β ; π ) = β. Similarly, we have V (cid:48) ( a β − ; π a β ,b ) = A ( a β ) g (cid:48) ( a β ) + V (cid:48) ( a β ; π ) = A ( a β ) g (cid:48) ( a β ) + β. As g (cid:48) ( a β ) > 0, if A ( a β ) ≤ 0, then Lemma 8.4 implies that there is an interval I (cid:40) ( c β,χ , a β ) where A ( b ) is decreasing, since A ( b ) is increasing on the neighbourhood of c β . Again from Lemma 8.4,40e can deduce that V (cid:48) ( b − ; π b,b ) > β for b ∈ I , which is what we have to show. If A ( a β ) > V (cid:48) ( a β − ; π a β ,b ) > β and hence by continuity, there is a c β,χ < b < a β such that V (cid:48) ( b − ; π b,b ) = β . For uniqueness, we will choose b to be the smallest oneif there are more than one such b . I. Proof of Lemma 8.6 We first need to establish that for κ > 0, it holds that − µγ + δ (1 − e s κ ) − κ < . Denote h the left hand side of the above inequality, then we have obviously h (0) = 0. By takingderivative with respect to κ and note r s ( µ/ ( γ + δ )) = r s , we can have h (cid:48) ( κ ) = − µγ + δ ( − s ) e s κ − e s κ − 1) + s r e s k < . From Lemma 8.3, we have V (cid:48) ( b − ; π b, ∞ ) = A ( b ) g (cid:48) ( b ) − γµ ( γ + δ ) s e s b + γγ + δ with A ( b ) = ( β − γγ + δ ) b − χ − γµ ( γ + δ ) (1 − e s b ) g ( b ) . This implies that there is a (cid:101) b ≥ b > (cid:101) b implies A ( b ) > 0, andlim b →∞ A ( b ) g (cid:48) ( b ) = + ∞ and hence we are done if V (cid:48) ( χβ − ; π χβ , ∞ ) ≤ β. In particular, this would be the case if A ( χ/β ) < 0, since g (cid:48) ( χ/β ) > V (cid:48) ( χβ − ; π χβ , ∞ ) = A ( χβ ) g (cid:48) ( χβ ) + V (cid:48) ( χβ ; π ) ≤ A ( χβ ) g (cid:48) ( χβ ) + γγ + δ < A ( χβ ) g (cid:48) ( χβ ) + β. This is indeed the case, thanks to very first inequality developed in this section, A ( χβ ) = γγ + δ ( − µγ + δ (1 − e s χβ ) − χβ ) g ( χβ ) < . J. Proof of Lemma 9.1 Suppose β ≤ γ/ ( γ + δ ), then from Theorem 4.1, we know that a periodic barrier strategy π b is optimal. Based on the observation in e.g. Avanzi et al. (2014), we can conclude that π is41ptimal for µ = 0 (we omit the proof here although a separate check is possible). Now, in view ofTheorem 8.2, we are in Case 1: χ ≥ β ( − µ/ ( γ + δ )) for small enough | µ | (note µ < β ≤ γ/ ( γ + δ ), we conclude that a periodic barrier strategy π is optimal. Hence, the continuity isestablished.For β > γ/ ( γ + δ ), again in view of Theorem 8.2 Case 1, we can conclude that a liquidation( b , ∞ ) is optimal, with the barrier b such that the derivative of the value function at the barrieris β . Note that such strategy is also the same as a hybrid ( a p , a c , b ) strategy with a p = a c = 0and b = b . Since the optimal hybrid ( a c , a p , b ) strategy imposes that the derivative of the valuefunction at the upper barrier b is β (which is unique by Corollary 7.2), the continuity of the barrierswould be established if we can show that for µ = 0, the hybrid (0 , , b ) strategy is optimal.The value function of a hybrid (0 , , b ) strategy, denoted by V (instead of V ( · ; π , ,b ) for conve-nience), is given by V ( x ) = (cid:40) Ag ( x ) + γγ + δ x, x ≤ bβx − χ, x > b , which can be derived with ease. When µ = 0, the function g ( x ) can be rewritten as g ( x ) = e r x − e − r x , which has the property g (cid:48)(cid:48) ( x ) = r g ( x ) > x > 0. This implies that g (cid:48) ( b ) > g (cid:48) (0) . (J.1)On the other hand, by direct computation, we have V (cid:48) ( b ) = Ag (cid:48) ( b ) + γγ + δ = β which shows that A > 0. Therefore, we have V (cid:48) (0) = Ag (cid:48) (0) + γγ + δ ≤ Ag (cid:48) ( b ) + γγ + δ = β. This shows that the hybrid (0 , , b ) strategy is optimal for µ = 0 (see Definition 5.2 and Theorem7.1) and completes the proof. K. Proof of Lemma 9.2 For a hybrid ( a p , a c , b ) strategy, if V (cid:48) ( a c ) = V (cid:48) ( b ) = β , we have V (cid:48) ( a ) − − s γγ + δ − s + q ( r + s ) = ( r − s ) I ( y + l, q ) = ( r − s ) I ( l, q ) , I ( x, q ) = β − γγ + δ + (cid:16) γγ + δ − − s γγ + δ − s + q ( r + s ) (cid:17) e s x g (cid:48) ( x ) + g ( x )( − r s ) µγ + δ (1 − q ) ,q := Q ( a ) = 1 − f ( a ) /f (cid:48) ( a ) µ/δ ∈ [0 , . When β ↓ γγ + δ , from the fact that q > y ↑ ∞ .This implies that I ( y + l, q ) ↓ 0. Suppose − s r γγ + δ > 1, we get V (cid:48) ( a p ) = − s γγ + δ − s + q ( r + s ) = 1, andTherefore, in order for I ( l, q ) = I ( y + l, q ), we must have l ↑ ∞ , a p = Q − (cid:16) s δγ + δ r + s (cid:17) , y ↑ ∞ . Similarly, if − s r γγ + δ ≤ 1, we have l ↑ ∞ , a p = 0 , y ↑ ∞ . If a > 0, after some tedious algebra, we can show that( r − s ) A + γγ + δ = γγ + δ , implying that A = 0.Otherwise, if a = 0, we have q = 1 and therefore − s γγ + δ − s + q ( r + s ) = γγ + δ − s r . Similar to the abovecalculation, we also have ( r − s ) A + γγ + δ = γγ + δ , implying that A = 0. Therefore, when β ↓ γ/ ( γ + δ ), the value function V ( x ; π a p ,a c ,b ↑∞ ) convergesto V ( x ; π a p ), the value function of the periodic barrier strategy with barrier level a p . We are leftto verify that a p is the optimal barrier in the pure periodic setting, e.g. Noba et al. (2018).Recall from Remark 5.3 that the condition for a = 0 are the same when β ↓ γ/ ( γ + δ ). For a > 0, we have Cf (cid:48) ( a ) = − s γγ + δ r + q ( r + s ) = 1, which is the same as f ( a ) f (cid:48) ( a ) = (1 − s δγ + δ r + s ) µδ . V (cid:48)(cid:48)(cid:48) ( a +) = Bs = δγ + δ Cf ( a ) s − γγ + δ µγ + δ s = δγ + δ f ( a ) f (cid:48) ( a ) Cf (cid:48) ( a ) s − γγ + δ µγ + δ s = δγ + δ s and f (cid:48)(cid:48)(cid:48) ( a ) = γ + δσ / (cid:16) δγ + δ f (cid:48) ( a ) − µγ + δ f (cid:48)(cid:48) ( a ) (cid:17) = ( − r s ) (cid:16) ( δγ + δ − ( r + s ) µγ + δ ) f (cid:48) ( a ) + ( r + s ) δγ + δ f ( a ) (cid:17) . Therefore, we have f (cid:48)(cid:48)(cid:48) ( a ) f (cid:48) ( a )( − r s ) = δγ + δ − ( r + s ) µγ + δ + ( r + s )(1 − s r + s δγ + δ ) µγ + δ = δγ + δ − s r , which further implies V (cid:48)(cid:48)(cid:48) ( a − ) = Cf (cid:48)(cid:48)(cid:48) ( a ) = Cf (cid:48) ( a ) f (cid:48)(cid:48)(cid:48) ( a ) f (cid:48) ( a ) = δγ + δ s = V (cid:48)(cid:48)(cid:48) ( a +) , which is the “smoothness condition”, equation (4.1) in Noba et al. (2018), which characterises theunique periodic barrier. L. Computational considerations In the computation of the barriers for hybrid ( a p , a c , b ) strategies, instead of using the max-imisation of V ( a c ) − βa c , we solved the derivative conditions directly. The barriers were efficientlycalculated without difficulties.1. For each l ≥ 0, denote the unique y such that V (cid:48) ( b ) = β as y ¯ a ( l ) and y ( l ) for a = ¯ a and a = 0 respectively.2. For each l ≥ y ∈ [ y ¯ a ( l ) , y ( l )], we can find a corresponding q = Q ( a ) such that V (cid:48) ( b ) = β . As Q is monotone in a , we can recover a .3. For each l ≥ 0, ( y, a ) pair in the previous step indexed by y , we evaluate the derivative ofthe value function at a . For each l ≥ 0, we can find a ( y, a ) pair such that both V (cid:48) ( a ) = 1(or a = 0 if V (cid:48) (0) ≤ 1) and V (cid:48) ( a + y + l ) = β .4. For l = 0, using the corresponding ( y, a ) pair to compute V (cid:48) ( a ). If V (cid:48) ( a ) ≤ β , then set l = 0. Otherwise, write a function to output the corresponding V (cid:48) ( a + l ) using the ( y, a )pair from the previous step. By increasing l following by a bisection method, we can find acorresponding l such that V (cid:48) ( a + l ) = β . 44ll equations can be solved by for example bisection method combining with some searchingtechnique. In case of numerical overflow, we can rescale the scale parameters ( µ, σ, χ ) to find thebarriers then scale back. It should be clear that rescaling should not change the optimality of anoptimal strategy.In order to make sure that the possibilities of multiple solutions for some equations would notresult in some disruptive impact to the numerical procedure, we can verify that the final outputindeed satisfies all conditions proposed.Finally, when β is close to the asymptote γ/ ( γ + δ ), we have numerical overflow as b ∗ ↑ ∞ .In this case, an approximation is used based on Proposition 9.2, where we treat b ∗ = ∞ andcalculate a ∗ p and a ∗ c − a ∗ p independently. Sometimes, b ∗ may not be large enough to validate suchapproximation and a “bias” is resulted. In this case, we will adjust the bias term such that theapproximation piece glues to the piece without approximation. We decreases the bias term (linearlyfor convenience) so that it eventually vanishes at the asymptote β = γ/ ( γ + δ ). The region wherewe employ such approximation is indicated between the dotted line (where the numerical overflowstarts) and the solid line (the asymptote).For µ < 0, it is straightforward from Theorem 8.2. Specifically, we proceed the following:1. First check whether χ ≥ β ( − µ/ ( γ + δ )).2. Suppose χ ≥ β ( − µ/ ( γ + δ )). If β ≤ γ/ ( γ + δ ), π is optimal. No numerical method isneeded. On the other hand, if β > γ/ ( γ + δ ), we express V (cid:48) ( b − ; π b, ∞ ) as a function of b andsearch for V (cid:48) ( b − ; π b, ∞ ) = β .3. Suppose χ < β ( − µ/ ( γ + δ )). Invert Λ at χ/β to output β = Λ − ( χ/β ). If β ≤ β , then π is optimal and no numerical method is needed. If β ∈ ( β , γ/ ( γ + δ )), first output c β,χ bysolving V (cid:48) ( a β ) = β . It is then followed by solving 2 equations: V (cid:48) ( b − ; π b ,b ) = β in b and V (cid:48) ( b +; π b ,b ) = β in b , respectively. Note that for both equations, the other parameter isnot used. Finally, if β ∈ [ γ/ ( γ + δ ) , V (cid:48) ( b − ; π b, ∞ ) as afunction of b and search for V (cid:48) ( b − ; π b, ∞ ) = β .Again, all equations can be solved by for example bisection method combining with some searchtechniques.To compute b ∗ , we do not use the results directly from Noba et al. (2018). Instead, we use theformula given by the third item in Proposition 9.2, which holds for any β . M. Continuity for different cases in Theorem 8.2 We consider here the 4 different cases enumerated in Theorem 8.2 sequentially. M.1. χβ < − µγ + δ , β ↑ γγ + δ . When β ↑ γ/ ( γ + δ ), we have (from V (cid:48) ( · ; π ) ↑ γ/ ( γ + δ )) that a β → ∞ which implies that b →∞ and therefore from Theorem 8.2 case 2b we can conclude that b is continuous at β = γ/ ( γ + δ ),i.e. π b ,b →∞ → π b , ∞ when β ↑ γ/ ( γ + δ ). 45 .2. χβ ≥ − µγ + δ , β ↓ γγ + δ . Recall the optimal strategy in the setting π b, ∞ is characterised by the derivative condition at b , i.e. V (cid:48) ( b ; π b, ∞ ) = β . In view of the first 2 equations in the proof of Lemma 8.6, we have V (cid:48) ( b − ; π b, ∞ ) = ( β − γγ + δ ) b − χ − γµ ( γ + δ ) (1 − e s b ) g ( b ) g (cid:48) ( b ) − γµ ( γ + δ ) s e s b + γγ + δ . (M.1)Therefore, we have ∂∂β V (cid:48) ( b − ; π b, ∞ ) = b g (cid:48) ( b ) g ( b ) > . This implies that when β decreases from β to β , the original b = b ( β ) yields V (cid:48) ( b − ; π b, ∞ ) < β and therefore from the proof of Lemma 8.6 we need to use a larger b . This implies that b ( β ) > b ( β ).In other words, when β ↓ γ/ ( γ + δ ), the corresponding b = b ( β ) is increasing. It remains to showthat b ( β ) is not converging so that we have π b ↑∞ , ∞ → π . Suppose b ( β ) ↑ b < ∞ as β ↓ γ/ ( γ + δ ). Then by taking the limit, we have V (cid:48) ( b − ; π b, ∞ ) = β = γγ + δ . Thus, from (M.1) we have − χ = γµ ( γ + δ ) (cid:16) s e s b + (1 − e s b ) g (cid:48) ( b ) g ( b ) (cid:17) = γµ ( γ + δ ) s , which is impossible since − χ is negative but the very last term is positive. M.3. χβ ↑ − µγ + δ , β ∈ [ β , γγ + δ ] . Here, we want to show that if β < γ/ ( γ + δ ) then when χ/β is “close” to − µ/ ( γ + δ ), we have β > β . This means the two conditions χ/β ↑ − µ/ ( γ + δ ) and β ∈ [ β , γ/ ( γ + δ )] cannot be satisfiedsimultaneously unless β = γ/ ( γ + δ ). In other words, the cells in the second row of the Table 1 arecontinuous only at β = γ/ ( γ + δ ), which has already been taken care of.Recall that β is defined by the inverse of the increasing function Λ at χ/β , i.e. β = Λ − ( χ/β ).In addition, Λ maps β ∈ [ V (cid:48) (0; π ) , γ/ ( γ + δ )) to [0 , − µ/ ( γ + δ )). Therefore, when χ/β ↑ γ/ ( γ + δ ),we have β ↑ γ/ ( γ + δ ), which implies that β → γ/ ( γ + δ ). M.4. χβ < − µγ + δ , β ↓ β . Recall β = Λ − ( χ/β ) and therefore β ↓ β implies that (8.1) is an equality at the limit and wehave a β − c β,χ → 0. Consequently, from c β,χ ≤ b ≤ a β we can conclude that b → a β .It remains to show b → a β . In view of Lemma 8.5, we can establish b → a β if we can showthat at β = β , we have (H.2) with b = a β , i.e. s ( β − γγ + δ ) a β − (cid:16) χs + s γµ ( γ + δ ) − ( γγ + δ − β ) (cid:17) = 0 . V ( x ; π ) = − γµ ( γ + δ ) e s x + γγ + δ (cid:16) x + µγ + δ (cid:17) , we can re-express V (cid:48) ( a β ; π ) = β as s − γµ ( γ + δ ) e s a β = β − γγ + δ (M.2)and (8.1) (with inequality replaced by equality) as s (cid:16) − γµ ( γ + δ ) e s a β + γγ + δ (cid:16) a β + µγ + δ (cid:17)(cid:17) = s βa β − s χ ⇐⇒ s ( β − γγ + δ ) a β − s χ − s γµ ( γ + δ ) = s − γµ ( γ + δ ) e s a β = β − γγ + δ , where the last equality is from (M.2) and the last line is essentially what we are trying to show,i.e. (H.2).Since we have b , b → a β when β ↓ β , we have π b → a β ,b → a β → π ..