Optimizing dividends and capital injections limited by bankruptcy, and practical approximations for the Cramér-Lundberg process
aa r X i v : . [ m a t h . O C ] F e b Optimizing dividends and capital injections limited bybankruptcy, and practical approximations for theCramér-Lundberg process
F. Avram ∗1 , D. Goreac , R. Adenane , and U. Solon Laboratoire de Mathématiques Appliquées, Université de Pau, France 64000 LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447Marne-la-Valleée, France School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, PRChina Département des Mathématiques, Université Ibn-Tofail, Kenitra, Maroc 14000 Institute of Mathematics, University of the Philippines Diliman, PhilippinesFebruary 23, 2021
Abstract
The recent papers Gajek-Kucinsky(2017) and Avram-Goreac-Li-Wu(2020) investigated the controlproblem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The firstpaper works under the spectrally negative Lévy model; the second works under the Cramér-Lundbergmodel with exponential jumps, where the results are considerably more explicit. The current paperhas three purposes. First, it illustrates the fact that quite reasonable approximations of the generalproblem may be obtained using the particular exponential case studied in Avram-Goreac-Li-Wu(2020).Secondly, it extends the results to the case when a final penalty P is taken into consideration as wellbesides a proportional cost k > for capital injections. This requires amending the “scale and Gerber-Shiu functions" already introduced in Gajek-Kucinsky(2017). Thirdly, in the exponential case, theresults will be made even more explicit by employing the Lambert-W function. This tool has particularimportance in computational aspects and can be employed in theoretical aspects such as asymptotics. Keywords: dividend problem, capital injections, penalty at default, scale functions, Lambert-W function,De Vylder-type approximations, rational Laplace transform
Contents ( − a, , b ) policies, for the spectrally negative Lévy case 5 a ∗ , b ∗ when F ( x ) := 1 − e − µx , P > − cq η ( b ) = 0 , δ k,P = 0 . . . . . . . . . . . . . . . . . . . . . 10 c W q of scale function . . . . . 14 ∗ Corresponding author. E-mail: [email protected] Examples of computations involving scale function and dividend value approximations 16 J along one parameter families ofCramér-Lundberg processes 237 The profit function when the claims are distributed according to a matrix exponentialjumps density 25 This paper concerns the approximate optimization of a new type of boundary mechanism, which emergedrecently in the actuarial literature [APY18, GK17, AGLW20], in the context of the optimal control ofdividends and capital injections.
The model . Consider the spectrally negative Lévy risk model: X t = x + ct − ¯ ξ t , c ≥ , where ¯ ξ t is a spectrally positive Lévy process, with Lévy measure ν ( x ) dx . The classic example is that of theperturbed Cramèr-Lundberg risk model with ¯ ξ t = N t X i =1 ξ i + σB t , where B t is a Brownian motion, where N t is an independent Poisson process of intensity λ > , and ( ξ i ) i ≥ isa family of i.i.d.r.v. whose distribution, density and moments are denoted respectively by F, f, m i , i ∈ { , ... } .Furthermore,• the process is modified by dividends and capital injection : π := ( D, I ) ⇒ X πt := X t − D t + I t , where D, I are adapted, non-decreasing and càdlàg processes, with D − = I − = 0 ;• the first time when we do not bail-out to positive reserves σ π := inf (cid:8) t > X πt − − △ ¯ ξ t + △ I t < (cid:9) is called bankruptcy/absolute ruin ;• prior to bankruptcy, dividends are limited by the available reserves: △ D t := D t − D t − ≤ X πt − − △ ¯ ξ t + △ I t , where ¯ ξ t := P N t i =1 ξ i . The set of “admissible" policies satisfying this constraint will be denotedby ˜Π( x ) .The objective is to maximize the profit : J πx := E x "Z [ ,σ π ] e − qt ( dD s − kdI s ) − P e − qσ π , k ≥ , P ≥ . The value function is V ( x ) := sup π ∈ ˜Π( x ) J πx , x ∈ R . Motivation.
The recent papers [GK17, AGLW20] investigated the above control problem of optimizingdividends and capital injections for processes with jumps, when bankruptcy is allowed as well. The secondpaper works under the Cramér-Lundberg model with exponential jumps, while the first works under thespectrally negative Lévy model, allowing also for the presence of Brownian motion and infinite activityjumps. It turns out that the optimal policy belongs to the class of ( − a, , b ) , a > , b ≥ “bounded bufferpolicies", which consist in allowing only capital injections smaller than a given a and declaring bankruptcyat the first time when the size of the overshoot below 0 exceeds a, and in paying dividends when the reserve2eaches an upper barrier b . These will briefly be described as ( − a, , b ) policies from now on. Furthermore,the optimal ( a ∗ , b ∗ ) are the roots of one variable equations with explicit solutions related to the Lambert-W(right) function (ProductLog in Mathematica).Below, our goal is to show numerically that exponential approximations provide quite reasonable results(as the de Vylder approximation provides for the ruin problem). We will focus in our examples on the case ofmatrix exponential jumps(which are known to be dense in the class of general nonnegative jumps, with evenerror bounds for completely monotone jumps being available [VAVZ14]), for two reasons. One is in orderto highlight certain exact equations which are similar to their exponential versions, and which may at theirturn be used to produce even more accurate approximations in the future, and, secondly, since numericalLaplace inversion for this class may easily tuned to have arbitrarily small errors. History of the problem : The case of no capital injections (also characterized by k = ∞ or absorptionbelow 0) is the dividend problem posed by De Finetti [DF57, Ger69] where dividends are paid above barrier b ∗ and a ∗ = 0 is imposed. “The challenge is to find the right compromise between paying early in view ofthe discounting or paying late in order not to reach ruin too early and thus profit from the positive safetyloading for a longer time" [AAM20].Forced injections and no bankruptcy at (also characterized by a reflection at ) is studied in Shreve[SLG84] where dividends are paid above barrier b ∗ and a ∗ = ∞ is imposed.From Lokka, Zervos, [LZ08] we know that in the Brownian motion case, it is optimal to either alwaysinject, if k ≤ k c , for some critical cost k c (i.e. use Shreve), or, stop at (use De Finetti). We proposeto call this the Lokka, Zervos alternative . The “proof" of this alternative starts by largely assuming itvia a heuristically justified border Ansatz [LZ08, (5.2)]: max {− V (0) , V ′ (0) − k } = 0 = ⇒ either V (0) =0 or V ′ (0) = k. Extensive literature on SLG forced bailouts (no bankruptcy) can be found at Avram et al., (2007)[APP07], Kulenko and Schmidli, (2008) [KS08], Eisenberg and Schmidli, (2011) [ES11], Pérez et al., (2018)[PYB18], Lindensjo, Lindskog (2019) [LL19], Noba et al., 2020 [NPY20].Articles [GK17, AGLW20] are the only papers which relate declaring bankruptcy to the size of jumps,with general and exponential jumps, respectively. [GK17] deals also with the presence of Brownian motionand infinite activity jumps, by conditioning at the first draw-down time; the optimality proof is quiteinvolved.In [AGLW20], it is also shown that neither V (0) = 0 nor V ′ (0) = k are possible: the Lokka-Zervosalternative disappears, but another interesting alternative holds. Above a certain critical k c the optimaldividends barrier switches from strictly positive to , and k c is related in (26) to the Lambert-W function .The results of [GK17, AGLW20] may be divided in three parts:1. Compute the value of bounded buffer policies. The key result is (5) below, an explicit determinationof the objective J = J a,b , which allows optimizing it numerically. REMARK 1.
Computing the value function is considerably simplified by the use of first passagerecipes available for spectrally negative Lévy processes [AKP04, Kyp14, KKR13, AGVA19], which arebuilt around two ingredients: the W q and Z q q -scale functions, defined respectively for x ≥ , q ≥ as:(a) the inverse Laplace transform of κ ( s ) − q , where κ ( s ) is the Laplace exponent (which characterizesa Lévy process) and(b) Z q ( x ) = 1 + q R x W q ( y ) dy – see the papers [Sup76, Ber98, AKP04] for the first appearance of these functions. The name q -scale/harmonic functions is justified by the fact that these functions are harmonic for the process X killed upon entering ( −∞ , , in the sense that { e − q min[ t,T ] W q ( X min[ t,T ] ) , e − q min[ t,T ] Z q ( X min[ t,T ] ) } , t ≥ are martingales, as shown in [Pis04, Prop. 3] (in the case of Z q , there is also a penalty of at ruin,generalizing to other penalties produces the so-called Gerber-Shiu function).
2. Equations determining candidates for the optimal a ∗ , b ∗ are obtained by differentiating the objective(which is expressed in terms of the scale functions W q , Z q ), and the optimal pair ( a ∗ , b ∗ ) is identified.As a result, the critical k c is related in (26) to the Lambert-W function.3. The optimality of the ( − a ∗ , , b ∗ ) policy is established.3ote that the last step is quite non-trivial and is achieved by different methods in [GK17] and [AGLW20].The latter paper starts by formulating a (new) HJB equation associated to this stochastic control problem– see (8). REMARK 2.
The objective may be optimized numerically using the first step only (the equation (10) for J a,b ). Exponential approximations may also be used, which are similar in spirit with the de Vylder-type approx-imations. Recall that the philosophy of the de Vylder approximation is to approximate a Cramèr-Lundbergprocess by a simpler process with exponential jumps, with cleverly chosen exponential rate µ , and theparameters λ, c may also be modified, if one desires to make the approximations exact at x = 0 –see forexample [AHPS19] for more details)The efficiency of the de Vylder approximation for approximating ruin probabilities is well documented[DV78]. The natural question of whether this type of techniques may work for other objectives, like forexample for optimizing dividends and/or reinsurance was already discussed in [Høj02, DD05, BDW07,GSS08, AHPS19]. In this paper, following on previous works [ACFH11, AP14, ABH18], we draw first theattention to the fact that we have not one, but three de Vylder-type approximations for W q ( x ) (as for theruin probability). The best approximation in our experiments when the loading coefficient θ is large turnout to be the classic de Vylder approximation. However, for approximating near the origin, the two pointPadé approximation which fixes both the values W q (0) = c , W ′ q (0) = q + λc works better. The end resulthere is simply replacing the inverse rate µ − by m , in the formula for the scale function of the Cramèr-Lundberg process with exponential jumps. In between x = 0 and x → ∞ , the winner is sometimes the“Renyi approximation" which replaces the inverse rate by m m , and modifies λ as well (for the de Vylderapproximation, µ − is replaced by m m , and both λ, c are modified).We end this introduction by highlighting in figure 1 the fact that for exponential jumps, the limitedcapital injections objective function J given by (14) for arbitrary b but optimal a = s ( b ) (via a complicatedformula) improves the value function with respect to de Finetti and Shreve, Lehoczky and Gaver, for any b . - - De Finetti Value = /( c W' [ b ]) SLG Value =( - k Z1' [ b ])/( q W [ b ]) Limited Injections Value
Figure 1: The value function J given by (14), for arbitrary b but optimal a . The inequality observed isa consequence of the properties of the Lambert function. The improvement with respect to de Finetti isconsiderable, of . (the SLG approach is not competitive in this case). Note also that the optimalbarrier b = 0 . is smaller than the de Finetti and SLG optima of . , . respectively. Contents and contributions . Section 2 offers a conjectured profit formula for ( − a, , b ) policies,where we include also a final penalty P . The theoretical result of the section 1 revisits [GK17, AGLW20]by linking the two formulations together and emphasizing the impact of the bankruptcy penalty P (via thescale function G ). Its proof is beyond the scope of the present (already lengthy enough) paper and it canbe inferred from either one of [GK17] and [AGLW20] through a three step argument:4. express the cost by conditioning on the reserve ( J x ) starting from ≤ x ≤ b hitting either 0 or b;2. get a further relationship on costs J b and J by conditioning on the first claim;3. finally, mix these conditions together in order to obtain the explicit formula for J x .We also wish to point out (in section 2.1) the link to an appropriate HJB variational inequality equation8and the definition 1 specifying the action regions and their computation starting from the regimes in the HJBsystem. To the best of our knowledge, this is new and the preliminary studies conducted on more complicatedproblems (involving reinsurance and reserve-dependent premium) seem to reinforce the relevance of this tool.In section 7 we provide an alternative matrix exponential form of the exact cost, in the case of matrixexponential jumps.An explicit determination of a ∗ , b ∗ and an equity cost dichotomy when dealing with exponential jumpsare given in section 3, taking also advantage of properties of the Lambert-W function, which were notexploited before. The two main novelties of the section are:• emphasizing the computations of the optimal buffer/barrier (from [AGLW20]) in relation with thescale-like quantities appearing in [GK17];• making explicit use of the (computation-ready) Lambert-W function to describe the dependency ofoptimal a ∗ b (in equation 14) and of the dichotomy-triggering cost k c in equation 26.Again, a further novelty is the presence of the bankruptcy cost P .Section 4 reviews, for completeness, the de Vylder approximation-type approximations. Section 4.1recalls, for warm-up, some of the oldest exponential approximations for ruin probabilities. Section 4.2recalls in Proposition 5, following [AP14, AHPS19] three approximations of the scale function W q ( x ) § ,obtained by approximating its Laplace transform. These amount finally to replacing our process by onewith exponential jumps and cleverly crafted parameters based on the first three moments of the claims.In section 5, we consider particular examples and obtain very good approximations for two fundamentalobjects of interest: the growth exponent Φ q of the scale function W q ( x ) , and the (last) global minimumof W ′ q ( x ) , which is fundamental in the de Finetti barrier problem. Proceeding afterwards to the problemof dividends and limited capital injections, concepts in section 3 are used to compute a straightforwardexponential approximation based on an exponential approximation of the claim density, and a new “correctingredients approximation" which consists of plugging into the objective function (10) for exponential claimsthe exact "non-exponential ingredients" (scale functions and, survival and mean functions) of the non-exponential densities. Both methods are observed to yield reasonable values in approximating the objective.This leads us to our conclusion that from a practical point of view, exponential approximations aretypically sufficient in the problems discussed in this paper. ( − a, , b ) policies, for the spectrally negativeLévy case In this section, we allow ¯ ξ t to be a spectrally positive Lévy process, with a Lévy measure admitting a density ν ( dy ) = ν ′ ( y ) dy . The simplest example is that of the perturbed Cramèr-Lundberg risk model with ¯ ξ t = N t X i =1 ξ i + σB t , where N t is a Poisson process of intensity λ > , ( ξ i ) i ≥ is an independent family of i.i.d.r.v. with density f ( y ) , and B t is an independent Brownian motion.We revisit here the problem of optimizing the value of "bounded buffer ( − a, , b ) policies", following[GK17, AGLW20] (in order to relate the results, one needs to replace γ in the objective of [GK17] by /k ),while taking into account also the bankruptcy penalty P .An important role in the results will be played by the expected scale after a jump C ( x ) = Z x W q ( x − y ) ν ( y ) dy = cW q ( x ) − Z q ( x ) + σ W ′ q ( x ) , (1) § essentially, this is the “dividend function with fixed barrier", which had been also extensively studied in previous literaturebefore the introduction of W q ( x ) ν ( y ) = R ∞ y ν ( u ) du is the tail of the Lévy measure and σ is the Brownian volatility (the identity abovefollows easily from the q -harmonicity of Z q , after an integration by parts of the convolution term and adivision by q ).The problem of limited reflection requires introducing a new "scale function S a ( x ) and Gerber-Shiufunction G a,σ ( x ) "– see Remark 4 for further comments on this terminology: ( S a ( x ) = Z q ( x ) + C a ( x ) , C a ( x ) = R x W q ( x − y ) ν ( a + y ) dyG a,σ ( x ) = G a ( x ) + k σ W q ( x ) (2)where G a ( x ) = Z x W q ( x − y ) ( k m a ( y ) + P ν ( a + y )) dy := kM a ( x ) + P C a ( x ) ,m a ( y ) = Z a zν ( y + z ) dz. EXAMPLE 1.
With exponential jumps and possibly σ > , using the identities ν ( y + a ) = e − µa ν ( y ) , m a ( y ) = λe − µy m ( a ) , m ( a ) = Z a y µe − µy dy = 1 − e − µa µ − ae − µa , we find that the functions (2) are expressible as products of C ( x ) and the survival or mean function ofthe jumps: C a ( x ) = C ( x ) e − µa = C ( x ) F ( a ) , F ( a ) = 1 − F ( a ) S a ( x ) = Z q ( x ) + e − µa C ( x ) G a ( x ) = (cid:0) km ( a ) + P F ( a ) (cid:1) C ( x ) (3) ([GK17] use s c , r c , instead of M a ( x ) := R x W q ( x − y ) m a ( y ) dy, C a ( x ) , respectively). When P = 0 = σ ,these reduce to quantities in [AGLW20].The formulas above will be used below as a heuristic approximation in non-exponential cases. REMARK 3.
Note that C a (0) = 0 , G a (0) = 0 , S a (0) = 1 , C (0) = 0 , C ′ (0) = ( λc σ = 00 σ > , (4) and that C ( x ) , G a ( x ) , S a ( x ) are increasing functions in x . We state now a generalization of [GK17, Thm. 4] for the value function J a,b of ( − a, , b ) policies, interms of S a ( x ) , G a ( x ) . In the Cramèr-Lundberg case illustrated below, the proof is straightforward, following[AGLW20]. In the other case, one needs to adapt the proof of [GK17]. THEOREM 1. Cost function for ( a, b ) policies For a spectrally negative Lévy processes, let J x = J a,b ( x ) := E x "Z T − a e − qt (d D t − k d I t ) − P e − qT − a denote the expected discounted dividends minus capital injections associated to policies consisting in payingcapital injections with proportional cost k ≥ , provided that the severity of ruin is smaller than a > , andpaying dividends as soon as the process reaches some upper level b . Put G a,σ ( x ) = G a ( x ) + k σ W q ( x ) . Then, it holds that J x = G a,σ ( x ) + J a,b S a ( x ) = G a,σ ( x ) + − G ′ a,σ ( b ) S ′ a ( b ) S a ( x ) , x ∈ [0 , b ] kx + J a,b x ∈ [ − a, x ≤ − a . (5)6 EMARK 4.
The first equality in (5) will be easily obtained by applying the strong Markov property atthe stopping time T = min[ T , − , T b, + ] , but it still contains the unknown J .This relation suggests a definition of the scale S a and the Gerber-Shiu function G a,σ , as the coefficientof J and the part independent of J , respectively.This equality is also equivalent to J a,b = J x − G a,σ ( x ) S a ( x ) = 1 − G ′ a,σ ( b ) S ′ a ( b ) , (6) which suggests another analytic definition of the scale and Gerber-Shiu function corresponding toan objective J x which involves reflection at b .The functions S a ( x ) , G a ( x ) may be shown to stay the same for problems which require only modifying theboundary condition at b , like the problem of capital injections for the process reflected at b , or the problem ofdividends for the process reflected at b , with proportional retention k D (this is in coherence with previouslystudied problems). COROLLARY 2.
Let us consider the Cramèr-Lundberg setting without diffusion (i.e. σ = 0 ), For fixed k ≥ , b ≥ , the optimality equation ∂∂a J a,b = 0 may be written as J a,b = ka − P ⇔ J a,b − a = − P. (7) REMARK 5.
The first equality in (7) provides a relation between the objective J and the variable a ; thesecond recognizes this as the smooth fit equation J − a = 0 . Proof:
Recalling the expressions of J a,b , G a ( x ) , in (6), in (2), and from [GK17, Lem. A.4] M a ′ ( x ) = − aC a ′ ( x ) , where C a ′ ( x ) , M a ′ ( x ) denote derivatives with respect to the subscript a , Whenever b > , if a achieves themaximum in J a,b , it is straightforward (think of the economic interpretation) that a achieves the maximumof a J a,bx for every x ∈ [0 , b ] . Therefore, we find ∂∂a J a,b = 0 ⇔ J a,b = − G a ′ ( x ) C a ′ ( x ) = − kM a ′ ( x ) − P C a ′ ( x ) C a ′ ( x ) = ka − P ⇔ J a,b − a = J a,b − ka ⇔ J a,b − a = − P. The optimality proof in [AGLW20] is based on showing that the function J x (5) with a ∗ , b ∗ defined in (18),(19) is the minimal AC-supersolution of the HJB system ( max { H ( x, V, V ′ ( x )) , − V ′ ( x ) , V ′ ( x ) − k } = 0 , ∀ x ∈ R + max { V ′ ( x ) − k, − P − V ( x ) } = 0 , ∀ x ∈ R − , (8)where the Hamiltonian H is given by H ( x, φ, v ) := cv + λ Z R + φ ( x − y ) µe − µy − ( q + λ ) φ ( x ) . (9)To discuss (8), it is useful to introduce the concept of dividend-limited injections strategies and barrier strategies . The following are also valid for its generalizations to mixed singular/continuous controlstaking into account reinsurance: DEFINITION 1.
Dividend-limited injections strategies are stationary strategies where the dividends arepaid according to a partition of the state space R in five sets A , B , C , C , D as follows:1. If the surplus is in A (absolute ruin), bankruptcy is declared and a penalty P is paid; . If the surplus is in B bailouts/capital injections are used for bringing the surplus to the closest pointof C .3. If the surplus is in the open set C (continuation/no action set), no controls are used.4. If the current surplus is in C ⊂ D (these are upper-accumulation points of C ), dividends are paid at apositive rate, in order to keep the surplus process from moving.5. If the current surplus is in D , a positive amount of money is paid as dividends in order to bring thesurplus process to C . Barrier strategies are stationary strategies for which A , B , C , D are four consecutive intervals. REMARK 6.
The four sets A , B , C , D correspond to the cases when equality in the HJB equation (8) is attained by at least one of the operators − V − P, V ′ − k, ( G − q ) V, and − V ′ , respectively. Note thisgeneralizes [AM14, Ch. 5.3], where only the last two operators are considered. REMARK 7.
One may conjecture that dividend-limited injections strategies are of a (recursive) multi-band nature. In the case of exponential jumps, [AGLW20] show that the four sets A , B , C , D in the optimalsolution are intervals, denoted respectively by ( −∞ , − a ) , [ − a, , (0 , b ) , [ b, ∞ ) .Cheap equity corresponds the case when C = ∅ , and the partition reduces to three sets. a ∗ , b ∗ when F ( x ) := 1 − e − µx , P > − cq In this section we turn to the exponential case, where explicit formulas for the optimizers a ∗ , b ∗ are available.In particular, we will take advantage of properties of the Lambert-W function, which were not exploitedin [AGLW20]. Subsequently, in sections 5, 6 we will show that exponential approximations work typicallyexcellently in the general case. Although these results have already been established in [AGLW20], thepresent formulations have two achievements:1. allow an unified formulation of [AGLW20] and [GK17] (via the previously introduced scale functions);2. make use of a numerical tool (Lambert-W function) to express the optimal quantities of interest a ∗ , b ∗ . PROPOSITION 3. Cost function and optimality equations in the exponential case J a,b = 1 − C ′ ( b ) (cid:0) k m ( a ) + P F ( a ) (cid:1) ( F ( a )) C ′ ( b ) + qW q ( b ) = γ ( b ) − k m ( a ) − P F ( a ) qθ ( b ) + F ( a ) , (10) where we put γ ( b ) = 1 C ′ ( b ) , θ ( b ) = W q ( b ) C ′ ( b ) .
2. Put j ( b ) := γ ′ ( b ) qθ ′ ( b ) . (11) For fixed a ≥ , the optimality equation ∂∂b J a,b = 0 may be written as J a,b = j ( b ) . (12)
3. For fixed k ≥ and b ≥ , at critical points with a ( b ) = a ( k,P ) ( b ) = 0 satisfies ∂∂a J a ( b ) ,b = 0 we musthave h J a,b − ( ka − P ) i a = a ( b ) = 0 . Explicitly, η ( b, a ) := γ ( b ) θ ( b ) − kµθ ( b ) F ( a ) − q ( ka − P ) . (13)8 . When P ≥ − cq and b ≥ is fixed, the solution of (13) may be expressed in terms of the principal valueof the “Lambert-W(right)" function (an inverse of L ( z ) = ze z ) [ − e − , ∞ ) ∋ L ( z ) , z ∈ [ − , ∞ ) [CGH +
96, Boy98, BFS08, Pak15, VLSHGG +
19] (this observation is missing in [AGLW20]). (0 , ∞ ) ∋ a ( b ) = µ − (cid:18) − h ( b ) + L (cid:18) e h ( b ) qθ ( b ) (cid:19)(cid:19) , h ( b ) = h ( b, P ) = 1 qθ ( b ) − µk (cid:18) γ ( b ) qθ ( b ) + P (cid:19) (14) It follows that J a ( b ) ,b = kµ (cid:18) − h ( b ) + L (cid:18) qθ ( b ) e h ( b ) (cid:19)(cid:19) − P. (15)
5. In the special case b = 0 , (13) implies that a = a ( k,P ) = a ( k,P ) (0) satisfies the simpler equation δ k,P ( a ) := λη (0 , a ) = e c − k (cid:18) aq + λµ (1 − e − µa ) (cid:19) , e c = c + qP > , (16) with solution µ a ( k,P ) = − g + L (cid:18) λq e g (cid:19) > , g = h (0) = λq − µkq e c. (17)
6. At a critical point ( a ∗ , b ∗ ) , a ∗ > , b ∗ > , we must have both J a ∗ ,b ∗ = j ( b ∗ ) = ka ∗ − P = ⇒ a ∗ = s ( b ∗ ) , s ( b ) := j ( b ) + Pk , (18) and η ( b ∗ ) , η ( b ) := η ( b, s ( b )) = γ ( b ) θ ( b ) − qj ( b ) − kµθ ( b ) F (cid:18) j ( b ) + Pk (cid:19) = 0 . (19)
7. The equation η ( b ) may be solved explicitly for P , yielding P = − kµ log " qθ ( b ) j ( b ) − γ ( b ) kµ − j ( b ) . (20) Proof:
1. follows from Theorem 1.1.2. Let M ( b ) , N ( b ) denote the numerator an denominator of J a,b := M ( b ) N ( b ) in (10). The optimality equation ∂∂b J a,b = N ′ ( b ) N ( b ) (cid:16) M ′ ( b ) N ′ ( b ) − J a,b (cid:17) = 0 simplifies to J a,b = M ′ ( b ) N ′ ( b ) = γ ′ ( b ) qθ ′ ( b ) = j ( b ) .
3. (13) is a consequence of 1 and of the smooth fit result Corollary 2.4. See the proof of the particular case 5; a ∈ (0 , ∞ ) holds since P ≥ − cq = ⇒ h ( b ) < qθ ( b ) .
5. (16) follows from W q (0) = c , θ (0) = λ − . To get (17), rewrite the equation (16) as ze z = λq e g , z = µa + g ; a ∈ (0 , ∞ ) holds since P ≥ − cq = ⇒ g < λq .
6. follows from 2. and .3.7. is straightforward.
REMARK 8.
Note that the de Finetti and Shreve, Lehoczky and Gaver solutions a ∗ = a ( b ∗ ) = ( ∞ are always non-optimal, when P ≥ − cq (see (14) ).However, as k → ∞ , h ( b ) → qθ ( b ) = 0 and, a ( b ) = µ − (cid:0) − h ( b ) + L (cid:0) h ( b ) e h ( b ) (cid:1)(cid:1) = 0 . This suffices toinfer that you get de Finetti case. n the other hand, P → ∞ = ⇒ h ( b ) → −∞ = ⇒ a ( b ) → ∞ = ⇒ J a ( b ) ,b → γ ( b ) − k/µqθ ( b ) = − kC ′ ( b ) /µqW q ( b ) = J ,SLG ( b ) η ( b ) → q (cid:16) J SLG ,k ( b ) − j ( b ) (cid:17) , ∀ b > . (21) Thus, these regimes can be recovered asymptotically. Let now b ∗ ,Sk , b ∗ ,DP denote the unique roots of η ( b ) = 0 in the two asymptotic cases, which coincide with the classic Shreve, Lehoczky and Gaver and de Finettibarriers.Then, it may be checked that b ∗ ≤ min[ b ∗ ,Sk , b ∗ ,DP ] . η ( b ) = 0 , δ k,P = 0 The following (new) result discusses the existence of the roots of the equations η ( b ) = 0 , δ k,P = 0 introducedin proposition (3) and relates them to the Lambert-W function. PROPOSITION 4. θ increases from θ (0) = /cλ/c = λ to θ ( ∞ ) = c Φ q − q , as we see it in the figurebelow. x Figure 2: Plot of θ with θ (0) = 2 and θ ( ∞ ) = 22 . , for µ = 2 , c = 3 / , λ = 1 / , q = 1 / , P = 1 and k = 3 / . γ is increasing-decreasing (from cλ to ), with a maximum at the unique root of C ′′ ( x ) = 0 given by ¯ b := 1Φ q − ρ − log (cid:18) ρ − Φ q (cid:19) , (22) where Φ q , ρ − denote the positive and negative roots of the Cramèr-Lundberg equation κ ( s ) = 0 .The figure below illustrates the plot of the function γ and j ( b ) in which the ¯ b is represented by the blackpoint. b - Figure 3: Plots of j ( b ) and γ ( b ) with ¯ b = 2 . and j (0) = 4 . , for µ = 2 , c = 3 / , λ = 1 / , q = 1 / , P = 1 and k = 3 / . 10 f cµ − ( q + λ ) > , then ¯ b > defined in (22) is the unique positive root of j ( b ) and η (cid:0) ¯ b (cid:1) = W q ( ¯ b ) > .See the figure (4) b - Figure 4: For µ = 2 , c = 3 / , λ = 1 / , q = 1 / , P = 1 and k = 3 / , the root of η ( b ) = 0 is at b = 0 . The function j ( b ) = γ ′ ( b ) qθ ′ ( b ) is nonnegative and decreasing to on [0 , ¯ b ] , with j (0) = λµq − C ′′ (0)( C ′ (0)) = cµ − ( q + λ ) µq . (23) See the figure (3) .2. Put δ k,P := δ k,P ( a ( k,P ) ) = δ k,P ( j (0) + Pk ) = δ k,P ( e cµ − ( q + λ ) kµq ) = λ + q − λk (cid:16) − e − e cµ − λ − qqk (cid:17) µ , (24) and assume lim k →∞ δ k,P = λ + qµ − λ ( µ e c − ( λ + q )) qµ = ( λ + q ) − λµ e cqµ < ⇔ e cµ > λ − ( λ + q ) . (25) Then, ∀ P > − cq , the function δ k,P is decreasing in k with δ ,P > , and has a unique root k c = k c ( P ) := q + λλ ff + L ( − f e − f ) > q + λλ , (26) where f := λq + λ e cµ − ( λ + q ) q > ⇔ e cµ > λ − ( λ + q ) ⇔ P > P l := q − (cid:16) µ − λ − ( λ + q ) − c (cid:17) (27) (note that the denominator f + L (cid:0) − f e − f (cid:1) does not equal since f > and L takes always valuesbigger than − ; or, note that − f = L − ( L ( − f )) , where L − is the other real branch of the Lambertfunction).Furthermore, δ k,P < ⇔ k > k c ( P ) . (28)
3. It follows that η ( b ) = 0 has at least one solution of in (0 , ¯ b ] iff η (0) = cλ − λ (cid:18) c − q + λµ (cid:19) − kµ F (cid:16) a ( k,P ) (cid:17) = 1 λµ (cid:16) λ + q − λkF (cid:16) a ( k,P ) (cid:17)(cid:17) = 1 λ δ k,P < ⇔ k > k c . (29) The first such solution will be denoted by b ∗ . roof: For 1. see [AGLW20, Proof of Theorem 11, A2].2. By using the assumption e cµ > λ − ( λ + q ) we get e cµ ≥ λ + q , and k ∈ [1 , ∞ ) → δ k,P is decreasing.Put d = e cµ − ( λ + q ) q . The inequality δ k,P < (see (29)) may be reduced to e − dk < − q + λλk ⇔ < e dk (cid:18) − ( q + λ ) dλ dk (cid:19) := e z (1 − z/f ) . Rewriting the latter as − f > e z ( z − f ) we recognize, by putting z = y + f , an inequality reducible to ye y < − f e − f . The solution is y < L (cid:0) − f e − f (cid:1) , where L is the principal branch of the Lambert-W function.The final solution is (28), where we may note that the variables k, P have been separated.3. is straightforward. REMARK 9.
The function [1 , ∞ ) ∋ f ff + L ( − fe − f ) ∈ (1 , ∞ ) blows up at f = 1 , and converges to when f → ∞ (or when either µ or e c = c + qP are large enough) as may be noticed in the figure below,which blows up at the value P l := − / . Note also that when f (or one of c, P, µ are large enough), k c givenby (26) stabilizes to the equilibrium λ + qλ = 6 / ; this is related to [APP07, Lemma 2], [KS08, Lemma 7],who obtain the same condition for b ∗ = 0 (without buffering capital injections). Intuitively, under theseconditions, buffering is not crucial.At the other end, as f tends to its lower limit and to the regime B, the notion of equity expensivenessvanishes, and k c → ∞ . - P k * as function of P Figure 5: k c as function of P , for several values of c , with the vertical asymptote at P l fixed.The next two figures illustrate how k c blows up at the critical values q l := (1 / − λ + P λµ + √ λ √ µ √ c − P λ + P λµ ) and λ l := (cid:16) cµ − p ( − cµ + µ ( − P ) q + 2 q ) − q + µP q − q (cid:17) (represented by red points inthe figures below). The dark (blue) parts correspond to the regime A .12 .5 1.0 1.5 2.0 q - - (a) k c defined in (26)as function of q , for µ = 2 , c =3 / , λ = 1 and P = 1 ; q l = √ . (cid:8)(cid:9)(cid:10)(cid:11) λ - (b) k c as function of λ , for µ = 2 , c = 3 / , q = 1 / and P = 1 . Figure 6: k c as a function of q , λ In the simplest case of exponential jumps of rate µ and σ = 0 , the formula for the ruin probability is Ψ( x ) = P x [ ∃ t ≥ X t <
0] = 11 + θ exp (cid:18) − xθµ θ (cid:19) = 11 + θ exp (cid:18) − xθm − θ (cid:19) , (30)where θ = c − λm λm is the loading coefficient. By plugging the correct mean of the claims in the second formulayields the simplest approximation for processes with finite mean claims.More sophisticated is the Renyi exponential approximation Ψ R ( x ) = 11 + θ exp (cid:18) − xθ b m − θ (cid:19) , b m = m m ; (31)This formula can be obtained as a two point Padé approximation of the Laplace transform, whichconserves also the value Ψ(0) = (1 + θ ) − [AP14]. It may be also derived heuristically from the first formulain (30), via replacing µ by the correct “excess mean" of the excess/severity density f e ( x ) = F ( x ) m = 1 − F ( x ) m , which is known to be b m . Heuristically, it makes more sense to approximate f e ( x ) instead of the original den-sity f ( x ) , since f e ( x ) is a monotone function, and also an important component of the Pollaczek-Khinchineformula for the Laplace transform b Ψ( s ) = R ∞ e − sx F ( dx ) – see [Ram92, AP14].More moments are put to work in the de Vylder approximation Ψ DV ( x ) = 11 + e θ exp − x e θ b m − e θ ! , b m := m m , e λ = 9 m m λ, e c = c − λm + e λ b m , e θ = 2 m m m θ = b m b m θ. (32)Interestingly, the result may be expressed in terms of the so-called "normalized moments" b m i = m i i m i − (33)introduced in [BHT05].The de Vylder approximation parameters above may be obtained either from1. equating the first three cumulants of our process to those of a process with exponentially distributedclaim sizes of mean b m , and modified λ, c [DV78] (however p = c − λm = E [ X ] must be conserved,since this is the first cumulant), or2. a Padé approximation of the Laplace transform of the ruin probabilities [ACFH11].13he second derivation via Padé shows that higher order approximations may be easily obtained as well.They might not be admissible, due to negative values, but packages for “repairing" the non-admissibility areavailable – see for example [DcSA16].The first derivation of the de Vylder approximation is a process approximation (i.e., independent of theproblem considered); as such, it may be applied to other functionals of interest besides ruin probabilities( W q ( x ) ,dividend barriers, etc), simply by plugging the modified parameters in the exact formula for the ruin prob-ability of the simpler process. c W q of scalefunction The simplest approximations for the scale function W q ( x ) will now be derived heuristically from the followingexample. EXAMPLE 2. The Cramér-Lundberg model with exponential jumps
Consider the Cramér-Lundberg model with exponential jump sizes with mean /µ , jump rate λ , premium rate c > , and Laplaceexponent κ ( s ) = s (cid:16) c − λµ + s (cid:17) . Solving κ ( s ) − q = 0 ⇔ cs + s ( cµ − λ − q ) − qµ = 0 for s yields two distinctsolutions γ ≤ ≤ γ = Φ q given by γ ( µ, λ, c ) = γ = 12 c (cid:18) − ( µc − λ − q ) + q ( µc − λ − q ) + 4 µqc (cid:19) ,γ ( µ, λ, c ) = γ = 12 c (cid:18) − ( µc − λ − q ) − q ( µc − λ − q ) + 4 µqc (cid:19) . The W scale function is: W q ( x ) = A e γ x − A e γ x c ( γ − γ ) ⇔ c W q ( s ) = s + µcs + s ( cµ − λ − q ) − qµ , (34) where A = µ + γ , A = µ + γ .Furthermore, it is well-known and easy to check that the function W ′ q ( x ) is in this case unimodal withglobal minimum at b DeF = 1 γ − γ ( log ( γ ) A ( γ ) A = log ( γ ) ( µ + γ )( γ ) ( µ + γ ) if W ′′ q (0) < ⇔ ( q + λ ) − cλµ < if W ′′ q (0) ≥ ⇔ ( q + λ ) − cλµ ≥ , (35) since W ′′ q (0) = ( γ ) ( µ + γ ) − ( γ ) ( µ + γ ) c ( γ − γ ) = ( q + λ ) − cλµc and that the optimal strategy for the de Finetti problemis the barrier strategy at level b DeF (see for example [APP07], [AGVA19, Sec. 3]).
Plugging now the respective parameters of the de Vylder type approximations in the exact formula (34)for the Cramèr-Lundberg process with exponential claims, we obtain three approximations for c W q :1. “Naive exponential" approximation obtained by plugging µ − → m in (34) (as was done, for a differentpurpose) in (30)2. Renyi § , obtained by plugging µ − → b m , λ R → λ m b m (since c is unchanged, the latter equation isequivalent to the conservation of ρ = λm c , and to the conservation of θ , so this coincides with theRenyi ruin approximation used in (31).)3. De Vylder, obtained by plugging µ − → b m , e λ → λ m m , e c = c − λm + e λ b m . REMARK 10.
In the case of exponential claims, these three approximations are exact, by definition (orcheck that for exponential claims all the normalized moments are equal to µ − ). REMARK 11.
The conditions for the non-negativity of the barrier is W ′′ q (0 + ) < ⇔ ( λ + qc ) < λc f (0) .Here, this condition is satisfied for the exact when θ > ( λ + q ) (1 − ρ ) λ f (0) m . § This is called DeVylder B) method in [GSS08, (5.6-5.7)], since it is the result of fitting the first two cumulants of the riskprocess.
14t is shown in [AHPS19, Prop. 1] that the three de Vylder type approximations are two-point Padéapproximations of the Laplace transform (hence higher order generalizations are immediately available).We recall that two-point Padé approximations incorporate into the Padé approximation two initial valuesof the function (which can be derived easily via the initial value theorem, from the Pollaczek-KhinchineLaplace transform): W q (0 + ) = lim s →∞ s c W q ( s ) = 1 c , (36) W ′ q (0 + ) = lim s →∞ s (cid:18) sκ ( s ) − q − W q (0 + ) (cid:19) = q + λc . (37)In our case, incorporating both W q (0 + ) , W ′ q (0 + ) leads to the natural exponential approximation whichis therefore the best near x = 0 . Incorporating none of them yields the de Vylder approximation, which isthe best asymptotically. Incorporating only W q (0 + ) leads to Renyi, which is expected to be the best in anintermediate regime.Note that when the jump distribution has a density f , it holds that : ‖ W ′′ q (0 + ) = lim s →∞ s (cid:18) s (cid:18) sκ ( s ) − q − W q (0 + ) (cid:19) − W ′ q (0 + ) (cid:19) = 1 c (cid:16) ( λ + qc ) − λc f (0) (cid:17) . (38)Thus, W ′′ q (0) already requires knowing f C (0) (which is a rather delicate task starting from real data);therefore we will not incorporate into the Padé approximation more than two initial values of the function.We recall below in Proposition 5 three types of two-point Padé approximations [AHPS19, Prop. 1], andparticularize them to the case when the denominator degree is n = 2 (which are further illustrated below). PROPOSITION 5. Three matrix exponential approximations for the scale function .1. To secure both the values of W q (0) and W ′ q (0) , take into account (36) and (37) , i.e. use the Padéapproximation c W q ( s ) ∼ P n − i =0 a i s i cs n + P n − i =0 b i s i , a n − = 1 , b n − = ca n − − λ − q. For n = 2 we recover the “natural exponential " approximation of plugging µ → m in (34) : c W q ( s ) ∼ m + scs + s (cid:16) cm − λ − q (cid:17) − qm , (39) used also (for a different purpose) in (30) .2. To ensure only W q (0) = c , we must use the Padé approximation c W q ( s ) ∼ P n − i =0 a i s i cs n + P n − i =0 b i s i , a n − = 1 . For n = 2 , we find c W q ( s ) ∼ m m + scs + s ( cm − λm − m q ) m − m qm = b m + scs + s (cid:16) c b m − λ m b m − q (cid:17) − q b m , (40) where b m = b m = m m is the first moment of the excess density f e ( x ) . Note that it equals the scalefunction of a process with exponential claims of rate b m − and with λ modified to λ R = λ m b m . Since c is unchanged, the latter equation is equivalent to the conservation of ρ = λm c , and to the conservationof θ , so this coincides with the Renyi approximation § used in (31) . ‖ This equation is important in establishing the nonnegativity of the optimal dividends barrier. § This is called DeVylder B) method in [GSS08, (5.6-5.7)], since it is the result of fitting the first two cumulants of the riskprocess. . The pure Padé approximation yields for n = 2 c W q ( s ) ∼ s + m m s (cid:16) c − λm + λ m m (cid:17) + s (cid:16) c m m − m m m λ − q (cid:17) − m m q = s + b m e cs + s (cid:16)e c b m − e λ − q (cid:17) − b m q , e c = c − λm + e λ b m , e λ = λ m m . Note that both the coefficient of s in the denominator coincides with the coefficient e c in the classic deVylder approximation, since e λ b m = λ m m m m = λ m m , and so does the coefficient of s , since c m m − m m m λ = e c b m − e λ = (cid:16) c − λm + e λ b m (cid:17) b m − e λ. Our goal in this section is to investigate whether exponential approximations are precise enough to yieldreasonable estimates for quantities important in control like1. the dominant exponent Φ q of W q ( x )
2. the last local minimum of W ′ q ( x ) , b DeF , which yields, when being the global minimum, the optimalDe Finetti barrier3. W ′′ q (0) , which determines if b DeF = 0
4. the functional J yielding the maximum dividends with capital injections.All the examples considered involve a Cramèr-Lundberg model with rational Laplace transform c W q ( s ) (since in this case, the computation of W q , Z q is fast and in principle arbitrarily large precision may beachieved with symbolic algebra systems).1. For the first three problems, we will use de Vylder type approximations. Graphs of W ′ q , W ′′ q and sometables summarizing the simulation results will be presented. We note that in most of the cases that weobserved, the de Vylder approximation of Φ q deviates from the exact value the least – see for exampleTable 2. For the De Finetti barrier, the "winner" depends on the size of b DeF . Unsurprisingly, whennear , the natural exponential approximation wins, and as b DeF increases, Renyi and subsequentlythe de Vylder approximation take the upper hand – see for example Table 3.2. For the computation of J , we provide, besides the exact value, also two approximations:(a) For a given density of claims f one computes an exponential density approximation f e ( x ) = m exp ( − xm ) where m is the first moment of f . Subsequently, W , Z , J and a, b are obtainedusing the exponential approximation f e . Quantities obtained by this method would be referredto with an affix ‘expo pure’.(b) For a given density of claims f , the value function is computed via the formula which assumesexponential claims in equation 3, but the "ingredients" W , Z, F and the mean function m are thecorrect ingredients corresponding to our original density f . Quantities obtained by this methodwould be referred to with an affix ‘expo CI’.It turns out that the pure expo approximation works better for large θ , and the correct ingredientsapproximation works better for small θ .Note that we only included tables illustrating approximating J for the first two examples, to keep thelength of the paper under control, but similar results were obtained for the other examples.16 .1 A Cramér-Lundberg process with hyperexponential claims of order 2 We take a look at a Cramér-Lundberg process with density function f ( x ) = e − x + e − x with λ = 1 , θ = 1 and q = . (cid:12)(cid:13)(cid:14)(cid:15) (a) W ′ q ( x ) - - - (b) W ′′ q ( x ) Figure 7: Exact and approximate plots of W ′ q ( x ) and W ′′ q ( x ) for f ( x ) = e − x + e − x , θ = 1 , q = .Dominant exponent Φ q Percent relative error( Φ q ) Optimal barrier b DeF
Percent relative error( b DeF )Exact 0.110113 0 3.45398 0Expo 0.110657 0.494313 3.51173 1.67191Dev 0.110115 0.00195933 3.48756 0.972251Renyi 0.110078 0.0321413 3.5323 2.26744Table 1: Exact and approximate values of Φ q and b DeF for f ( x ) = e − x + e − x , theta = 1 , q = , aswell as percent relative errors, computed as the absolute value of the difference between the approximationand the exact, divided by the exact, times 100. Relative errors for Φ q are less than . , with the pureexponential approximation proving to be the worst and the DeVylder the best approximations, respectively.The optimal barrier b DeF is also best approximated by DeVylder, with Renyi being the worst at . . θ Closest approximation Φ q exact Φ q approximation % error Φ q Φ q for f ( x ) = e − x + e − x when θ varies.17 Closest approximation Barrier exact Barrier approx % error Barrier1 Dev 3.45398 3.48756 0.9722510.9 Dev 3.20191 3.23103 0.9094870.8 Dev 2.90951 2.93074 0.7296280.7 Dev 2.57043 2.57742 0.2720880.6 Dev 2.1804 2.16054 0.9106660.5 Ren 1.74216 1.75266 0.602780.4 Expo 1.2735 1.29456 1.653780.3 Expo 0.81068 0.652264 19.5412Table 3: Exact and approximate values of b DeF for for f ( x ) = e − x + e − x when θ varies. As b DeF approaches , errors of all the approximations increase dramatically, with the pure exponential approximationperforming better than the rest. Meanwhile, as b DeF increases, Renyi and subsequently the de Vylderapproximation take the upper hand. For θ = 0 . and θ = 0 . , all the approximations yield a 0 barrierapproximate for exact barrier values of . and . respectively, hence failing to predict thenon-zero barrier. J0 θ J0 exact J0 expo pure J0 expo pure error J0 expo CI J0 expo CI error1 5.95034 5.99151 0.691856 6.26009 5.205510.9 5.15579 5.17573 0.386663 5.45269 5.75840.8 4.39383 4.38494 0.202205 4.67042 6.294890.7 3.68299 3.63933 1.18555 3.92937 6.689580.6 3.04577 2.96728 2.57704 3.25112 6.74230.5 2.50331 2.39942 4.15022 2.65901 6.219740.4 2.06833 1.9585 5.31006 2.17044 4.937250.3 1.74095 1.65616 4.86984 1.78984 2.808780.2 1.50439 1.44242 4.11969 1.50871 0.2866720.1 1.30271 1.25324 3.79748 1.30271 0Table 4: Values of J compared with approximations using all exponential inputs ( J expo pure) and actualinputs but computed using the exponential formula ( J expo CI). The pure exponential approximation doesa good job of approximating J for higher values of θ considered, while the exponential CI approximationseemed to fair better for lower θ values a θ a exact a expo pure a expo pure error a expo CI a expo CI error1 3.9669 3.99434 0.691861 4.17339 5.205510.9 3.4372 3.45049 0.386665 3.63512 5.75840.8 2.92922 2.9233 0.202204 3.11361 6.294890.7 2.45533 2.42622 1.18555 2.61958 6.689580.6 2.03051 1.97818 2.57704 2.16741 6.74230.5 1.66888 1.59961 4.15022 1.77268 6.219740.4 1.37888 1.30566 5.31006 1.44696 4.937250.3 1.16063 1.10411 4.86983 1.19323 2.808780.2 1.00293 0.961612 4.11969 1.0058 0.2866720.1 0.868476 0.835496 3.79748 0.868476 0Table 5: Values of a compared with approximations using all exponential inputs ( a expo pure) and actualinputs but computed using the exponential formula ( a expo CI)18 θ b exact b expo pure b expo pure error b expo CI b expo CI error1 1.41036 1.46188 3.65293 1.25374 11.10450.9 1.37645 1.44439 4.93621 1.23362 10.37610.8 1.31492 1.40417 6.78809 1.19529 9.097810.7 1.21057 1.32258 9.25207 1.12775 6.841780.6 1.04634 1.17215 12.0245 1.01753 2.75290.5 0.810767 0.920406 13.5229 0.853397 5.258050.4 0.510085 0.538725 5.61475 0.634716 24.43350.3 0.17425 0.0105496 93.9457 0.376872 116.2820.2 0 0 0 0.105322 1000.1 0 0 0 0 0Table 6: Values of b compared with approximations using all exponential inputs ( b expo pure) and actualinputs but computed using the exponential formula ( b expo CI) Consider a Cramér-Lundberg process with density function f ( x ) = e − x + e − x + e − x , and c = 1 , λ = , θ = , p = , q = .The Laplace exponent of this process is κ ( s ) = s − s s +1) − s s +2) − s s +3) and from this one caninvert κ ( s ) − q = c W q ( s ) to obtain the scale function W q ( x ) = − . e ( − . x ) − . e ( − . x ) − . e ( − . x ) + 1 . e (0 . x ) . From this, we see that the dominant exponent is Φ q = 0 . .Figure 8 shows the exact and approximate plots of the first two derivatives of W q . The exact plots arelabelled Wxexact, and coloured as the darkest. The plots of W ′ q exhibit noticeable unique minima around x = 2 , with the exact one being at b DeF = 1 . , which is the optimal barrier that maximizes dividendshere. Note that the approximations are practically indistinguishable from the exact around this point (whichis our main object of interest here). (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)1(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !" (a) W ′ q ( x ) - &’( - )*+ (b) W ′′ q ( x ) Figure 8: Exact and approximate plots of W ′ q ( x ) and W ′′ q ( x ) for f ( x ) = e − x + e − x + e − x , c = 1 , q = . Laplace inversion done via Mathematica; coefficients and exponents are decimal approximations of the real values. Φ q Percent relative error( Φ q ) Optimal barrier b DeF
Percent relative error( b DeF )Exact 0.18198 0 1.89732 0Expo 0.184095 1.162215628 2.04608 7.840532962Renyi 0.181708 0.149466974 2.08136 9.699997892Dev 0.182011 0.017034839 1.91233 0.79111589Table 7: Exact and approximate values of Φ q and b DeF for f ( x ) = e − x + e − x + e − x , c = 1 , q = ,as well as percent relative errors, computed as the absolute value of the difference between the approximationand the exact, divided by the exact, times 100. Relative errors for the Φ q value are less than . , even forthe worse natural exponential approximation, and the DeVylder approximation is the winner. The optimalbarrier b DeF is well approximated only by DeVylder. θ Closest approximation Φ q exact Φ q approximation % error Φ q Φ q , for f ( x ) = e − x + e − x + e − x , θ varies. θ Closest approximation Barrier exact Barrier approx % error Barrier263/235 Dev 1.89732 1.91233 0.791183243/235 Dev 1.79954 1.78002 1.08482183/235 Ren 1.45224 1.52484 4.9989163/235 Ren 1.31579 1.33691 1.60463143/235 Ren 1.16804 1.12368 3.79796123/235 Expo 1.00898 1.04123 3.19653103/235 Expo 0.839228 0.794964 5.2744483/235 Expo 0.660338 0.513179 22.285463/235 Expo 0.474896 0.196234 58.678543/235 Expo 0.286563 0 10023/235 Expo 0.0998863 0 1003/235 All 0 0 0Table 9: Exact and approximate values of b DeF for f ( x ) = e − x + e − x + e − x , when θ varies.Unsurprisingly, when b DeF is near , the natural exponential approximation wins, but has poor performance,and as b DeF increases, Renyi and subsequently the de Vylder approximation take the upper hand. For thesmallest three values of θ , all of the approximations yielded e b DeF = 0 as the optimal barrier, with this beingtrue only for θ = 3 / .We move now to the dividend problem with capital injections with cost k ≥ as in Theorem 1. One cancompute the value function J at x = 0 in terms of W , Z , C , S , and G – see equation 5.20o provide a more concrete example, fixing q = , P = 0 , k = 3 / as input parameters we compute forvalues of J as a function of θ , with results summarized in the tables below. The tables provide comparisonsof the computed optimal quantities J , a , and b to an approximation using all exponential inputs (referredto as J , a , and b expo pure) and to an approximation which uses actual inputs but computed using theexponential formula as described in equation 3 (referred to as J , a , and b expo CI).J0 θ J J expo pure J expo pure error J expo CI J expo CI error263/235 3.7747 3.76883 0.155556 4.11784 9.09041243/235 3.41491 3.38603 0.845606 3.74156 9.5654223/235 3.0636 3.00802 1.81444 3.36985 9.99637203/235 2.72335 2.63828 3.1238 3.00466 10.3296183/235 2.39737 2.28225 4.80185 2.64879 10.4871163/235 2.08958 1.94765 6.79226 2.3062 10.3665143/235 1.80446 1.64396 8.8946 1.9823 9.85516123/235 1.54668 1.38072 10.73 1.68379 8.86472103/235 1.32041 1.16526 11.7499 1.4178 7.3758783/235 1.12864 1.00194 11.2258 1.19022 5.4555563/235 0.972835 0.88785 8.73585 1.00404 3.2079843/235 0.852739 0.789923 7.36635 0.859039 0.73883723/235 0.751597 0.701299 6.69218 0.751597 03/235 0.660372 0.620567 6.02761 0.660372 0Table 10: Values of J compared with approximations using all exponential inputs ( J expo pure) and actualinputs but computed using the exponential formula ( J expo CI). The pure exponential approximation doesa good job of approximating J for higher values of θ considered, while the exponential CI approximationseemed to fair better for lower θ values a θ a a expo pure a expo pure error a expo CI a expo CI error263/235 2.51647 2.74523 0.155536 2.51255 9.09042243/235 2.27661 2.49437 0.845596 2.25735 9.56541223/235 2.0424 2.24657 1.81443 2.00535 9.99638203/235 1.81557 2.00311 3.1238 1.75885 10.3296183/235 1.59825 1.76586 4.80185 1.5215 10.4871163/235 1.39306 1.53747 6.79226 1.29844 10.3665143/235 1.20298 1.32153 8.8946 1.09598 9.85516123/235 1.03112 1.12252 10.73 0.920479 8.86472103/235 0.880271 0.945198 11.7499 0.77684 7.3758783/235 0.752428 0.793477 11.2258 0.667962 5.4555563/235 0.648557 0.669362 8.73585 0.5919 3.2079843/235 0.568493 0.572693 7.36635 0.526616 0.73883823/235 0.501065 0.501065 6.69218 0.467532 03/235 0.440248 0.440248 6.02761 0.413711 0Table 11: Values of a compared with approximations using all exponential inputs ( a expo pure) and actualinputs but computed using the exponential formula ( a expo CI)21 θ b b expo pure b expo pure error b expo CI b expo CI error263/235 0.709355 0.805116 13.4997 0.677918 4.43179243/235 0.695874 0.801936 15.2416 0.671779 3.46259223/235 0.677601 0.794377 17.2337 0.662801 2.18425203/235 0.653005 0.779265 19.3352 0.649805 0.490147183/235 0.620126 0.751601 21.2012 0.631097 1.76912163/235 0.576553 0.704104 22.123 0.604293 4.81128143/235 0.519526 0.627369 20.7579 0.566198 8.98366123/235 0.446259 0.511076 14.5246 0.512961 14.947103/235 0.354524 0.346046 2.39143 0.440755 24.323183/235 0.243362 0.126054 48.2032 0.347059 42.609963/235 0.113593 0 100 0.231975 104.21643/235 0 0 0 0.0987484 023/235 0 0 0 0 03/235 0 0 0 0 0Table 12: Values of b compared with approximations using all exponential inputs ( b expo pure) and actualinputs but computed using the exponential formula ( b expo CI)To provide a point of comparison, we fix q = , and compute the de Finetti barrier to be b DeF = 1 . and the corresponding dividend value function when starting at x = 0 to be J DeF = 1 . . k J % deviation J − J DeF J b % deviation b J and b in presence of capital injections compared to the case where capital injections arenon-existent, J DeF = 1 . and b DeF = 1 . . As k is increased one can see that J and b approaches J DeF and b DeF . This is expected since higher costs of injecting capital makes it less viable, hence it istreated like the concept does not exist.
In the following example, we study a Cramèr-Lundberg model with density of claims given by f ( x ) = ue − ax (cid:18) ωx + φ (cid:19) = ue − ax (1 + cos( ωx + φ )) == e − ax ( u + u cos( φ ) cos( ωx ) − u sin( φ ) sin( ωx )) where u = a (cid:0) a + ω (cid:1) a + ω + a cos( φ ) − aω sin( φ ) . Assuming further that a = 1 , φ = 2 , ω = 20 , and that θ = 1 , q = 1 / , the Laplace exponent for this22rocess is κ ( s ) = s ( . s +5 . s +843 . s +420 . ) ( s +1 . )( s +2 .s +401 . ) and the scale function is W q ( x ) = 0 . e . x − . e − . x + e − . x cos(19 . x ) (cid:16) − (0 . . i ) sin(39 . x ) − (0 . . i ) + ( − . . i ) cos(39 . x ) (cid:17) + e − . x sin(19 . x ) (cid:16) − (0 . − . i ) sin(39 . x )+ (0 . . i ) cos(39 . x ) − (0 . − . i ) (cid:17) . WxDevWxRen1 2 3 4 5 60.140.160.180.200.220.240.26 (a) W ′ q ( x ) - - (b) W ′′ q ( x ) Figure 9: Plots of W ′ q ( x ) , and W ′′ q ( x ) of the exact solution and the approximations for f ( x ) = ue − ax (cid:16) ωx + φ (cid:17) , θ = 1 , q = .Dominant exponent Φ q Percent relative error( Φ q ) Optimal barrier b DeF
Percent relative error( b DeF )Exact 0.0881484 0 4.38201 0Expo 0.0878658 0.32053 4.42263 0.927122Renyi 0.0881481 0.000314617 4.39788 0.362284Dev 0.0881484 6.11743*10^-6 4.39745 0.352331Table 14: Exact and approximate values of Φ q and b DeF for f ( x ) = ue − ax (cid:16) ωx + φ (cid:17) , θ = 1 , q = .The DeVylder approximation wins on both fronts.Clearly, our completely monotone approximation cannot fully reproduce functions like W ′ q ( x ) , W ′′ q ( x ) in examples like this where oscillations occur (note however that the de Finetti optimal barrier is wellapproximated here). If a more exact reproduction is necessary, higher order approximations should be used. J along oneparameter families of Cramér-Lundberg processes In this section, we provide the two approximations for the dividend value with capital injections J , andthe dividend barrier b , for two one parameter families of Cramér-Lundberg processes, with densities givenrespectively by: f ( x ) = k ǫ (cid:2) e − x + ǫe − x (cid:3) (41) f ( x ) = k ǫ (cid:20) e − x + ǫ (cid:18) e − x + 15083 e − x (cid:19)(cid:21) , (42)where k ǫ is the normalization constant, and compute the maximal error of approximation when ǫ ∈ (0 , ∞ ) and θ ≈ . For this choice, the pure exponential approximation works considerably better, ∀ ǫ .230J0 exact J0 expo pure J0 expo pure error J0 expo CI J0 expo CI error0.001 7.1879 7.18802 0.0016603 7.18849 0.008279670.01 7.17075 7.17193 0.0164663 7.17666 0.08247820.1 7.008 7.01863 0.151653 7.06358 0.7930411 5.95034 5.99151 0.691856 6.26009 5.2055110 4.20175 4.19406 0.183122 4.40089 4.73941100 3.66909 3.6654 0.100631 3.69555 0.7212281000 3.6025 3.60208 0.0117065 3.60523 0.075585Table 15: λ = 1 , θ = 1 , q = k = 3 / and P = 0 . As expected, the errors decrease both as ǫ goes to zeroand infinity since the densities approach an exponential density. ϵ ϵ J0 % eABCD Figure 10: J values and errors plotted against ǫ . Errors peak at ǫ = 1 .We do the same thing for the family of densities given by f ( x ) = k ǫ (cid:2) e − x + ǫ (cid:0) e − x + e − x (cid:1)(cid:3) .J0 ǫ J0 exact J0 expo pure J0 expo pure error J0 expo CI J0 expo CI error0.001 7.95508 7.95771 0.0330111 7.96565 0.1327960.01 7.68765 7.71127 0.307292 7.78772 1.301660.1 6.06176 6.15381 1.5186 6.62641 9.315071 3.7747 3.76883 0.155556 4.11784 9.0904110 3.1382 3.1379 0.00959692 3.23354 3.03813100 3.05894 3.06427 0.174306 3.12284 2.089211000 3.0508 3.05678 0.196219 3.11149 1.98956Table 16: λ = 1 , c = 1 , q = k = 3 / and P = 0 . As ǫ goes to zero, the density becomes exponentialhence the decrease in errors. As ǫ goes to infinity, the density approaches a hyper exponential density oforder 2, but still both methods of approximating J yield reasonable results.24 .010 0.100 1 10 100 1000 ϵ J0 ϵ J0 % EFGHI
Figure 11: J values and errors plotted against ǫ . Errors peak at ǫ = 0 . . Consider now the more general case when the claims are distributed according to a matrix exponentialdensity generated by a row vector ~β and by an invertible matrix B of order n , which are such that the vector ~βe xB is decreasing componentwise to , and ~β. = 0 , with a column vector. As customary, we restrictw.l.o.g. to the case when ~β is a probability vector, and ~β. = 1 , so that F ( x ) = ~βe xB is a valid survival function.The matrix versions of our functions are: C a ( x ) = λ R x W q ( x − y ) F ( y + a ) dy = λ~β R x W q ( x − y ) e yB dy e aB = ~C ( x ) e aB m a ( y ) = R a zf ( y + z ) dz = ~β e yB R a z e zB ( − B ) dz = ~β e yB M ( a ) G a ( x ) = λ R x W q ( x − y ) m a ( y ) dy = ~C ( x ) M ( a ) , (43)where ( C ( x ) = λ R x W q ( x − y ) e yB dy~C ( x ) = λ~β R x W q ( x − y ) e yB dy . (44)The product formulas (43) may also be established directly in the phase-type case, using the conditionalindependence of the ruin probability of the overshoot size.We derive first these extensions from scratch for ( ~β, B ) phase-type densities, in order to highlight theirprobabilistic interpretation. Later, we will show that the matrix exponential jumps case follows as a partic-ular case of [GK17].Recall first [AA10] that Ψ q ( x ) = ~ Ψ q ( x ) , where ~ Ψ q ( x ) is a vector whose components represent theprobability that ruin occurs during a certain phase, and that the conditional independence of ruin andovershoots translates into the product formula Ψ q ( x, y ) := P x [ T , − < ∞ , X T , − < − y ] = ~ Ψ q ( x ) e yB . (45)To take advantage of this, it is convenient to replace from the beginning Z q ( x ) by Ψ q ( x ) , taking advantageof the formula [AKP04, Kyp14] Z q ( x ) = Ψ q ( x ) + W q ( x ) q Φ q = ⇒ C ( x ) = ( c − q Φ q ) W q ( x ) − Ψ q ( x ) . (46)Alternatively, one may introduce a vector function ~Z q ( x ) := ~ Ψ q ( x ) + W q ( x ) q Φ q . (47)25n the other hand, the mean function may be written as m a = Z a y F (d y ) = − aF ( a ) + Z a F ( x ) dx = ~βM ( a ) , M ( a ) = − B − − e aB (cid:0) aI n − B − (cid:1) . The following result follows in the phase-type case just as in the exponential case [AGLW20]; in thematrix exponential jumps case, it may be obtained from [GK17]:
PROPOSITION 6.
For a Cramèr-Lundberg process (compound Poisson ) with matrix exponential jumpsof type ( ~β, B ) , it holds that1. J x = kG a ( x ) + J S a ( x ) = kG a ( x ) + − kG ′ ( b ) S ′ ( b ) S a ( x ) , x ∈ [0 , b ] kx + J x ∈ [ − a, x ≤ − a , (48) where C ( x ) = λ R x W q ( x − y ) e yB dy~C ( x ) = λ~β R x W q ( x − y ) e yB dy = cW q ( x ) ~ − ~Z q ( x ) = ( c − q Φ q ) W q ( x ) ~ − ~ Ψ q ( x ) G a ( x ) = ~C q ( x ) M ( a ) R a ( x ) = S a ( x ) − Z q ( x ) = ~C q ( x ) e aB , (49) and J = 1 − k ~C ′ q ( b ) M ( a ) qW q ( b ) + ~C ′ q ( b ) e aB . (50)
2. For fixed a , the optimality equation ∂∂b J a,b = 0 simplifies to J = k ~C ′′ q ( b ) M ( a ) qW ′ q ( b ) + ~C ′′ q ( b ) e aB . (51) REMARK 12.
The additive separation of a, b which was the basis of proving optimality in the exponentialcase does not seem possible anymore, but (50) allows the numeric computation of the optimum.
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F. Avram ∗1 , D. Goreac , R. Adenane , and U. Solon Laboratoire de Mathématiques Appliquées, Université de Pau, France 64000 LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447Marne-la-Valleée, France School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, PRChina Département des Mathématiques, Université Ibn-Tofail, Kenitra, Maroc 14000 Institute of Mathematics, University of the Philippines Diliman, PhilippinesFebruary 23, 2021
Abstract