Power identities for Lévy risk models under taxation and capital injections
PPOWER IDENTITIES FOR L ´EVY RISK MODELS UNDER TAXATIONAND CAPITAL INJECTIONS
HANSJ ¨ORG ALBRECHER AND JEVGENIJS IVANOVS
Abstract.
In this paper we study a spectrally negative L´evy process which is refracted atits running maximum and at the same time reflected from below at a certain level. Such aprocess can for instance be used to model an insurance surplus process subject to tax paymentsaccording to a loss-carry-forward scheme together with the flow of minimal capital injectionsrequired to keep the surplus process non-negative. We characterize the first passage time overan arbitrary level and the cumulative amount of injected capital up to this time by theirjoint Laplace transform, and show that it satisfies a simple power relation to the case withoutrefraction, generalizing results by [3] and [6]. It turns out that this identity can also be extendedto a certain type of refraction from below. The net present value of tax collected before thecumulative injected capital exceeds a certain amount is determined, and a numerical illustrationis provided. Introduction
The aim of this paper is to study certain power relations of level crossing quantities forspectrally negative L´evy processes, which are motivated by insurance applications. Concretely,assume that the surplus process of an insurance portfolio is modeled by a spectrally negativeL´evy process, and tax payments on profits according to a loss-carry-forward scheme are imple-mented by paying a certain proportion γ of the premium income, whenever the surplus processis at its running maximum. For a constant value of γ , it was shown by [3] and [6] that theprobability of the resulting process to stay positive is intimately connected to the one withouttax payments by a simple power relation (see also [1, 16, 18] for extensions). The implementedtax rule can alternatively be seen as a general profit participation scheme for shareholders,which for the special case of γ = 1 reduces to a horizontal dividend barrier strategy. Whereasin classical models business is stopped as soon as the surplus is negative, it is natural to con-sider the amount of capital needed to bring the surplus back to zero whenever it turns negativeand henceforth continue the business operations. Under horizontal dividend payments and acompound Poisson model for insurance claims, this question was considered by [10], and [15]showed that it can be optimal for shareholders to “save” the insurance business in this way (foranother injection scheme see [20]).In this paper we consider capital injections below zero for the general case γ ≤
1. Thisamounts to study level crossing events for a spectrally negative L´evy process refracted at itsrunning maximum and at the same time reflected at zero. We characterize the first passage timeover an arbitrary level and the cumulative amount of injected capital up to this time by theirjoint Laplace transform, and establish a simple power relation to the case without refraction.
Key words and phrases. spectrally-negative L´evy process, exit problems, collective risk theory, insurance,capital injections, dividends, alternative ruin concepts.Supported by the Swiss National Science Foundation Project 200020-143889. a r X i v : . [ q -f i n . P R ] M a r H. ALBRECHER AND J. IVANOVS
From the proof it becomes clear that such a power identity can not hold, if reflection frombelow is generalized to refraction at the running minimum. However, if refraction always startsat the same fixed level, a power identity still holds.In Section 2, we discuss simultaneous refraction and reflection. Section 3 then states themain results, which are proved in Section 4. In Section 5 we consider an application of theobtained formula to determine the net present value of tax collected before the cumulativeinjected capital exceeds an exponential amount, and give a concrete numerical example for acompound Poisson risk model. Finally, in Section 6 we illustrate with yet another example thatpower identities hold in wide generality. Concretely, we use our proof technique to extend thepower tax identity for first passage times (without capital injections) to a relaxed concept ofruin which was considered recently in the literature.2.
Refraction and reflection
For a c`adl`ag sample path X t of any stochastic process, consider reflection of X t at a level b (from above) defined by Y t = X t − U t ≤ b , where U t is a non-decreasing c`adl`ag function with U = 0 ∨ ( X − b ), whose points of increase are contained in the set { t ≥ Y t = b } . Thisidentifies U t in a unique way, and implies that U t = 0 ∨ ( X t − b ), where X t = sup { X s : 0 ≤ s ≤ t } ,see e.g. [13]. Essentially, U t evolves as the supremum process.For an arbitrary γ ∈ R we call the process X t − γU t a refraction from above, which has someinterpretations in insurance risk theory. For γ = 1 we retrieve the reflected process, which canmodel an insurance surplus process with dividends paid out according to a barrier strategywith barrier b , whereas γ ∈ (0 ,
1) refers to an insurance surplus process taxed according toa loss-carried forward scheme (see e.g. [3, 6]). A value γ < γ > γ couldbe allowed to depend on the current value of U t (or on the running maximum of the refractionitself), which leads to a more general process of the form X t − (cid:82) U t γ ( x )d x . For simplicity, wewill however assume throughout this work that γ is a constant, and only give some commentsin Remark 4.1.This paper focuses on processes refracted from above with rate γ ≤ a, b ], where a is the level for reflection from below, and b is the initial level for refraction from above. First, we consider a free process X t until it exits[ a, b ], at which moment we start either reflection from below (it exits through a ) or refractionfrom above (it exits through b ). Assuming (w.l.o.g.) the latter, we consider the time at whichthe corresponding refraction goes below a , and then start reflection from below. When thisreflection goes above the running maximum, the refraction from above starts, and so on, seeFigure 1 for an illustration of such a process.The above procedure is described rigorously in the form of an algorithm in the Appendix,where we also allow for two-sided refraction. For the present model it results in a representation(1) Y t = X t + L t − γU t , AXATION AND CAPITAL INJECTIONS 3 ba tt t t b (2) Figure 1.
A sample path refracted from above and reflected from below.where it is assumed that X ∈ [ a, b ], and γ ≤ L t and U t are non-decreasing c`adl`ag functions, and the points of increase of L t and U t are containedin the sets { t ≥ Y t = a } and { t ≥ Y t = Y t ∨ b } respectively. Finally, note that L t and U t are interrelated and both depend on the parameter γ .3. A power identity
Throughout this work we assume that X t is a spectrally negative L´evy process with Laplaceexponent ψ ( α ) so that E e αX t = e ψ ( α ) t for α ≥
0. Define the first passage times τ ± y = inf { t ≥ ± X t > y } and recall that for all q ≥ W q : [0 , ∞ ) → R + , suchthat W q ( y ) > y > E x [ e − qτ + y ; τ + y < τ − ] = W q ( x ) /W q ( y ) for y ≥ x ≥ , y > , and (cid:82) ∞ e − αy W q ( y )d y = 1 / ( ψ ( α ) − q ) for α larger than the rightmost zero of ψ ( α ) − q . This W q is called a scale function.For a L´evy risk model with tax, it was shown by [6] that certain probabilities and transformscan be related to their analogues under no taxation by power identities. We will now gener-alize such power identities to the setting of a refraction from above and reflection from below.Consider a process Y t given by (1), where X = x >
0, the reflection barrier is placed at thelevel a = 0, and the refraction from above at rate γ ≤ b = x (itis straightforward to extend our result to b > x using identities for reflected L´evy processes).Let also T y = inf { t ≥ Y t > y } be the first passage time of the refraction above the level y . Theorem 3.1.
For γ < and q, θ ≥ it holds that (3) E γx e − qT y − θL Ty = (cid:0) E x e − qT y − θL Ty (cid:1) − γ , where y ≥ x > and E γx denotes the expectation operator for the model defined by (1) with a = 0 and b = x . It should be noted that the right hand side of (3) can be identified using results on reflectedL´evy processes. In particular, [12] shows that(4) E x e − qT y − θL Ty = Z q,θ ( x ) /Z q,θ ( y ) , H. ALBRECHER AND J. IVANOVS where Z q,θ ( x ) is a so-called second scale function given by Z q,θ ( x ) = e θx [1 − ( ψ ( θ ) − q ) (cid:90) x e − θy W q ( y )d y ] , see also [21] for the case when θ = 0. Observe thatlim θ →∞ E x e − qT y − θL Ty = E x [ e − qT y ; L T y = 0] = E x [ e − qτ + y ; τ + y < τ − ] = W q ( x ) W q ( y ) . Similarly, for θ → ∞ the left-hand side of (3) becomes the transform of the first passage time T y on the event that it precedes ruin, hence we recover the tax identity (3.1) of [6] as a specialcase.In the case γ = 1 (corresponding to payments of dividends according to a barrier strategy atthe level x ) we have T y = ∞ for all y ≥ x . Instead we look at(5) ρ y = inf { t ≥ U t > y } , which is the first time that the amount of accumulated dividends (or taxes) exceeds a level y . Theorem 3.2.
For q, θ ≥ and x > , y ≥ it holds that (6) E x e − qρ y − θL ρy = e − λ q,θ ( x ) y , where λ q,θ ( x ) = Z q,θ (cid:48) ( x ) /Z q,θ ( x ) = θ − ( ψ ( θ ) − q ) W q ( x ) Z q,θ ( x ) . In a somewhat different form this formula appears also in [12]. We note that for θ = ∞ onehas to take λ q ( x ) = W q (cid:48) + ( x ) /W q ( x ), which is intimately related to the excursion measure, seee.g. [17, Lem. 8.2]. Remark 3.1.
The power identity (3) fails to hold for a two-sided refraction (defined in Ap-pendix) with γ L < . The case of reflection γ L = 1 is special because in this case we know thedistance to the (lower) reflection barrier at the first passage time T y (in other words, a ( n ) in thealgorithm defining the two-sided refraction is constant, see Appendix).Nevertheless, from the proof in Section 4 it becomes clear that if one modifies the modeland considers either refraction from below always starting at a fixed level a or always startingat a fixed distance from the running maximum (rather than starting at the current runningminimum), then the power identity (3) is preserved also in the case γ L < . Proofs
In this section we prove Theorem 3.1 and Theorem 3.2. We construct an auxiliary processby a certain modification of paths of the simultaneously refracted and reflected process. Thismodification preserves excursions from the maximum, but leads to the same ‘behavior at themaximum’ as the one of the free process. Furthermore, the auxiliary process corresponding to γ = 1 exhibits a lack of memory property at its first passage times, because the lower reflectionbarrier is always placed at a constant distance from the maximum. This gives rise to a certainexponent λ ( x ), and allows to relate this process to the processes corresponding to different γ ,see Lemma 4.1. Subsequently the strong Markov property is applied to establish a differentialequation for the quantity of interest, which then yields the results. AXATION AND CAPITAL INJECTIONS 5
It is convenient to shift our process, so that X = 0 and reflection from below is applied atthe level − x <
0. Recall also that refraction from above is applied immediately. Note that E γ e − qT y can be written as P γ ( T y < ∞ ) for an exponentially killed process, i.e. when X t is sentto an additional absorbing state at an independent exponentially distributed time e q with rate q ≥
0. The double transform E γ e − qT y − θL Ty is obtained by additional killing at the time when L t surpasses an independent exponentially distributed e θ . Hence it suffices to analyze P γ ( T y < ∞ )for a doubly killed process.Let us fix some terminology and notation concerning the paths of Y t . Segments of a path ofthe process Y t − Y t in the intervals when this difference is strictly negative are called excursionsof Y t (from the maximum). The starting level of an excursion is the corresponding value of Y t .Next, consider a triplet ( Y t , L t , U t ) of paths (where each component depends on the choice of γ ) and define ˜ Y t = X t + L t = Y t + γU t . From the construction of Y t one can see that Y t = (1 − γ ) U t , which immediately yields ˜ Y t = U t .Letting ˜ T y = inf { t ≥ Y t > y } we see that ˜ T y = ρ y and for γ < T y = T (1 − γ ) y . It is noted that we could have avoided constructing the auxiliary process, since it is possible touse the stopping time ρ y instead of ˜ T y . But then the following arguments would become lessvisually appealing.When γ = 1 the reflecting barrier is always placed at a constant distance x from the maxi-mum, which together with the strong Markov property of X t implies that(8) P ( ˜ T y + z < ∞| ˜ T y < ∞ ) = P ( ˜ T z < ∞ )for all y, z > e q and e θ is essentialhere). From (8) it follows that there exists a λ ( x ) ≥ P ( ˜ T y < ∞ ) = e − λ ( x ) y , where x denotes the distance between the reflecting barriers. This provides the proof of Theo-rem 3.2 up to the identification of λ ( x ). Lemma 4.1.
It holds for all γ ≤ that P γ ( ˜ T h < ∞ ) = P ( ˜ T h < ∞ ) + o ( h ) as h ↓ . Proof.
In the following we will need to compare the sample paths of ˜ Y t processes for different γ , hence throughout this proof we write ˜ Y γt and ˜ T γy to make their dependence on γ explicit.For the ease of exposition, consider first the case γ = 0, where ˜ Y t is a process X t reflected atthe level − x . Let δ ≥ X t from the maximumexceeding height x ; this is also the starting level of the first excursion of ˜ Y t leading to reflection(i.e. an increase of L t ). Note that on the event { δ > h } the times ˜ T h and ˜ T h coincide. In thefollowing we exclusively work on the complementary event { δ ≤ h } .The lack of memory of ˜ Y t at its first passage times implies that the number of excursionsof ˜ Y t starting in [0 , h ] and leading to reflection defines a (killed) L´evy process indexed by h . H. ALBRECHER AND J. IVANOVS − x xδ ˜ T h ˜ T h h Figure 2.
A schematic sample path of ˜ Y t and ˜ Y t (with a dashed line).Hence on the event { ˜ T h < ∞} this number is Poisson distributed. Using the lack of memoryof ˜ Y t at ˜ T h we see that P ( δ ≤ h, ˜ T h < ∞ , ˜ T h = ∞ ) = P ( δ ≤ h, ˜ T h < ∞ ) P ( ˜ T h = ∞ )= O ( h )( λ ( x ) h + o ( h )) = o ( h ) . Hence considering { δ ≤ h, ˜ T h < ∞} we can assume that ˜ T h < ∞ and also there is only oneexcursion of ˜ Y t starting in [0 , h ] and leading to reflection. Comparison of the sample paths of˜ Y t and ˜ Y t , see Figure 2, reveals that ˜ T h < ∞ , because the difference between them is boundedby h . For an arbitrary γ ≤ − γ ) h , hence one can take h + (1 − γ ) h insteadof 2 h to finish this part of the proof.Let us now consider { ˜ T h < ∞} . Note that ˜ T h = ∞ can only happen as a consequence ofkilling according to e θ . Hence it is only required to show that this happens with probability o ( h ). In fact, it is enough to show that for a non-killed process ˜ Y t it holds that P ( δ ≤ h, e θ ∈ ( L ˜ T δ − h, L ˜ T δ )) = o ( h ) , which follows from the independence of e θ . Again, for general γ , h in the above display isreplaced by (1 − γ ) h . (cid:3) Combining Lemma 4.1, (9), and (7) we get for γ < P γ ( T h < ∞ ) = P ( ˜ T h/ (1 − γ ) < ∞ ) + o ( h ) = 1 − λ ( x )1 − γ h + o ( h ) as h ↓ . Let us now return to the original set-up, where X = x and the reflecting barrier is placed atthe level 0; we use P x to denote the corresponding law. Proof of Theorem 3.1.
Assume that γ < P γx ( T y < ∞ ) = P γx ( T x + h < ∞ ) P γx + h ( T y < ∞ ) . According to (10) we have P γx ( T x + h < ∞ ) = 1 − λ ( x )1 − γ h + o ( h ) as h ↓
0. Hence P γx + h ( T y < ∞ ) → P γx ( T y < ∞ ), and moreover(11) ∂∂x P γx ( T y < ∞ ) = λ ( x )1 − γ P γx ( T y < ∞ ) . Formally, this computation gives only the right derivative.
AXATION AND CAPITAL INJECTIONS 7
Let us identify λ ( x ) using the existing theory. In particular (4) states that P x ( T y < ∞ ) = Z ( x ) /Z ( y ). Hence we obtain Z (cid:48) ( x ) /Z ( y ) = λ ( x ) Z ( x ) /Z ( y ) yielding(12) λ ( x ) = Z (cid:48) ( x ) /Z ( x ) for x > , which also shows that λ ( x ) is continuous on (0 , ∞ ).It is not hard to see that for any γ < y > P γx ( T y < ∞ ) , x ∈ (0 , y ]is continuous and non-zero. Hence for all x ∈ (0 , y ) we have the following right derivative: ∂∂x ln P γx ( T y < ∞ ) = λ ( x )1 − γ , which together with P γy ( T y < ∞ ) = 1 yieldsln P γx ( T y < ∞ ) = − − γ (cid:90) yx λ ( u )d u. Uniqueness of the solution is based on the fact that a continuous function with right derivative0 at every point of an interval is constant on this interval. So we have(13) P γx ( T y < ∞ ) = e − − γ (cid:82) yx λ ( u )d u , which immediately yields the power relation of Theorem 3.1. (cid:3) Finally, Theorem 3.2 is a direct consequence of (9) and (12).
Remark 4.1.
When the refraction rate γ ( x ) depends on the level, assuming some regularityconditions (e.g. γ ( x ) is continuous and bounded away from 1), one can still apply Lemma 4.1to derive the differential equation (11) . In this case the solution takes the form P γx ( T y < ∞ ) = e − (cid:82) yx λ ( u ) / (1 − γ ( u ))d u . An application: Profit participation and capital injection
As an application of Theorem 3.1, interpret Y t in (1) as an insurance surplus process at time t , where γU t is a profit participation scheme for an investor (a proportion γ of the profits ispaid out to the investor) and, in turn, if needed the investor injects a minimal flow of capitalinto the company to prevent its bankruptcy, i.e. to keep the surplus non-negative, with L t being the total amount injected up to time t . Alternatively, one can think of γU t as taxpayments for profits up to time t according to a loss-carry forward scheme with constant taxrate 0 < γ < L t would then be the necessary amount of capital up to time t to bail out the insurance company to prevent bankruptcy. Consider an upper limit e θ for thecumulative amount that the investor is willing to inject, which is assumed to be an independentexponential random variable with rate parameter θ ≥ h , the investor willstop payments with probability θh (independently of everything else). This concept extendsthe notion of classical ruin (which is retrieved for θ = ∞ ), and leads to an interesting trade-offbetween collected profits (or tax) and injected capital.The expected discounted profit (tax) payments for this model can be written as V ( γ ) = γ − γ E γx (cid:90) ∞ e − qt { L t Let us consider a concrete simple example, for which the scalefunction W ( x ) has an explicit form, and hence the expected discounted profit (tax) payments V ( γ ) as identified in (14) can be easily evaluated. We assume that the driving process is aCram`er-Lundberg risk process X t = x + ct − (cid:80) N ( t ) n =1 M i , where N ( t ) is a homogeneous Poissonprocess with rate 1, the insurance claims M i are independent and identically distributed expo-nential random variables with mean m and the constant premium intensity is chosen as c = 1,so that the drift of X is then given by E X (1) = 1 − m . Choose further the initial capital x = 1,the discount factor q = 0 . 01, and the investor impatience parameter θ = 1.Figure 3 depicts V as a function of γ for different values of the drift. Essentially, the shapeof these functions is the same as in the case of classical ruin ( θ = ∞ ), but higher in absolutevalue due to the longer life-time of the process. This shape reflects that overly large values of γ may lead to an early ruin resulting in a smaller profit.In Figure 4(a), this is visualized by comparing V ( γ ) for θ = 1 and θ = ∞ for a fixeddrift of E X (1) = 0 . 3, and Figure 4(b) depicts the increase of V ( γ ) as compared to the caseof classical ruin. This expected increase of profit comes at the cost of the capital injections,whose expected value does not exceed E e θ = 1. The latter is in fact a crude upper bound,because of two reasons: no discounting, and the fact that cumulative injections may neverreach the threshold e θ . These results show that on average it can be quite advantageous for aninvestor to perform these capital injections, in particular for those γ for which the difference V ( γ ) − V ∞ ( γ ) is larger than 1. If one would compare this difference to the actual expected AXATION AND CAPITAL INJECTIONS 9 Figure 3. V ( γ ) for drift = (0 . , . , . , . , . , , − . (a) V ( γ ) (thick) and V ∞ ( γ ) (b) V ( γ ) − V ∞ ( γ ) Figure 4. V ( γ ) for θ = 1 and θ = ∞ .discounted investments, the effect would be even more pronounced. The analysis of the netpresent value of injections is, however, considerably more involved, and could be an interestingdirection for future work.6. Power identities under a relaxed ruin concept It turns out that power relations similar to (3) hold in quite wide generality. Essentially, it isonly required that killing and modification (such as reflection) of excursions of the (non-taxed)process is done in a memoryless way (in other words, what happens after the first passage time T y is independent from the past and has the same law as the original process started in y ).Of course, one still has to handle model-specific technical details similar to those contained inLemma 4.1.For illustration, let us consider an example from [2] and [5], where bankruptcy is declaredat some rate θ > X t spent below zero surpasses an in-dependent exponential random variable e θ (one can also introduce dependence of θ on the level,but for clarity we refrain from doing so, and only note that generalizations of power identitiesto arbitrary measurable, locally bounded functions θ ( x ) do not cause additional problems). Asbefore we assume that X t is a spectrally negative L´evy process (no reflection from below). The concept of occupation times plays an important role in this setting. Let M ( A, t ) = (cid:90) t { X s ∈ A } d s be the time X spends in a Borel set A up to time t . Theorem 6.1. Consider the model (1) without reflection from below ( a = −∞ , b = x ≥ ),and let ν θ be the time of bankruptcy: ν θ = inf { t ≥ M (( −∞ , , t ) > e θ } . Then for all γ < and q ≥ it holds that E γx [ e − qT y ; T y < ν θ ] = (cid:0) E x [ e − qT y ; T y < ν θ ] (cid:1) − γ . Proof. Without real loss of generality one can assume that q = 0. One can repeat the argumentsfrom the previous section. In fact, many things simplify since there is no process L t . Inparticular, paths of the processes ˜ Y γt (and X t ) are the same, but the intervals of times whenthe processes are in danger of bankruptcy are different for different γ , and so the killing pointsare different. In order to (re-)establish Lemma 4.1, we have to show that the differences between‘in danger’ sets up to the time τ + h are small in certain sense. It is enough to show that(16) P ( M ([ − x + γh, − x + h ) , τ + h ) > e θ ) = o ( h )as h ↓ 0. Moreover, to establish the differential equation (11) we have to show (for the reasonof continuity) that(17) M ( { x } , t ) = 0 a.s. for any t, x. The latter fact is well-known, see [9, Prop. I.15]. So it is only left to show that (16) holds.The probability in (16) can be bounded from above by P ( τ − x − h < τ + h ) P ( M ([ − (1 − γ ) h, (1 − γ ) h ] , τ + x +(1 − γ ) h ) > e θ ) . In short, the process must go below the upper boundary of the interval, then we start it at thelower boundary and make the strip twice as large, so that it starts in the middle. The firstprobability is given by 1 − W ( x − h ) /W ( x ) = W (cid:48)− ( x ) /W ( x ) h + o ( h ), and the second decreasesto 0 as h ↓ 0, because M ([ − h, h ] , τ + y ) → y > X t → ∞ a.s. or X t → −∞ a.s.). This concludes the proof. (cid:3) Corollary 6.1. For the model of Theorem 6.1 and q ≥ it holds that E γx [ e − qT y ; T y < ν θ ] = (cid:18) Z q, Φ ( x ) Z q, Φ ( y ) (cid:19) − γ , γ < , E γx [ e − qρ y ; ρ y < ν θ ] = exp (cid:32) − Z q, Φ (cid:48) ( x ) Z q, Φ ( x ) y (cid:33) , γ = 1 , where Φ is the unique positive solution of φ (Φ) = q + θ .Proof. It holds that E x [ e − qτ + y ; τ + y < ν θ ] = Z q, Φ ( x ) /Z q, Φ ( y ) , which can be easily deduced from the results by [19] or [4]. The rest follows from Theorem 6.1and its proof which employs the ideas of Section 4. (cid:3) AXATION AND CAPITAL INJECTIONS 11 Appendix In the following we present an algorithm defining a two-sided refraction of a c`adl`ag samplepath X t corresponding to the interval [ a, b ]. It is assumed that X ∈ [ a, b ], and γ L , γ U ≤ Y t , L t , U t ) is defined iteratively as follows(cf. Figure 1 depicting refraction from above at b and reflection from below at a ). Algorithm : Initialization ( n = 0): Y (0) t = X t , U (0) t = 0 , L (0) t = 0 , t = 0 and a (1) = a, b (1) = b,t = inf { t ≥ X t / ∈ [ a, b ] } . Step ( n = n + 1): X ( n ) t = Y ( n − t n + X t n + t − X t n for t ≥ If X ( n )0 ≥ b ( n ) : L ( n ) t = 0 and Y ( n ) t = X ( n ) t − γ U U ( n ) t is the refraction of X ( n ) t , t ≥ b ( n ) . Put∆ n = inf { t ≥ Y ( n ) t < a ( n ) } and t n +1 = t n + ∆ n , a ( n +1) = a ( n ) , b ( n +1) = Y ( n )∆ n . If X ( n )0 ≤ a ( n ) : U ( n ) t = 0 and Y ( n ) t = X ( n ) t + γ L L ( n ) t is the refraction of X ( n ) t , t ≥ a ( n ) . Put∆ n = inf { t ≥ Y ( n ) t > b ( n ) } and t n +1 = t n + ∆ n , a ( n +1) = Y ( n )∆ n , b ( n +1) = b ( n ) .Finally, we set Y t = Y ( n ) t − t n , L t = n − (cid:88) i =0 L ( i )∆ i + L ( n ) t − t n , U t = n − (cid:88) i =0 U ( i )∆ i + U ( n ) t − t n for t ∈ [ t n , t n +1 ) . Observe that the above procedure defines the process Y t for all t ≥ 0, i.e. t n → ∞ as n → ∞ ,because a c`adl`ag function can not cross the interval [ a, b ] infinitely many times in finite time;here we use the fact that the intervals [ a ( n ) , b ( n ) ] are increasing. Careful examination of theabove algorithm (together with known properties of a one-sided refraction) shows that Y t = X t + γ L L t − γ U U t , where L t and U t are non-decreasing c`adl`ag functions. Moreover, the points of increase of L t and U t are contained in the sets { t ≥ Y t = Y t ∧ a } and { t ≥ Y t = Y t ∨ b } respectively. Itmay be interesting to find an explicit representation of the two-sided refraction similar to thosegiven by [7] and [14] for the two-sided reflection. References [1] H. Albrecher, S. Borst, O. Boxma, and J. Resing. The tax identity in risk theory—a simple proof and anextension. Insurance Math. Econom. , 44(2):304–306, 2009.[2] H. Albrecher, H. U. Gerber, and E. S. W. Shiu. The optimal dividend barrier in the Gamma-Omega model. Eur. Actuar. J. , 1(1):43–55, 2011.[3] H. Albrecher and C. Hipp. Lundberg’s risk process with tax. Bl. DGVFM , 28(1):13–28, 2007.[4] H. Albrecher and J. Ivanovs. A risk model with an observer in a Markov environment. Risks , 1(3):148–161,2013. [5] H. Albrecher and V. Lautscham. From ruin to bankruptcy for compound Poisson surplus processes. ASTINBull. , 43(2):213–243, 2013.[6] H. Albrecher, J.-F. Renaud, and X. Zhou. A L´evy insurance risk process with tax. J. Appl. Probab. ,45(2):363–375, 2008.[7] L. N. Andersen and M. Mandjes. Structural properties of reflected L´evy processes. Queueing Syst. , 63(1-4):301–322, 2009.[8] S. Asmussen. Applied probability and queues , volume 51 of Applications of Mathematics (New York) .Springer-Verlag, New York, second edition, 2003. Stochastic Modelling and Applied Probability.[9] J. Bertoin. L´evy processes , volume 121 of Cambridge Tracts in Mathematics . Cambridge University Press,Cambridge, 1996.[10] D. C. M. Dickson and H. R. Waters. Some optimal dividends problems. ASTIN Bull. , 34(1):49–74, 2004.[11] F. Hubalek and A. Kyprianou. Old and new examples of scale functions for spectrally negative L´evyprocesses. In Seminar on Stochastic Analysis, Random Fields and Applications VI , volume 63 of Progr.Probab. , pages 119–145. Birkh¨auser/Springer Basel AG, Basel, 2011.[12] J. Ivanovs. A new approach to fluctuations of reflected L´evy processes. Technical report, Eurandom, Eind-hoven University of Technology, 2011. arXiv:1004.3857v1.[13] O. Kella. Reflecting thoughts. Statist. Probab. Lett. , 76(16):1808–1811, 2006.[14] L. Kruk, J. Lehoczky, K. Ramanan, and S. Shreve. An explicit formula for the Skorokhod map on [0 , a ]. Ann. Probab. , 35(5):1740–1768, 2007.[15] N. Kulenko and H. Schmidli. Optimal dividend strategies in a Cram´er-Lundberg model with capital injec-tions. Insurance Math. Econom. , 43(2):270–278, 2008.[16] A. Kyprianou and C. Ott. Spectrally negative L´evy processes perturbed by functionals of their runningsupremum. J. App. Probab. , 49(4):1005–1014, 2012.[17] A. E. Kyprianou. Introductory lectures on fluctuations of L´evy processes with applications . Universitext.Springer-Verlag, Berlin, 2006.[18] A. E. Kyprianou and X. Zhou. General tax structures and the L´evy insurance risk model. J. Appl. Probab. ,46(4):1146–1156, 2009.[19] R. L. Loeffen, J.-F. Renaud, and X. Zhou. Occupation times of intervals until first passage times forspectrally negative L´evy processes. Stochastic Process. Appl. , 124(3):1408–1435, 2014.[20] C. Nie, D. Dickson, and S. Li. Minimizing the ruin probability through capital injections. Annals of Actu-arial Science , 5(2):195–209, 2011.[21] M. R. Pistorius. On exit and ergodicity of the spectrally one-sided L´evy process reflected at its infimum. J. Theoret. Probab. , 17(1):183–220, 2004.[22] J.-F. Renaud and X. Zhou. Distribution of the present value of dividend payments in a L´evy risk model. J. Appl. Probab. , 44(2):420–427, 2007., 44(2):420–427, 2007.