The equivalence of two tax processes

Transcription

1 The equivalence of two ta processes Dalal Al Ghanim Ronnie Loeffen Ale Watson 6th November 218 arxiv: v1 [math.pr] 5 Nov 218 We introduce two models of taation, the latent and natural ta processes, which have both been used to represent loss-carry-forward taation on the capital of an insurance company. In the natural ta process, the ta rate is a function of the current level of capital, whereas in the latent ta process, the ta rate is a function of the capital that would have resulted if no ta had been paid. Whereas up to now these two types of ta processes have been treated separately, we show that, in fact, they are essentially equivalent. This allows a unified treatment, translating results from one model to the other. Significantly, we solve the question of eistence and uniqueness for the natural ta process, which is defined via an integral equation. Our results clarify the eisting literature on processes with ta. Key words and phrases. Risk process, ta process, ta rate, spectrally negative Lévy process, ruin probability, ta identity, optimal control. MSC21 classification. 6G51, 91B3, 93E2, 91G8. 1 Introduction and main results Risk processes are a model for the evolution in time of the (economic) capital or surplus of an insurance company. Suppose that we have some model X = (X t ) t for the risk process, in which X t represents the capital of the company at time t; for instance, a common choice is for X to be a Lévy process with negative jumps. Any such model can be modified in order to incorporate desirable features. For instance, reflecting the path at a given barrier models the situation where the insurance company pays out any capital in ecess of the barrier as dividends to shareholders. Similarly, refracting the path at a given level and with a given angle corresponds to the case where dividends are paid out at a certain fied rate whenever the capital is above the level or, equivalently, corresponds to a two-step premium rate. These modifications are described in more detail in Chapter 1 of Kyprianou [8], in the Lévy process case. Between the reflected and refracted processes are a class of processes where partial reflection occurs whenever the process reaches a new maimum. The motivation in risk theory for these processes is that the times of partial reflection can be understood to 1

2 correspond to ta payments associated with a so-called loss-carry-forward regime in which taes are paid only when the insurance company is in a profitable situation. In this paper we study ta processes of this kind. Before we define rigorously the type of ta processes that we are interested in, we make some assumptions on X that are in place throughout the paper. We assume that X is a stochastic process with càdlàg paths (i.e., right-continuous paths with left-limits) and without upward jumps (that is, X t lim s t X s for all t ). We also assume X = for some fied R. For eample, these conditions are satisfied if X is a Lévy process without upward jumps. In fact, the main results presented in this work hold pathwise, in the sense that they apply to each individual path of the stochastic process. A random model is strictly only required for the study of specific eamples; however, given the applications we have in mind, it seems appropriate to phrase everything in terms of stochastic processes. One way to incorporate a loss-carry-forward taation regime for the risk process X is to introduce the ta process U γ := (U γ t ) t with U γ t = X t + γ(x s ) dx s, t, (1) where γ : [, ) [, 1) is a measurable function and X t = sup s t X s is the running maimum of X. Note that, here and later, = denotes the integral over (, t]. + (,t] Since every path t X t is increasing (in the weak sense), and is further continuous due to the assumptions on X, the integral in (1) is a well-defined Lebesgue-Stieltjes integral. We call U γ a latent ta process or the ta process with latent ta rate γ. For this latent ta process we have that, roughly speaking, in the time interval [t, t + h] with h > small, a fraction γ(x t ) of the increment X t+h X t is paid as ta. In particular, ta contributions are made whenever X reaches a new maimum (which is whenever U γ reaches a new maimum; see Lemma 4 below), which is why the taation structure in (1) can be seen to be of the loss-carry-forward type. Since γ < 1, this can be seen as partial reflection; setting γ = 1 [b, ) would correspond to fully reflecting the path at the barrier b. A great deal of literature has emerged in the study of this ta process. It was introduced by Albrecher and Hipp [1] in the case where X is a Cramér Lundberg process and γ is a constant, and in that work the authors studied the ruin probabilities, proving a strikingly simple relation between ruin probabilities with and without ta, the so-called ta identity. This work was etended by Albrecher, Renaud and Zhou [2], using ecursion theory, to the case where X is a general spectrally negative Lévy process, with γ still constant. In [1], Kyprianou and Zhou took γ to be a function, and studied problems related to the two-sided eit problem and the net present value of the taes paid before ruin. In the same setting Renaud [13] provided results on the distribution of the (present) value of the taes paid before ruin. Wang and Hu [15] studied a problem of optimal control of latent ta processes in which one seeks to maimise the net present value of the taes paid before ruin. A variation of the latent ta process in which the ta rate eceeds the value 1 can be found in Kyprianou and Ott [9]. An unusual property of the process U γ is that the taation at time t depends, not on the running maimum U γ t = sup s t Us γ of the process U γ itself, but on the running maimum 2

3 of X, i.e., X t. In other words, the amount of ta the company pays out at time t is not determined by the amount of capital the company has at that time but it depends on a latent capital level, namely X t, which is the amount of capital that the company would have at time t if no taes were paid out at all. Besides being somewhat unnatural, this also means that in the common case where X is modelled by a Markov process, the process (U γ, U γ ) is not Markov; one must consider the three-dimensional process (U γ, U γ, X) in order to maintain the Markov property. For these reasons, it may be more suitable to use another ta process V = (Vt ) t, satisfying the equation V t = X t + (V s) dx s, (2) where V t = sup s t Vs and : [, ) [, 1) is a measurable function. We call V a natural ta process or a ta process with natural ta rate. Since (2) is an integral equation, it is not immediately clear whether such a process V eists and if so if it is uniquely defined. We will shortly give a simple condition for eistence and uniqueness. Assuming that eistence and uniqueness holds and that X is a Markov process, the natural ta process V has the advantage that the two-dimensional process (V, V ) is Markov. Albrecher, Borst, Boma and Resing [3] looked at ta processes with a natural ta rate in the case where X is a Cramér-Lundberg risk process and studied the ruin probability, though they do not provide a definition of the ta process in terms of an integral equation and in particular do not discuss eistence and uniqueness. In the setting where X is a Cramér-Lundberg risk process, Wei [16] and Cheung and Landriault [6] considered a more general class of natural ta processes than ours in which the associated premium rate is allowed to be surplus-dependent. Although [16] and [6] do contain the definition (2) for the natural ta process in the case where X is a Cramér-Lundberg risk process (see [16, Section 1] with = and [6, Equation (1.2)] with the function c( ) being constant), neither paper addresses the question of eistence and uniqueness. The purpose of this work is to clarify the relationship between these two ta processes. Whereas latent and natural ta processes appear quite different when considering their definitions, it emerges that these two classes of ta processes are essentially equivalent, an observation which has seemingly gone unnoticed in the literature. This equivalence allows us to deal in a rather straightforward way with the eistence and uniqueness of the natural ta process, which is something that has not been dealt with before. Before presenting our main theorem, we emphasise that our assumptions allow for a large class of stochastic processes for X that includes, amongst others, spectrally negative Lévy processes, spectrally negative Markov additive processes (see [4]), diffusion processes (see [11]) and fractional Brownian motion. However, from a practical modelling point of view (1) and (2) might not in all cases be the right way to define a taed process. For instance, when one considers a Cramér-Lundberg risk process where the company earns interest on its capital as well as pays ta according to a loss-carry-forward scheme, then one should not work with a process of the form (1) or (2), but define the ta process differently, as in [16]. Our definitions (1) and (2) are practically suitable for modelling ta processes when the underlying risk process without ta X has a spatial homogeneity property, which is the case for, for instance, spectrally negative Lévy processes, spectrally negative Markov additive processes or Sparre Andersen risk processes. 3

4 In order to present our main result, we will need to consider the following ordinary differential equation, for a given measurable function : [, ) [, 1): dy(t) = 1 ( y dt (t) ), t, y() =. (3) We say that y : [, ) R is a solution of this ODE if it is an absolutely continuous function and satisfies (3) for almost every t. Theorem 1. Recall that X =. (i) Let U γ be the ta process with latent ta rate γ, where γ : [, ) [, 1) is a measurable function. Define γ : [, ) R by γ (s) = + and consider its inverse γ 1 s (1 γ(y))dy, s, (4) : [, ] [, ], with the convention that γ 1 (s)). Then, when s γ ( ). Define γ : [, γ ( )) [, 1) by γ (s) = γ( γ 1 and U γ is a natural ta process with natural ta rate γ. (s) = U γ t = γ (X t ), t, (5) (ii) Let : [, ) [, 1) be a measurable function and assume that there eists a unique solution y (t) of (3). Define γ : [, ) [, 1) by γ (s) = ( y (s ) ). Then, the integral equation (2) defining the natural ta process has a unique solution V = (V t ) t. Moreover, V t = y (X t ), t, (6) and so the solution V to (2) is a latent ta process with latent ta rate given by γ. This theorem is the main contribution of the article. It states that a sufficient condition for eistence and uniqueness of solutions to (2) can be given in terms of a simple ODE. From the proofs given in Section 3 below, it is not difficult to see that the eistence and uniqueness of the ODE (3) is also a necessary condition for eistence and uniqueness of a solution to (2). Theorem 1 also gives a precise relationship between the two types of ta processes. In particular, every latent ta process is a natural ta process, though the corresponding latent and natural ta rates may differ. Conversely, every well-defined natural ta process is also a latent ta process. The net eample illustrates this equivalence for piecewise constant ta rates. Eample 2. Define the piecewise constant function f b by { f b α, z b, (z) = β, z > b, (7) 4

5 (a) = 7, b = 2 and b = (b) = 1, b = 2 and b = 16. Figure 1: Plots of the risk process X (dashed line) and the associated latent ta process U f b or equivalently natural ta process V f b (solid line), where f b is the piecewise constant function defined by (7) with α =.4 and β =.9. The dashed-dot lines mark the values of b and b. where b > = X and α β < 1. Note that the ODE (3) with = f b has a unique solution, see e.g. Section 2. It is clear that the ta process with latent ta rate f b differs from the ta process with natural ta rate f b, unless α = β or α =. However, from Theorem 1 we deduce that the ta process with latent ta rate f b is equal to the ta process with natural ta rate f b for b = (1 α)b + α. Note that b depends on the starting point, unless α =. Figure 1 contains two plots in which an eample of X and the corresponding ta process U f b, or equivalently V f b, are drawn. From this figure we see that indeed the first time X reaches the level b is equal to the first time the ta process reaches the level b. The theorem allows us to very easily translate results derived for the latent ta process to 5

6 results on the natural ta process, or vice versa. As an eample, by using the corresponding result derived in [1] for the latent ta processes, we provide below an analytical epression of the so-called two-sided eit problem of the natural ta process in the case where X is a spectrally negative Lévy process. For an introduction to spectrally negative Lévy processes and their scale functions we refer to Chapter 8 in [8]. Corollary 3. Let X be a spectrally negative Lévy process on the probability space (Ω, F, P ) such that P (X = ) = 1. Let : [, ) [, 1) be a measurable function such that there eists a unique solution y to (3). Let V be the ta process with natural rate associated with the spectrally negative Lévy process X. Define the first passage times τ a = inf{t : Vt < a} and τ a + = inf{t : Vt > a}, where a R. Let q and let W (q) : R [, ) be the scale function of X defined by W (q) (z) = for z < and characterised on [, ) as the continuous function whose Laplace transform is given by e λy W (q) (z)dz = ( log ( E [ e λx 1 ]) q ) 1, for λ > sufficiently large. Then, for < a < y ( ), we have [ ] E e qτ + a 1 + {τ a <τ } = ep { a } W (q) (y) W (q) (y)(1 (y)) dy, (8) where [ W (q) denotes ] a density of W (q) on (, ). On the other hand, if a y( ), then E e qτ + a 1 + {τ a <τ } =. The rest of this article is organised as follows. In section 2, we eplain the consequences of our results and make further connections with the literature. Section 3 is devoted to the proofs of Theorem 1 and Corollary 3. 2 Related work and further applications Markov property Assume X is a Markov process. As we have already commented in the previous section, it follows from the integral equation (2) for the natural ta process V that the process (V, V ) is Markov. One might epect that the equivalence between the two types of ta processes should imply the same for (U γ, U γ ) where U γ is an arbitrary latent ta process, since we know by Theorem 1(i) that U γ is also a natural ta process. However, the corresponding natural ta rate is γ = γ X, which depends upon the initial value of X. For this reason, we do not obtain the Markov property for (U γ, U γ ) in general. Eistence of the ta process with progressive natural ta rates When the ta rate increases with the amount of capital one has, the taation regime is typically called progressive. We will show that, when is an increasing (in the weak sense) measurable function : [, ) [, 1), then the ODE (3) has a unique solution, which implies the eistence and uniqueness of the natural ta process with ta rate. 6

7 For eistence, since is an increasing function, we have that g(z) := 1 1 (z), z, is a strictly positive, increasing measurable function, and hence integrable, so G(y) := y g(z) dz, y, is absolutely continuous. Moreover, since G is continuous and strictly increasing, G 1 eists and, as G > a.e., G 1 is absolutely continuous [5, Vol. I, p. 389]. Thus, (G 1 ) (t) eists for almost every t, and it follows that a solution to (3) is given by y (t) = G 1 (t). This is because, by the inverse function theorem [12, Theorem 31.1], it holds that dg 1 (t) dt = 1 g(g 1 (t)) = 1 (G 1 (t)), for a.e. t >, and since G() =, we have G 1 () =. For uniqueness, since is increasing, the right hand side of (3) is decreasing. guarantees uniqueness, as can be proved using, for instance, [7, Theorem 1.3.8]. This Optimal control In the case where X is a spectrally negative Lévy process, Wang and Hu [15] studied a very interesting optimal control problem for the latent ta process U γ, given by [ σ E sup γ Π e qt γ(x t ) dx t ], (9) where σ = inf{t : U γ t < }, and Π is the set of measurable functions γ : [, ) [α, β], where α β < 1 are fied. Denote by γ the function γ Π which maimises (9), if it eists. A remarkable feature of Wang and Hu s work is that they obtain a natural ta process as the optimal solution to the problem of controlling a latent ta process, as we will now eplain. In their Theorem 3.1, Wang and Hu state that γ should satisfy the equation ( ) γ (X t ) = η + Xt (1 γ (y)) dy = η(u γ t ), for some function η which they call the optimal decision rule. On the other hand, if is a function satisfying the assumptions of Theorem 1(ii), γ is defined as in that result, and if we let ξ = γ, then by definition of γ, we have from respectively (6) and (5) the relation ( ) ξ(x t ) = + Xt (1 ξ(y)) dy = (U ξ t). It follows from Theorem 1 that the relationship between Wang and Hu s optimal decision rule η and optimal ta rate γ is nothing more than the relationship between a particular natural ta rate and the equivalent latent ta rate γ. Our results clarify that this 7

8 connection is a sensible one even outside of the optimal control contet, and make clear under eactly which conditions this connection is valid. Wang and Hu go on to show that η must be piecewise constant, and in particular η = f b, as defined in (7), where b is specified in terms of scale functions of the Lévy process but is independent of ; see section 4 and equation (5.15) in their work (where b is denoted u ). Combining this with our result, we see that Wang and Hu s solution of the optimal control problem (9) is actually a ta process with the piecewise constant natural ta rate f b, or equivalently the piecewise constant latent ta rate f b(), where b() depends on ; see also Eample 7. Ta identity Assume we are in the setting of Corollary 3 where in particular X is a spectrally negative Lévy process. We are interested here in the ta identity: a relationship between the survival probability of the natural ta process V and the one of the risk process with out ta X. To this end, let ) ) φ () = P (inf V t and φ () = P (inf X t t t be the survival probability in the risk model with and without taation, respectively. If y( ) <, the process V cannot eceed the level y( ). Since from every starting level (and thus in particular from y( )), there is a strictly positive probability of V going below zero, a standard renewal argument shows that the survival probability φ () is zero in this case. On the other hand, if y ( ) =, then we can apply Corollary 3 to get a relation between the two survival probabilities. Namely, by letting q and a in (8) and using the well-known epression for φ () (see e.g. [8, Equation 8.18]), we have that φ () = ep { W (y) W (y)(1 (y)) dy } = ep { } d dy ln(φ 1 (y)) (1 (y)) dy. This agrees with [3, Proposition 3.1] for the special case where X is a Cramér-Lundberg risk process, which confirms that in [3] natural ta processes are considered. 3 Proofs We start with a lemma generalising a result from [1]. Lemma 4. Let K = (K t ) t be a stochastic process for which every path is measurable as a function of time and such that K t < 1 for every t. Define Then, H t = X t H t = X t + K s dx s, t. + K s dx s, where H t = sup s t H s. Moreover, {t : H t = H t } = {t : X t = X t }. 8

9 Proof. Since K t < 1 for all t, the proof in [1, Lemma 2.1] works without alteration. Net we prove part (ii) of Theorem 1 ecept for the eistence of the integral equation. Lemma 5. Let : [, ) [, 1) be a measurable function and assume that there eists a unique solution y of (3). Define γ : [, ) [, 1) by γ(s) = ( y(s ) ). If there eists a solution V = (Vt ) t to the integral equation V t = X t + (V r)dx r, t, (1) then V t = y (X t ) and hence V is a latent ta process with latent ta rate given by γ. Proof. Suppose that V solves (1). By Lemma 4, V t = X t We define L t = X t and we let L 1 a L 1 a := + (V r) dx r, t. (11) be its right-inverse, i.e. { inf{t > : L t > a} = inf{t > : X t > a + }, if a < L,, if a L. As X does not have upward jumps, t X t is continuous, which implies X L 1 a = + (a L ). (12) Using respectively (11) for t = L 1 a = L 1 a L, (12) and the change of variables formula with r = L 1 b (see, for instance, [14, foot of p. 8]), we have for a, V L 1 a L L 1 a L = X L 1 (V r) dx r a L + = + (a L ) = + (a L ) = + a L( ) 1 1 {r L + ( 1 a L } (V r) dx r 1 1 {<L b L 1 ( V L 1 b a L } (V L 1 ) db b )) db, where for the last equality we used that L 1 b is strictly increasing on [, L ], which follows because t X t is continuous. By the hypothesis that (3) has a unique solution y, we deduce, V L 1 a L = y (a L ) = y ( ) X L 1, a, (13) a L where the last equality follows by (12). As t X t does not jump upwards, X L 1 = X t L t for all t, which implies via (11) that V L 1 = V L t for all t. So by invoking (13) for t a = L t, we conclude that V t = y (X t ) for all t. 9

10 We are now ready to prove Theorem 1 and Corollary 3. Proof of Theorem 1. (i) Fi t. By Lemma 4, we have U γ t = X t + γ(x r )dx r. By applying the change of variable y = X r, we obtain U γ t = X t Xt γ(y)dy = γ (X t ), where we recall that γ (s) = + s (1 γ(y)) dy. Hence, γ 1 (U γ t ) = X t, and so γ(x t ) = γ( γ 1 (U γ t )) = (U γ γ t ). It follows that U γ is a natural ta process with natural ta rate. γ (ii) The uniqueness of a solution to (2), and the equality (6), follow directly from Lemma 5. So it remains to prove the eistence of a solution to (2). By hypothesis there eists a unique solution y to (3). With γ(z) = ( y(z ) ), we define : [, ) [, 1) by (z) = γ( ( γ ) 1 (z) ) ( = y where ( γ ) 1 is the inverse function of γ (z) = + z ( ) ( γ 1 )) (z), (1 γ (y))dy. (14) By part (i) of Theorem 1, the ta process with latent ta rate γ is a natural ta process with natural ta rate. Thus, it remains to show that (z) = (z) for z. Note that γ is an absolutely continuous function and hence ( γ ) eists almost everywhere. By (3) we have that, for z such that ( γ ) (z) eists, d ( y dz (( γ ) 1 (z) ) ) = [ 1 ( ( y ( γ ) 1 (z) ))] d ( ( γ dz ) 1 (z) ) = [ ( 1 γ ( γ ) 1 (z) )] d ( ( γ dz ) 1 (z) ). Since by the inverse function theorem [12, Theorem 31.1], d ( ( γ dz ) 1 (z) ) = 1 ( γ ) (( γ ) 1 (z)) = 1 1 γ (( γ ) 1 (z)), we see that d ( y dz (( γ ) 1 (z) ) ) = 1 a.e., and therefore, by absolute continuity, for some constant c, we have that y (( γ ) 1 (z) ) = z + c, z. Since ( γ ()) 1 = = y (), we get that c =. We conclude that (z) = (z) for z, and this completes the proof. 1

11 Proof of Corollary 3. From (6) we see that τ a + = when a y ( ). Hence we can assume without loss of generality that a < y( ). By part (ii) of Theorem 1 we know that V is a latent ta process with latent ta rate γ. Hence we can use Theorem 1.1 in [1] to conclude that, [ ] { a } E e qτ + a W (q) (y) 1 + {τ a <τ } = ep W (q) (y) (1 γ (( γ ) 1 (y))) dy, where ( γ ) 1 is the inverse of the function γ given by (14). Note that in [1] the additional assumption (1 γ (z))dz = is made on the latent ta rate, but from the proof of Theorem 1.1 in [1] it is clear that this assumption ( is unnecessary when a < y( ). In the proof of Theorem 1(ii) we showed that γ ( γ ) 1 (y) ) = (y) for all y, which finishes the proof. References [1] H. Albrecher and C. Hipp. Lundberg s risk process with ta. Bl. DGVFM, 28(1): 13 28, 27. ISSN doi:1.17/s [2] H. Albrecher, J.-F. Renaud, and X. Zhou. A Lévy insurance risk process with ta. J. Appl. Probab., 45(2): , 28. ISSN doi:1.1239/jap/ [3] H. Albrecher, S. Borst, O. Boma, and J. Resing. The ta identity in risk theory a simple proof and an etension. Insurance Math. Econom., 44(2):34 36, 29. ISSN doi:1.116/j.insmatheco [4] H. Albrecher, F. Avram, C. Constantinescu, and J. Ivanovs. The ta identity for Markov additive risk processes. Methodol. Comput. Appl. Probab., 16(1): , 214. ISSN doi:1.17/s y. [5] V. I. Bogachev. Measure theory. Vol. I, II. Springer-Verlag, Berlin, 27. ISBN ; doi:1.17/ [6] E. C. K. Cheung and D. Landriault. On a risk model with surplus-dependent premium and ta rates. Methodol. Comput. Appl. Probab., 14(2): , 212. ISSN doi:1.17/s [7] Z. Jackiewicz. General linear methods for ordinary differential equations. John Wiley & Sons, Inc., Hoboken, NJ, 29. ISBN doi:1.12/ [8] A. E. Kyprianou. Fluctuations of Lévy processes with applications. Universitet. Springer, Heidelberg, second edition, 214. ISBN ; doi:1.17/ Introductory lectures. [9] A. E. Kyprianou and C. Ott. Spectrally negative Lévy processes perturbed by functionals of their running supremum. J. Appl. Probab., 49(4):15 114, 212. ISSN

12 [1] A. E. Kyprianou and X. Zhou. General ta structures and the Lévy insurance risk model. J. Appl. Probab., 46(4): , 29. ISSN doi:1.1239/jap/ [11] B. Li, Q. Tang, and X. Zhou. A time-homogeneous diffusion model with ta. J. Appl. Probab., 5(1):195 27, 213. ISSN doi:1.1239/jap/ [12] G. B. Price. Multivariable analysis. Springer-Verlag, New York, ISBN doi:1.17/ [13] J.-F. Renaud. The distribution of ta payments in a Lévy insurance risk model with a surplus-dependent taation structure. Insurance Math. Econom., 45(2): , 29. ISSN doi:1.116/j.insmatheco [14] D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, ISBN doi:1.17/ [15] W. Wang and Y. Hu. Optimal loss-carry-forward taation for the Lévy risk model. Insurance Math. Econom., 5(1):121 13, 212. ISSN doi:1.116/j.insmatheco [16] L. Wei. Ruin probability in the presence of interest earnings and ta payments. Insurance Math. Econom., 45(1): , 29. ISSN doi:1.116/j.insmatheco