François-Gaël Gaël Michalec

Sami Souissi
email: sami.souissi@univ-lille1.fr

Gaël Dur

Mohamed-Sofiane Sofiene Mahjoub

François G Schmitt

Jiang-Shiou Hwang

Differences in behavioral responses of Eurytemora affinis (Copepoda, Calanoida) adults to salinity variations

Keywords: Eurytemora affinis, estuarine copepod, salinity, behaviour, symbolic analysis Eurytemora affinis, estuarine copepod, salinity, swimming behavior, symbolic analysis

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION 20 21

Maintenance and fitness optimization of copepod populations in macrotidal estuaries is a complex 22 process because of the highly dynamic environment, leading to frequent variations in environmental 23 conditions which can be considered as real selection pressures. Alternating tides reconfigure the 24 local estuarine hydrodynamics and are responsible for variations in freshwater and seawater inputs, 25 leading to important and rapid changes in salinity which local populations of copepods have to face.

26

Salinity is a major factor affecting the ecology of estuarine copepods, driving the spatial distribution 27 of species (Collins and Williams, 198;Roddie et al., 1984;Soetaert and Van Rijswijk, 1993; 28 [START_REF] Laprise | Environmental variability as a factor controlling spatial patterns in 377 distribution and species diversity of zooplankton in the St. Lawrence Estuary[END_REF]Mouny and Dauvin, 2002;David et al, 2005) and affecting life history 29 traits [START_REF] Holste | The effects of temperature and salinity on egg production and 363 hatching success of Baltic Acartia tonsa (Copepoda: Calanoida): a laboratory investigation[END_REF]Calliari et al., 2006;Chen et al., 2006;Devreker et al., 2009; 30 Beyrend-Dur et al. 2009).

31

The calanoid copepod Eurytemora affinis (Poppe, 1880) is one of the commonest species in most 32 European and North American estuaries (Winkler and Grieve, 2004) and often dominates the 33 zooplankton community in the low to medium salinity zone [START_REF] Lawrence | Estuarine calanoid copepod abundance in relation[END_REF]. In the Seine 34 estuary, E. affinis is the dominant species in the oligohaline and mesohaline zones throughout the were made for each reproductive stage (male, non ovigerous female, ovigerous female) at each 92 salinity condition, to compensate for inter-individual variability (Seuront et al., 2004). During of trajectories ranging from 60 to 143 were considered for the data analysis (Table I). Such an 106 important difference in the trajectory numbers came from the different overall activities observed 107 between the replicates, with concordance neither with salinity nor sex. Total data number (i.e.

108 number of values of instantaneous speed between two successive positions) ranged from 90433 to 109 302478 (Table I). The instantaneous speed was estimated as the distance between two successive positions of a 116 copepod divided by the time step of the camera. Its value v i was thus computed as: Gamma PDF, corresponding rather to a heavy-tailed distribution. The probability density functions 167 all exhibited a clear power-law trend and supported significant power-law fits (Table II). 4). Non-ovigerous females 201 exhibited lower (0.14 sec) and higher (0.24 sec) durations at salinity 5 and 30, respectively, 202 corresponding to higher and lower instantaneous speeds. Males showed an increasing duration (0.13 203 to 0.4 sec) from minimal to maximal salinity, inversely correlated with their swimming speed.

204

Ovigerous females did not show a minimal duration at salinity 5 but at 0.5 (0.13 sec). However,

228

We found a marked effect of salinity on the swimming behavior of the three reproductive stages.

229

Maximum speed and activity were exhibited at low salinities then decreased at higher salinities. E. et al., 1984;Devreker et al., 2004Devreker et al., , 2007Devreker et al., , 2009))

  50 cm away from a Sony DCR camera (25 frames/sec) facing the tank. After 15 min of 90 acclimation, the recording began and finished after 30 min of filming. Four replicate observations 91

  93acclimation and after experiments, the temperature of the experimental vessel water was carefully 94 monitored and was almost constant (18-19°C).

  transferred to computer and cut into sequences of 5 min each (Adobe® 99 Premiere® Pro 2.0), allowing better running of the software used to extract the trajectories 100 (Labtrack v.2.1 ©Bioras). Only swimming paths in which the individuals were at least three body 101 lengths away from the walls parallel to the camera axis and from the surface and bottom were 102 considered. Trajectories with a duration shorter than 10 sec were similarly rejected. The coordinates 103 of the trajectories were extracted and combined into a 2D picture. Errors and artifacts resulting from 104 the extraction process were cleaned up. For each sex and at each salinity condition, a total number 105

  152p=0.002 for non-ovigerous females) (Figure1). The swimming speed of non-ovigerous females, 153 ovigerous females and males decreased when salinity increased between 5 and 30. At the extreme 154 condition of salinity 0.5, average speed dropped off for non-ovigerous females and ovigerous 155 females but not for males. Maximum speed was observed at salinity 0.5 for males and at salinity 5 156 for non-ovigerous females and ovigerous females.

Figure

  Figure2Ashows, as an example, the PDF of instantaneous speeds of E. affinis non-ovigerous

168Figure

  Figure2Bshows the PDFs of instantaneous speeds of E. affinis males at the five salinity conditions

205

  Figure5Ashows, as an exemple, the duration of successive break events for E. affinis males at

Table I .

 I Van Rijswijk, P. (1993) Spatial and temporal patterns of the zooplankton in the 399 Westerschelde estuary. Mar. Ecol. Prog. Ser., 97, 47-59. Neomysis integer, and their predation on the calanoid copepod Eurytemora Number of trajectories -data for Eurytemora affinis non-ovigerous females, ovigerous 410 females and males, for the five salinity conditions tested.

	280 408	an analysis process which greatly differed from the one applied in Seuront (2006), with differences all S. Souissi's students who contributed to the maintenance of copepod cultures in the Marine Table and Figure legends
	281 409	in the rearing period, number of individuals per recording, feeding during experiments, salinity and
	282	temperature acclimation time, and trajectory numbers. Acclimation to salinity and temperature is
	283 411	known to be a critical element in osmoregulation capabilities and tolerance to salinity (Lance, 1963;
	284 412	Roddie et al., 1984), and differences in the acclimation process may have led to different responses
	285	to salinity variation. It is also difficult to compare behavioral and physiological responses of
	286	copepods originating from a continuous culture (Seuront, 2006) and from field (this study).
	287	Beyrend-Dur et al. (2009) discussed the effect of the number of generations (i.e. age of the copepod
	288	culture) on the life cycle traits of E. affinis maintained at different salinities and indicated a possible
	289	selection pressure in laboratory continuous cultures. Since copepods are known to behave
	395 396	F o r F o r differently in the presence of food (Tiselius, 1992; van Duren and Videler, 1995; Seuront and F o r F o r F o r variability in the swimming behavior of the sub-tropical copepod Oncaea venusta (Copepoda: Poecilostomatoida). Mar. Ecol. Prog. Ser., 283, 199-217. F o r
		P P P P P Seuront, L., Vincent, D. (2008) Increased seawater viscosity, Phaeocystis globosa spring bloom and P
		e e e e e e
	400 401	e r e r e r e r e r Tiselius, P. (1992) Behaviour of Acartia tonsa in patchy food environments. Limnol. Oceanogr., 37, e r
	402	1640-1651.	R R R R R R
	403	e e e e e van Duren, L.A., Videler, J.J. (1995) Swimming behaviour of developmental stages of the calanoid e
	301 302 303 404 406	v i e v i e v i e v i e v i e copepod Temora longicornis at different food concentrations. Mar. Ecol. Prog. Ser., 126, 153-161. Winkler, G., Greve, W. (2004) Trophodynamic of two interacting species of estuarine mysids, v i e
	304 407	affinis. J. Exp. Mar. Biol. Ecol., 308, 127-146.	w w w w w w
	305	ACKNOWLEDGEMENTS
	306
	307	This paper was funded by the Seine-Aval program. It is a contribution to ZOOSEINE project
	238 308	. Using E. affinis financed by Seine-Aval IV scientific program. This paper is a contribution to the bilateral CNRS-
	239 309	as an experimental model to assess contaminant impacts on copepods, Cailleaud et al. (2007) NSC Taiwan project n° 17473 entitled "Study of behavioral activity and spatial and temporal
	240 310	investigated the effect of salinity on two biomarker enzymatic activities (i.e. acetylcholinesterase distribution of copepods in two contrasting ecosystems: temperate (France) and sub-tropical
		and glutathione S-transferase) and found maximal expression related to an optimal salinity range (Taiwan)". The authors are grateful to Michel Priem, Daniel Hilde and Dominique Menu for
	244 314
			10 11 14

241

(i.e. at salinity 5 and 10), showing a physiological preference of E. affinis. The authors assumed an 242 increasing energy devoted to osmotic regulation and a decrease in energetic cost allocated to other 243 biological processes due to stressful salinity conditions. Modifications in metabolism when salinity 290

Vincent, 2008)

and in order to avoid any possible synergistic effects of food and salinity, copepods

291

were not fed during our experiments. We assume that the behavioral response observed in this study 292 reflects the effect of salinity. Finally, because of the high inter-individual variability in copepod 293 behavior and also due to its sensitivity to experimental conditions, a high number of trajectories 294 should be considered.

295

In conlusion, the proposed metrics used in this study can be routinely applied to study the effects of 296 different external factors on copepod behavior. Further investigations are needed to compare E. 297 affinis responses with those exhibited by other species, contributing to a better understanding of the 298 ecology of copepods within estuaries. Such studies offer the hope of elucidating some of the key 299 factors driving the amazing inter and intra-specific behavior diversity displayed under different 300 environmental conditions. 311 helping with E. affinis sampling and culture maintenance, and to David Devreker for helping with 312 copepod sorting. The authors thank James E. Bron, Adam J. Brooker and Ted Wu for English 313 correction, and the two anonymous reviewers for their constructive comments. We are grateful to 397 Temora longicornis feeding and swimming behaviours. Mar. Ecol. Prog. Ser., 363, 131-145. 398 Soetaert, K., 405 Praunus flexuosus and

Table II .

 II Parameter values of the power-law fits adjusted to instantaneous speed probability density

	413
	414	functions for non-ovigerous females, ovigerous females and males of Eurytemora affinis, for the
	415	five salinity conditions tested, of the type:	.
	416

Table III .

 III Parameter values of the power-law fits adjusted to the probability density functions of 417 event durations for non-ovigerous females, ovigerous females and males of Eurytemora affinis, for Plot of the time duration of 10000 successive break events for Eurytemora affinis males

		F o r Fig. 5. (A.) F o r
		P P
		e e
		e r e r
		R R
		e e
		v i e v i e
	432	w w
	433
	434	Fig. 3. Swimming state frequencies of Eurytemora affinis non-ovigerous females (A.), ovigerous
	435	female (B.) and male (C.), for the five salinity conditions tested. Break state: black; cruising state:
	436	grey.
	437
		Fig. 4. Mean duration of break state (A.) and cruising state (B.) for Eurytemora affinis non-
	440
	441
		15 16

418

the five salinity conditions tested, of the type:

. Fig. 1. Mean instantaneous speed (mm/sec) of Eurytemora affinis non-ovigerous females (black 421 line, triangular symbol), ovigerous females (grey line, circular symbol) and males (light grey line, 422 cross symbol), for the five salinity conditions tested. 423 424 Fig. 2. (A.) Probability density functions of a Gaussian distribution (solid line, black), a Log-normal 425 distribution (solid line, dark grey), a Gamma distribution (solid line, light grey), and experimental 426 PDF of instantaneous speeds (dashed line, black) of Eurytemora affinis non-ovigerous females at 427 salinity 5. Theoretical PDFs were computed from distributions following the same parameters as the 428 experimental distribution. (B.) Probability density functions of instantaneous speeds, ranging from 429 6 to 40 mm.s -1 , of Eurytemora affinis males for the five salinity conditions tested (salinity 0.5: solid 430 line, black; salinity 5: dashed line, black; salinity 15: solid line, dark grey; salinity 25: dashed line, 431 dark grey; salinity 30: solid line, light grey). 438 ovigerous females (black, cross symbol), ovigerous females (grey, triangular symbol) and males 439 (light grey, circular symbol), for the five salinity conditions tested.

Table 1 .

 1

	Salinity	0.5	5	15	25	30
	Non-Ovigerous Females	60 -96217	117 -131802	80 -151353	143 -302478	84 -132478
	Ovigerous Females	65 -90433	90 -118574	99 -127342	72 -155664	118 -260441
	Males	141 -11885	123 -146715	125 -178812	108 -243986	89 -213564

Table II .

 II

			Salinity	a	b	R 2
	Non-Ovigerous	Females	0.5 5 15 25	2.7 2.9 3.1 3	1.4 2.2 2.4 2.3	0.95 0.96 0.95 0.91
			30	2.8	1.6	0.97
			0.5	3.2	3.3	0.95
	Ovigerous	Females	5 15 25 30 0.5	2.7 3.2 3.2 3.3 3.5 F 2 3.2 3.2 2.8 4.3 o r 0.97 0.95 0.94 0.96 0.93
	Males	5 15 25	2.9 2.7 2.9	2.8 1.3 2.1	0.96 0.75 0.92 P e
			30	3.4	2.6	0.96	e r
							R
							e
							v i e
							w
							18

Table III .

 III

					Break			Cruising
			Salinity	a	b	R 2	a	b	R 2
	Non-Ovigerous	Females	0.5 5 15 25	2.1 2.3 2.3 2.2	6.2 6.5 6.3 6.2	0.99 0.99 0.99 0.99	3.2 3.1 3.2 4.3	8.3 8.0 8.3 10.5	0.99 0.99 0.99 0.94
			30	2.2	5.9	0.99	3.4	8.6	0.99
	Ovigerous	Females	0.5 5 15 25	2.3 2.2 2.1 2.2 F 6.4 6.3 6.1 5.9 o F o F o F o F o r 0.99 0.99 0.99 0.99 r r r r	3.4 2.9 2.8 3.8	8.6 7.5 7.4 9.5	0.98 0.98 0.99 0.99
	Males	30 0.5 5 15 25 30	2.2 2.3 2.1 2.1 2.1 2.0	5.9 6.5 6.3 6.1 6.0 5.8	0.99 0.99 0.99 P P 420x186mm (259 x 259 DPI) 3.0 8.0 0.99 2.8 7.2 0.99 2.9 7.5 P P 390x191mm (300 x 300 DPI) P 0.99 0.99 2.6 6.9 0.98 0.99 3.2 8.4 0.99 0.99 2.8 7.6 0.99 e e r e e r e e r e e r e e r
						R e v i e w 218x188mm (259 x 259 DPI) R e v i e w R e v i e w 381x380mm (300 x 300 DPI) R e v i e w R e v i e w

400x186mm (300 x 300 DPI)

http://mc.manuscriptcentral.com/jplankt Journal of Plankton Research

maintenance in the Columbia River estuarine turbidity-maxima region.Mar. Biol., 129,[309][310][311][312][313][314][315][316][317]

Mouny, P., Dauvin, J.C. (2002) Environmental control of mesozooplankton community structure in

temperature and individual variability on the reproduction of Eurytemora affinis (Copepoda; 353 Calanoida) from the Seine estuary: A laboratory study. J. Exp. Mar. Biol. Ecol.,368,[113][114][115][116][117][118][119][120][121][122][123] 354 Gaudy, R., Cervetto, G., Pagano, M. (2000) Comparison of the metabolism of Acartia clausi and A.


C Constantinescu

D Kortschak

V Maume-Deschamps

Ruin probabilities in models with a Markov chain dependence structure

Keywords: Ruin probability, dependence, Markov chain, random walk, rational Laplace transform, density satisfying ODE with constant coefficients

In this paper we derive explicit expressions for the probability of ruin in a renewal risk model with dependence described-by/incorporated-in the real-valued random variable Z k = -cτ k + X k , namely the loss between the (k -1)-th and the k-th claim. Here c represents the constant premium rate, τ k the inter-arrival time between the (k -1)-th and the k-th claim and X k is the size of the k-th claim. The dependence structure among (Z k ) k>0 is given/driven by a Markov chain with a transition kernel satisfying an ordinary differential equation with constant coefficients.

Introduction

Since the classical discrete time risk model (seen as a random walk) due to de [START_REF] De | Su un'impostazione alternativa dell teoria collecttiva del rischio[END_REF], the ruin probability is an established attempt to evaluate the solvency of an insurance company. In this context, we are interested in providing explicit expressions for the probability of ruin.

While most classical models assume independence between inter-arrival times and claim amounts, we consider a special dependence between the positive random variable τ k representing the k-th inter-arrival time and the subsequent claim size denoted by the positive random variable X k . More specifically, we consider a random walk like structure of the real-valued random variable

Z k = X k -cτ k , k = 1, 2, . . . ( 1
)
where c represents the premium rate and E{Z} < 0. The sum of increments (random walk if independent increments)

W n = n k=1 Z k
Email address: corina.constantinescu@unil.ch, dominik.kortschak, veronique.maume@univ-lyon1.fr (C. Constantinescu a , D. Kortschak a , V. Maume-Deschamps b ) represents the loss of the company after n claims. For an initial capital amount u > 0, the company is considered to be insolvent if W n > u, and the probability ψ(u) of this event is called the ruin probability. W.l.o.g., in this paper P(Z = 0) = 0.

By characterizing the dependence in renewal risk models via the series of identical distributed losses (Z k ) k>0 , one has two possible scenarios: S1. Z 1 , Z 2 , . . . are independent (random walk structure) S2. Z 1 , Z 2 , . . . are not independent (with multiple possible dependence structures among Z k s).

Since in risk theory literature several cases preserving (Z k ) k>0 independent (scenario S1) have already been addressed (see [START_REF] Albrecher | Exponential behavior in the presence of dependence in risk theory[END_REF], [START_REF] Rohana | Ultimate ruin probability in the Sparre Andersen model with dependent claim sizes and claim occurrence times[END_REF] or [START_REF] Badescu | Dependent risk models with bivariate phase-type distributions[END_REF]), we will concentrate on a case pertaining to the second scenario. For a recent and more exhaustive survey of other dependence structures in risk theory see [START_REF] Asmussen | Ruin probabilities[END_REF][Chapter XIII].

The challenging part of choosing (Z k ) k>0 to be dependent is the impossibility of relying on standard random walk theory. However, we show that one can still use the decomposition into positive-negative parts

Z = IZ + + (1 -I)Z -, I ∼ Bernoulli(p), (2)
with Z + = {Z | Z > 0} and Z -= {Z | Z < 0}, introduced in the random walk treatment, to find closed form solutions for the probability of ruin ψ(u). Similarly to [START_REF] Albrecher | An algebraic operator approach to the analysis of Gerber-Shiu functions[END_REF], we derive the exact forms via solving ordinary differential equations. The difference is that while in [START_REF] Albrecher | An algebraic operator approach to the analysis of Gerber-Shiu functions[END_REF] one exploits the independence of, and between τ k and X k , here one focuses on the dependence structure of the losses Z k , whereas the pair τ k , X k can have any dependent structure (even independence). Thus the necessary conditions to be imposed for calculating exactly the ruin probability are no longer on the densities of τ and X as in [START_REF] Albrecher | An algebraic operator approach to the analysis of Gerber-Shiu functions[END_REF], but on densities pertinent to Z or Z + /Z -.

More specifically, in this paper we consider a dependence structure for the dependent losses (Z k ) k>0 given by a Markov chain with a transition kernel p k (x, y) (starting at x, jumps from state x to state y at time k with probability p k ) that can be written as a product of two functions on each quadrant. The condition needed is that combinations of these functions further satisfy certain ODEs with constant coefficients (see Assumption 1). This condition is not intuitively evident. However, besides the mathematical amenability, the motivation/interpretation of such a choice of dependent model is the following. The weight p = P(Z > 0) not being constant over time, but dependent on the previous step in a deterministic way, influences the tendency of the future behavior of the steps Z. Thus, by the choice of the mechanism to determine the next p, we can generate models where if one period was bad (we earn less than we spend i.e. Z > 0) it is more likely than in the independent case that the next period will also be bad.

To justify further our Markov chain structure choice, we present as a particular/ limiting case of it the case of independent (Z k ) k>0 , with densities of Z + and Z -satisfying ODEs with constant coefficients. A further particular case of it is the case present in risk theory literature, of the density of Z satisfying an ODE with constant coefficients, as in [START_REF] Albrecher | Exponential behavior in the presence of dependence in risk theory[END_REF]; Ambagaspitiya (2009); [START_REF] Badescu | Dependent risk models with bivariate phase-type distributions[END_REF].

Similarly to [START_REF] Albrecher | An algebraic operator approach to the analysis of Gerber-Shiu functions[END_REF], where one transforms an integral equation for the probability of ruin into an ordinary differential equation, here we transform a system of integral equations into a system of ordinary differential equations that one can solve explicitly. This way we show that even under this dependence scenario, the probability of ruin has still a rational Laplace transform.

Thus, the classes of distributions for which one can identify the probability of ruin as having a rational Laplace transform are the following:

A. In the dependent case: Markov chain structure on Z k with a kernel satisfying certain ODE conditions, as in Propositions 1, 2, and 3.

B. In the independent case: the classical result in random-walk theory of Z + with a rational Laplace transform, as in Proposition 4.

C. In the independent case: the classical case in ruin theory literature of Z with rational moment generating function as in Proposition 5.

Note that for the distribution class C one can always find a dependence structure where the marginal distributions of τ and X have rational Laplace transform (see Lemma 1 in Appendix A). Equally natural once can identify a dependence structure with marginals (of τ and X) not having a rational Laplace transform. The paper is organized as follows. In Section 2 we introduce a renewal risk model via a random walk with real-valued steps Z k , and define the probability of ruin in this context. In Section 3, we analyze a risk model where the steps Z k are dependent via an order one Markov chain. We consider a Markov chain structure with a transition kernel expressed-through/described-by some ODE with constant coefficients and show that the ruin probability has a rational Laplace transform. In Section 4, we present the case of independent steps Z k as a particular, limiting case of the Markov chain dependence from Section 3. We reiterate/reformulate in our ODE language well-known results from random walk and risk literature. In Section 5, we look at a Markov chain dependent case with a transition kernel expressed through exponential densities, and for a numerical example we compare the values of probability of ruin under a Markov chain dependence structure with the independent increments one. After the conclusions from Section 6, in the Appendix we present some additional lemmas and some deferred proofs.

Note regarding notation and terminology. Depending on relevance and purpose, we will switch back and forth between the following equivalent concepts:

• the density function f X of the random variable X satisfies an ODE with constant coefficients

• the moment generating function (mgf) M X (s) of the random variable X is rational; or, equivalently, X has a rational Laplace transform.

These are both equivalent to matrix exponentials [START_REF] Bladt | Multivariate matrix-exponential distributions[END_REF].

Generic random variable Z

Consider the collective risk model

U (t) = u + ct - Nt k=1 X k , (3)
where X k is the k-th claim amount and N t is the number of claims that occurred up to time t. The ruin can occur only at the time of a claim. Immediately after the k-th claim, for u ≥ 0, the reserve is

U (T k ) = u + cT k - k i=1 X i = u + c k i=1 τ i - k i=1 X i = u - k i=1 Z i .
The probability of ruin in infinite time is defined as

ψ(u) = P(sup k≥1 k i=1 Z i > u). 3. (Z k ) k≥1 dependent
In this section we study the case where (Z k ) k≥0 are dependent and assume to have an order one Markov chain structure. For a given Z 0 (= x), we define the probability of ruin by

ψ(u, x) = P sup k≥1 k i=1 Z i > u | Z 0 = x . (4)
We denote the transition density of Z by p(x, y)dx

:= P(Z i+1 ∈ dy|Z i = x) = P (Z i+1 ∈ dy|Z i = x, Z i-1 = x, . . .) .
For u ≥ 0, by conditioning on Z 1 one has

ψ(u, x) = P (Z 1 > u | Z 0 = x) + P Z 1 ≤ u, sup k≥1 k i=2 Z i > u -Z 1 | Z 0 = x = P(Z 1 > u | Z 0 = x) + u -∞ P sup k≥1 k i=2 Z i > u -Z 1 | Z 0 = x, Z 1 = y p(x, y) dy = P(Z 1 > u | Z 0 = x) + u -∞ P sup k≥1 k i=2 Z i > u -Z 1 | Z 1 = y p(x, y) dy = ∞ u p(x, y) dy + u -∞ ψ(u -y, y)p(x, y) dy. (5)
To obtain a system of ODEs for the ruin probability we will assume a rather special structure for the transition density, namely:

Assumption 1. Let p(x, y) =          g 1 (x)h 1 (y), (x, y) ∈ I 1 g 2 (x)h 2 (y), (x, y) ∈ I 2 g 3 (x)h 3 (y), (x, y) ∈ I 3 g 4 (x)h 4 (y), (x, y) ∈ I 4 , (6)
where I k denotes the k-th quadrant of the cartesian plane. Further let g 1 (x) > 0, g 4 (x) > 0 for x > 0, g 1 (x) = g 4 (x) = 0 for x < 0, g 1 (x), g 4 (x) are linearly independent and g 1 (x) + g 4 (x) = 1. Similarly g 2 (x) > 0, g 3 (x) > 0 for x < 0, g 2 (x) = g 3 (x) = 0 for x > 0, g 2 (x), g 3 (x) are linearly independent and g 2 (x) + g 3 (x) = 1. Further assume that for (k, m) ∈ {(1, 1), (1, 2), (4, 1), (4, 2), (2, 3), (2, 4), (3, 3), (3, 4)} there exists polynomials q k,m (x) :

= n i=0 q k,m i x i with q k,m d du g k (u)h m (u) = 0 u = 0
An interpretation of this kind of dependence is given in Section 5.

Remark 1. One can easily extend the results of this paper to a transition density of the form

p(x, y) =          n1 i=1 g 1 i (x)h 1 i (y), (x, y) ∈ I 1 n2 i=1 g 2 i (x)h 2 i (y), (x, y) ∈ I 2 n3 i=1 g 3 i (x)h 3 i (y), (x, y) ∈ I 3 n4 i=1 g 4 i (x)h 4 i (y), (x, y) ∈ I 4 (7)
with the obvious extension of the conditions to g 1 i (x) and h 1 i (y). Nevertheless note that this generalization is straight-forward (one needs to replace all matrices by corresponding block matrices). Hence, for ease of notation we consider only the case n = 1.

In order to evaluate the ruin probability we first need to check that for a given set of parameters the process tends to -∞. Therefore one needs to evaluate the invariant distribution f Z (x) of p(x, y) (see Lemma 4 in the Appendix), although this is not needed further for the derivation of the ruin probability. To find explicit expressions for ψ(u, x) a key observation is: Proposition 1. If Assumption 1 is fulfilled, then the probability of ruin has the form

ψ(u, x) = g 1 (x)ψ 1 (u) + g 4 (x)ψ 4 (u), x > 0 g 2 (x)ψ 2 (u) + g 3 (x)ψ 3 (u), x < 0, ( 8
)
where

ψ i = ∞ u h i (y)dy + u 0 ψ(u-y, y)h i (y)dy, for i = 1, 3 and ψ i = 0 -∞ ψ(u- y, y)h 4 (y)dy, for i = 2, 4. Further, for i = 1, . . . , 4, lim u→∞ ψ i (u) = 0.
Remark 2. Note that for x > 0, ψ 1 (u) is the probability of ruin given that the first step (jump) is positive and ψ 4 (u) is the ruin probability given that the first step is negative. Similarly, for x < 0, ψ 2 (u) is the probability of ruin given that the first step is positive and ψ 3 (u) is the ruin probability given that the first step is negative.

Next we provide a system of ODEs for the functions ψ i . Proposition 2. If Assumption 1 is fulfilled, then ψ i (u), i = 1, . . . , 4 are a solution of the system of ODEs

A ψ = 0, where ψ = (ψ 1 , ψ 2 , ψ 3 , ψ 4 ) T , A =      
(r 1,1 -q 1,1 )q 4,1 gcd(q 1,1 ,q 4,1 ) 0 0 q 1,1 r 4,1 gcd(q 1,1 ,q 4,1 ) r 1,2 q 4,2 gcd(q 1,2 ,q 4,2 ) -q 1,2 q 4,2 gcd(q 1,2 ,q 4,2 ) 0 q 1,2 r 4,2 gcd(q 1,2 ,q 4,2 ) 0 r 2,3 q 3,3 gcd(q 2,3 ,q 3,3 ) (r 3,3 -q 3,3 )q 2,3 gcd(q 2,3 ,q 3,3 ) 0 0 r 2,4 q 3,4 gcd(q 2,4 ,q 3,4 ) q 2,4 r 3,4 gcd(q 2,4 ,q 3,4 ) -q 2,4 q 3,4 gcd(q 2,4 ,q 3,4 )       , gcd(a, b) denotes the greatest common divisor of the polynomials a and b and r k,m is defined by

M k,m (-s) := M g k h m (-s) := ∞ -∞ e -sx g k (x)h m (x)dx = r k,m (s) q k,m (s) . ( 9
)
Further if h 1 (x) = h 2 (x) then ψ 1 (u) = ψ 2 (u) and if h 3 (x) = h 4 (x) then ψ 3 (u) = ψ 4 (u).
Proof. Substituting (8) into the last two lines of the matrix A leads to

g 1 (x) ∞ u h 1 (y)dy + u 0 ψ 1 (u -y)g 1 (y) + ψ 4 (u -y)g 4 (y) h 1 (y)dy + g 4 (x) 0 -∞ ψ 2 (u -y)g 2 (y) + ψ 3 (u -y)g 3 (y) h 4 (y)dy = g 1 (x)ψ 1 (u) + g 4 (x)ψ 4 (u)
Since g 1 (x) and g 4 (x) are linearly independent we get the equations

ψ 1 (u) - u 0 ψ 1 (u -y)g 1 (y) + ψ 4 (u -y)g 4 (y) h 1 (y)dy = ∞ u h 1 (y)dy ψ 4 (u) - 0 -∞ ψ 2 (u -y)g 2 (y) + ψ 3 (u -y)g 3 (y) h 4 (y)dy = 0
Analogously, using the linear independence of g 2 (x) and g 3 (x) from the last two lines of equation ( .3) one has the system

ψ 2 (u) - u 0 ψ 1 (u -y)g 1 (y) + ψ 4 (u -y)g 4 (y) h 2 (y)dy = ∞ u h 2 (y)dy ψ 3 (u) - 0 -∞ ψ 2 (u -y)g 2 (y) + ψ 3 (u -y)g 3 (y) h 3 (y)dy = 0
In order to write this system of equations in matrix form, we introduce the operators

K l,m + ψ := u 0 ψ(u-y)g l (y)h m (y)dy and K l,m -ψ := 0 -∞ ψ(u-y)g l (y)h m (y)dy
(note that g l , h m vanish either on the positive or negative half-line). Further, denoting by 1 the identity operator, the above system of equations can be written as

    K 1,1 + -1 0 0 K 4,1 + K 1,2 + -1 0 K 4,2 + 0 K 2,3 - K 3,3 --1 0 0 K 2,4 - K 3,4 - -1         ψ 1 ψ 2 ψ 3 ψ 4     =     - ∞ u h 1 (y)dy - ∞ u h 2 (y)dy 0 0     . (10) If h 1 (x) = h 2 (x) then K 1,1 + = K 1,2 + and K 4,1 + = K 4,2
+ , hence it follows by subtracting the second line from the first line that ψ 1 (u) = ψ 2 (u). Similarly if h 3 (x) = h 4 (x), then we get by subtracting the fourth line from the third line that ψ 3 (u) = ψ 4 (u), Hence the second part of the proposition follows.

For the first part we want to reduce this system of IDEs to a system of ODEs that one can solve. It is enough to act on the integral system with a matrix of the form       q 1,1 q 4,1 gcd(q 1,1 ,q 4,1 ) 0 0 0 0 q 1,2 q 4,2 gcd(q 1,2 ,q 4,2 ) 0 0 0 0 q 2,3 q 3,3 gcd(q 2,3 ,q 3,3 ) 0 0 0 0 q 2,4 q 3,4 gcd(q 2,4 ,q 3,4 )

     
formed by operators of the type q k,m d du that annihilate each K k,m ± . Applying this matrix to the right-hand side of the equation leads to a homogeneous system, since the left-hand side becomes zero after differentiation. More specifically, for i = 1, 2,

∞ u h i (y)dy = ∞ u g 1 (y)h i (y)dy + ∞ u g 4 (y)h i (y)dy = ∞ 0 g 1 (u + y)h i (u + y)dy + ∞ 0 g 4 (u + y)h i (u + y)dy.
Each term in the sum is annihilated by the corresponding operator,

q 1,i ( d du ) ∞ 0 g 1 (u + y)h i (u + y)dy = 0, respectively q 4,i ( d du ) ∞ 0 g 4 (u + y)h i (u + y)dy = 0.
Now the result follows from Lemma 2 and 3 in the Appendix.

From Proposition 2 we get that ψ i (u) is a solution of a system of linear ODEs. The following theorem gives the explicit form of ψ. Its proof is given in the Appendix.

Proposition 3. If Assumption 1 is fulfilled, then ψ i (u) = n j=1 m k=0 c i j,k u k e -λj u ,
where -λ j are either the negative roots of the polynomial q 1,1 q 1,2 q 2,3 q 2,4 q 3,3 q 3,4 q 4,1 q 4,2 gcd(q 1,1 , q 4,1 )gcd(q 1,2 , q 4,2 )gcd(q 2,3 , q 3,3 )gcd(q 2,4 , q 3,4 ) or the roots of

1 = -M 1,1 (-s)M 3,3 (-s) -M 1,1 (-s) -M 3,3 (-s) -M 2,3 (-s)M 3,4 (-s)M 4,2 (-s) (M 1,1 (-s) -1) + M 1,2 (-s)M 2,3 (-s)M 3,4 (-s)M 4,1 (-s) + M 2,4 (-s)M 4,2 (M 1,1 (-s)M 3,3 (-s) -M 1,1 (-s) -M 3,3 (-s) + 1) -M 1,2 (-s)M 2,4 (-s)M 4,1 (-s) (M 3,3 (-s) -1) ,
with M k,m are defined as in (9).

Note that for a fixed j, c i j,k are linearly dependent. To get the c i j,k we can plug ψ i (u) into Equation (10), which finally results in a linear system of equations for c i j,k . Further note that by setting u = 0 in the first two lines of Equation ( 10) we get the conditions

ψ 1 (0) = ∞ 0 h 1 (y)dy = 1 and ψ 2 (0) = ∞ 0 h 1 (y)dy = 1.
An intuitive meaning of these expressions is that given that we have a positive jump at zero, if u = 0 then ruin is certain.

The connection with (Z k ) k≥1 independent

If we set h 1 (x) = h 2 (x) as the density of Z + satisfying an ODE with constant coefficients (e.g. phase-type distributions), h 3 (x) = h 4 (x) as the density of Z -, g 1 (x) = g 2 (x) = P(Z > 0) and g 3 (x) = g 4 (x) = P(Z < 0), then our model corresponds to the independent (Z k ) k≥1 case. However, note that in this case Assumptions 1 are not fullfilled, since g 1 (x) and g 4 (x) are now not independent. A way around this problem is to set

g 1 (x) = g 2 (x) = P(Z > 0)e -θx and g 3 (x) = g 4 (x) = 1 -g 1 (x).
Thus, letting θ → 0 one recovers the independent case.

Moreover, taking θ → 0 in (10) and then applying the appropriate differential operator (to eliminate the integrals on the right-hand side), one obtains a new (simpler) system of ODEs

pr + -q + pr + (1 -p)r -(1 -p)r --q - ψ 1 ψ 4 = 0 0
, where p = P(Z > 0). Let * stand for either + or -. Then, here q * is polynomial (as in characteristic equation) describing the ODE with constant coefficients that the densities of Z * , whether r * is defined through the rational moment generating functions of Z * , M Z * (-s) = r * (s) q * (s) .

Standard ordinary differential equation theory says that ψ 1 (u), ψ 4 (u) and implicitely ψ(u), are a linear combination of exponential functions, where the exponents are solutions of the polynomial equations

(pr + -q + )((1 -p)r --q -) -p(1 -p)r + r - = q + q --(1 -p)r -q + -pr + q -= 0. (11)
Equation ( 11) is equivalent to M Z (s) = 1, further equivalent to ψ(u) having a rational Laplace transform, which coincides with the result of [START_REF] Asmussen | Phase-type representations in random walk and queueing problems[END_REF][Corollary 5.1.] for Z + phase-type distributed. In the following proposition, we (re)state a slight extension of this corollary (from phase-type to rational Laplace transform class). Using our notations, we make further connections with other existing results from the (Z k ) k≥1 i.i.d. literature.

Proposition 4 (Corollary 5.1, [START_REF] Asmussen | Phase-type representations in random walk and queueing problems[END_REF]). Assume that the density function of Z + satisfies an ODE with constant coefficients described by the polynomial q + (x) = q 0 + q 1 x + • • • + q n x n , with non-homogeneous boundary conditions, and denote by M Z -(s) the moment generating function of Z -. Define r + (s) through

M Z + (-s) = r + (s) q + (s) ,
and assume that for x 0 with q + (x 0 ) = 0 we have r + (x 0 ) = 0. Then

ψ(u) = n i=1 C i u mi e -λiu , (12)
where C i , and λ i ∈ C, are the solutions of the Lundberg equation

M Z -(s) = q + (-s) -pr + (-s) (1 -p)q + (-s) or equivalently M Z (s) = 1. ( 13
)
Further, m i ∈ {0, . . . n} is the multiplicity of the root λ i minus 1.

Thus, for the slightly less general case, namely when Z itself has a rational moment generating function, one has the following result (see e.g page 270 in [START_REF] Asmussen | Ruin probabilities[END_REF]).

Proposition 5. Assume that the moment generating function of Z is rational and

M Z (-s) = r(s) q(s)
and also assume that for x 0 with q(x 0 ) = 0 we have r(x 0 ) = 0. Then the probability of ruin is a solution of the ODE

q d du -r d du ψ(u) = 0.
Namely it has form (12), where λ i are the solutions of the characteristic equation

q(s) -r(s) = 0, ( 14
)
with multiplicity m i plus 1. This is equivalent to saying that ψ has a rational Laplace transform,

ψ(s) = ((q -r) + (s) -dq + (s)) s(q -r) + (s) (15)
where (qr) + (s) is the polynomial that has as only roots the positive roots of the characteristic equation ( 14) and d = (qr) + (0)/q + (0).

Remark 3. After partial fractions decomposition and inversion of the Laplace transform, one obtains the explicit form of the probability of ruin this time down to constants, not only up to constants as in the theorem.

As examples of the Z-rational-mgf scenario, one can revisit all the models considered by Ambagaspitiya ( 2009) and derive exact expressions for the ruin probability. For exemplification, we consider in the following the Kibble and Moran's bivariate Gamma joint density for τ and X, defined through

M Z (s) = E(e sZ ) = E(e -cτ +X ) = M τ,X (s) = 1 (1 + cs β1 )(1 -s β2 ) + ρ cs 2 β1β2 m ,
where ρ ≥ 0 is the correlation coefficient between τ and X.

Example 1 (Kibble and Moran's bivariate Gamma joint density). According to Proposition 5 one needs to solve first the Lundberg equation

M Z (s) = 1 or equivalentely (1 + cs β 1 )(1 - s β 2 ) + ρ cs 2 β 1 β 2 m = 1.
Assuming that cβ 2 > β 1 , we get the roots (compare Ambagaspitiya ( 2009))

λ i ± = cβ 2 -β 1 ± (cβ 2 -β 1 ) 2 + 4c(1 -ρ)β 1 β 2 (1 -e 4πk/m ) 2c(1 -ρ) ,
where Re(λ i + ) > 0 and Re(λ i -) ≤ 0. Hence ψ satisfies the ODE

1 - c β 1 d du 1 + 1 β 2 d du + ρ c 2 β 1 β 2 d du 2 m -1 ψ(u) = 0, with solution ψ(u) = m i=1 c i e -λ i + u .
Further, note that for i = 1, . . . , m -1,

c i = s + λ i s (q -r) + (s) -dq + (s) (q -r) + (s) s=-λi
.

(Z k ) k≥1 dependent with exponential steps/densities

In this section we return to the dependent case endowed with a Markov chain structure and provide an example for the results derived. We are again in the set-up of g i (x), i = 1, . . . 4 being non-constant functions of the initial step

Z 0 = x. Let          g 1 (x) = 1 -g 4 (x) = 1 -e -θx (I) g 2 (x) = 1 -g 3 (x) = 1 -e ηx (1 -I) h 1 (y) = λ 1 e -λ1y , h 2 (y) = λ 2 e -λ2y (Z + ) h 3 (y) = λ 3 e λ3y , h 4 (y) = λ 4 e λ4y (Z -).
More precisely we will further assume that λ 1 = λ 2 and λ 3 = λ 4 . The interpretation of such a model is the following: The size of the jumps are exponentially distributed, having different parameters for positive and negative jumps. Thus, the functions h 1 , h 2 and respectively h 3 , h 4 play the same role as the densities of Z + and Z -of (2) in the independent random-walk case of [START_REF] Asmussen | Phase-type representations in random walk and queueing problems[END_REF].

The parameters θ or η control the probability of going up or down in the next step of the random walk. Functions g 1 , g 4 (reps. g 2 , g 3 ) correspond to 1 -I, I for x > 0 (respectively x < 0) as in the decomposition (2). If θ is large, then it is more likely that a positive jump is followed by another positive jump. On the contrary, if η is close to zero then it is more likely that a negative jump is followed by another negative jump (see Figure 1 compare the graph to paths when sampled from the invariant distribution in Figure 2). Since the lucky events are when the random walk goes to -∞, choosing a large θ one would emphasize the bad tendencies. On the other hand, choosing a η close to zero would emphasize the good tendencies. Also, remark that choosing a small θ would lead to some counter-cyclical, tamed behavior (see Figure 4 the invariant distribution is the same as the one of the model in Figure 1).

Next we give a numerical example. We use the parameters λ 1 = λ 2 = 1, λ 3 = λ 4 = 4/5, θ = 2 and η = 0.4 (see Figure 3 for a plot of some paths of this process). Using Lemma 4 from the Appendix we derive the density of the invariant distribution,

f Z (x) = 1 2 e -x x ≥ 0 1 2 4 5 e 4 5 x
x < 0.

With plots 1 and 4 we emphasize the difference in the behavior of the paths when θ and η have different values, even though they have the same invariant distribution. Since ψ 1 (u) = ψ 2 (u) and ψ 3 (u) = ψ 4 (u), to calculate ψ i (u) one needs to solve the system of ODEs

A ψ 1 ψ 4 = 0 0 .
In this case, the matrix A is A =

(r 1,1 -q 1,1 )q 4,1 gcd(q 1,1 ,q 4,1 ) q 1,1 r 4,1 gcd(q 1,1 ,q 4,1 ) r 2,3 q 3,3 gcd(q 2,3 ,q 3,3 ) (r 3,3 -q 3,3 )q 2,3 gcd(q 2,3 ,q 3,3 ) , with the determinant

det(A) = s s 3 + 14s 2 5 - 87s 25 - 6 25 .
Given the infinity condition, one needs to consider only the negative roots of the polynomial det(A) = 0. Thus, the solution of the system of ODEs has the form ψ i (u) = c i 2 e -3.7185u + c i 2 e -0.0655856u , which once plugged back into Equation (10) leads to

ψ 1 (u) = ψ 2 (u) = 0.0488089e -3.7185u + 0.951191e -0.0655856u ψ 3 (u) = ψ 4 (u) = 0.000839295e -3.7185u + 0.75528e -0.0655856u
and thus

ψ(u, x) = (1 -e -2x )ψ 1 (u) + e -2x ψ 3 (u) x > 0 (1 -e 4 10 x )ψ 1 (u) + e -4 10 x ψ 3 (u) x < 0.
Further, if we assume that Z 0 is distributed according to the invariant distribution, then ψ(u) = 0.0248241e -3.7185u + 0.853236e -0.0655856u .

We compare this to the ruin probability of a process with iid increments, which follows the same invariant distribution,

ψ indep (u) = 9 10 e -u 10 .
Numerical values of these ruin probabilities are plotted in Figures 5 and6. One can clearly see that, in this case, dependence causes an increase of the risk of ruin.

Conclusions

One of the fundamental problems in random walk theory is the computation of the ladder height distributions. More specifically, for a random walk with increments having the distribution F Z and ladder epochs defined as τ -= inf{n ≥ 1|W n ≥ 0} respectively τ + = inf{n ≥ 1|W n > 0}, the ladder height distributions are G -(u) = P(Z τ-≤ u) and G + (u) = P(Z τ+ ≤ u). These quantities are relevant in sequential, queuing and risk theory. Results have been derived for increments having a structure Z = pZ 1 + (1p)Z 2 [START_REF] Feller | An introduction to probability theory and its applications[END_REF] or a difference structure Z = τ -X [START_REF] Asmussen | Phase-type representations in random walk and queueing problems[END_REF].

W n = n k=1 Z k ,
Focusing on a risk theory model, one could analyze the probability of ruin in this setting. In this paper we base our analysis on the real-valued random variable Z k = -cτ k + X k describing the difference between the inter-arrival time τ k and its consecutive claim size X k . In the classical model, τ and X are assumed to be independent. If regardless of the dependence structure of τ and X, Z k are independent, we are in the random walk setting. A natural condition for Z in order to be able to derive explicit forms of the ruin probability is that its mgf is rational. Or, the weaker/more general version of it, that only the mgf of Z + = {Z | Z > 0} is rational.

Once we step away from the independence of Z k , there are a multitude of dependence structures possible. We introduced a Markov chain dependence which still permits the exact calculation of the probability of ruin. A motivation for such a model, besides the mathematical amenability, consists in the fact that one can implement a counter-cyclic behavior of the sample paths. Although a step away from the classical random walk setting, the model and the method can be presented as an immediate alteration/extension of it.

1. If the joint moment generating function M τ,X (t, s) of the vector (τ, X) is rational, then the moment generating functions of Z + and Z -, M Z + and respectively M Z -are also rational.

2. If M Z + and M Z -are rational, then there exists a vector (τ, X) such that M τ,X (t, s) is rational.

Proof. Ad 1. Since M τ,X (t, s) is rational, the density F Z of Z = Xcτ = IZ + + (1 -I)Z -has a rational Fourier transform, implying

f Z (x) = n i=1 c i x mi e -λix 1 {sign(Re(λi))[0,∞]} (x). Now 1 follows from pf Z + (x) = n i=1,Re(λi)>0 c i x mi e -λix 1 {(0,∞)} (x)
and

(1 -p)f Z -(x) = n i=1,Re(λi)<0 c i x mi e -λix 1 {(-∞,0)} (x).
For the second part of the lemma note that by induction

d du n u 0 ψ(u -y)g(y)dy = d du n u 0 ψ(y)g(u -y)dy = n-1 k=0 ψ (k) (u)g (n-k-1) (0) + u 0 ψ(y)g (n) (u -y)dy.
It follows

q d du u 0 ψ(u -y)g(y)dy = n l=0 q l l-1 k=0 ψ (k) (u)g (n-k-1) (0) + u 0 ψ(y)q d du g(u -y)dy = r d du ψ(u).
Lemma 3. Let g(y) and ψ(y) be two functions that are sufficiently often differentiable, where g(y) is bounded for y < 0 and ψ(y) is bounded for y > 0.

Assume that there exists a polynomial q(x) = q 0 + q 1 x + • • • + q n x n with q d dy g(y) = 0, y < 0.

Further define the polynomial

r(x) = - n-1 l=0 n i=l+1 q i g (i-l-1) (0)x l .
Then 0 -∞ e -sy g(y)dy = r(s) q(s)

and

q d du 0 -∞ ψ(u -y)g(y)dy = r d du ψ(u).
Proof. At first note that by partial integration 0 -∞ e -sy g(y)dy = -

l k=1 1 s k g (k-1) (0) + 0 -∞ 1 s l e -sx g (l) (x)dx.
Multiplying with q(s) leads to

q(s) 0 -∞ e -sy g(y)dy = - n l=0 q l l k=1 s l-k g (k-1) (0) - 0 -∞ e -sx q l g (l) (x)dx = - n-1 k=0 n l=k+1 q l g (l-k-1) (0)s k + ∞ 0 e -sx q d dx g(x)dx = r(s).
For the second part of the lemma note that by induction

d du n 0 -∞ ψ(u -y)g(y)dy = d du n ∞ u ψ(y)g(u -y)dy = - n-1 k=0 ψ (k) (u)g (n-k-1) (0) + ∞ u ψ(y)g (n) (u -y)dy. It follows q d du ∞ u ψ(u -y)g(y)dy = - n l=0 q l l-1 k=0 ψ (k) (u)g (n-k-1) (0) + ∞ u ψ(y)q d du g(u -y)dy = r d du ψ(u).
Lemma 4. Let Z be a Markov chain with transition density satisfying the Assumption 1. Denote with

K l,m = ∞ 0 g l (x)h m (x)dx, if l ∈ {1, 4}, m ∈ {1, 2} 0 -∞ g l (x)h m (x)dx, if l ∈ {2, 3}, m ∈ {3, 4} .
Then the invariant distribution has the density

f Z (y) = K 1,2 K 2,3 h 1 (y) + K 2,3 K 4,1 h 2 (y) K 1,2 K 2,3 + 2K 2,3 K 4,1 + K 3,4 K 4,1 , y > 0, f Z (y) = K 3,4 K 4,1 h 3 (y) + K 2,3 K 4,1 h 4 (y) K 1,2 K 2,3 + 2K 2,3 K 4,1 + K 3,4 K 4,1 , y < 0,
Proof. The invariant distribution is defined through

f Z (y) = ∞ 0 p(x, y)f Z (x)dx.
We can split this equation into two equations, for y > 0 and y < 0,

f Z (y) =h 1 (y) ∞ 0 g 1 (x)f Z (x)dx + h 2 (y) 0 -∞ g 2 (x)f Z (x)dx, y > 0, f Z (y) =h 4 (y) ∞ 0 g 4 (x)f Z (x)dx + h 3 (y) 0 -∞ g 3 (x)f Z (x)dx, y < 0.
It follows that the invariant distribution can be written as

f Z (y) = c 1 h 1 (y) + c 2 h 2 (y), y > 0, f Z (y) = c 3 h 3 (y) + c 4 h 4 (y), y < 0,
and we are left with determining the constants c l , which are the solution of the system of linear equations

         c 1 = K 1,1 c 1 + K 1,2 c 2 c 2 = K 2,3 c 4 + K 2,4 c 4 c 3 = K 3,3 c 4 + K 3,4 c 4 c 4 = K 4,1 c 1 + K 4,2 c 2 .
In our matrix notation

    K 1,1 -1 K 1,2 0 0 0 -1 K 2,3 K 2,4 0 0 K 3,3 -1 K 3,4 K 4,1 K 4,2 0 -1         c 1 c 2 c 3 c 4     =     0 0 0 0     (.1)
Using the fact that h i (x)dx = 1 and 1g i (x) = g 5-i (x) one can rewrite the matrix in (.1) as

    -K 4,1 1 -K 4,2 0 0 0 -1 K 2,3 K 2,4 0 0 -K 2,3 1 -K 2,4 K 4,1 K 4,2 0 -1    
Moreover, by adding all other rows to the last one, it becomes

    -K 4,1 1 -K 4,2 0 0 0 -1 K 2,3 K 2,4 0 0 -K 2,3 1 -K 2,4 0 0 0 0         c 1 c 2 c 3 c 4     =     0 0 0 0    
It follows that

c 1 = tK 1,2 K 3,4 + K 2,4 K 4,1 = t K 1,2 K 4,1
c 2 = t K 3,4 + K 2,4 = t c 3 = t K 3,4 K 2,3 c 4 = t.

Since f Z (x) is a density function, t -1 = K 1,2 K 4,1 + K 3,4 K 2,3 + 2. Remark 4. In the general case we have that h i (x)dx = 1 and 1j g i j (x) = j g 5-i j (x) and hence for fixed j and m, i K l,m i,j + K 5-l,m i,j

= 1 (K l,m i,j is defined as K l,m by replacing h m by h m j and g l by g l i ). If we interpret the matrix (.1) as a block matrix where K l,m corresponds to the matrix with elements K l,m i,j and 1 to the identity matrix, then the sum of all rows is 0. Which means that we get non-trivial candidates for f Z .

Proof of Proposition 1. Equation (5) for ψ(u, x) will have two expressions, de-pending on the sign of x. For x ∈ I 1 ∪ I 4 , (x > 0) ψ(u, x) = Proof of Theorem 3. From the fact that ψ i solve a system of ODEs with boundary conditions lim u→∞ ψ i (u) = 0, we have (r 1,1 -q 1,1 )q 4,1 gcd(q 1,1 ,q 4,1 ) 0 0 q 1,1 r 4,1 gcd(q 1,1 ,q 4,1 ) r 1,2 q 4,2 gcd(q 1,2 ,q 4,2 ) -q 1,2 q 4,2 gcd(q 1,2 ,q 4,2 ) 0 q 1,2 r 4,2 gcd(q 1,2 ,q 4,2 ) 0 r 2,3 q 3,3 gcd(q 2,3 ,q 3,3 ) (r 3,3 -q 3,3 )q 2,3 gcd(q 2,3 ,q 3,3 ) 0 0 r 2,4 q 3,4 gcd(q 2,4 ,q 3,4 ) q 2,4 r 3,4 gcd(q 2,4 ,q 3,4 ) -q 2,4 q 3,4 gcd(q 2,4 ,q 3,4 ) ,       = 0 (.4) equivalent to 0 = (r 1,1q 1,1 )(r 3,3q 3,3 )q 1,2 q 2,3 q 2,4 q 3,4 q 4,1 q 4,2 gcd(q 1,1 , q 4,1 )gcd(q 1,2 , q 4,2 )gcd(q 2,3 , q 3,3 )gcd(q 2,4 , q 3,4 ) + r 2,3 r 3,4 q 2,4 q 3,3r 2,4 (r 3,3q 3,3 )q 2,3 q 3,4 r 4,2 (r 1,1q 1,1 )q 1,2 q 4,1r 1,2 r 4,1 q 1,1 q 4,2 gcd(q 1,1 , q 4,1 )gcd(q 1,2 , q 4,2 )gcd(q 2,3 , q 3,3 )gcd(q 2,4 , q 3,4 ) .

ψ i (u) =

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: Paths of the risk process when λ 1 = λ 2 = 1, λ 3 = λ 4 = 4/5, θ = 100 and η = 0.008

Figure 6 :

 6 Figure 6: Comparison of the ruin probability (-log 10 of the values) of the Markov chain with λ 1 = λ 2 = 1, λ 3 = λ 4 = 4/5, θ = 2 and η = 0.4, and the corresponding process of independent Z with the invariant distribution.

  y, y)h 4 (y)dy = g 1 (x)ψ 1 (u) + g 4 (x)ψ 4 (u).(.2)Analogously for x ∈ I 2 ∪ I 3 , (x < 0) ψ(u, x) = ∞ u g 2 (x)h 2 (y)dy + 0 -∞ ψ(uy, y)g 3 (x)h 3 (y)dy + u 0 ψ(uy, y)g 2 (x)h 2 (y)dy = g 2 (x) y, y)h 3 (y)dy = g 2 (x)ψ 2 (u) + g 3 (x)ψ 3 (u). (.3)Thus the first part of the theorem follows. The second part of the theorem follows from lim u→∞ ψ(u, x) = 0 for every x.

  u k e -λj u , where -λ j are the negative roots of the polynomial det

  Y 1 and Y 2 have a rational Laplace transform, but Y 3 has not. Actually, Y 3 can have any positive distribution function, even heavy-tailed.

	While the classes B and C are present in the applied probability literature, this
	paper brings forth class A and shows that as in the other two classes of distri-
	butions, the probability of ruin can be derived by means of ODEs with constant
	coefficients.

For instance, one can have τ = Y 1 + Y 3 and X = Y 2 + Y 3 , where Y 1 , Y 2 and Y 3 are independent,

Acknowledgments. This work has been partially supported by the French Research National Agency (ANR) under the reference ANR-08-BLAN-0314-01 and the Swiss National Science Foundation Project 200021-124635/1. Lemma 1.

Ad 2. We have

We use

To show that this is the moment generating function of a random vector, let I be Bernoulli(p) and Y be Exp(1). Then the random vector (τ, X)

has M τ,X (t, s) as moment generating function.

Lemma 2. Let g(y) and ψ(y) be two functions that are sufficiently often differentiable and bounded for y > 0. Assume that there exists a polynomial q(x)

Further define the polynomial

Then the Laplace transform of g ∞ 0 e -sy g(y)dy = r(s) q(s) , and

Proof. Note that for s > 0, using integration be parts

Multiplying with q(s) leads to q(s)

∞ 0 e -sy g(y)dy

Dividing by q 1,1 q 1,2 q 2,3 q 2,4 q 3,3 q 3,4 q 4,1 q 4,2 gcd(q 1,1 , q 4,1 )gcd(q 1,2 , q 4,2 )gcd(q 2,3 , q 3,3 )gcd(q 2,4 , q 3,4 ) leads to the equation 0 = (r 1,1q 1,1 )(r 3,3q 3,3 ) q 1,1 q 3,3 + r 2,3 r 3,4 r 4,2 (r 1,1q 1,1 ) q 1,1 q 2,3 q 3,4 q 4,2 -r 1,2 r 2,3 r 3,4 r 4,1 q 1,2 q 2,3 q 3,4 q 4,1 -(r 1,1q 1,1 )(r 3,3q 3,3 )r 2,4 r 4,2 q 1,1 q 2,4 q 3,3 q 4,2 + (r 3,3q 3,3 )r 1,2 r 2,4 r 4,1 q 1,2 q 2,4 q 3,3 q 4,1

The claim follows from the definition of M k,m .