A risk model with an observer in a Markov environment

Albrecher, Hansjörg; Ivanovs, Jevgenijs Article A risk model with an observer in a Markov environment Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Albrecher, Hansjörg; Ivanovs, Jevgenijs (2013) : A risk model with an observer in a Markov environment, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 1, Iss. 3, pp. 148-161, https://doi.org/10.3390/risks1030148 This Version is available at: https://hdl.handle.net/10419/103601 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/3.0/ Risks 2013,1, 148-161; doi:10.3390/risks1030148 OPEN ACCESS risks ISSN 2227-9091 www.mdpi.com/journal/risks Article A Risk Model with an Observer in a Markov Environment Hansj¨ org Albrecher 1,2,* and Jevgenijs Ivanovs 1,* 1Department of Actuarial Science, University of Lausanne, Lausanne CH-1015, Switzerland 2Swiss Finance Institute, University of Lausanne, Lausanne CH-1015, Switzerland *Authors to whom correspondence should be addressed; E-Mails: hansjoer[email protected] (H.A.), jevgenijs.ivano[email protected] (J.I.). Received: 11 October 2013; in revised form: 5 November 2013 / Accepted: 5 November 2013 / Published: 11 November 2013 Abstract: We consider a spectrally-negative Markov additive process as a model of a risk process in a random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process as negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory, we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration. Keywords: Markov additive process; level-crossing probabilities; Poissonian observation; ruin probability; occupation times 1. Introduction In classical risk theory, the ruin of an insurance portfolio is defined as the event that the surplus process becomes negative. In practice, it may be more reasonable to assume that the surplus value is not checked continuously, but at certain times only. If these times are not fixed deterministically, but are assumed to be epochs of a certain independent renewal process, then one often still has sufficient analytical structure to obtain explicit expressions for ruin probabilities and related quantities; see [1,2] for corresponding studies in the framework of the Cram´ er–Lundberg risk model and Erlang inter-observation times. An alternative ruin concept is studied in [3], where negative surplus does not necessarily lead to bankruptcy, but bankruptcy is declared at the first instance of an inhomogeneous Poisson process with a rate depending on the surplus value, whenever it is negative. When this rate is constant, this bankruptcy Risks 2013,1149 concept corresponds to the one in [1,2] for exponential inter-observation times. Yet another related concept is the one of Parisian ruin, where ruin is only reported if the surplus process stays negative for a certain amount of time (see, e.g., [4,5]). If this time is assumed to be an independent exponential random variable instead of a deterministic value, one recovers the former models with exponential inter-observation times and a constant bankruptcy rate function, respectively. Recently, simple expressions for the corresponding ruin probability have been derived when the surplus process follows a spectrally-negative L´ evy process, see [6]. In this paper, we extend the above model and allow the surplus process to be a spectrally-negative Markov additive process. The dynamics of such a process change according to an external environment process, modeled by a Markov chain, and changes of the latter may also cause a jump in the surplus process. We assume that the value of the surplus process is only observed at epochs of a Poisson process, and ruin occurs when at any such observation time, the surplus process is negative. We also allow the rate of observations to depend on the current state of the environment (one possible interpretation being that if the environment states refer to different economic conditions, a regulator may increase the observation rates in states of distress). Using an approximation argument and the spectral theory for Markov additive processes, we explicitly calculate for any initial capital the survival probability and the probability to reach a given level before ruin in this model. The resulting formulas turn out to be quite simple. At the same time, these formulas provide information on certain occupation times of the process, which may be of independent theoretical interest. In Section 2, we introduce the model and the considered quantities in more detail. Section 3gives a brief summary of general fluctuation results for Markov additive processes that are needed later on. In Section 4, we state our main results and discuss their relation with previous results, and the proofs are given in Section 5. In Section 6, we reconsider the classical ruin concept and show how the present results implicitly extend the classical simple formula for the ruin probability with zero initial capital to the case of a Markov additive surplus process. Finally, in Section 7, we give a numerical illustration of the results for our relaxed ruin concept in a Markov-modulated Cram´ er–Lundberg model. 2. The Model Let (X(t), J(t)), t ≥0be a Markov additive process (MAP), where X(t)is a surplus process and J(t) is an irreducible Markov chain on nstates representing the environment; see, e.g., [7]. While J(t) = i, X(t)evolves as some L´ evy process Xi(t), and X(t)has a jump distributed as Uij when J(t)switches from ito j. Consequently, X(t)has stationary and independent increments given the corresponding states of the environment. We assume that X(t)has no positive jumps and that none of the processes, Xi(t), is a non-increasing L´ evy process. The latter assumption is not a real restriction, because one can always remove the corresponding states of J(t)and replace non-increasing processes by the appropriate negative jumps. Note that the Markov-modulated Cram´ er–Lundberg risk model with X(t) = u+Zt 0 cJ(v)dv − N(t) X j=1 Yj(1) is a particular case of the present framework, where uis the initial capital of an insurance portfolio, ci>0is the premium density in state i,N(t)is an inhomogeneous Poisson process with claim arrival Risks 2013,1150 intensity βiin state iand Yjare independent claim sizes with distribution function Fi, if, at the time of occurrence, the environment is in state i(in this case, Uij ≡0for all i, j); see [7]. Write Eu[Y;J(t)] for a matrix with ijth element E(Y1{J(t)=j}|J(0) = i, X(0) = u), where Yis an arbitrary random variable, and Pu[A, J(t)] = Eu[1A;J(t)] for the probability matrix corresponding to an event A. If u= 0, then we simply drop the subscript. We write I,O,1,0for an identity matrix, a zero matrix, a column vector of ones and a column vector of zeros of dimension n, respectively. For x≥0, define the first passage time above x(below −x) by: τ± x= inf{t≥0 : ±X(t)> x} As in [2], we assume that ruin occurs when an independent Poissonian observer sees negative X(t), where, in our setup, the rate of observations depends on the state of J(t), i.e., the rate is ωJ(t)≥0for given ω1, . . . , ωn. Recall that a Poisson process of rate ωhas no jumps (observations) in some Borel set B⊂[0,∞)with probability exp(−ωRBdt). Hence, the probability of survival (non-ruin) in our model with initial capital uis given by the column vector: φ(u) = Eue−PjωjAj,where Aj:= Z∞ 0 1{X(t)<0,J(t)=j}dt(2) which follows by conditioning on the Ajs. The ith component of this vector refers to the probability of survival with initial state J(0) = i. Define for any u≤xthe n×nmatrix: R(u, x) := Eu[e−PjωjAj(x);J(τ+ x)],with Aj(x) := Zτ+ x 0 1{X(s)<0,J(s)=j}ds(3) so R(u, x)is the matrix of probabilities of reaching level xwithout ruin, when starting at level u. It is known that X(t)/t converges to a deterministic constant, µ(the asymptotic drift of X(t)) a.s. as t→ ∞, independently of the initial state, J(0). If µ < 0, then X(t)→ −∞ a.s.; so Aj→ ∞ a.s. for all j, and consequently, ruin is certain (unless all ωj= 0). If µ≥0, then τ+ x<∞a.s. for all x, and so: φ(u) = lim x→∞ R(u, x)1 Finally, note that R(u, x)can be interpreted as a joint transform of the occupation times, Aj(x). Moreover, with the definition R(x) := R(0, x), the strong Markov property and the absence of positive jumps give: R(x)R(x, y) = R(y)(4) for 0≤x≤y(see, also, [8]). Hence, R(x, y)can be expressed in terms of R(x)and R(y), given that these matrices are invertible. That is, it suffices to study the matrix-valued function, R(x). Remark 2.1. The present framework can be extended to include positive jumps of phase type, cf. [7]. One can convert a MAP with positive jumps of phase type into a spectrally-negative MAP using the so-called fluid embedding, which amounts to an expansion of the state space of J(t)and to putting Xi(t) = tfor each auxiliary state, i; see e.g., [9] and [10] (Section 2.7). Next, we set ωi= 0 for all the new auxiliary states, i, and compute the corresponding survival probability vector for the new model, which, when restricted to the original states, yields the survival probabilities of interest. Risks 2013,1151 3. Review of Exit Theory for MAPs Let us quickly recall the recently established exit theory for spectrally-negative MAPs, which is an extension of the one for scalar L´ evy processes (see, e.g., [11], Section 8). A spectrally-negative MAP, (X(t), J(t)), is characterized by a matrix-valued function, F(θ), via E[eθX(t);J(t)] = eF(θ)tfor θ≥0. We let πbe the stationary distribution of J(t). It is not hard to see that J(τ+ x), x ≥0is a Markov chain, and thus, P(J(τ+ x) = j|J(0) = i)=(eΛx)ij for a certain n×ntransition rate matrix, Λ, which can be computed using an iterative procedure or a spectral method; see [12,13] and the references therein. It is easy to see that J(τ+ x), x ≥0is non-defective (with a stationary distribution, πΛ), if and only if µ≥0. The two-sided exit problem for MAPs without positive jumps was solved in [14], where it is shown that: Pu[τ+ x< τ− 0, J(τ+ x)] = W(u)W(x)−1 for 0≤u≤xand x > 0, where W(x), x ≥0is a continuous matrix-valued function (called the scale function) characterized by the transform: Z∞ 0 e−θxW(x)dx=F(θ)−1(5) for sufficiently large θ. It is known that W(x)is non-singular for x > 0and so is F(θ)in the domain of interest. In addition: W(x) = e−ΛxL(x)(6) where L(x)is a positive matrix increasing (as x→ ∞) to L, a matrix of expected occupation times at zero (note that in the case of the Markov modulated Cram´ er–Lundberg model, Equation (1), cjLij provides the expected number of times when the surplus is zero in state jgiven J(0) = iand X(0) = 0). If µ6= 0, then Lhas finite entries and is invertible. Finally: Eu[eθX(τ− 0);τ− 0< τ+ x, J(τ− 0)] = Z(θ, u)−W(u)W(x)−1Z(θ, x)(7) where Z(θ, x) = eθx I−Zx 0 e−θyW(y)dyF(θ) is analytic in θfor fixed x≥0in the domain <(θ)>0. Importantly, all the above identities hold for defective (killed) MAPs, as well, i.e., when the state space of J(t)is complemented by an absorbing ‘cemetery’ state; the original states of J(t)then form a transient communicating class, and the (killing) rate from a state, i, into the absorbing state is ωi≥0. We refer to [15] for applications of the killing concept in risk theory. Note that killed MAPs preserve the stationarity and independence of increments given the environment state. Furthermore, we get probabilistic identities of the following type: eˆ Λx=ˆ P[J(τ+ x)] = E[e−PjωjRτ+ x 01{Jt=j}dt;J(τ+ x)] (8) Risks 2013,1152 where ˆ Pand ˆ Λrefer to the killed process, and we are still concerned with the original nstates only. The right-hand side of Equation (8) is similar to the definition of the matrix, R(x), in Equation (3); it is also the joint transform of certain occupation times. However, R(x)is more complicated, as there, the killing is only applied when the surplus process is below zero; so with the setup of this paper, one leaves the class of defective MAPs (the increments now depend on the current value of X(t)). Let us recall the relation between F(θ)and its killed analogue ˆ F(θ): ˆ F(θ) = F(θ)−∆,∆ = diag(ω1, . . . , ωn)(9) Letting ∆πbe a diagonal matrix with the stationary distribution vector, π, of Jon the diagonal, we note that ˜ F(θ)=∆−1 πF(θ)T∆πcorresponds to a time-reversed process, which is, again, a spectrally-negative MAP (with no non-increasing L´ evy processes as building blocks) with the same asymptotic drift, µ; see [7]. Using the characterization Equation (5), one can see that the corresponding scale function is given by f W(x)=∆−1 πW(x)T∆π. 4. Results The following main result determines the matrix of probabilities of reaching a level, x, without ruin: Theorem 4.1. For x≥0, we have: R(x) = E[e−PjωjAj(x);J(τ+ x)] = eˆ ΛxI−Zx 0 W(y)∆eˆ Λydy−1 where ˆ Λcorresponds to the killed process with killing rates, ωi≥0, identified by ˆ F(θ)in Equation (9). The vector of survival probabilities according to our relaxed ruin concept has the following simple form: Theorem 4.2. Assume that the asymptotic drift µ > 0, all obervation rates, ωi, are positive and Λand ˆ Λdo not have a common eigenvalue. Then, the vector of survival probabilities is given by: φ(0) = lim x→∞ R(x)1=U−11 where Uis the unique solution of: ΛU−Uˆ Λ = L∆(10) Equation (4) then immediately gives: Corollary 4.1. For 0≤u≤xit holds that R(u, x) = I−Zu 0 W(y)∆eˆ Λydyeˆ Λ(x−u)I−Zx 0 W(y)∆eˆ Λydy−1 Under the conditions of Theorem 4.2, we have for every u≥0 : φ(u) = R(u)−1φ(0) Risks 2013,1153 Equation (10) is known as the Sylvester equation in control theory. Under the conditions of Theorem 4.2, it has a unique solution [16], which has full rank, because L∆has full rank, see Theorem 2 in [17]. Moreover, the solution, U, can be found by solving a system of linear equations with n2 unknowns. With regard to coefficient matrices, there are two methods to compute Λand ˆ Λ; see Section 3. In principle, the matrix, L, can be obtained from W(x),cf. Equation (6). This method, however, is ineffective and numerically unstable. In the following, we give a more direct way of evaluating L. Proposition 4.1. Let µ6= 0. Then, for a left eigen pair (γ, h)of −Λ, i.e., −hΛ = γh, it holds that: hL= lim q↓0qhF(γ+q)−1 More generally, if h1,...,hjis a left Jordan chain of −Λcorresponding to an eigenvalue γ, i.e., −h1Λ = γh1and −hiΛ = γhi+hi−1for i= 2,...j, then: hjL= lim q↓0q j−1 X i=0 1 i!hj−i[F(q+γ)−1](i) Remark 4.1. Consider the special case n= 1, i.e., X(t)is a spectrally-negative L´ evy process with Laplace exponent F(θ) = log EeθX(1), with observation rate ω. Then, ˆ Λ = −Φ(ω), where Φ(·)is the right-inverse of F(θ), i.e., F(Φ(ω)) = ω. According to Theorem 4.1, we have: R(x) = e−Φ(ω)x/1−ωZx 0 e−Φ(ω)yW(y)dy= 1/Z(Φ(ω), x)(11) Note that 1/Z(θ, x)is a certain transform corresponding to X(t)reflected at zero at the time of passage over level x; see [14], which may lead one to an alternative direct probabilistic derivation of Equation (11). Finally, if µ=EX(1) >0, then Λ=0, and hence, L= 1/F0(0) = 1/µ, according to Proposition 4.1. Accordingly, in this case, Theorem 4.2 reduces to: φ(0) = Eexp −ωZ∞ 0 1{X(t)<0}dt=Φ(ω) ωµ which coincides with Theorem 1 in [6]. Remark 4.2. In the case when X(t)has no jumps, i.e., it is a Markov-modulated Brownian motion (MMBM), the matrices, W(x)and L, can be given in an explicit form; see [10,18,19]. Furthermore, the same is true for a model with arbitrary phase-type jumps downwards, because such a model can be reduced to an MMBM using fluid embedding. Note that the fluid embedding is only used to determine the matrices, W(x)and L, corresponding to the original process (the setup of this paper would not allow for processes Xi(t) = −totherwise); compare this to Remark 2.1 discussing phase-type jumps upwards. 5. Proofs The proofs rely on a spectral representation of the matrix, ˆ Λ, which we quickly review in the following. Let v1,...,vjbe a Jordan chain of −ˆ Λcorresponding to an eigenvalue γ, i.e., −ˆ Λv1=γv1 and −Λvi=γvi+vi−1for i= 2,...j. From the classical theory of Jordan chains, we know that: e−ˆ Λxvj= j−1 X i=0 xi i!eγxvj−i(12) Risks 2013,1154 for any x∈Rand j= 1, . . . , k and, in particular, e−ˆ Λxv1=eγxv1. Moreover, this Jordan chain turns out to be a generalized Jordan chain of an analytic matrix function ˆ F(θ),<(θ)>0corresponding to a generalized eigenvalue, γ, i.e., for any j= 1, . . . , k, it holds that: j−1 X i=0 1 i!ˆ F(i)(γ)vj−i= j−1 X i=0 1 i!F(i)(γ)vj−i−∆vj=0(13) and, in particular, F(γ)v1= ∆v1; see [13] for details. Proof of Proposition 4.1.Observe that he−Λx=eγxh, and so Equations (5) and (6) yield: hF(θ)−1=Z∞ 0 e−θxeγxhL(x)dx for large enough θ. Since L(x)is bounded from above by L, this equation can be analytically continued to <(θ)><(γ)with non-singular F(θ). Hence, for small enough q > 0, we can write: qhF(q+γ)−1=qZ∞ 0 e−qxhL(x)dx=hEL(eq) where eqis an exponentially distributed r.v. with parameter q. Letting q↓0completes the proof of the first part. According to Equation (12), we have hj−ie−Λx=Pj−i−1 k=0 xk k!eγxhj−i−k. Next, consider: hj−i[F(θ)−1](i)=Z∞ 0 (−x)ie−θxhj−ie−ΛxL(x)dx= j−1 X k=i (−1)i (k−i)!hj−kZ∞ 0 xke−θx+γxL(x)dx where differentiation under the integral sign can be justified using standard arguments. Finally: j−1 X i=0 1 i!hj−i[F(θ)−1](i)= j−1 X k=0 k X i=0 (−1)i i!(k−i)!hj−kZ∞ 0 xke−θx+γxL(x)dx=hjZ∞ 0 e−θx+γxL(x)dx because the second sum is (1 −1)k= 0 for k≥1. The final step of the proof is the same as in the case of j= 1. The proof of Theorem 4.1 relies on an approximation idea, which has already appeared in various papers; see, e.g., [6,20,21]. We consider an approximation, R(x), of the matrix, R(x). When computing the occupation times, we start the clock when X(t)goes below −(rather than zero), but stop it when X(t)reaches the level of zero. Mathematically, we write, using the strong Markov property: R(x) = P[τ+ x< τ− , J(τ+ x)] +Z− −∞ P[τ− < τ+ x, X(τ− )∈dy, J(τ− )]Ey[e−PjωjRτ+ 0 01{Jt=j}dt;J(τ+ 0)]R(x) Using the exit theory for MAPs discussed in Section 3, we note that the first term on the right is W()W(x+)−1, and the second, according to Equation (8), is: Z∞ 0P[τ− 0< τ+ x+,−X(τ− 0)∈dy, J(τ− 0)]eˆ Λ(y+)R(x) Risks 2013,1155 By the monotone convergence theorem, the approximating occupation times converge to Aj(x) as ↓0, and then, the dominated convergence theorem implies convergence of the transforms: R(x)→R(x)as ↓0for any x > 0. Hence, we have: W(x) lim ↓0W()−1I−Z∞ 0P[τ− 0< τ+ x+,−X(τ− 0)∈dy, J(τ− 0)]eˆ Λ(y+)R(x) = I(14) where we also used the continuity of W(x). We will need the following auxiliary result for the analysis of the above limit. Lemma 5.1. Let f(y), y ≥0be a Borel function bounded around zero. Then: lim ↓0W()−1Z 0 f(y)W(y)dy=O Proof. Consider a scale function of the time-reversed process: ˜ W(x)=∆−1 πW(x)T∆π. It is enough to show that: lim↓0R 0f(y)˜ W(y)dy˜ W()−1= 0,but: Z 0 f(y)˜ W(y)dy˜ W()−1=Z 0 f(y)˜ Py(τ+ < τ− 0;J(τ+ ))dy which clearly converges to the zero matrix. Proof of Theorem 4.1.First, we provide a proof under a simplifying assumption, and then, we deal with the general case. Part I: Assume that −ˆ Λhas nlinearly-independent eigenvectors v:−ˆ Λv=γv. Considering Equation (14), we observe that the integral multiplied by vis given by: Z∞ 0e−γ(y+)P[τ− 0< τ+ x+,−X(τ− 0)∈dy, J(τ− 0)]v= e−γE[eγX(τ− 0);τ− 0< τ+ x+, J(τ− 0)]v=e−γ(Z(γ, )−W()W(x+)−1Z(γ, x +))v according to Equation (7). Hence, the limit in Equation (14) multiplied by vis given by: lim ↓0W()−1Z 0 e−γyW(y)dyF(γ)v+W(x)−1Z(γ, x)v=W(x)−1Z(γ, x)v according to the form of Z(γ, )and Lemma 5.1. Finally, from Equation (13), we have: Z(γ, x)v=eγxv−Zx 0 W(y)∆eγ(x−y)vdy=e−ˆ Λx−Zx 0 W(y)∆eˆ Λ(y−x)dyv which, under the assumption that there are nlinearly-independent eigenvectors, shows that: e−ˆ Λx−Zx 0 W(y)∆eˆ Λ(y−x)dyR(x) = I completing the proof. Part II: In general, we consider a Jordan chain, v1,...,vj, of −ˆ Λcorresponding to an eigenvalue, γ. Using Equation (12), we see that the integral in Equation (14) multiplied by vjis given by: j−1 X i=0 1 i!E[(X(τ− 0)−)ieγ(X(τ− 0)−);τ− 0< τ+ x+, J(τ− 0)]vj−i