The ruin probabilities of a multidimensional perturbed risk model

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1 MATHEMATICAL COMMUNICATIONS 231 Math. Commun. 18(2013, The ruin probabilities of a multidimensional perturbed risk model Tatjana Slijepčević-Manger 1, 1 Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, HR Zagreb, Croatia Received June 27, 2012; accepted January 30, 2013 Abstract. In this paper we consider the ruin probabilities of a multidimensional insurance risk model perturbed by Brownian motion. A Lundberg-type upper bound is derived for the infinite-time ruin probability when claims are light-tailed. The proof is based on the theory of martingales. An explicit asymptotic estimate is obtained for the finite-time ruin probability in the heavy-tailed claims case. AMS subject classifications: 60G50, 60J65, 60G44 Key words: multidimensional risk model, martingale, Poisson process, ruin probability 1. The model Multidimensional models with a common arrival process describe situations where each claim event usually produces more than one type of claim. One common example is natural catastrophe insurance where an accident could cause claims for different types of bodily injuries and property damages. The same situations exist in motor insurance. We consider a multidimensional insurance risk process R(t = (R1 (t,..., R n (t T perturbed by a multidimensional Brownian motion u 1 R 1 (t = + t R n (t u n c 1 c n N(t X 1i σ 1 B 1 (t +, t 0. (1 σ n B n (t Here u = (u 1,..., u n T stands for the initial surplus vector, c = (c 1,..., c n T for the premium rate vector, while X i = (X 1i,..., X ni T, i = 1, 2,... denote n-tuples of claims whose common arrival times constitute a counting process {N(t, t 0. The process {N(t, t 0 is a Poisson process with intensity λ > 0 and { X i, i = 1, 2,... is a sequence of independent copies of the random n-tuple X = (X 1,..., X n T with a joint distribution function F (x 1,..., x n and marginal distribution functions F 1 (x 1,..., F n (x n. The vector B (t = (B 1 (t,..., B n (t T denotes a standard multidimensional Brownian motion with constant correlation coefficients r ij [ 1, 1], i = 1,..., n 1, j = i + 1,..., n, while σ i 0, i = 1,..., n denote the X ni Corresponding author. address: tmanger@grad.hr (T. Slijepčević-Manger c 2013 Department of Mathematics, University of Osijek

2 232 T. Slijepčević-Manger marginal volatility coefficients of B (t. All vectors X i for i = 1, 2,..., u and c consist of only nonnegative components. The random processes { X i, i = 1, 2,..., {N(t, t 0 and { B (t, t 0 are all mutually independent. Let x = (x 1,..., x n T and y = (y 1,..., y n T be two n-dimensional vectors. Then we write x < y if x i < y i, i = 1,..., n and in the same way we define other inequalities. The ruin time of the model (1 can be defined in two different ways: or T min = inf{t > 0 min{r 1 (t,..., R n (t < 0 T max = inf{t > 0 R(t < 0 = inf{t > 0 max{r 1 (t,..., R n (t < 0. Here we assume that inf =. T max is the first time when all R i (t, i = 1,..., n go below zero. At time T min the insurance company may be able to survive more easily because probably only one of its subsidiary companies gets ruined. That means that T max represents a more critical time than T min. We also define the infinite-time ruin probability of the model (1 in two ways: or ψ( u = P (T min < R(0 = u (2 ψ( u = P (T max < R(0 = u, respectively. Finally, we define the finite-time ruin probability ψ( u ; T = P (T max T R(0 = u, T > 0. (3 In Section 2, we derive a Lundberg-type upper bound for the case of light-tailed claims and for the infinite-time ruin probability ψ( u. We use the techniques from martingale theory with no restrictions on the dependence structure of the process X. In Section 3, we derive an explicit asymptotic estimate for the finite-time ruin probability ψ( u ; T for the case of heavy-tailed claims, where we do assume that X 1,..., X n are independent. 2. A Lundberg-type upper bound for the ruin probability of light-tailed claims Throughout this section we consider only the claims with light tails. We also assume that the claim vector X has a finite mean vector µ = (µ 1,..., µ n T and that the safety loading condition c > λ µ holds. Our main result - an upper bound for the infinite-time ruin probability is given by the following theorem: Theorem 1. Let

3 The ruin probabilities of a multidimensional perturbed risk model 233 (i ˆm(s 1,..., s n = E[exp{s 1 X s n X n ]; (ii f(s 1,..., s n = λ ˆm(s 1,..., s n λ n c is i [ n σ2 i s2 i + 2 n 1 n j=i+1 r ijσ i σ j s i s j ]; (iii s 0 1 = sup{s 1 ˆm(s 1, 0,..., 0 <,..., s 0 n = sup{s n ˆm(0,..., 0, s n < ; (iv G 0 = {(s 1,..., s n s 1 0,..., s n 0, ˆm(s 1,..., s n < \ (0,..., 0; (v 0 = {(s 1,..., s n G 0 f(s 1,..., s n = 0. If s 0 1 > 0,...,s 0 n > 0 and sup (s1,...,s n G 0 f(s 1,..., s n > 0, then ψ( { u inf exp (s 1,...,s n 0 s i u i. (4 Hölder inequality gives that the set G 0 is non-empty provided that s 0 1 > 0,..., s 0 n > 0. First we will prove a proposition: Proposition 1. Let s 0 1 > 0,..., s 0 n > 0 and sup (s1,...,s n G 0 f(s 1,..., s n > 0. Then the following statements hold: (a The set 0 is non-empty. (b If v > 0 solves the equation f(s 1,..., ls 1,... = 0 for given l 0, then f(s 1,..., ls 1,... > 0 for every s 1 > v and f(s 1,..., ls 1,... < 0 for every 0 < s 1 < v. Here ls 1 comes in the i-th position and s j = s 1 for j i, i = 1,..., n. Proof. (a: For some given l 0 and s i = ls 1, i = 1,..., n, we calculate df(s 1,..., ls 1,... ds 1 [ n = λ +2l j=i+1 ˆm(s 1,..., s n + l ˆm(s ] 1,..., s i,... s j s i c j lc i + r ij σ i σ j s i 1 σj 2 s 1 + 2l r ji σ i σ j s 1 n 1 j=1 k=j+1,k i s i=ls 1,s j=s 1,j i r jk σ j σ k s 1 + l 2 σ 2 i s 1, so that df(s 1,..., ls 1,... ds 1 = s1=0 (c j λµ j l(c i λµ i < 0, because of the safety loading conditions. This means that the function s 1 f(s 1,..., ls 1,... decreases when s 1 > 0 is sufficiently close to the point s 1 = 0. For l =, the equation s i = ls 1 represents the line s 1 = 0 and in this case we can easily show that the function f(0,..., s i,... takes smaller values than f(0,..., 0

4 234 T. Slijepčević-Manger when s i > 0. Now we conclude that f(s 1,..., s i,... < 0 holds for all (s 1,..., s n sufficiently close to the origin, because f(0,..., 0 = 0 and 0 l can be arbitrary. By this and the condition sup f(s (s1,...,s n G 1,..., s 0 n > 0 we see that (a holds. (b: Let s 1 > 0 and l 0. We have [ n d 2 f(s 1,..., ls 1,... ds 2 1 = λ +2l + λ +2 2 ˆm(s 1,..., s n s 2 j 2 ] ˆm(s 1,..., s n +l 2 2 ˆm(s 1,..., s n s i s j n 1 i 1 σj 2 + 2l r ji σ i σ j + 2l j=1 k=j+1,k i E[(X j + lx i 2 ] + r jk σ j σ k + l 2 σ 2 i s 2 i j=i+1 r ij σ i σ j [σ j lσ i ] 2 > 0, s i=ls 1,s j=s 1,j i where s i = ls 1, i = 1,..., n. We conclude that the function s 1 f(s 1,..., ls 1,... is convex on (0, s 0 1, so the equation f(s 1,..., ls 1,... = 0 can have only one root in (0, s 0 1 and the result (b obviously follows. Now we will prove the theorem using Proposition 1. Proof of the theorem. First we are going to construct a martingale based on the surplus process { R(t, t 0. This martingale is needed for establishing a Lundbergtype upper bound for the ruin probability. Let s 1,..., s n be real numbers such that ˆm(s 1,..., s n <. We will show that the process M( { R(t = exp s i R i (t f(s 1,..., s n t, t 0, is an F-martingale, where F = {F t, t 0 represents the natural filtration of { R(t, t 0. For every t, h 0 we have [ E exp { = exp exp { h { 1 2 ] s i (R i (t + h R i (t s i c i exp{λ ˆm(s 1,..., s n h λh [ n = exp{f(s 1,..., s n h, n 1 σi 2 s 2 i + 2 j=i+1 r ij σ i σ j s i s j ]h

5 The ruin probabilities of a multidimensional perturbed risk model 235 since the Poisson process {N(t, t 0 has stationary independent increments. This gives that E[M( [ { ] R(t + h F t ] = E exp s i R i (t + h f(s 1,..., s n (t + h F t { = exp s i R i (t f(s 1,..., s n t = M( R(t and we conclude that M( R(t is a martingale with respect to F. Now the equation E[M( { R(t] = exp s i u i, t 0 (5 follows from M( { R(0 = exp n s iu i and the definition of a martingale. Next we will show that T min and M( R(t are a stopping time and a martingale, respectively, with respect to a common filtration F = {F t, t 0. Let {F t, t 0 be a complete σ-algebra of {F t, t 0 with respect to P and let F t+ = s>t F s. M( R(t is an right-continuous F-martingale, so it is also a martingale with respect to {F t+, t 0 (see [1, Theorem VI.1.3]. The definition of T min and the fact that { R(t, t 0 is a cádlág process, gives that T min is an {F t+, t 0-stopping time, hence an {F t+, t 0-stopping time since F t+ F t+ (see [3, I.1.28 Proposition]. So, if we select F = {F t+, t 0, we get that T min is an F -stopping time and M( R(t is an F -martingale, respectively. Let 1 A be the indicator function of an event A. By equality (5 and by the fact that T min and M( R(t are a stopping time and a martingale, for every (s 1,..., s n such that ˆm(s 1,..., s n < we have { exp s i u i = E[M( R(t] E[M( R(t1 (Tmin t] = E{E[M( R(t F Tmin+]1 (Tmin t = E{M( R(T min T min tp (T min t. Since there is at least one i {1,..., n such that R i (T min < 0, we can find (s 1,..., s n G 0 for which { exp s i R i (T min 1. By rearranging inequality (6 using the definition of M( R(t and the above inequality we get { P (T min t exp s i u i sup exp{f(s 1,..., s n h. 0<h<t (6

6 236 T. Slijepčević-Manger We define = {(s 1,..., s n G 0 f(s 1,..., s n < 0 and + = {(s 1,..., s n G 0 f(s 1,..., s n > 0. For (s 1,..., s n +, the right-hand side of the above relation tends to as t so this case makes no sense. We conclude that { P (T min t inf exp s i u i. (s 1,...,s n 0 By Proposition 1(a we know that the equation f(s 1,..., s n = 0 has at least one root in G 0. Applying Proposition 1(b it is easy to see that the infimum in the above inequality can be attained on 0. It follows that { P (T min t inf exp s i u i (s 1,...,s n 0 and for t we get a Lundberg-type upper bound for the infinite-time ruin probability (4 when the ruin time equals T min. In view of the obvious inequality P (T max < t P (T min < t we can see that the relation (4 also holds when we take T max to be the ruin time of the process. 3. Asymptotics for the finite-time ruin probability In this section we consider the risk process (1 with heavy-tailed claims. We further assume that the claim vector X and the multidimensional Brownian motion B (t consist of independent components. Here we do not assume the safety loading condition. A well-known class of heavy-tailed distribution functions is the subexponential class. A distribution function F on [0, is said to be subexponential if for some (or, equivalently, for all n = 2, 3,... the relation F n (x nf (x, x (7 holds, where F n denotes the n-fold convolution of F and if F (x > 0 for all x 0. Here means that the quotient of the left-hand and the right-hand side tends to 1 according to the indicated limit procedure. We write F S. More on subexponential distributions can be found in [6, 2.5]. In the following theorem we derive an asymptotic estimate for the finite-time ruin probability ψ( u ; T defined in (3. The limit procedure used in this theorem is always (u 1,..., u n (,...,. Theorem 2. Let F 1,..., F n be in S. Then ψ( u ; T f(nf 1 (u 1... F n (u n, (8 for every fixed time ( T > 0 and for each positive integer n, where f(0 = 1, f(1 = λt n 1 and f(n = λt f(i. i=0 ( n 1 i

7 The ruin probabilities of a multidimensional perturbed risk model 237 Proof. First we define B j (T = inf B j(t, B j (T = sup B j (t, j = 1,..., n. 0 t T 0 t T By the reflection principle (see [4, 2.6] there is so because F j S P (B j (T < x = P (B j (T > x = 2P (B j (T > x, P (B j (T < x = P (B j (T > x = o(f j (x for every x > 0 and j = 1,..., n. Obviously, ψ( u ; T = P ( R(t < 0 for some 0 < t T R(0 = u ( N(t =P X i t c (σ 1 B 1 (t,..., σ n B n (t T > u for some 0<t T. (9 First we will find an asymptotic upper bound for ψ( u ; T. independence of random vector components we get ψ( u ; T P k=0 ( N(T X i (σ 1 B 1 (T,..., σ n B n (T T > u n = P (N(T = k j=1 ( k P Now we need the result from [2, Lemma 1.3.5]: X ji σ j B j (T > u j. From the assumed (10 If F is a subexponential distribution, then for every ɛ > 0 there exists a constant C ɛ > 0 such that holds for all n = 1, 2,... and all x 0. F n (x C ɛ (1 + ɛ n F (x (11 By inequality (11 for every ɛ > 0 there exist constants C ɛ (1, C ɛ (2 > 0 such that for all k = 1, 2,..., ( k P X 1i σ 1 B 1 (T > u 1 0 = u 1 P ( k X 1i x > u 1 P (σ 1 B 1 (T = dx + P (σ 1 B 1 (T < u 1 C (1 ɛ (1 + ɛ k 0 u 1 P (X 1 x > u 1 P (σ 1 B 1 (T = dx + P (σ 1 B 1 (T < u 1 C ɛ (1 (1 + ɛ k P (X 1 σ 1 B 1 (T > u 1 + P (σ 1 B 1 (T < u 1 C ɛ (1 C ɛ (2 (1 + ɛ k F 1 (u 1.

8 238 T. Slijepčević-Manger In the last step we used P (X 1 σ 1 B 1 (T > u 1 F 1 (u 1 which follows from the fact that for nonnegative independent random variables X and Y with X distributed by F S it holds that P (X Y > x F (x, x (12 [7, Lemma 4.2]. We also used P (σ 1 B 1 (T < u 1 = o(1f 1 (u 1. For every fixed k = 1, 2,..., by (7 and from the fact that for distribution functions F, G S on [0, satisfying G(x = o(f (x it holds that [6, Lemma 2.5.2], we have F G(x F (x; (13 ( k P X 1i σ 1 B 1 (T > u 1 kf 1 (u 1. ( k The same relations also hold for P X ji σ j B j (T > u j, where k = 1, 2... and j = 1,..., n. Now using the dominated convergence theorem, we can see that the right-hand side of (10 is asymptotic to P (N(T = kk n F 1 (u 1... F n (u n = f(nf 1 (u 1... F n (u n, k=0 where f(0 = 1, f(1 = λt and f(n = λt This proves that ( n 1 i=0 ( n 1 i f(i. ψ( u ; T (1 + o(1f(nf 1 (u 1... F n (u n. (14 Next, we derive asymptotic lower bound for the ruin probability ψ( u ; T. From relation (9 we have ψ( u ; T P k=0 ( N(T X i T c (σ 1 B 1 (T,..., σ n B n (t T > u n = P (N(T = k j=1 ( k P As in the first part of the proof we can see that X ji σ j B j (T > u j + c j T. ( k P X ji σ j B j (T > u j + c j T kf j (u j for every j = 1,..., n and for each fixed k = 1, Therefore, using the dominated convergence theorem, the right-hand side of (16 is also asymptotic to P (N(T = kk n F 1 (u 1... F n (u n = f(nf 1 (u 1... F n (u n, k=0 (15

9 The ruin probabilities of a multidimensional perturbed risk model 239 where f(0 = 1, f(1 = λt i f(n = λt ( n 1 i=0 ( n 1 i f(i. This proves that ψ( u ; T (1 + o(1f(nf 1 (u 1... F n (u n. (16 Finaly, using inequalities (14 and (16 we obtain the required relation (8. References [1] C. Dellacherie, P. Meyer, Probabilities and potential B (Theory of martingales, North-Holland Mathematic Studies, Amsterdam, [2] P. Embrects, C. Klüppelberg, T. Mikosch, Modelling extremal events for insurance and finance, Springer-Verlag, Berlin, [3] J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, Second edition, Springer-Verlag, Berlin, [4] I. Karatzas, S. E. Shreve, Brownian motion and stochastic calculus, Springer- Verlag, Berlin, [5] J. Li, Z. Liu, Q. Tang, On the ruin probabilities of a bidimensional perturbed risk model, Insurance: Mathematics and Economics 41(2007, [6] T. Rolski, H. Schmidli, V. Schmidt, J. Teugels, Stochastic processes for insurance and finance, J. Wiley & Sons Inc., New York, [7] Q. Tang, The ruin probability of a discrete time risk model under constant interest rate with heavy tails, Scandinavian Actuarial Journal 3(2004,