Asymptotic finite-time ruin probabilities in a dependent bidimensional renewal risk model with subexponential claims
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Asymptotic finite‑time ruin probabilities in a dependent bidimensional renewal risk model with subexponential claims Dongya Cheng1 · Yang Yang2 · Xinzhi Wang2 Received: 22 August 2019 / Revised: 8 February 2020 © The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2020
Abstract This paper considers a bidimensional continuous-time renewal risk model, in which the two components of each pair of claim sizes are linked via the strongly asymptotic independence structure and the two claim-number processes from different lines of business are (almost) arbitrarily dependent. Precise asymptotic formulas for three kinds of finite-time ruin probabilities are established when the claim sizes have heavy-tailed tails. Keywords Bidimensional renewal risk model · Ruin probability · Subexponential distribution · Strongly asymptotic independence Mathematics Subject Classification Primary 62P05 · Secondary 62E10
1 Introduction Consider a bidimensional continuous-time risk model, in which the discounted value of the surplus process is described as
* Yang Yang [email protected] Dongya Cheng [email protected] Xinzhi Wang [email protected] 1
School of Mathematical Sciences, Soochow University, Suzhou 215006, China
2
Department of Statistics, Nanjing Audit University, Nanjing 211815, China
13
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�
R1 (t) R2 (t)
� =
� � � t −rs � � t −rs � ∫0− e C1 (ds) ∫0− e D1 (ds) x + − t t y ∫0− e−rs C2 (ds) ∫0− e−rs D2 (ds) �∑ � (1) N1 (t) Xk e−r𝜏k − ∑k=1 , t ≥ 0, (2) N2 (t) Yk e−r𝜏k k=1
(1.1)
where (x, y) denotes the vector of the initial surpluses; r ≥ 0 is the constant interest rate; {(C1 (t), C2 (t));t ≥ 0} represents the premium accumulation process with the nondecreasing and right-continuous paths satisfying (C1 (0), C2 (0)) = (0, 0) ; {(Xk , Yk );k ≥ 1} constitutes a sequence of independent and identically distributed (i.i.d.) claim-size vectors with generic vector (X, Y) following the marginal distributions F1 and F2 both on [0, ∞) , respectively; {(𝜏k(1) , 𝜏k(2) );k ≥ 1} is ∑ a sequence of claim-arrival time vectors with 𝜏k(i) = kl=1 𝜃l(i) , i = 1, 2 , which constitutes the renewal claim-number vector process {(N1 (t), N2 (t));t ≥ 0} ; and {(D1 (t), D2 (t));t ≥ 0} is a real-valued stochastic vector process reflecting the additional net losses of the two lines of business (equal to the total additional expenses minus income not including claims and premiums) satisfying (D1 (0), D2 (0)) = (0, 0) . Assume that {(Xk , Yk );k ≥ 1} , {(C1 (t), C2 (t));t ≥ 0} , {D1 (t);t ≥ 0} , {D2 (t);t ≥ 0} , (𝜃1(1) , 𝜃1(2) ) and {(𝜃k(1) , 𝜃k(2) );k ≥ 2} are mutually independent; in addition, 𝜃1(1) and 𝜃1(2) are independent of each other. Denote the two finite renewal functions by ∑∞ 𝜆i (t) = E[Ni (t)] = k=1 P(𝜏k(i) ≤ t), i = 1, 2. In the above continuous-time bidimensional risk model (1.1), for any T ≥ 0 , define the three kinds of finite-time ruin probabilities by ( ) | | 𝜓sim (x, y) = P inf {R1 (t) ∨ R2 (t)} < 0|(R1 (0), R2 (0)) = (x, y) 0≤t≤T | ( ) | | = P Ri (t) < 0,
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