Large Deviations In Finance And Insurance

Risk theory and ruin probabilities in particular are traditionally considered as parts of insurance mathematics, and have been an active area of research from the days of Lundberg all the way up to today. Large deviation probabilities occur quite naturally in the context of large claim insurance, in particular, reinsurance. This problem has aroused general concern. More and more mathematicans and staff members of financial circles are devoting their time to it. The papers concerning the large deviations problem has obtained many good achievements in various risk models and various class of light or heavy-tailed distributions. In this paper, we extend one-sided probability to two-sided probability and enlarge the applicable scope. On what has already been achieved we obtain some new results. The paper is divided into four chapters according to contents.In the first chapter, by using a basic result on the class of regularly varying functionsP(S_n - ES_n > x) P(max(X₁,…, X_n) > x) nF(x), nâ†'∞and two important assumptions, essential for our purposes, on the process (N(t))_t≥0, which are assumed to hold for λ(t) = EN(t) nâ†'∞ .Assumption N1.Assumption N2. There exist small positive ε and δ such that we shall study the power of large deviations for the random sums S(t) =∑X_i, t ≥ 0 i.e. P(|S(t) - μ(t)|ⁿ > ε) when F ∈ ERV(-α, -β) for some1 < α < β ≥∞ and give its upper bound and lower bound. This is thefurther improvement on the probability of {(S(t) - μ(t)) > ε}, giving a view of the power of it i.e. the probability of |S(t) - μ(t)|ⁿ > ε.We obtain the following results:Theorem 1.2.1 Let F G ERV(α,β) for some 1 < α ≤ β <∞ , then for every fixed γ > 0,holds uniformly for x ≥ (γ +μ)λ(t)Theorem 1.2.2 Let F e ERV(-α, -β) for some 1 < α α ≤ β <∞, then for any ε> 0,δ> 0, we haveIn the second chapter, we use some properties of the class V of heavy-tailed claim distributions in [7].Theorem K^[7] Let F D and has a finite mean μ. Then we have: (i)For any γ > 0, there exists a constant C(γ) > 0 such thatF^n(?)(x) ≥ C(γ)nF{x)holds for all n ≥ 1 and all x≥ γn;(ii) For any γ > μ⁺ = max(μ, 0), there exists a constant D(γ) > 0 such thatholds for all n ≥ 1 and all x ≥γn.Corollary K^[7]. Let F G V and has a finite mean , then for any , there exist some positive constants C(γ) and D(γ) such thatholds for all n ≥ 1 and all x ≥γn.We obtain the following results:Theorem 2.2.1 Let F D and has a finite mean , then for any , there exist some positive constants C(γ) and D(γ) such thatholds for all n > 1 and all x > (7 - /i)n.Theorem 2.2.2 Let F e V and has a finite mean n, then for any 7 > ï¿¡t, there exist some positive constants C(7) and ^(7) such thatC(j)\(t)F(x) < P(S(t) - nit) >x)< D(-y)X(t)F(x)holds for all x > (7 —In the third chapter, we assume the two important assumptions Nl, N2 and use some results for F E S which are proved by Cline, Hsing(1991) for the class of subexponential distributions:P(,Sn -ESn > x) ^P(max(Xi,--- ,Xn) > x) nF(x),n -> 00holds uniformly for every fixed 7 > 0 and for all x > 771. In this chapter we consider the large deviations problem in the classical risk model and gain the following result:Theorem 3.2.1 Let the process (iV(ï¿¡))t>o consists of integer-valued r.v.s N(t) on R+ and (Xn)nejv constitutes a sequence of i.i.d. non-negative r.v.s with d.f.F, independent of (N(t))t>o. Assume (N(t))t>o satisfies conditions Nl, N2, furthermore, let F G 0,P{S{t) - n(t) > x) \(t)F(x)holds uniformly for all x > 7A(t).In the forth chapter, we consider the general compound poisson risk model and the generalized compound poisson risk model perturbed by diffusion. An asymptotic estimation of large deviations for its tail distribution and the joint distributions of the maximum and the minimum of the surpluses first recovered from negative to zero, and last recovered from negative are obtained. We obtain the following results:Theorem 4.2.1 For the GCPRM perturbed by diffusion, S(t) =N(t)]T Xi - cM(t) - aW{t), let F € ERV(-a, -/?), 1< a < (3 < 00, then fort=iany fixed 7 > 0 such that 7A > c<5,...