Applications Of Large Deviations In Financial Risks

Financial risk theory is a hot topic in actuarial field recently. As a part of actuarial mathematics, financial risk theory has 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 there chapters according to contents.The first chapter: Large deviations inequality in the class of V.The second chapter: The problem of large deviations in S(γ).The third chapter: The large deviations of risk model for two-type-risk insurance perturbed by diffusion.In the first chapter, Kai W. Ng and Qihe Tang have weaken the two assumptions N₁, N₂ to one. Thus, the renewal model and compound renewal model all can satisfy it. On this base, we consider and get an inequiality of the large deviations in the class of V.Assumption N1.Assumption N2. There exist small positive s and S such thatP{N(t) >k)(l+e)k^O, A(t)-?oo, â– Jfe>(l+<5)A(t)Assumption B : For some p > 7F,for fixed 5 > 0,there is ENp{t)I{m>il+s)Ht)) = o(\(t)).This chapter discuss the large deviations of V about Sn,S(t) and ST on the condition B.And we obtain the following results:Theorem 1.2.1 Let F eT> and has a finite mean fi, then for any 7 > fi, there exists some positive constants C{^) and D(-y) such thatC{i)nF{x) < P{Sn ⁿfi> x) < D(j)nF(x)holds for all n > 1 and all x > (7 - n)n.Theorem 1.2.2 Let F ï¿¡ V and has a finite mean n, then for any 7 > //, there exists some positive constants ï¿¡(7) and D{i) such thatC(j)X{t)F(x) < P(S(t) - ii(t) >x)< D(j)X(t)F(x)holds for all x > (7 —Theorem 1.2.3: If F G CnV, andr satisfide P(r > x) = o(F(:r)).ThenP(ST - EST >x) EtF(x), x ->â–  00.In the second chapter, asumme satisfy assumptions Ni and JV2. We obtain the following results:Theorem 2.2.1 If F G ï¿¡(7), then for every 7' > fj,, we haveP(Q — f?Q >, r\ r^ nf(—'y\n^]-p^lnlJ-F(r\ r —>â–  no â– *â–  V ^ ^n ^ ) J \ 1) V/i L^j-Theorem 2.2.2 If F G 5(7), and N(t)satisfies assumptions iVi and A^2, then we haveP(S(t) - ES(t) > x) A(i)e-7A{t)"/(-7)A(t)F(a;), \(t) ->? oo.for every fixed 7' > 0, holds uniformly for x>Theorem 2.2.3 If F e 5(7), r is non-negative random variable, and it has finite value Er, and independent with (Xn)n>i. If there exist e > 0 makesï¿¡'[(/(—7) + e)T] < 00, thenP{ST - ESr > x) e-^Er^[r/(-7)r-1]F(x),x ^ 00.In the third chapter, we obtion the large deviations of risk model for two-type-risk insurance perturbed by diffusion which F G ï¿¡7ZV.Theorem 3.2.1 For GGCPRM with diffusion, if F e STlV{-a, -/?), thenP{S{t) - ES{t) > x) XtF(x).For fixed 7 > 0, holds uniformly for x > jXt.Theorem 3.2.2 For GGCPRM with diffusion, we define ruin time,r(u) :=inf{t;5(t) > u).If F G ï¿¡HV{-a, -P), for some 1 < a < (3 < 00, then (1) For all x > 0 #1 y > 0, we have:liminf------logP(r(u) < yux) > x — /3 max{l, x\\u*oo log U(2) For x = 1 ft! 0 < y < ^{^} or 0 < x < 1 ffl y > 0, we have:Hmsup------log P(r(u) < yux) < x — a.u->oo lOgU...