Random Walk And Its Applications In Risk Theory

This dissertation is devoted to dealing with the relations between random walk and risk process. We apply it in the risk theory to resolve some ruin problems.Firstly, we give several definitions.Definition 1.1 Suppose X₁,X₂,…, are independent random variables, X_i,i≥ 2 are i.i.d. with a common distribution function F on (—∞, +∞), F(0) < 1; X₁ has a distribution function G, which is possibly different from F. Define for all n ≥ 1, S_n = X₁ + … + X_n, whereas S₀ = 0. S = (S_n)_n≥0 is called the non-standard random walk. In particular, if F = G, then S is the random walk.In Chapt 1, we get some results for the maximum of non-standard random walk with heavy tails, then we apply it in the risk theory and get some results. Theorem 1.2.1 (i) In non-stand random walk, if F_I ∈ S 'and (G(x))/(F_I(x))â†' 0 then,(ii) In non-standard random walk, if F_I ∈ S and (G(x))/(F_I(x))â†'λ,0 < λ < ∞ then,(iii) In non-standard random walk, if F_I∈ S,G ∈ S and (G(x))/(F_I(x))â†' ∞ then,Ï€(x) G(x).Theorem 1.2.2 (i) If F ∈ S^* and lim_{xâ†' ∞}(G(x))/(F_I(x)) = β [0,∞), then for each z > 0(ii) If F ∈ S^*, G ∈ L and lim_{xâ†' ∞}(G(x))/(F_I(x)) =∞ , then for each z > 0In Chapt 2, we study the maximum of non-standard random walk when it is of 5(7) kind, and get the following theorems.Theorem 2.2.1 In non-standard random walk, supposse F ∈ S(γ) and f(-γ) < 1. (i) If and G S(γ), then(ii) If 0 , then (iii) If and G S(γ), then Particularly, if G S, then Where Theorem 2.2.2 In non-standard random walk, suppose F S(γ), and f(-γ) < 1.(i) If and G S(γ), then for any z > 0,(ii) If and G S(γ), then for any z > 0,Particularly, if G S, then Where Similarly we also apply these results in risk theory to resolve some ruin problems.In Chapt 3, we give a new definition.Definition 3.1 Suppose X_i, X_j^' be independent identical distribution variables respectively, where X_j^' G(x), X_i F(x), then we call S = S₁+S₂ double random walk, where both are random walk.Now, we give several models. Model 3.1.1 In the double random walk, S₂ is dependent with S₁ having relation:...