Bankruptcy Theory, Discrete-time Model

This dissertation mainly discussed the problem of some probability in the discrete model. The results are obtained by caculating the expected discounted penalty. The expected discounted penalty is a function of initial captal and depend on r.v. such as the ruin time, the surplus immediately befor ruin. We found a renwal function of the expected discounted penalty from which we derive the recursion solution of f(u;x),g(u;y) and Ψ(u). By taking transform to the renewal function we obtain the transform solution. The proper discrete renewal equation is obtained by multiplying the defective discrete renewal equation of Ψ(u;ω) with R~u. When the initial surplus is large enough, we use the limit theorem of discrete renewal equotion to obtain the asympototic solution of f(u;x),g(u; y] and ip(u) respectly,given the adjustment coefficient exist.Risk model is divided to continual model and discrete model according to the means of premium income. Classical risk model is Poisson risk model. The study about it is perfect. Feller renewal proof and Gerber martingale skill give a strict and concise proof to the result intheclassical risk model, we discuss in the compound binomial model by using the two methods.In chapter one. we introduce Poisson model and some important results.(1)(2)Lundberg inequality(3)Lundberg - Cramer approximation:there exist positive C, such thatthat isWe also introduce discrete risk model, discrete renewal equation and a limit theorem. We give the definition and relevant theorem of martingale in section 4. Adjustment coefficient plays a key role in small number claim. Finally we give the definition and some equation about it.In chapter two. We obtain the defective discrete renewal equation of expected discounted penalty.We now apply control convergence theorem to the equation and take limit. Then yield:Explicit solution are obtained for f(0;x).g(0: y) and Ψ(0) respectly by substituting w with different value, from the explicit solution of Ψ(u, ω).Rearranging the defective discrete renewal equation of Ψ(u;ω). then we have the recursive solution for f(u; x),g(u; y) and Ψ(u} respectly. Mutiplying the defectivediscrete renewal equation of Ψ(u;ω) with R u. then we have the transform solution for f(u:x), g(u:y) and Ψ(u) respectly.In chapter three. We obtain the proper discrete renewal equation by multiplying the defective discrete renewal equation for fy(u:u;) with R~u. When the initial surplus is large enough, we use the limit theorem of discrete renewal equation to obtain the asympototic solution for Ψ(u:u;), according to an equation repect to adjustment coefficient.whereSubstituting u with different value, yield:/(u; x) ~ CXR u,u â†'∞. whereThe asympototic solution for g(u;y): g(u;y) ~ CyR u,u Ψ∞. whereThe asympototic solution for Ψ(u): Ψ(u) simC*R-u,u â†' oo. whereIn chapter four. We proved {vnR u-n} is a positive martinggale, according to the independ presumption in in the compound binomial model and the equation which adjustment coefficient satisfies. Applying option sampal theorem and control convergence theorem to {vnR-u-n}. Then obtain a equation for Ψ(u):and Lundberg inequalityWe have a aplliation to 0 < p < l,one solution of r.Finally we present the work that we will do.