Bankruptcy Probability Of The Discrete Risk Model

Based on the classical risk model, the paper discusses several discrete-time risk models. These models assume that the amounts of premium in everyperiods are random and the claim sequence is a binomial process or negativebinomial process. Then we obtain their properties of surplus and the formulaof the ruin probability with two kinds of methods. we also obtain Lundberginequality of ruin probability and the distribution of the surplus immediatelybefore ruin and the time in the red.The thesis is divided into four chapters according to contents:In Chapter 1, in this chapter, the history and developing process of risktheory are discussed. Then the conclusion of this paper is given.In Chapter 2, we consider the discrete binomial risk model:We get its properties and ruin probability. The properties are as follows: theprocess is with stationary and independent increments,E[U(n)] = u + nμ_X-npμ_Y, (?)U(n) = +∞,E[e^-rU(n)]=e^-ru(M_X(-r))ⁿ(pM_Y(r)+q)ⁿ,the equation M_X{-r)(pM_Y{r) + q) = 1 has a solution r=R and we call thissolution adjustment coefficient. Then we get the theorem of(?)(u)=e^-Ru/E[exp(-RU(T))|T<∞]and Lundberg inequality(?)(u)-Ru In Chapter 3, we continue to consider the discrete binomial risk modeland get the distribution of the surplus immediately before ruin :其中The distribution of the time in the red is gotten:The integration equation of ruin probability is given:其中(?)₁(u)=∫₀^∞(pF_X(y-u)+qF_X(-u))dF_Y(y).In Chapter 4, we consider the discrete negative binomial risk model:Then we get its properties and ruin probability. The properties are as follows:the process is with stationary and independent increments, E[U(n)] = u +nμ_X-nq/pμ_Y,lim_{n→∞}U(n) =+∞, the equation M_X(-r)p/1-qM_Y(r)= 1 has asolution r = R and we call this solution as adjustment coefficient. Then weget the theorem of(?)(u)=e^-Ru/E[exp(-RU(T))|T<∞ and Lundberg inequality(?) (u) < e^-Ru.