Transient and Stationary Analyses of the Surplus in a Risk Model

Abstract

The surplus process in a risk model is stochastically analyzed. We obtain the characteristic function of the level of the surplus at a finite time, by establishing and solving an integro-differential equation for the distribution function of the surplus. The characteristic function of the stationary distribution of the surplus is also obtained by assuming that an investment of the surplus is made to other business when the surplus reaches a sufficient level. As a consequence, we obtain the first and second moments of the surplus both at a finite time and in an infinite horizon (in the long-run).

An optimal continuous type investment policy for the surplus in a risk model

“…Cho et al (2013) introduced a risk model where the surplus process continues to move even though the level of the surplus becomes negative and an investment of the surplus is made, by a fixed amount, to other business jump-wise and instantly, if the level of the surplus reaches a given level. They obtained the characteristic functions of the transient and stationary distributions of the surplus process.…”

“…They obtained the characteristic functions of the transient and stationary distributions of the surplus process. Lim et al (2016) studied an optimal investment policy in the risk model introduced by Cho et al (2013). After assigning, to the risk model, the reward of the investment, the penalty of the surplus being short and the opportunity cost of keeping the surplus, they showed that there exists a unique amount of the surplus being invested, which minimizes the long-run average cost per unit time.…”

Stationary distribution of the surplus process in a risk model with a continuous type investment

“…However, most works, in the above, have been concentrated on the ruin probability of the surplus and its related characteristics, until Cho et al (2013) analyze some transient and stationary behaviors of the surplus process in the risk model with investments. Thereafter, Kim and Lee (2015) adopted a level crossing approach to obtain the stationary distribution of the surplus process in the risk model with dividends and reinvestments.…”

Surplus Process Perturbed by Diffusion and Subject to Two Types of Claim

“…Dickson and Willmot (2005) calculated the density function of the time to ruin by an inversion of its Laplace transform. Jeong and Lee (2010) suggested an optimal control policy for the surplus process and Cho et al (2013) obtained the transient and stationary distributions of the surplus process. Dufresne and Gerber (1991) generalized the classical surplus process by assuming that the surplus is perturbed by diffusion between the time points of occurrence of claim and studied the ruin probabilities of the surplus process.…”