In this paper, we study the optimal control problem for a company whose surplus process evolves as an upward jump diffusion with random return on investment. Three types of practical optimization problems faced by a company that can control its liquid reserves by paying dividends and injecting capital. In the first problem, we consider the classical dividend problem without capital injections. The second problem aims at maximizing the expected discounted dividend payments minus the expected discounted costs of capital injections over strategies with positive surplus at all times. The third problem has the same objective as the second one, but without the constraints on capital injections. Under the assumption of proportional transaction costs, we identify the value function and the optimal strategies for any distribution of gains.

In this paper, we consider a risk model in which each main claim induces a delayed claim called a by-claim. The time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. From martingale theory, an expression for the ultimate ruin probability can be derived using the Lundberg exponent of the associated nondelayed risk model. It can be shown that the Lundberg exponent of the proposed risk model is the same as that of the nondelayed one. Brownian motion approximations for ruin probabilities are also discussed.

Consider a bivariate Lévy-driven risk model in which the loss process of an insurance company and the investment return process are two independent Lévy processes. Under the assumptions that the loss process has a Lévy measure of consistent variation and the return process fulfills a certain condition, we investigate the asymptotic behavior of the finite-time ruin probability. Further, we derive two asymptotic formulas for the finite-time and infinite-time ruin probabilities in a single Lévy-driven risk model, in which the loss process is still a Lévy process, whereas the investment return process reduces to a deterministic linear function. In such a special model, we relax the loss process with jumps whose common distribution is long tailed and of dominated variation.

This paper considers the compound Markov binomial risk model proposed by Cossette et al. ( 2003 Cossette, H. , Landriault, D. and Marceau, É. 2003. Ruin probabilities in the compound Markov binomial model. Scandinavian Actuarial Journal, 2003(4): 301–323. [Taylor & Francis Online] [Google Scholar] 2004 Cossette, H. , Landriault, D. and Marceau, É. 2004. Exact expressions and upper bound for ruin probabilities in the compound Markov binomial model. Insurance: Mathematics and Economics, 34: 449–466. [Crossref], [Web of Science ®] [Google Scholar] ). Two discrete-time renewal (ordinary renewal and delayed renewal) risk processes associated with the compound Markov binomial risk model are analyzed. Based on the associated ordinary renewal process, a defective renewal equation for the conditional Gerber–Shiu expected discounted penalty function is obtained. The relationship between the conditional expected discounted penalty function in the ordinary renewal case and that in the delayed renewal case is then established. From these results, the conditional ultimate probability of ruin as well as the conditional joint distribution of the surplus just prior to ruin and the deficit at ruin are studied. Finally, it is shown that a modified version of the compound Markov binomial risk model is a special case of the discrete-time semi-Markov risk model introduced by Reinhard and Snoussi ( 2001 Reinhard, J.M. and Snoussi, M. 2001. On the distribution of the surplus prior to ruin in a discrete semi-Markov risk model. ASTIN Bulletin, 31: 255–276. [Google Scholar] 2002 Reinhard, J.M. and Snoussi, M. 2002. The severity of ruin in a discrete semi-Markov risk model. Stochastic Models, 18: 85–107. [Google Scholar] ).