This paper revisits the optimal capital structure model with endogenous bankruptcy, first studied by Leland (J. Finance 49:1213–1252, 1994) and Leland and Toft (J. Finance 51:987–1019, 1996). Unlike in the standard case where shareholders continuously observe the asset value and bankruptcy is executed instantaneously without delay, the information of the asset value is assumed to be updated periodically at the jump times of an independent Poisson process. Under a spectrally negative Lévy model, we obtain the optimal bankruptcy strategy and the corresponding capital structure. A series of numerical studies provide an analysis of the sensitivity, with respect to the observation frequency, of the optimal strategies, optimal leverage and credit spreads.

This paper analyses the mixture of two continuous-time Markov chains with absorption on the same state space moving at different speeds, where the mixture occurs at a random time. Variety of associated distributional properties of the Markov mixture process are discussed, for example the transition matrix, the distribution of its lifetime and the corresponding forward intensity. Identities are given explicitly in terms of the Bayesian updates of switching probability and the intensity matrices of the underlying Markov chains despite the fact that the mixture process is not Markovian. They form nonstationary functions of time and duration and have two appealing features: the ability to capture heterogeneity and path dependence when conditioning on the available information (either full or partial) about the past history of the process until current time. In particular, the unconditional lifetime distribution forms a generalized mixture of phase-type distributions. The distribution has dense and closure properties under finite convex mixtures and finite convolutions. When the underlying Markov chains move at the same speed, in which case the mixture process reduces to a simple Markov chain, the heterogeneity and path dependence are removed, and the lifetime has the usual phase-type distribution, Neuts [Neuts MF (1975) Probability distributions of phase-type. Liber Amicorum Prof. Emiritus H. Florin (University of Louvain, Belgium), 173–206]. Some numerical examples are given to illustrate the main results.

This paper discusses Parisian ruin problem under drawdown with capital injection when the underlying source of randomness of the surplus is modeled by a general Lévy insurance risk process. Capital injection is provided at the first instance the surplus drops below the drawdown level that is a pre-specified function of its current maximum. The capital is continuously injected to keep the surplus above the drawdown level until either it goes above its current maximum or a Parisian type ruin occurs, which is announced at the first time the surplus process has undergone an excursion below its current maximum for an independent exponential period of time consecutively since the most recent drawdown time. Some distributional identities concerning this excursion are presented. Firstly, we give the Parisian ruin probability and the joint Laplace transform (possibly killed at the first passage time above a fixed level for the surplus process) of the ruin time, the surplus position at ruin, and the total capital injection at ruin. Secondly, we obtain the q-potential measure of the surplus process killed at Parisian ruin. Finally, we give the expected present value of the total discounted capitals injected up to the Parisian ruin time. The results are derived using recent developments in fluctuation and excursion theory of spectrally negative Lévy process and are presented semi-explicitly in terms of the scale function of the Lévy process. Some numerical examples are given to facilitate the analysis of the impact of initial surplus and frequency of observation periods to the ruin probability and to the expected total discounted capital injection.