The Markov Additive risk process under an Erlangized dividend barrier strategy

In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of Albrecher et al. (2011). Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang($n$) distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. (2011), it is assumed that the event of ruin is monitored continuously (Avanzi et al. (2013) and Zhang (2014)), i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases.postprin

On the expected discounted dividends in the Cramér-Lundberg risk model with more frequent ruin monitoring than dividend decisions

In this paper, we further extend the insurance risk model in Albrecher et al. (2011b), who proposed to only intervene in the compound Poisson risk process at the discrete time points ${L_k}_{k=0}^infty$ where the event of ruin is checked and dividend decisions are made. In practice, an insurance company typically balances its books (and monitors its solvency) more frequently than deciding on dividend payments. This motivates us to propose a generalization in which ruin is monitored at ${L_k}_{k=0}^infty$ whereas dividend decisions are only made at ${L_{jk}}_{k=0}^infty$ for some positive integer $j$. Assuming that the intervals between the time points ${L_k}_{k=0}^infty$ are Erlang($n$) distributed, the Erlangization technique (e.g. Asmussen et al. (2002)) allows us to model the more realistic situation with the books balanced e.g. monthly and dividend decisions made e.g. quarterly or semi-annually. Under a dividend barrier strategy with the above randomized interventions, we derive the expected discounted dividends paid until ruin. Numerical examples about dividend maximization with respect to the barrier $b$ and/or the value of $j$ are given.postprin

On the Optimal Stochastic Control of Dividend and Penalty Payments in an Insurance Company

In this thesis we consider the surplus of a non-life insurance company and assume that it follows either the classical Cramér-Lundberg model or its diffusion approximation. That is, we consider a continuous time model, where premiums are cashed at a constant rate and claims occur randomly with random sizes modelled by a compound Poisson process. In actuarial mathematics the risk of an insurance company is traditionally measured by the probability of ruin, where the time of ruin is defined as the first time when the surplus becomes negative. Using the ruin probability as a risk measure has been criticised because it is unrealistic to assume that an insurance company is ruined as soon as the surplus becomes negative. In this thesis, we assume that the insurer is not ruined although the surplus becomes negative. In order to avoid ruin, penalty payments occur, depending on the level of the surplus. For example, penalty payments occur if the insurance company needs to borrow money. In the first part of this thesis we consider the diffusion approximation to the Cramér-Lundberg model and we aim to determine a dividend strategy that maximises the difference between the expected discounted dividend and penalty payments, where penalty payments are either modelled by an exponential, linear or quadratic function. We show that the optimal strategy is a so-called barrier strategy and calculate the optimal barrier. The second part studies the analogous problem where the surplus process of an insurance company is given by a Cramér-Lundberg model. Here, similar results are obtained. In conclusion, we consider the problem where we have to determine an optimal investment and reinsurance strategy and the surplus follows the diffusion approximation. The insurance company can invest in several risky assets and reduce the insurance risk either by excess of loss or proportional reinsurance. The aim is to find a strategy which minimises the penalty payments that are necessary to avoid ruin. Various penalty functions are considered and closed form solutions are derived

Randomized observation times for the compound Poisson risk model: The discounted penalty function

In the framework of collective risk theory, we consider a compound Poisson risk model for the surplus process where the process (and hence ruin) can only be observed at random observation times. For Erlang(n) distributed inter-observation times, explicit expressions for the discounted penalty function at ruin are derived. The resulting model contains both the usual continuous-time and the discrete-time risk model as limiting cases, and can be used as an effective approximation scheme for the latter. Numerical examples are given that illustrate the effect of random observation times on various ruin-related quantities

Log-concavity of compound distributions with applications in operational and actuarial models

We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s, S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters. Copyrigh

Exit Problems for Lévy and Markov Processes with One-Sided Jumps and Related Topics

Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps)

Algorithmic Analysis of a General Class of Discrete-based Insurance Risk Models

The aim of this thesis is to develop algorithmic methods for computing particular performance measures of interest for a general class of discrete-based insurance risk models. We build upon and generalize the insurance risk models considered by Drekic and Mera (2011) and Alfa and Drekic (2007), by incorporating a threshold-based dividend system in which dividends only get paid provided some period of good financial health is sustained above a pre-specified threshold level. We employ two fundamental methods for calculating the performance measures under the more general framework. The first method adopts the matrix-analytic approach originally used by Alfa and Drekic (2007) to calculate various ruin-related probabilities of interest such as the trivariate distribution of the time of ruin, the surplus prior to ruin, and the deficit at ruin. Specifically, we begin by introducing a particular trivariate Markov process and then expressing its transition probability matrix in a block-matrix form. From this characterization, we next identify an initial probability vector for the process, from which certain important conditional probability vectors are defined. For these vectors to be computed efficiently, we derive recursive expressions for each of them. Subsequently, using these probability vectors, we derive expressions which enable the calculation of conditional ruin probabilities and, from which, their unconditional counterparts naturally follow. The second method used involves the first claim conditioning approach (i.e., condition on knowing the time the first claim occurs and its size) employed in many ruin theoretic articles including Drekic and Mera (2011). We derive expressions for the finite-ruin time based Gerber-Shiu function as well as the moments of the total dividends paid by a finite time horizon or before ruin occurs, whichever happens first. It turns out that both functions can be expressed in elegant, albeit long, recursive formulas. With the algorithmic derivations obtained from the two fundamental methods, we next focus on computational aspects of the model class by comparing six different types of models belonging to this class and providing numerical calculations for several parametric examples, highlighting the robustness and versatility of our model class. Finally, we identify several potential areas for future research and possible ways to optimize numerical calculations

Mathematical control theory and Finance

Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, running from ”pure” branches of mathematics, like geometry, to more applied areas where the objective is to find solutions to ”real life” problems, as is the case in robotics, control of industrial processes or finance. The ”high tech” character of modern business has increased the need for advanced methods. These rely heavily on mathematical techniques and seem indispensable for competitiveness of modern enterprises. It became essential for the financial analyst to possess a high level of mathematical skills. Conversely, the complex challenges posed by the problems and models relevant to finance have, for a long time, been an important source of new research topics for mathematicians. The use of techniques from stochastic optimal control constitutes a well established and important branch of mathematical finance. Up to now, other branches of control theory have found comparatively less application in financial problems. To some extent, deterministic and stochastic control theories developed as different branches of mathematics. However, there are many points of contact between them and in recent years the exchange of ideas between these fields has intensified. Some concepts from stochastic calculus (e.g., rough paths) have drawn the attention of the deterministic control theory community. Also, some ideas and tools usual in deterministic control (e.g., geometric, algebraic or functional-analytic methods) can be successfully applied to stochastic control. We strongly believe in the possibility of a fruitful collaboration between specialists of deterministic and stochastic control theory and specialists in finance, both from academic and business backgrounds. It is this kind of collaboration that the organizers of the Workshop on Mathematical Control Theory and Finance wished to foster. This volume collects a set of original papers based on plenary lectures and selected contributed talks presented at the Workshop. They cover a wide range of current research topics on the mathematics of control systems and applications to finance. They should appeal to all those who are interested in research at the junction of these three important fields as well as those who seek special topics within this scope.info:eu-repo/semantics/publishedVersio