On a Sparre Andersen risk model perturbed by a spectrally negative Lévy process

Asymptotic and numerical analysis of the optimal investment strategy for an insurer

Optimal investment for insurers

The multiple disorder problem consists of finding a sequence of stopping times which are as close as possible to the (unknown) times of ’disorder’ when the distribution of an observed process changes its probability characteristics. We present a formulation and solution of the multiple disorder problem for a Wiener and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial optimal switching problems to the corresponding coupled optimal stopping problems and solving the equivalent coupled free-boundary problems by means of the smooth- and continuous-fit conditions.

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

On non-negative modeling with CARMA processes

In this thesis, we present the new results found in exploring the asymptotic tail probability of the phase-type scale mixture distributions, and apply a subclass of the phase-type scale mixtures, the class of Erlangized scale mixture distributions, to the approximation of the probability of ruin.We consider the class of phase-type scale mixture distributions, which can be written as MellinnStieltjes convolution P*G of a phase-type distribution G and a nonnegative but otherwise ar-bitrary distribution P. Such a class can also be seen as the class of distributions of the product of two independent random variables: a phase-type random variable Y s G and a nonnegative random variable S s P. We call S the scaling random variable and its corresponding distribution P the scaling distribution. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction.In this thesis, particular focus is on phase-type scale mixture distributions where the scaling distribution has discrete support m such a class has been recently used in risk applications to approximate heavy-tailed distributions. We extend an existing scheme for numerically calculating the probability of ruin of a classical CramernLundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a subclass of the phase-type scale mixtures, which is also a dense class of distributions that we denominate Erlangized scale mixtures (ESM). Such a class corresponds to nonnegative and absolutely continuous distributions which can be written as a MellinnStieltjes convolution P*Gmnof a discrete nonnegative distribution P with an Erlang distribution Gmn. A distinctive feature of such a class is that it contains heavy-tailed distributions when the scaling distributions have unbounded support, and it provides sharp approximations to heavy-tailed distributions. We suggest a simple methodology for constructing a sequence of distributions having the form P*Gmnwith the purpose of approximating the integrated tail distribution of the claim sizes. Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.

This chapter complements the material of Chap. 4 in the sense that explicit identification of requisite quantities is possible with stronger parametric assumptions about either the (marginal) distribution of the claim sizes or that of the interclaim times. Convenient parametric assumptions involve Erlang (or Coxian) distributions, due to the generality and mathematical tractability inherent in the use of such models. Sect. 5.1 directly generalizes the classical Poisson model of Chap. 3 by essentially assuming a (marginal) Erlang form for the interclaim times. A Lagrange polynomial approach together with Laplace transform arguments is used to explicitly identify all components of the Gerber-Shiu function. The independent model with an arbitrary interclaim time distribution and exponential claim sizes is then considered in Sect. 5.2. The conditioning argument of Sect. 5.2 is then extended in Sect. 5.3 to a model where the (marginal) claim size distribution is assumed to be of Coxian form. In particular, identification of the Gerber-Shiu function is shown to be possible for special choices of the penalty function, and more generally for an arbitrary penalty function of the deficit. Finally, the Coxian assumption of Sect. 5.3 is replaced by a (possibly infinite) mixture of Erlang distributions, resulting in tractable results for ruin probabilities, the deficit at ruin, and related quantities.

This work deals with an optimal dividend payment problem for a piecewise-deterministic compound Poisson insurance risk model. The objective is to maximize the expected discounted dividend payout up to the time of ruin. When the dividend payment rate is restricted, the value function is shown to be a solution of the corresponding Hamilton-Jacobi-Bellman equation, which in turn leads to a tractable methodology to find an optimal threshold dividend payment policy. For the case of unrestricted payment rate, the value function and an optimal barrier strategy are determined explicitly with exponential claim size distributions. A comparison of two examples is provided to illustrate the main results.

Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates

A stochastic model describing the impact of daily rainfall depth distribution on the soil water balance

In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distribution -- Pareto distribution. The ruin probability for finite-horizon 5 and 10 was determined from recurrence equations. Moreover for exponential distribution the upper bound of ruin probability by Lundberg adjustment coefficient is given. For Pareto distribution the adjustment coefficient does not exist, hence an asymptotic approximation of the ruin probability if an initial capital tends to infinity is given. Obtained numerical results are given as tables and they are illustrated as graphs.

In the classical optimal execution problem, the basic assumption of underlying asset price is Arithmetic Brownian Motion (ABM) or Geometric Brownian Motion (GBM). However, many empirical researches show that the return distribution of assets may have heavy tails than those of normal distribution. The uncertain information impact on financial market may be considered as one of the main reasons for heavy tails of return distribution. To introduce this information impact, our paper proposes a Jump Diffusion model for optimal execution problem. The jumps in our model are described by the compound Poisson process where random jump amplitude depicts the information impact on price process. In particular, the model is simple enough to derive closed-form strategies under risk neutral and Mean-VaR criterion. Simulation analysis of the model is also presented.

Methods of analysis are described which determine the two parameters in an assumed exponential hailstone size distribution from the Doppler radar spectrum. When applied to the data of Battan and Theiss it is found that these methods yield computed exponential distributions and experimental size spectra which are in very good agreement, and that the various methods of analysis are in substantive agreement when the maximum hailstone diameter is at least 2 cm.

The present chapter derives the reliability functions and hazard functions, when the threshold for deterioration is an increasing function of time. Four cases are considered. Case I: The threshold is a step function with K jumps at known points. Case II: The threshold is a step function with K jumps, where the location of jumps are random, following a renewal process. Case III: The controlled case. The jumps occur after the first crossing of a control limit. Case IV: The threshold increases linearly. The theory is illustrated in all four cases with a Markovian deterioration process, i.e., a compound Poisson process with i.i.d. jumps following an exponential distribution.

In this paper we will build a bank model using Poisson measures and Jackson queueing networks. We take into account the relationship between the Poisson and the exponential distributions, and we consider for each credit/deposit type a node where shocks are modeled as the compound Poisson processes. The transmissions of the shocks are modeled as moving between nodes in Jackson queueing networks, the external shocks are modeled as external arrivals, and the absorption of shocks as departures from the network.