Classical Surplus Process Research Articles

Time-inconsistent view on a dividend problem with penalty

We consider the dividend maximization problem including a ruin penalty in a diffusion environment. The additional penalty term is motivated by a constraint on dividend strategies. Intentionally, we use different discount rates for the dividends and the penalty, which causes time-inconsistency. This allows to study different types of constraints. For the diffusion approximation of the classical surplus process we derive an explicit equilibrium dividend strategy and the associated value function. Inspired by duality arguments, we can identify a particular equilibrium strategy such that for a given initial surplus the imposed constraint is fulfilled. Furthermore, we illustrate our findings with a numerical example.

Quantum Computing in Insurance Capital Modelling

This paper proposes a quantum computing approach for insurance capital modelling. Using an open-source software development kit, Qiskit, an algorithm for working on a superconducting type IBM quantum computer is developed and implemented to predict the capital of insurance companies in the classical surplus process. With the fundamental properties of quantum mechanics, Dirac notation and Feynman’s path calculation are shown. Furthermore, custom quantum insurance premium and claim gates are investigated in order to build a quantum circuit with respect to initial reserve, premium and claim amounts. Some numerical results are presented and discussed at the end of the paper.

Ruin probability via Quantum Mechanics Approach

The finite time ruin probability in the classical surplus process setup with additional capital injections and withdrawals is investigated via the Quantum Mechanics Approach. The results are compared with the Picard–Lefevre Appell Polynomial approach and the traditional Markov Chain approach. In addition, several optimization problems in the insurance market are numerically solved by applying the Quantum Mechanics Approach.

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton–Jacobi–Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.

Some Optimal Dividend Problems for a Surplus Process with Stochastic Interest

In this paper, some results on the dividend payments prior to ruin in the classical surplus process with stochastic interest are derived. An integro-differential equation with a boundary conditions satisfied by the expected present value of dividend payments is derived and solved. Furthermore, closed-form expressions for exponential claims are given.

Constant Dividend Barrier in a Risk Model with a Generalized Farlie-Gumbel-Morgenstern Copula

In this paper, we consider the classical surplus process with a constant dividend barrier and a dependence structure between the claim amounts and the inter-claim times. We derive an integro-differential equation with boundary conditions. Its solution is expressed as the Gerber-Shiu discounted penalty function in the absence of a dividend barrier plus a linear combination of a finite number of linearly independent particular solutions to the associated homogeneous integro-differential equation. Finally, we obtain an explicit solution when the claim amounts are exponentially distributed and we investigate the effects of dependence on ruin quantities.

On barrier strategy dividends with Parisian implementation delay for classical surplus processes

In this paper, we apply a single barrier strategy to optimise dividend payments in the situation where there is a time lag d > 0 between decision and implementation. Using a classical surplus process with exponentially distributed jumps, we obtain the optimal barrier b ∗ which maximises the expected present value of dividends.

Optimal Dynamic Reinsurance

We consider a classical surplus process where the insurer can choose a different level of reinsurance at the start of each year. We assume the insurer’s objective is to minimise the probability of ruin up to some given time horizon, either in discrete or continuous time. We develop formulae for ruin probabilities under the optimal reinsurance strategy, i.e. the optimal retention each year as the surplus changes and the period until the time horizon shortens. For our compound Poisson process, it is not feasible to evaluate these formulae, and hence determine the optimal strategies, in any but the simplest cases. We show how we can determine the optimal strategies by approximating the (compound Poisson) aggregate claims distributions by translated gamma distributions, and, alternatively, by approximating the compound Poisson process by a translated gamma process.

Optimal Dynamic Reinsurance

We consider a classical surplus process where the insurer can choose a different level of reinsurance at the start of each year. We assume the insurer’s objective is to minimise the probability of ruin up to some given time horizon, either in discrete or continuous time. We develop formulae for ruin probabilities under the optimal reinsurance strategy, i.e. the optimal retention each year as the surplus changes and the period until the time horizon shortens. For our compound Poisson process, it is not feasible to evaluate these formulae, and hence determine the optimal strategies, in any but the simplest cases. We show how we can determine the optimal strategies by approximating the (compound Poisson) aggregate claims distributions by translated gamma distributions, and, alternatively, by approximating the compound Poisson process by a translated gamma process.

Optimal Dividends Under a Ruin Probability Constraint

ABSTRACTWe consider a classical surplus process modified by the payment of dividends when the insurer's surplus exceeds a threshold. We use a probabilistic argument to obtain general expressions for the expected present value of dividend payments, and show how these expressions can be applied for certain individual claim amount distributions. We then consider the question of maximising the expected present value of dividend payments subject to a constraint on the insurer's ruin probability.

The Gerber–Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier

In this paper, we consider the classical surplus process with interest and a constant dividend barrier. Under constant interest, we derive an integro-differential equation for the Gerber–Shiu expected discounted penalty function. Following an idea of Lin, Willmot and Drekic [Lin, X.S., Willmot, G.E., Drekic, S., 2003. The classical risk model with a constant dividend barrier: Analysis of the Gerber–Shiu discounted penalty function. Insurance: Math. Econom. 33, 551–566], we obtain the solution to the integro-differential equation which is in the form of an infinite series. In some special cases with exponential claims, we are able to find closed-form expressions for the Gerber–Shiu expected discounted penalty function. Finally, we extend the integro-differential equation to the case where the surplus is invested in an investment portfolio with stochastic return on investments.

Flow of Dividends under a Constant Force of Interest

This study addresses the issue of maximization of dividends of an insurer whose portfolio is exposed to insurance risk. The insurance risk arises from the classical surplus process commonly known as the Cramer-Lundberg model in the insurance literature. To enhance his financial base, the insurer invests in a risk free asset whose price dynamics are governed by a constant force of interest. We derive a linear Volterra integral equation of the second kind and apply an order four Block-by-block method of Paulsen et al.[1] in conjunction with the Simpson rule to solve the Volterra integral equations for each chosen barrier thus generating corresponding dividend value functions. We have obtained the optimal barrier that maximizes the dividends. In the absence of the financial world, the analytical solution has been used to assess the accuracy of our results.

Numerical Ultimate Ruin Probabilities under Interest Force

This work addresses the issue of ruin of an insurer whose portfolio is exposed to insurance risk arising from the classical surplus process. Availability of a positive interest rate in the financial world forces the insurer to invest into a risk free asset. We derive a linear Volterra integral equation of the second kind and apply an order four Block-by-block method in conjuction with the Simpson rule to solve the Volterra equation for ultimate ruin. This probability is arrived at by taking a linear combination of some two solutions to the Volterra integral equation. The several numerical examples given show that our results are excellent and reliable.

How many claims does it take to get ruined and recovered?

We consider in the classical surplus process the number of claims occurring up to ruin, by a different method presented by Stanford and Stroiński [Astin Bulletin 24 (2) (1994) 235]. We consider the computation of Laplace transforms (LTs) which can allow the computation of the probability function. Formulae presented are general. The method uses the computation of the probability function of the number of claims during a negative excursion of the surplus process, in case it gets ruined. When initial surplus is zero this probability function allows us to completely define the recursion for the transform above. This uses the fact that in this particular case, conditional time to ruin has the same distribution as the time to recovery, given that ruin occurs. We consider also the computation of moments of the number of claims during recovery time, which with initial surplus zero allows us to compute the moments of the number of claims up to ruin. We end this work by giving some insight on the shapes of the two types of probability functions involved.

On the expected discounted penalty function at ruin of a surplus process with interest

In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.

Ruin probabilities with compounding assets

We consider a classical surplus process modified by the action of a constant force of interest. We derive recursive algorithms for the calculation of the probability of ruin in finite time. We also discuss the numerical evaluation of the probability of ultimate ruin using methods proposed by De Vylder [De Vylder, F., 1996. Advanced Risk Theory. Editions de l’Université de Bruxelles.] and Sundt and Teugels [Sundt, B., Teugels, J.L., 1995. Insurance: Mathematics and Economics 16, 7–22]. Finally, we consider the problem of recovery from ruin.

Reinsurance and ruin

We study the effect of reinsurance on the probability of ultimate ruin in the classical surplus process and consider a retention level as optimal if it minimises the ruin probability. We show that optimal retention levels can be found when the reinsurer's premium loading depends on the retention level. We also show that when the aggregate claims process is approximated by a translated Gamma process, very good approximations to both optimal retention levels and ruin probabilities can be obtained. Finally, we discuss the effect of reinsurance on the probability of ruin in finite time.

Ruin problems and dual events

Dickson (1992) uses dual events to explain results relating to the distribution of the surplus immediately prior to ruin in the classical surplus process. In this paper we show that dual events can be used to explain other results in ruin theory. In particular we prove and explain the relationship between the density of the surplus immediately prior to ruin, and the joint density of the surplus immediately prior to ruin and the severity of ruin.

Inversed martingales in risk theory

The theory of inversed martingales is used in order to prove a generalization of a result of H. Cramér on the probability of non-ruin for a classical surplus process if the initial reserve is positive.