Surplus Process Research Articles

Fluctuation identities for Omega-killed spectrally negative Markov additive processes and dividend problem

AbstractIn this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of new generalizations of classical scale matrices for MAPs. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the Omega model, where bankruptcy takes place at rate $\omega(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider Markov-modulated Brownian motion (MMBM) as a special case and present analytical and numerical results for a particular choice of piecewise intensity function $\omega(\cdot,\cdot)$ .

Optimality of Impulse Control Problem in Refracted L\xe9vy Model with Parisian Ruin and Transaction Costs

Here, we investigate an optimal dividend problem with transaction costs, in which the surplus process is modeled by a refracted Lévy process and the ruin time is considered with Parisian delay. The presence of the transaction costs implies that the impulse control problem needs to be considered as a control strategy in such a model. An impulse policy which involves reducing the reserves to some fixed level, whenever they are above another, is an important strategy for the impulse control problem. Therefore, we provide sufficient conditions under which the above described impulse policy is optimal. Furthermore, we provide new analytical formulae for the Parisian refracted q-scale functions in the case of the linear Brownian motion and the Crámer–Lundberg process with exponential claims. Using these formulae, we show that, for these models, there exists a unique policy, which is optimal for the impulse control problem. Numerical examples are also provided.

Optimal Investment Policy for Insurers under the Constant Elasticity of Variance Model with a Correlated Random Risk Process

This paper investigates the optimal portfolio choice problem for a large insurer with negative exponential utility over terminal wealth under the constant elasticity of variance (CEV) model. The surplus process is assumed to follow a diffusion approximation model with the Brownian motion in which is correlated with that driving the price of the risky asset. We first derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation and then obtain explicit solutions to the value function as well as the optimal control by applying a variable change technique and the Feynman–Kac formula. Finally, we discuss the economic implications of the optimal policy.

On the cumulative Parisian ruin of multi-dimensional Brownian motion risk models

Consider a multi-dimensional Brownian motion which models the surplus processes of multiple lines of business of an insurance company. Our main result gives exact asymptotics for the cumulative Parisian ruin probability as the initial capital tends to infinity. An asymptotic distribution for the conditional cumulative Parisian ruin time is also derived. The obtained results on the cumulative Parisian ruin can be seen as generalisations of some of the results derived in Dbicki et al. [(2018). Extremal behavior of hitting a cone by correlated Brownian motion with drift. Stochastic Processes and Their Applications 128, 4171–4206]. As a particular interesting case, the two-dimensional Brownian motion risk model is discussed in detail.

Liquidation risk in insurance under contemporary regulatory frameworks

In traditional research in insurance and finance, a firm is subject to immediate liquidation when its asset value process drops to an absorbing low barrier. This treatment greatly simplifies research but largely ignores the complexity of the liquidation procedure in the real world. In banking and finance, many researchers have taken into account the features of Chapter 7 liquidation and Chapter 11 reorganization of the U.S. Bankruptcy Code. Also, there have been similar discussions in insurance regulation, but few works have been done to achieve a quantitative understanding of the liquidation risk in insurance under contemporary regulatory frameworks. In this paper, we quantify the rehabilitation proceeding in insurance, which is akin to Chapter 11 reorganization of the U.S. Bankruptcy Code, and we conduct a probabilistic analysis of the liquidation risk of an insurance company having the option of rehabilitation. In doing so, we construct a three-barrier model to describe the solvent and insolvent states in which the surplus process follows different time-homogeneous diffusions. We derive analytical expressions for the liquidation probability and the Laplace transform of the liquidation time with a fixed grace period and then extend the study to the case with independent exponentially distributed grace periods. If further restricted to the constant elasticity of variance (CEV) framework, the obtained formulas become completely explicit.

How long does the surplus stay close to its historical high?

In this paper we find the Laplace transforms of the weighted occupation times for a spectrally negative Lévy surplus process to spend below its running maximum up to the first exit times. The results are expressed in terms of generalized scale functions. For step weight functions, the Laplace transforms can be further expressed in terms of scale functions.

A Lévy Risk Model with Ratcheting Dividend Strategy and Historic High-Related Stopping

This paper focuses on the De Finetti’s dividend problem for the spectrally negative Lévy risk process, where the dividend is deducted from the surplus process according to the racheting dividend strategy which was firstly introduced in Albrecher et al. (2018). A major feature of the racheting strategy lies in which the dividend rate never decreases. Unlike the conventional studies, the closed form expression for the expected, accumulated, and discounted dividend payments until the draw-down time (rather than the ruin time) is obtained in terms of the scale functions corresponding to the underlying Lévy process. The optimal barrier for the ratcheting strategy is also studied, where the dividend rate can be increased. Finally, two special cases, where the scale functions are explicitly known, i.e., the Brownian motion with drift and the compound Poisson model, are considered to illustrate the main result.

Ruin probabilities for the phase-type dual model perturbed by diffusion

In risk theory, ruin probabilities and dividend strategies have drawn lots of attentions. The dual risk model assumes that the surplus process decreases at a constant rate over time and gains by means of random jumps at random times, which is in accordance with real-life scenes such as life insurance. The dual model is widely used in describing security portfolio, pension funds, profits of enterprises whose income-generating depends on inventions or found, etc. In this paper, we consider a phase-type dual model perturbed by diffusion, whose inter-claim times are continuously distributed in phase-type distribution. We derive integro-differential equations, boundary conditions for ruin probability and obtain an explicit expression of the ruin probability when the phase-type distribution degenerates into the generalized Erlang (n) distribution. Finally, we obtain the integro-differential equations for the expected discounted dividend function under dividend payment with a threshold strategy.

Optimal mean-variance reinsurance in a financial market with stochastic rate of return

In this paper, we investigate the optimal investment and reinsurance strategies for a mean-variance insurer when the surplus process is represented by a Cramer-Lundberg model. It is assumed that the instantaneous rate of investment return is stochastic and follows an Ornstein-Uhlenbeck (OU) process, which could describe the features of bull and bear markets. To solve the mean-variance optimization problem, we adopt a backward stochastic differential equation (BSDE) approach and derive explicit expressions for both the efficient strategy and efficient frontier. Finally, numerical examples are presented to illustrate our results.

On the dual risk model with diffusion under a mixed dividend strategy

Inspired by the work of Zhang and Han (2017), this paper investigates a dual model with diffusion where dividends are paid under a mixed strategy. This strategy is composed of two parts: dividends will be paid continuously at a fixed rate α > 0 as long as the surplus process is above a fixed threshold level b > 0; for a pre-specified sequence of strictly increasing periodic dividend decision times {Zj}j ≥ 1, whenever the surplus level observed at Zj is above b, the excess value will also be paid out as dividend. In addition, ruin is declared when the observed surplus equals to 0 for the first time. The integro-differential equations satisfied by the expected present value of dividends paid up to ruin (i.e., V(x; b)) and the Laplace transform of the ruin time (i.e., Φ(x; b)) are derived. The solutions of V and Φ are constructed by the method of inverse Laplace transform and through some auxiliary functions. Finally, several numerical examples are provided to illustrate our results.

Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments

In this paper, we study the optimal dividend and capital injection problem with the penalty payment at ruin. The dividend strategy is assumed to be restricted to a small class of absolutely continuous strategies with bounded dividend density. By considering the surplus process killed at the time of ruin, we transform the problem to a combined stochastic and impulse control one up to ruin with a free boundary at zero. We illustrate the theoretical verifications for different types of capital injection strategies comparing to the conventional results given in the literature, where the capital injections are made before the time of ruin. Under the assumption of restricted dividend density, the value function is proved as the unique increasing, bounded, Lipschitz continuous and upper semi-continuous at zero viscosity solution to the corresponding quasi-variational Hamilton–Jacobi–Bellman (HJB) equation. The uniqueness of such class of viscosity solutions is shown by considering its boundary condition at infinity. The optimality of a specific band-type strategy is proved for the case when the premium rate is (i) greater than or (ii) less than the ceiling dividend rate respectively. Some numerical examples are presented under the exponential and gamma claim size assumptions.

Optimal reinsurance and investment in danger‐zone and safe‐region

SummaryThis paper studies the optimal reinsurance‐investment problems for an insurance company where the claim process follows a Brownian motion with drift. It turns out that there is a region where the probability of drawdown, namely, the probability that the value of the insurer's surplus process reaches some fixed fractional value of its maximum value to date is positive. Then in the complementary region, drawdown can be avoided with certainty. For this reason, we call the former region the “danger‐zone” and “safe‐region” for the latter. In the danger‐zone, we consider the problem of minimizing the probability of drawdown; and in the safe‐region, we turn our attention to the optimization problem of minimizing the expected time to reach a given capital level. Using the technique of stochastic control theory and the corresponding Hamilton‐Jacobi‐Bellman equation, explicit expressions of the optimal reinsurance‐investment strategies and the associated value functions are derived for the two optimization problems. Moreover, we provide several detailed comparisons to investigate the impact of some important parameters on the optimal strategies and illustrate the observation from behavior finance of view.

Generalized expected discounted penalty function at general drawdown for Lévy risk processes

This paper considers an insurance surplus process modeled by a spectrally negative Lévy process. Instead of the time of ruin in the traditional setting, we apply the time of drawdown as the risk indicator in this paper. We study the joint distribution of the time of drawdown, the running maximum at drawdown, the last minimum before drawdown, the surplus before drawdown and the surplus at drawdown (may not be deficit in this case), which generalizes the known results on the classical expected discounted penalty function in Gerber and Shiu (1998). The results have semi-explicit expressions in terms of the q-scale functions and the Lévy measure associated with the Lévy process. As applications, the obtained result is applied to recover results in the literature and to obtain new results for the Gerber–Shiu function at ruin for risk processes embedded with a loss-carry-forward taxation system or a barrier dividend strategy. Moreover, numerical examples are provided to illustrate the results.

Optimal Ratcheting of Dividends in Insurance

We address a long-standing open problem in risk theory, namely finding the optimal strategy to pay out dividends from an insurance surplus process if the dividends are paid according to a dividend ...

On the Gerber-Shiu function of a MAP risk model with possible delayed Phase-type By-claims

This paper introduces the delayed claim settlement feature in a MAP risk model. Surplus process of the model incorporates two kinds of phase-type claims, the main claim and the by-claim. The delay ...

Uniform asymptotics for the finite-time ruin probability of a generalized bidimensional risk model with Brownian perturbations

This paper considers a generalized bidimensional continuous-time risk model with heavy-tailed claims and Brownian perturbations. In this model, the claim sizes from different lines of business are tail asymptotically independent, while the claim-number processes from different lines of business are arbitrarily dependent. Under some moment conditions on the claim-number processes, an asymptotic formula is established for the finite-time ruin probability defined as the probability that the aggregate surplus process goes below zero over the time horizon . The asymptotics holds uniformly for t in some kind of interval. The moment conditions on the claim-number processes are automatically satisfied when the claim-number processes are renewal counting processes. The obtained results confirm that the Brownian perturbations have no influence on the asymptotics of the ruin probability under consideration.

Weak limits of random coefficient autoregressive processes and their application in ruin theory

We prove that a large class of discrete-time insurance surplus processes converge weakly to a generalized Ornstein–Uhlenbeck process, under a suitable re-normalization and when the time-step goes to 0. Motivated by ruin theory, we use this result to obtain approximations for the moments, the ultimate ruin probability and the discounted penalty function of the discrete-time process.

Optimal deterministic reinsurance and investment for an insurer under mean–variance criterion

This article is devoted to the study of a mean–variance problem for an insurer with deterministic reinsurance and investment strategy. The surplus process of the insurer and the financial risky asset process are described by general jump diffusion processes with random parameters. We use Malliavin calculus to obtain sufficient and necessary conditions for optimal strategy to satisfy. A particular case is discussed in which explicit expressions for optimal strategy can be derived.

Proportional reinsurance and investment in multiple risky assets under borrowing constraint

ABSTRACTIn this paper, we consider the ruin probability minimization of an insurance company that buys proportional reinsurance and invests in markets where borrowing is constrained. We use a diffusion approximation model for the surplus process of this company and assume that the company invests its surplus into a riskless and multiple risk assets that are modeled as geometric Brownian motions. To find the results, we introduce an auxiliary market parametrized with fictitious processes to relax the borrowing constraint and apply the techniques of stochastic optimal control. In this way, we find the optimal proportional reinsurance and investment strategy of an insurance company investing into multiple risky assets to minimize its ruin probability under the borrowing constraint. Furthermore, from our solutions, we show how our results connect to economic survival analysis and how investment and reinsurance strategies are related.

Time-consistent reinsurance and investment strategy combining quota-share and excess of loss for mean-variance insurers with jump-diffusion price process

This article investigates an optimal time-consistent reinsurance and investment problem under the mean-variance criterion for an insurer whose surplus process is governed by a compound Poisson risk model. The insurer is allowed to purchase combining quota-share and excess of loss reinsurance for claims and to invest in financial market to increase his wealth. The financial market consists of a risk-free asset and a risky asset whose price process follows a jump-diffusion process. The insurer wants to find an optimal time-consistent reinsurance-investment strategy so as to maximize the expected terminal surplus while minimizing the variance of the terminal surplus. By using the stochastic control technique, we obtain the explicit solutions for the optimal reinsurance-investment strategy and the corresponding optimal value function. Finally, numerical experiments are carried out to illustrate the effects of model parameters on the optimal time-consistent reinsurance-investment strategy, and to compare the pure excess of loss reinsurance with the pure quota-share reinsurance through analyzing the optimal value function.