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In this paper, we are interested in the optimal investment and reinsurance strategies of an insurer who wishes to maximize the expected power utility of its terminal wealth on finite time horizon. We are also interested in the problem of maximizing the growth rate of expected power utility per unit time on the infinite time horizon. The risk process of the insurer is described by an approximation of the classical Cramér–Lundberg process. The insurer invests in a market consisting of a bank account and multiple risky assets. The mean returns of the risky assets depend linearly on economic factors that are formulated as the solutions of linear stochastic differential equations. With this setting, Hamilton–Jacobi–Bellman equations that are derived via a dynamic programming approach have explicit solution obtained by solving a matrix Riccati equation. Hence, the optimal investment and reinsurance strategies can be constructed explicitly. Finally, we present some numerical results related to properties of our optimal strategy and the ruin probability using the optimal strategy. [ABSTRACT FROM AUTHOR]

In this paper, we define a compound generalized fractional counting process (CGFCP) which is a generalization of the compound versions of several well-known fractional counting processes. We obtain its mean, variance, and the fractional differential equation governing the probability law. Motivated by Kumar et al. (Math Financ Econ 14:43-65, 2020), we introduce a fractional risk process by considering CGFCP as the claim process and call it generalized fractional risk process (GFRP). We study the martingale property of the GFRP and show that GFRP and the associated increment process exhibit the long-range dependence (LRD) and the short-range dependence (SRD) property, respectively. We also define an alternative to GFRP, namely alternative generalized fractional risk process (AGFRP) which is premium wise different from the GFRP. Finally, the asymptotic structure of the ruin probability for the AGFRP is established in case of light-tailed and heavy-tailed claim sizes. [ABSTRACT FROM AUTHOR]

This study aims to improve the predictive performance for the event time through the machine learning model and find informative variables in the time-to-event data, simultaneously. To address this issue, after regarding the time-to-event data as the dichotomized counting processes data for predicting survival time, we consider the time-dependent support vector machine (SVM) framework for the dichotomized counting process data, where the decision function in this framework consists of the time-independent risk score and time-dependent intercept. Also, we consider the empirical partial derivative of the risk score function with respect to each marginal predictor as the indicator for the important predictor. Through this approach, it is possible to predict survival time and find variables that affect on the survival time at the same time. Simulation studies were conducted to confirm the performance of the model, and real data analysis was conducted by predicting the survival time of the lung cancer after the diagnosis and selecting genes associate with lung cancer through human gene data. [ABSTRACT FROM AUTHOR]

In this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the optimal investment and reinsurance strategies of an insurer with delay under the $ 4/2 $ stochastic volatility model. Indeed, the objective of the insurer is to maximize the expected power utility of the terminal wealth and the average wealth on finite time horizon. The other objective is to maximize the growth rate of expected power utility per unit time on infinite time horizon. The wealth of the insurer is described by an approximation of the classical Cramér–Lundberg process. Then, these problems can be formulated as stochastic control problems with delay. A pair of forward-backward stochastic differential equations that are derived via the stochastic maximum principle has an explicit solution obtained by solving a Riccati differential equation. So, the optimal strategy of the finite time horizon problem can be constructed explicitly. And, by investigating asymptotics of the Riccati equation, the infinite time horizon problem can be solved explicitly. Finally, we present some numerical results to illustrate our model, optimal strategies and sensitivities of some parameters. [ABSTRACT FROM AUTHOR]

Introduction Insurance is a means of protection from random and untoward events, which can lead to significant financial losses. It is a method of risk management, that is used in order to hedge against a contingent loss risk. Risk theory came into being and was developed in order to provide a basis for the existence and significance of the insurance system. One of the most fundamental problems in risk theory is the ruin problem. It analyzes the behavior of a stochastic process, that represents the evolution of the capital of an insurance company. The objective is to estimate the ruin probability, so that the surplus at some time becomes negative. From an insurers point of view, the surplus can be defined as initial surplus + premium income-claim payment. The ruin probabilities of insurance risk models have been a hot topic of risk theory and actuarial mathematics. The study on the asymptotic estimates for ruin probabilities in insurance risk models with constant force of interest has achieved fruitful results. In traditional investigation of insurance risk models, interest rate is assumed to be deterministic. Material and Methods In this article, for the collective risk model of an insurance company with a fixed premium and a compound Poisson process in a period of time, regardless of the statistical distribution of the loss sizes, a formula for the probability of bankruptcy in finite time using the probability of offer transfer and Then, the matrix form of the proposed formula was rewritten by the transition matrix and the generator matrix of the continuous-time Markov process. Results and discussion Assuming that the damage sizes have exponential, normal and Pareto distributions, many paths of the collective risk process were simulated and in addition to calculating the finite time bankruptcy probabilities, unbiased confidence intervals with 95% confidence coefficient were estimated as well as bankruptcy probabilities. We calculated for the transfer and generator matrices of the Markov process. Conclusion In the process of collective risk for all mentioned distributions, it was observed that with the increase of initial capital, the probability of bankruptcy decreases. Also, for Pareto's heavy tail distribution, the probability of bankruptcy is higher than the other two distributions. [ABSTRACT FROM AUTHOR]

We investigate the problem of minimizing ruin probability by joint decisions of excess-of-loss reinsurance and investment in a financial market. The insurer's reserve is modeled by a diffusion process and may be invested in a financial market consisting of a risky asset with the price process following the geometric Brownian motion and a risk-free asset with a fixed return rate. Borrowing is allowed, but with an interest rate higher than the interest rate of the risk-free investment. Meanwhile, an excess-of-loss reinsurance may be purchased to alleviate the risk of ruin. We apply stochastic control theory and find the optimal strategy of joint reinsurance and investment decisions, and derive the closed form expression of the minimum ruin probability function. Results are illustrated numerically. [ABSTRACT FROM AUTHOR]

In this paper, we study a Lévy risk process consisting of Brownian component together with premiums and claims that are phase type with many phases. Our aim is to approximate the probability of ruin without touching an upper barrier a. In line with this, the study demonstrates that the described Lévy risk process can essentially be replaced with a simpler risk process in which both premiums and claims are phase-type with just few phases. [ABSTRACT FROM AUTHOR]

We find the asymptotics of the value function maximizing the expected utility of discounted dividend payments of an insurance company whose reserves are modeled as a classical Cramér risk process, with exponentially distributed claims, when the initial reserves tend to infinity. We focus on the power and logarithmic utility functions. We also perform some numerical analysis. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The problem of designing an optimal insurance strategy in a modification of the risk process with discrete time is investigated. This model introduces stage-by-stage probabilistic constraints (Value-at-Risk (VaR) constraints) on the insurer's capital increments during each stage. Also, the set of admissible insurances is determined by a safety level reflecting a 'good' or 'bad' capital increment at the previous stage. The mathematical expectation of the insurer's final capital is used as the objective functional. The total loss of the insurer at each stage is modeled by the Gaussian (normal) distribution with parameters depending on a seded loss function (or, in other words, an insurance policy) selected. In contrast to traditional dynamic optimization models for insurance strategies, the proposed approach allows to construct the value functions (and hence the optimal insurance policies) by simply solving a sequence of static insurance optimization problems. It is demonstrated that the optimal seded loss function at each stage depends on the prescribed value of the safety level: it is either a stop-loss insurance or conditional deductible insurance having a discontinuous point. In order to reduce ex post moral hazard, we also investigate the case, where both parties in an insurance contract are obligated to pay more for a larger realization of loss. This leads to that the optimal seeded loss functions are either stop-loss insurances or unconditional deductible insurances. [ABSTRACT FROM AUTHOR]

Asymptotic Ruin Probabilities in the Cramér-Lundberg model has been widely studied when the claims have light-tailed or heavy-tailed distributions. However, it has not been studied for extreme values over some threshold. In this paper, we investigate the use of the peaks over threshold method, to construct a new estimator of the ruin probability for a risk process with heavy tails claims amounts with infinite variance for the stationary arrival claims in an infinite time. Our approach is based on approximating the sample over some threshold by the generalized Pareto distribution. We prove that the proposed estimator is consistent and asymptotically normal. The performance of our new estimator is illustrated by some results of simulations for some loss models and provides an extensive example application to Danish data on large fire insurance. [ABSTRACT FROM AUTHOR]

We find the asymptotics of the value function maximizing the expected utility of discounted dividend payments of an insurance company whose reserves are modeled as a classical Cramér risk process, with exponentially distributed claims, when the initial reserves tend to infinity. We focus on the power and logarithmic utility functions. We also perform some numerical analysis.

Based on a discrete version of the Pollaczeck–Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber–Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.

The paper considers a multivariate distribution whose coordinates are compounds. The number of the summands is itself also a multivariate compound with one and the same univariate Power series distributed number of summands and negative multinomially distributed summands. In the total claims amount process the summands are independent identically distributed random vectors. We provide the first full characterization of this distribution. We show that considered as a mixture this distribution would be Mixed Negative multinomial distribution having the possibly scale changed power series distributed THE first parameter. We provide an interesting application to risk theory.

In this paper, the insurance reserve period will be classified according to the claim frequency domain, such as high- or low-frequency periods. We use the clustering method to create and group claims data according to their frequency period. Meanwhile, we use a risk process to mimic and predict the movement of the reserve from time to time in each group of claim period that is formed. The risk process model used here is the Hawkes process, which is a one-dimensional simple point process and a special type of self-exciting process. Based on this process, we will estimate the reserve at a certain date in the future and the average historical reserve for each group period.

In this paper, we study a spectrally negative Lévy process that is reflected at its drawdown level whenever a drawdown time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are also obtained. The results are expressed in terms of scale functions for the spectrally negative Lévy process. [ABSTRACT FROM AUTHOR]

In this paper, we consider a perturbed risk model with stochastic return on investments. After discretizing of the renewal risk process, using the elementary properties of the classical conditional expectation, the Markov property and considerating the ordinary differential equation of a common density function of the inter-arrival times, we introduce a general integro-differential equation for the Laplace transform of the time of ruin with a positive initial surplus. The special cases, for different inter-arrival times distributions, are given in some details in the discrete renewal risk process. [ABSTRACT FROM AUTHOR]

Draw-down time for a stochastic process is the first passage time of a draw-down level that depends on the previous maximum of the process. In this paper we study the draw-down-related Parisian ruin problem for spectrally negative Lévy risk processes. Intuitively, a draw-down Parisian ruin occurs when the surplus process has continuously stayed below the dynamic draw-down level for a fixed amount of time. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit problems via excursion theory. We also find an expression for the potential measure for the process killed at the draw-down Parisian time. As applications, we obtain new results for spectrally negative Lévy risk processes with dividend barrier and with Parisian ruin. [ABSTRACT FROM AUTHOR]

Since the practical production is not continuously available and sometimes suffers unexpected breakdowns, this paper applies uncertainty theory to introducing an uncertain production risk process with breakdowns to handle the production problem with uncertain cycle times (consisting of uncertain on-times and uncertain off-times) and uncertain production amounts. The concept of shortage index of the uncertain production risk process with breakdowns is provided and some formulas for the calculation are given. Furthermore, the shortage time of the uncertain production risk process with breakdowns is proposed and its uncertainty distribution is obtained. Finally, some numerical examples are revealed. [ABSTRACT FROM AUTHOR]

The problem of designing an optimal insurance strategy in a new multistep insurance model is investigated. This model introduces stepwise probabilistic constraints (Value-at-Risk constraints) on the insurer's capital, i.e., probabilistic constraints on the insurer's capital increments during one step. As the objective functional the mathematical expectation of the insurer's final capital is used. The total damage to the insurer at each step is modeled by the Gaussian distribution with parameters depending on a risk sharing function selected. In contrast to traditional dynamic optimization models for insurance strategies, the approach proposed below takes into account stepwise constraints; within this approach, the Bellman functions are constructed (and hence the optimal risk sharing is found) by simply solving a sequence of static insurance optimization problems. It is demonstrated that the optimal risk sharing is the so-called stop-loss insurance. [ABSTRACT FROM AUTHOR]

Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations. [ABSTRACT FROM AUTHOR]

We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finite-time horizon. Numerical results will also be given. [ABSTRACT FROM AUTHOR]

Parisian ruin occurs once the surplus stays continuously below zero for a given period. We consider the spectrally negative Lévy risk process where ruin is declared either at the first time that the reserve stays continuously below zero for an exponentially or mixed Erlang distributed random variable, or once it reaches a given negative threshold. We consider the Laplace transform of the time to ruin and the Laplace transform of the time that the process is negative. [ABSTRACT FROM AUTHOR]

The paper overviews stochastic optimization models of actuarial mathematics and methods for their solution from the point of view of the methodology of multicriteria stochastic programming and optimal control. The evolution of the capital of an insurance company is considered in discrete time. The main random parameters of the models are insurance payouts, i.e., the ratios of paid insurance claims to the corresponding premiums per unit time. Optimization variables are the structure of the insurance portfolio (gross premium structure) and amount of dividends. As efficiency criteria, indicators of the profitability of the insurance business are used, and, as risk indicators the ruin probability and the recourse capital necessary to prevent the ruin are taken. The goal of the optimization is to find Pareto-optimal solutions. Methods for finding these solutions are proposed. [ABSTRACT FROM AUTHOR]

The Poisson process suitably models the time of successive events and thus has numerous applications in statistics, in economics, it is also fundamental in queueing theory. Economic applications include trading and nowadays particularly high frequency trading. Of outstanding importance are applications in insurance, where arrival times of successive claims are of vital importance. It turns out, however, that real data do not always support the genuine Poisson process. This has lead to variants and augmentations such as time dependent and varying intensities, for example. This paper investigates the fractional Poisson process. We introduce the process and elaborate its main characteristics. The exemplary application considered here is the Carmér–Lundberg theory and the Sparre Andersen model. The fractional regime leads to initial economic stress. On the other hand we demonstrate that the average capital required to recover a company after ruin does not change when switching to the fractional Poisson regime. We finally address particular risk measures, which allow simple evaluations in an environment governed by the fractional Poisson process. [ABSTRACT FROM AUTHOR]

Copyright of Cybernetics & Systems Analysis / Kibernetiki i Sistemnyj Analiz is the property of V.M. Glushkov Institute of Cybernetics of NAS of Ukraine and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we generate boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and a hypoexponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations, we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are linearly dependent on reserves, representing, for instance, returns on risk-free investments of the insurance capital, we firstly derive explicit solutions of the ordinary differential equations under considerations, in terms of special mathematical functions and integrals, from which we can further determine their asymptotics. This allows us to recover the ruin probabilities obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve owned by the insurance company.

We show how to solve Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance (Propositions 1 and 2), i.e., for a wealth portfolio X(t) consisting of a bond and a stock price described by general compound Hawkes process (GCHP), and for a capital R(t) (risk process) of an insurance company with the amount of claims described by the risk model based on GCHP. The main approach in both cases is to use functional central limit theorem for the GCHP to approximate it with a diffusion process. Then we construct and solve Hamilton–Jacobi–Bellman (HJB) equation for the expected utility function. The novelty of the results consists of the new Hawkes-based models and in the new optimal investment results in finance and insurance for those models.

Copyright of Alkalmazott matematikai lapok is the property of János Bolyai Mathematical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We consider an insurance risk model with extended flexibility, under which claims arrive according to a point process with an order statistics (OS) property, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous real valued function. We generalize the definition of an OS point process, assuming it is generated by an arbitrary cdf allowing jump discontinuities, which corresponds to an arbitrary (possibly discontinuous) claim arrival cumulative intensity function. The latter feature is appealing for insurance applications since it allows to consider clusters of claims arriving instantaneously. Under these general assumptions, a closed form expression for the joint distribution of the time to ruin and the deficit at ruin is derived, which remarkably involves classical Appell polynomials. Corollaries of our main result generalize previous non-ruin formulas e.g., those obtained by Ignatov and Kaishev (Scand Actuar J 2000(1):46–62, 2000; J Appl Probab 41(2):570–578, 2004; J Appl Probab 43:535–551, 2006) and Lefèvre and Loisel (Methodol Comput Appl Probab 11(3):425–441, 2009) for the case of stationary Poisson claim arrivals and by Lefèvre and Picard (Insurance Math Econom 49:512–519, 2011; Methodol Comput Appl Probab 16:885–905, 2014), for OS claim arrivals. [ABSTRACT FROM AUTHOR]

This article studies a risk process with constant dividend barrier where the delayed claims are considered, and two types of individual claims, main claims and by-claims, are defined. Each by-claim is induced by a main claim and may be delayed with a certain probability. Both the expected discounted penalty functions and the moment generating functions of the total dividend payments until ruin are derived. The moments of the total dividend payments until ruin and the dividends-penalty identity are given. Finally, numerical examples are provided to illustrate the applicability of the results and the effects of the delay probabilities on the expectations of the total dividend payments until ruin. [ABSTRACT FROM AUTHOR]