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This paper analyses the Gerber-Shiu penalty function of a Markov modulated risk model with delayed by-claims and random incomes. It is assumed that each main claim will also generate a by-claim and the occurrence of the by-claim may be delayed depending on associated main claim amount. We derive the system of integral equations satisfied by the penalty function of the model. Further, assuming that the premium size is exponentially distributed, an explicit expression for the Laplace transform of the expected discounted penalty function is derived. For a two-state model with exponential claim sizes, we present the explicit formula for the probability of ruin. Finally we numerically illustrate the influence of the initial capital on the ruin probabilities of the risk model using a specific example. An example for the risk model without any external environment is also provided with numerical results.

Analyzing a risk model using the Gerber-Shiu discounted function largely promoted the theory and provided a useful tool for the computation of many performance measures. As a classical risk model is too idealistic, in fact there are a lot of distracters, it has become necessary to study risk models having parameters governed by the external environment. Recently many authors considered risk models having Markov modulated environment or Markovian regime-switching models. The purpose of this generalization is to enhance the flexibility of the model parameters for the classical risk process. For a Markov modulated Poisson process, the arrival rate varies according to a given Markov process. The risk models managed by an insurance company are a long-term program and system parameters such as interest rates, premium rates, claim arrival rates, etc. may need to change whenever economic or political environment changes. So it is always preferable to regulate the model according to the external environment.

The assumption on independence among claims is an important condition used in the study of risk models. However, in many practical situations, this assumption is inconsistent with the operation of insurance companies. In reality, claims may be time- correlated for various reasons, and it is important to study risk models which can also depict this phenomenon. Two types of individual claims, main claims and associated by-claims are introduced, where every by-claim is induced by the main claim and could be delayed for one time period depending on the amount to be paid towards the main claim. Further, we discuss the model in the presence of random incomes in order to accommodate insurance companies having lump sums of income occurring time to time based on their business and other related activities.

The idea of delayed claims is gaining importance due to its relevance in many real world situations. Xie et al. [

Yu [

J. Gao and L. Wu [

The rest of this paper is organized as follows. In Section 2, we describe the risk model considered. In Section 3, the integral equations for the expected discounted penalty function are obtained. Section 4 deals with the case with exponential random incomes and Laplace transforms of the discounted penalty function derived. In Section 5, we illustrate the usefulness of the model by computing probability of ruin for a model having only two states and in Section 6 a risk model without any external environment. Section 7 concludes the paper.

Here we consider a continuous time risk model with random incomes, two types of insurance claims, namely the main claims and the by-claims, and where the parameters are depending on the external environment. Let

Let

The processes

In this paper, we consider the risk model having the following claim occurrence process. There will be a main claim

In this set up, the surplus process

where u is the initial capital and

where

The safety loading condition is

Now let us consider an auxiliary risk model, which is same as the one described above with a slight change assumed at the first claim epoch. Instead of having one main claim

where Y denotes the other by-claim amount added at the first claim epoch and let

We are interested in the Gerber-Shiu discounted penalty function of the model. Analyzing the surplus process

1) During

2) During the time interval

3) One main claim and a by-claim occurs in

4) No claim occurs in

5) No claim occurs, no premium arrival in

6) All other events having total probability

The Gerber-Shiu discounted penalty function of the model satisfies equation,

Similarly, for the auxiliary model we have

Expanding

Substituting

and

in the Equations (5) and (6). They reduce to,

For the auxiliary model, it is

where

and

Remark 1: Letting

Remark 2: Letting

Remark 3: Letting

This section assumes that the random premium amounts are exponentially distributed and we derive the Laplace transform of the Gerber-Shiu function.

Writing

i.e.

we have,

where

Similarly for the auxiliary model we have,

where

Suppose that the random income

Then we have,

Hence,

and

Further simplifying we have,

and

We consider the case where all the by-claims are delayed to the next claim epoch and both claim amounts are exponentially distributed, i.e.; the distribution functions are

The probability of ruin is obtained by putting

We have,

Numerical example 1: Let

Then we have

u | ||||
---|---|---|---|---|

0 | 0.3049 | 0.3462 | 0.1845 | 0.2267 |

0.2 | 0.1542 | 0.1287 | 0.09018 | 0.07562 |

0.4 | 0.08786 | 0.06297 | 0.05118 | 0.03569 |

0.6 | 0.05433 | 0.03407 | 0.03183 | 0.01927 |

0.8 | 0.03708 | 0.02066 | 0.02184 | 0.01175 |

1 | 0.0283 | 0.01446 | 0.01671 | 0.008264 |

1.2 | 0.02393 | 0.01163 | 0.01414 | 0.006668 |

1.4 | 0.02181 | 0.01038 | 0.01288 | 0.005958 |

1.6 | 0.02082 | 0.009839 | 0.0123 | 0.005653 |

1.8 | 0.02039 | 0.0062 | 0.01204 | 0.005529 |

2 | 0.02021 | 0.009538 | 0.01193 | 0.005482 |

2.2 | 002015 | 0.009513 | 0.0119 | 0.005468 |

2.4 | 0.02014 | 0.00951 | 0.01189 | 0.005467 |

2.6 | 0.02015 | 0.009514 | 0.0119 | 0.005469 |

2.8 | 0.02016 | 0.0052 | 0.0119 | 0.005472 |

3 | 0.02017 | 0.009525 | 0.0119 | 0.005475 |

One can note from the graph that in

In this section, we consider the risk model without external environment (i.e.;

where

Numerical example 2: Let

In this paper, we investigated a Markov-modulated risk model with random incomes

u | ||
---|---|---|

0 | 0.2432 | 0.1643 |

0.2 | 0.1371 | 0.0853 |

0.4 | 0.0816 | 0.0492 |

0.6 | 0.0515 | 0.0307 |

0.8 | 0.0354 | 0.0210 |

1 | 0.0269 | 0.0159 |

1.2 | 0.0227 | 0.0134 |

1.4 | 0.0206 | 0.0121 |

1.6 | 0.0196 | 0.0116 |

1.8 | 0.0192 | 0.0113 |

2 | 0.0190 | 0.0112 |

2.2 | 0.0189 | 0.0112 |

2.4 | 0.0189 | 0.0112 |

2.6 | 0.0189 | 0.0112 |

2.8 | 0.0190 | 0.0112 |

3 | 0.0190 | 0.0112 |

and two types of claims (i.e., main claims and by-claims) and where the by-claims may be delayed to the next claim point. We assume that the by-claim can be delayed depending on the corresponding main claim amount; whether it is exceeding the random threshold. All system parameters are assumed to be depending on the state of the external environment. System of integral equations for the Gerber-Shiu penalty function was obtained. Then we obtained Laplace transforms of the penalty function under the assumption that the random incomes follow an exponential distribution. Next for a simplified model with exponential claim amounts, we presented expressions for the probability of ruin and some numerical illustrations included. Finally we considered another simplified model in the absence of external environment and numerically illustrated the influence of initial capital on the ruin probabilities.

Future research includes investigation of the risk model with generalized distributions. It would be also interesting to find other ruin related parameters like surplus prior to ruin, deficit at ruin, etc.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Shija, G. and Jacob, M.J. (2016) Gerber Shiu Function of Markov Modulated Delayed By-Claim Type Risk Model with Random Incomes. Journal of Mathematical Finance, 6, 489-501. http://dx.doi.org/10.4236/jmf.2016.64039