## 1. Introduction

## 2. The Model

#### 2.1. The Claim Incurral Process

#### 2.2. The Claim Reporting and Handling Process

## 3. The Joint Distribution of Claims in Different States

**Theorem 1.**

**Remark 1.**It can be seen from Equation (7) that

## 4. The Joint Moments

**Remark 2.**Let ${\left\{{\mathbf{m}}_{IBNR}^{l}\left(t\right)\right\}}_{i}$ be the lth moment of the amount of IBNR claims conditional on $J\left(0\right)=i$. Since ${S}_{IBNR}=\{1,\cdots ,{K}_{1}\}$, we have

**Remark 3.**Let ${\mathbf{m}}^{l}\left(t\right)$ denote the vector of conditional moments of the total amount of incurred claims during $(0,t]$, regardless of the claim reporing/handling status. Then it can be calculated by setting ${p}_{k}\left(t\right)=1$ in equation (11). In the equilibrium case, we have that $\pi \mathbf{m}\left(t\right)=\pi {\mathbf{D}}_{\mu ,1}\mathbf{e}t.$ This result will be used in the next section.

**Remark 4.**Similar to Section 8 of Ramaswami and Neuts (1980) [9], it can be shown that when $t\to \infty $, the distribution of ${\mathbf{X}}_{k}\left(t\right),k\in \{{S}_{IBNR}\cup {S}_{RBNS}\}$ has an asymptotical limit and the joint moments ${\mathbf{m}}_{{k}_{1},{k}_{2}}^{{l}_{1},{l}_{2}}\left(t\right)$ for ${k}_{1},{k}_{2}\in \{{S}_{IBNR}\cup {S}_{RBNS}\}$ converges to a finite vector. This fact is actually illustrated in the numerical examples in the next section.

**Remark 5.**Since the calculation of the joint moments of $\mathbf{X}\left(t\right)$ only requires the moments of the claim sizes – the exact form of the claim size distribution is not needed. In the following numerical examples, exponential claim sizes are assumed for presentational convenience only.

## 5. Numerical Examples

**Case 1**: Claims arrive according to a Poisson process with inter claim arrival times following an $Exp\left(140\right)$ distribution. The claim sizes follow an $Exp(140/190)$ distribution. In terms of the MAP representation, we have

**Case 2**: The claims arrive according to a Markov modulated Poisson process and claim sizes are modulated by the states of the underlying Markov process. Specifically, assume that an external environment evolves according to a continuous time Markov chain ${\left\{E\left(t\right)\right\}}_{t\ge 0}$ with a state space $\{N,R\}$, where the two elements standing for normal and risky environment respectively. The environment process is assumed to have the infinitesimal generator

**Case 3**: This case is similar to Case 2, but with different parameter values. Here it is assumed that the environment process has state space $\{N,R\}$ and have the infinitesimal generator

## 6. Conclusion and Discussions

## Acknowledgments

## Conflicts of Interest

## Appendix

## References

- E. Arjas. “The claims reserving problem in non-life insurance: Some structural ideas.” Astin Bull. 19 (1989): 139–152. [Google Scholar] [CrossRef]
- R. Noberg. “Prediction of Outstanding Liabilities in Non-Life Insurance.” Astin Bull. 23 (1993): 95–117. [Google Scholar] [CrossRef]
- T. Mikosch. Non-Life Insurance Mathematics. Berlin, Germany: Springer-Verlag, Berlin, 2009. [Google Scholar]
- C. Hachemeister. “A stochastic model for loss reserving.” In Proceedings of the International Congress of Actuaries, Zurich and Lausanne, Switzerland; 1980, pp. 185–194. [Google Scholar]
- O. Hesselager. “A Markov Model for Loss Reserving.” Astin Bull. 24 (1994): 19–32. [Google Scholar] [CrossRef]
- W. Neuhaus. “On the estimation of outstanding claims.” Aust. Actuar. J. 10 (2004): 485–518. [Google Scholar]
- S. Asmussen. Applied Probability and Queues. New York, NY, USA: Springer, 2003. [Google Scholar]
- G.E. Willmot. “A Queueing Theoretic Approach to the Analysis of the Claims Payment Process.” Actuar. Res. Clear. House 1 (1990): 261–318. [Google Scholar]
- V. Ramaswami, and M.F. Neuts. “Some Explicit Formulas and Computational Methods for Infinite-Server Queues with Phase-Type Arrivals.” J. Appl. Probab. 17 (1980): 498–514. [Google Scholar] [CrossRef]
- H. Masuyama, and T. Takine. “Analysis of an Infinite-Server Queue with Batch Markovian Arrival Streams.” Queueing Syst. 42 (2002): 269–296. [Google Scholar] [CrossRef]
- M.F. Neuts, and J. Li. “An Algorithm for the P(n, t) matrices of a continuous BMAP.” In Matrix Analytic Methods in Stochastic Models. Edited by S. Chakravarthy and A. Alfa. New York, NY, USA: Marcel Dekker, 1997. [Google Scholar]
- M. Huynh, D. Landriault, T. Shi, and G.E. Willmot. “On a risk model with claim investigation.” Insur. Math. Econ. 65 (2015): 37–45. [Google Scholar] [CrossRef]

**Figure 2.**Correlation between Incurred But Not Reported (IBNR) and Reported But Not Settled (RBNS) claims.

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