A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of

We here study the classical Cramér–Lundberg risk process

The model (

The ultimate ruin probability, denoted by

Our aim here is to give an alternative approach to finding

We first define the Erlang mixture distributions of our interest and then use the known formula (

The density of the Erlang distribution is denoted by

We use the letters e.d. for the term equilibrium distribution. It is straightforward to show that

As in our previous work [

For the method proposed in this work to successfully operate it is essential to assume that

Suppose that

Our goal is to solve the recurrence relation (

Suppose the solutions of

Taking complex conjugate of (

In Examples

This claim will be numerically verified for some examples in Section

The equation (

Here we consider the case

Consider the case when all the roots have multiplicity 1, i.e.

We are now ready to state and prove our main result. This is a general version of (

We make use of the following extension of Philipson’s formula, see [

Formula (

As an example, it is shown in [

Formula (

This is a Lundberg-type approximation and, as mentioned before, is the contribution of the unique positive root

It is interesting to observe that in the case of

In this section, we show some numerical results of the calculation of the exact values of

Let

(Example

0 | 0 | 0 | |||

1 | |||||

2 | |||||

3 | |||||

4 | |||||

5 | |||||

6 | |||||

7 | |||||

8 | |||||

9 | |||||

10 | |||||

11 | |||||

12 | |||||

13 | |||||

14 | |||||

15 | |||||

16 | |||||

17 | |||||

18 | |||||

19 | |||||

20 |

(Example

Let

(Example

0 | 0 | 0 | 0.391461 | 0.46031 | 0.39146 |

1 | 0.366639 | 0.423505 | 0.36691 | ||

2 | 0.342903 | 0.389644 | 0.34390 | ||

3 | 0.320266 | 0.358489 | 0.32233 | ||

4 | 0.298728 | 0.329826 | 0.30211 | ||

5 | 0.278286 | 0.303455 | 0.28316 | ||

6 | 0.258928 | 0.279192 | 0.26540 | ||

7 | 0.240640 | 0.256869 | 0.24876 | ||

8 | 0.223402 | 0.236331 | 0.23316 | ||

9 | 0.207190 | 0.217435 | 0.21853 | ||

10 | 0.191975 | 0.200050 | 0.20483 | ||

11 | 0.177725 | 0.184055 | 0.19198 | ||

12 | 0.164405 | 0.169338 | 0.17994 | ||

13 | 0.151975 | 0.155799 | 0.16865 | ||

14 | 0.140396 | 0.143342 | 0.15808 | ||

15 | 0.129625 | 0.131881 | 0.14816 | ||

16 | 0.119620 | 0.121336 | 0.13887 | ||

17 | 0.110338 | 0.111635 | 0.13016 | ||

18 | 0.101737 | 0.102709 | 0.12200 | ||

19 | 0.093774 | 0.094497 | 0.11434 | ||

20 | 0.086408 | 0.086941 | 0.10717 |

(Example

As a subjective comparison of the two approximations

We have given an alternative approach to calculating

It must be emphasized that the present work is a sequel of [

Since finding the roots of polynomials of the form (

The following result is an extension of Philipson’s formula [

By Philipson’s formula [

We are grateful for the comments and suggestions from anonymous reviewers and editors. Their corrections helped improve the presentation of our article.