We deal with a generalization of the classical risk model when an insurance company gets additional funds whenever a claim arrives and consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. We compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.

Let

In the classical risk model (see, e.g., [

In addition to the classical risk model, we suppose that the insurance company gets additional funds

Let

The ruin time is defined as

Note that the ruin never occurs if

In this paper, we consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. Moreover, we compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.

Paper [

It is justified in the proof of Theorem

It is well known that even for the classical risk model, there are only a few cases where an analytic expression for the survival probability can be found. So numerous approximations have been considered and investigated for the classical risk model (see, e.g., [

We deal with the case where the claim sizes have a light-tailed distribution. The rest of the paper is organized as follows. In Section

To get an upper exponential bound for the ruin probability, we use the martingale approach introduced by Gerber [

Let

For all

For all

If there is

Thus,

It is easily seen that

Let the random variables

If

Furthermore, it is easy to check that, in this case,

If

To construct an analogue to the De Vylder approximation, we replace the process

Since the process

Let

Let

Applying similar arguments to the process

By (

Substituting

Dividing (

Consequently, we have

Substituting (

Substituting (

Note that since

By (

Thus, we get the following result.

Note that

If claim sizes and additional funds are exponentially distributed, then it is easily seen from (

Let

The assertion of Proposition

Formula (

Let the probability density functions of

In what follows,

An easy computation shows that

Moreover, for all

Thus, condition (

Hence, we get

Let

The results of computations are given in Table

Results of computations: Erlang distributions for claim sizes and additional funds

0 | 0.634149 | 0.612268 | −3.45 |
1.000000 | 57.69 |

1 | 0.492768 | 0.439087 | −10.89 |
0.705327 | 43.14 |

2 | 0.355769 | 0.314891 | −11.49 |
0.497487 | 39.83 |

5 | 0.137737 | 0.116142 | −15.68 |
0.174564 | 26.74 |

10 | 0.023224 | 0.022031 | −5.14 |
0.030473 | 31.21 |

Let

It is easy to check that

Furthermore, for all

Hence, condition (

Consequently, we get

Let

The results of computations are given in Table

Results of computations: hyperexponential distributions for claim sizes and additional funds

0 | 0.647560 | 0.581428 | −10.21 |
1.000000 | 54.43 |

1 | 0.540924 | 0.521454 | −3.60 |
0.895291 | 65.51 |

2 | 0.488597 | 0.467667 | −4.28 |
0.801545 | 64.05 |

5 | 0.346390 | 0.337363 | −2.61 |
0.575201 | 66.06 |

10 | 0.202323 | 0.195749 | −3.25 |
0.330856 | 63.53 |

20 | 0.067802 | 0.065903 | −2.80 |
0.109466 | 61.45 |

25 | 0.038194 | 0.038239 | 0.12 |
0.062965 | 64.86 |

Let

If

Since

Let

The results of computations are given in Table

Results of computations: exponential distribution for claim sizes and degenerate distribution additional funds

0 | 0.637998 | 0.582498 | −8.70 |
1.000000 | 56.74 |

1 | 0.549737 | 0.482780 | −12.18 |
0.822610 | 49.64 |

2 | 0.465171 | 0.400133 | −13.98 |
0.676687 | 45.47 |

5 | 0.277026 | 0.227808 | −17.77 |
0.376678 | 37.97 |

10 | 0.113399 | 0.089093 | −21.43 |
0.141886 | 25.12 |

Tables