# Approximation of the probability of ruin within a finite time horizon using saddlepoint methods

### Description

This function calculates an approximation to the probability of ruin within a finite time horizon for a compound Poisson risk process that is perturbed by a Wiener process. The approximation makes use of saddlepoint methods.

### Usage

1 2 | ```
ruinprob.finite.sdp(mgf, mgf.d1, mgf.d2, premium, freq, variance,
endpoint, verbose = FALSE)
``` |

### Arguments

`mgf` |
The moment-generating function of the individual claim amounts |

`mgf.d1` |
The first derivative of |

`mgf.d2` |
The second derivative of |

`premium` |
The premium force |

`freq` |
Frequency of the claims |

`variance` |
The variance of the Wiener process by which the risk process is perturbed |

`endpoint` |
The upper endpoint of |

`verbose` |
Return additional diagnostic information as an attribute of the output |

### Details

If neither or only the first derivative of `mgf`

is provided,
a numerical approximation to the missing derivative(s) will be used
instead (see grad and hessian).

The argument `endpoint`

is the (smallest) positive pole of
`mgf`

. Omitting this information will issue a warning and the value
1.0e+6 will be used instead, possibly yielding unexpected and unreliable
output or leading to further errors.

### Value

A function `psi(x, t)`

taking as inputs the initial capital `x`

and the time horizon `t`

. This function returns a list, the first
element of which contains a Lugannani–Rice-type approximation, the second
one contains a Skovgaard-type approximation.

### References

Gatto, R. and Baumgartner, B. (2013, to appear) *Saddlepoint
approximations to the probability of ruin in finite time for the compound
Poisson risk process perturbed by diffusion*. Methodology and Computing in
Applied Probability **TBA**(TBA), pp. 1–19.

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