Computation of the Probability of Ruin Within a Finite Time Horizon

Description

In the Cramér–Lundberg risk process perturbed by a Wiener process, this packages provides approximations to the probability of ruin within a finite time horizon. Currently, there are three methods implemented: The first one uses saddlepoint approximation (two variants are provided), the second one uses importance sampling and the third one is based on the simulation of a dual process. This last method is not very accurate and only given here for completeness.

Author(s)

Benjamin Baumgartner <benjamin@baumgrt.com>

References

Gatto, R. and Baumgartner, B. (2016) 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 18(1), pp. 217-235.

Simulation of a risk process that is perturbed by a Wiener process

Description

This function simulates paths of a compound Poisson risk process that is perturbed by a Wiener process. Multiple paths can be simulated simultaneously.

Usage

rriskproc(m = 1001, window = c(0, 1), num = 1,
          sigma = 1, freq = 1, drift = 0, jumpdist, ...)

Arguments

Details

Possible choices for jumpdist include rexp, rgamma and rlnorm.

It is assumed that the function specified for jumpdist interprets its first argument as the vector length of its return value, i. e. the number of simultaneously generated random variables.

The path realizations of the Wiener process are generated using the circulant embedding method (see references).

Value

A time-series object/time-series object containing the simulated sample path(s).

References

Dietrich, C. and Newsam, G. (1997) Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix. SIAM Journal on Scientific Computing 18(4), pp. 1088-1107

See Also

rhypoexp

Examples

    require(sdprisk)

    rriskproc(m        = 1001,
              window   = c(0, 5),
              num      = 1,
              sigma    = sqrt(0.4),
              freq     = 1,
              drift    = 2,
              jumpdist = rhypoexp,
              rate     = c(1, 10))

    # The same can be achieved using
    #   jumpdist = function(n) rexp(n, 1) + rexp(n, 10)

    rriskproc(window = c(0, 10),
              jumpdist = function(n) {
                  rexp(n, 1) + rexp(n, 10)
              })

Computation of the probability of ruin within a finite time horizon using a dual process

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 is based on a dual process to the risk process.

Usage

ruinprob.finite.dsim(Z)

Arguments

Details

This function computes an approximation to the probability of ruin within a finite time horizon using a dual process. See the references for more details.

Value

A function taking one numeric argument, the initial capital. This function returns the approximation for the specified initial reserve and for all values of the time horizon that are sampling points of Z.

References

Gatto, R. and Baumgartner, B. (2016) 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 18(1), pp. 217-235.

Approximation of the probability of ruin within a finite time horizon using importance sampling

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 is based on importance sampling.

Usage

    ruinprob.finite.imps()

Value

This function is not yet fully implemented. At the moment it invisibly returns NULL.

References

Gatto, R. and Baumgartner, B. (2016) 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 18(1), pp. 217-235.

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

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

Arguments

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. (2016) 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 18(1), pp. 217-235.