Academic journal article Journal of Risk and Insurance

# Martingale Approach for Moments of Discounted Aggregate Claims

Academic journal article Journal of Risk and Insurance

# Martingale Approach for Moments of Discounted Aggregate Claims

## Article excerpt

ABSTRACT

We examine the Laplace transform of the distribution of the shot noise process using the martingale. Applying the piecewise deterministic Markov processes theory and using the relationship between the shot noise process and the accumulated/discounted aggregate claims process, the Laplace transform of the distribution of the accumulated aggregate claims is obtained. Assuming that the claim arrival process follows the Poisson process and claim sizes are assumed to be exponential and mixture of exponential, we derive the explicit expressions of the actuarial net premiums and variances of the discounted aggregate claims, which are the annuities paid continuously. Numerical examples are also provided based on them.

INTRODUCTION

Let [X.sub.i], i = 1, 2, ..., be the claim amount, which are assumed to be independent and identically distributed with distribution function H(x) (x > 0). The accumulated value of aggregate claims up to time t, [L.sub.t] is given by

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [delta] is the instantaneous rate of net interest, [s.sub.i]'s are time points at which claims occur ([s.sub.i] < t < [infinity) and [N.sub.t] is the number of claims up to time t. If we multiply [e.sup.-[delta]t] both sides in Equation (1), the discounted value of aggregate claims up to time t, [L.sup.0.sub.t] is given by

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [L.sup.0.sub.t] = [e.sup.-[delta]t][L.sub.t].

If we ignore the effect of the rate of net interest [delta], considering the claim inflation experienced cancels out interest earned, Equations (1) and (2) are equivalent to the classical risk model (Buhlmann, 1970; Gerber, 1979; Beard et al., 1984). However, in practice, the instantaneous rate of interest might be more variable than the claims themselves, and inflation does not merely cancel the interest earned. Hence, it is of interest to obtain the distribution of the accumulated aggregate claims up to time t, [L.sub.t].

Unfortunately, it is known that it is not possible for us to obtain the distribution of the accumulated/discounted aggregate claims explicitly. Dufresne (1990) and Milevsky (1997) proved that the continuous temporary annuity has an inverted Gamma distribution when the time horizon goes to infinity using a stochastic Weiner rate of interest. Goovaerts et al. (2000) also presented a computable approximation for the distribution function of the present value of a sequence of cash flows that are discounted using a stochastic return process. Therefore in this article, applying the piecewise deterministic Markov processes theory and using the relationship between the shot noise process and accumulated/discounted aggregate claims process, we find the Laplace transform of the distribution of the accumulated aggregate claims.

Assuming that the claim arrival process follows the Poisson process and claim sizes are assumed to be exponential and mixture of exponential, we also obtain the explicit expressions of the mean (i.e., the actuarial net premium) and variance of the discounted aggregate claims, i.e., E([L.sup.0.sub.t]) and Var([L.sup.0.sub.t]). Interestingly, we witness that they can be expressed in terms of an annuity paid continuously. Numerical examples are also provided based on the explicit expressions of the actuarial net premiums and variances of the discounted aggregate claims.

SHOT NOISE PROCESS AND ITS GENERATOR

The shot noise process can be used in many diverse fields. In particular, it attracts us as it can be applied in financial and insurance field. The shot noise process is particularly useful as it measures the frequency, magnitude, and time period needed to determine the effect of primary events. As time passes, the shot noise process decreases until another event occurs which will result in a positive jump in the shot noise process. …

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