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The estimation of claims reserves is usually done by applying techniques called IBNR techniques within a stochastic framework. The main objective of this paper is to predict the partial reserve and to estimate the error rate of prediction distributions by using the stochastic model proposed in [1].

The calculation of the provisions for disaster payments is intended to allow the integral payment of the commitments to the policy-holders and the recipients of the contract. The provisions measure the commitments that the insurer still has to honor. Nevertheless, this countable concept requires a subjacent probabilistic model since it allows one to define the ultimate claim, taking into account the disasters not yet declared, but which have occurred, the disasters not sufficiently funded. Reserves are given by evaluating the provisions for each contract, IBNR (sinister not yet declared) and IBNER (sinister not sufficiently funded). Traditional methods of provisioning (by triangulation) rest on the assumption that the data are homogeneous and in sufficient quantity to ensure a certain stability and a certain credibility. The purpose of this paper is to propose a stochastic extension of the Chain-Ladder model concerning the partial prediction reserve and to estimate the error rate of prediction distributions, which seems to be closer to reality for us than the existing methods of Schnieper [

Models in which parameters move between a fixed number of regimes with switching controlled by an unobserved stochastic process, are very popular in a great variety of domains (Finance, Biology, Meteorology, Networks, etc.). This is notably due to the fact that this additional flexibility allows the model to account for random regime changes in the environment. In this paper we consider the prediction of partial reserve and consider the estimation of error rate of prediction distributions for a model described by a stochastic differential equation (SDE) with Markov regime-switching (MRS), i.e., with parameters controlled by a finite state continuous-time Markov chain (CTMC) [

The rest of the paper is structured as follows. In Section 2, we present the stochastic model for our problem. Section 3 is devoted to predicting the claims reserves variance. We conclude with a summary in the last section.

We suppose that the available data have a triangular form indexed by the year of accident, i, and the development time, t. Given a triangle, on T years, the goal is to consider models using a minimum of parameters, in order to envisage the best possible amounts of payments of future disasters. We note the evolution of the amounts of payments of the cumulated real disasters obtained by

We indicate

To simulate the future claims, it is supposed that the not sufficiently funded claims

ferential equation of Black and Scholes with jump. This assumption on the probability density function of

Conditionally with

where

and

Conditionally with

where

For any state

where

Let us define the log-returns

Given a path

Then, conditional on the path X, the solutions of the Equations (2) and (3) are respectively given by :

・ the solution of the first Equation (2) is

and,

・

The main reason for using stochastic models is to estimate the error rate of prediction distributions. It is useful for solvency, capital modelling and measurement of risk. We begin by proving how the predicted error rate can be calculated. The predicted error rate is obtained from the predicted variance of the partial reserve of loss. We

remember that

the partial reserve by:

The mean square error prediction of

Theorem 3.1 Let ^{th} contract, verifying Equations (2) and (3), then the mean square error prediction of partial reserve is

Proof. The mean square error prediction of the partial reserve can be written as follows

Using the fact that

and

As a first step, we calculate

We calculate the second term (12), taking into account

and

Then

We have studied the Bayesian approach for the regime switching geometric Brownian motion proposed by [

We would like to thank the reviews for the comments.