A review of the numerical calculation of ruin probabilities by means of recursions

Steenackers, A; Goovaerts, Marc

doi:10.1002/asm.3150070108

Cited by 2 publications

(4 citation statements)

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“…Precisely, first of all, Problems (21)(22)(23) Remark 3.1. Clearly, when solving Problem 2 (i.e., [10][11][12], the operator splitting approach is no longer needed, since Equation (10) does not contain any integral term. Therefore, the Steps ( 27) and ( 28) is not done at all and we simply compute u i+1 (𝑦) = u(𝑦).…”

Section: Time Discretizationmentioning

confidence: 99%

“…Problems (10)(11)(12) do not have a closed-form solution and thus some numerical approximation is required. In particular, an analytic approximation based on an infinite series of functions is derived in Leung et al 26 However, such a formula has some disadvantages.…”

Section: Problemmentioning

confidence: 99%

“…However, in the former case, exact closed-form solutions are seldom available, and the ruin probability has to be computed by means of some numerical approximation. In particular, a quite common procedure is to discretize the time arrival of claims and/or the single claim amount (see, e.g., previous studies [3][4][5][6][7][8][9][10] ), or to directly approximate the stochastic process modeling the insurer surplus (see, e.g., previous studies [11][12][13] ). Other methods are based on asymptotic approximations (see Biard et al 14 ), on integral equations (see previous works [15][16][17][18] ), on the Prabhu formula (see previous studies 29,20 ), on embedded Markov chains (see Dickson and Gray 21 ), or on modeling the interoccurrence times between successive claims and the single claim amounts by means of special probability distributions (see Dickson and Gray 22 ).…”

Section: Introductionmentioning

confidence: 99%

“…T. Clearly, for Problems (10)(11)(12), no domain truncation is needed, as the spatial domain [L, U] of Problem 2 is already bounded. Then, by applying the following change of variables:…”

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confidence: 99%

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The finite element method: A high‐performing approach for computing the probability of ruin and solving other ruin‐related problems

Ahmadian

Ballestra

2021

Math Methods in App Sciences

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The finite element method (FEM), despite being a very powerful tool for solving partial differential equations, has never been used to compute the probability of ruin, nor it has been applied to other kinds of ruin-related problems, at least to the best of our knowledge. In this paper, in order to explore possible applications of the FEM in insurance mathematics, a high-order FEM is proposed to solve problems related to ruin. Specifically, a FEM based on either quadratic, cubic, or quartic polynomials is developed to compute the probability of ruin and the (discounted) expected value of the assets of a firm in two previously published ruin models. Moreover, when dealing with the calculation of the probability of ruin, a suitable operator splitting technique is employed, such that the FEM variational formulation is used to approximate only spatial derivatives. Numerical experiments are presented showing that the FEM performs extraordinarily well. In fact, the solutions of the two problems considered can be obtained with an error (in the maximum norm) of order 10 −5 or even smaller in a time smaller than a second. Finally, in accordance with already established theoretical results, the FEM reveals to be superconvergent at the boundaries of the finite elements.

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Section: Time Discretizationmentioning

confidence: 99%

Section: Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…T. Clearly, for Problems (10)(11)(12), no domain truncation is needed, as the spatial domain [L, U] of Problem 2 is already bounded. Then, by applying the following change of variables:…”

mentioning

confidence: 99%

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