Sensitivity Analysis and Density Estimation for Finite-Time Ruin Probabilities Stéphane Loisel, Nicolas Privault

Sensitivity analysis and density estimation for finite-time ruin probabilities Stéphane Loisel, Nicolas Privault To cite this version: Stéphane Loisel, Nicolas Privault. Sensitivity analysis and density estimation for finite-time ruin probabilities. 2007. hal-00201347v2 HAL Id: hal-00201347 https://hal.archives-ouvertes.fr/hal-00201347v2 Preprint submitted on 3 Jan 2008 (v2), last revised 1 Apr 2008 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sensitivity analysis and density estimation for finite-time ruin probabilities Stéphane Loisel Universitéde lyon, UniversitéClaude Bernard Lyon 1, Institut de Science Financière et d’Assurances, 50, avenue Tony Garnier, F-69366 Lyon Cedex 07 Nicolas Privault Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong Abstract The goal of this paper is to obtain probabilistic representation formulas that are suitable for the numerical computation of the (possibly non-continuous) density functions of infima of reserve processes commonly used in insurance. In particular we show, using Monte Carlo simulations, that these representation formulas perform better than standard finite difference methods. Our approach differs from standard Malliavin probabilistic representation techniques which generally require more smoothness on random variables, entailing the continuity of their density functions. Key words: Ruin probability, Malliavin calculus, insurance, integration by parts. MSC Classification codes: 60J75, 60H07, 91B30. 1 Introduction In ruin theory, computational methods for finite-time ruin probabilities have received considerable attention in the last decade. The reader is referred to the books by Gerber [8], Grandell [10], Panjer and Willmot [17], Asmussen [2], and Kaas et al. [12] for general results on ruin-related issues; see also, e.g., Gerber and Shiu [9], Albrecher et al. [1] for more recent results. Email addresses: [email protected] (Stéphane Loisel), [email protected] (Nicolas Privault). Consider the classical compound Poisson risk model, in which the surplus process (Rx(t))t≥0 is defined as R (t)= x + f(t) S(t), t 0, (1.1) x − ≥ where x 0 is the amount of initial reserves and f(t) is the premium income ≥ received between time 0 and time t> 0. Here, the aggregate claim amount up to time t is described by the compound Poisson process Nt S(t)= Wk, kX=1 where the claim amounts W , k 1, are non-negative independent, identically- k ≥ distributed random variables, with the convention S(t)=0if Nt = 0. The number of claims N until t 0 is modeled by an homogeneous Poisson t ≥ process (Nt)t≥0 with intensity λ > 0, and claim amounts are assumed to be independent of arrival times. However we do not make any assumption on the claim amount distribution. Our results are general and operational for light or heavy-tailed, discrete or continuous claim amount distributions, as illustrated in Section 5. Given T > 0 a finite time horizon, a formula for the finite-time ruin probability ψ(x, T )= P ( t [0, T ] : R (t) < 0) ∃ ∈ x has been proposed by Picard and Lefèvre [18], discussed by De Vylder [24] and Ignatov et al. [11], and compared to a Prabhu or Seal-type formula by Rullière and Loisel [23]. Its influence function and estimation risk (related to the jump size distribution) have been studied by Loisel, Mazza and Rullière in [14] and [13]. Another important practical problem is to obtain numerical values of the sensitivity of the finite-time ruin probability with respect to the initial reserve ∂ψ (x, T ), ∂x in particular due to new solvency regulations in Europe. In [22] Privault and Wei used the Malliavin calculus to compute the sensitivity of the probability P (Rx(T ) < 0) that the terminal surplus is negative with respect to parameters such as the initial reserve x or the interest rate of the model. 2 However the problem of computing the corresponding sensitivity for the finite- time ruin probability ψ(x, T ) has not been covered in [22] because Rx(T ) does not satisfy the smoothness conditions imposed therein. For this reason, in this paper we develop an alternative solution to this problem, based on direct integration by parts techniques. This paper is organized as follows. We start in Section 2 by a review of the principles of Malliavin’s calculus applied to density estimation. In Section 3 we present the particular setting of Malliavin’s calculus on the Poisson space and we show in particular why the conditions imposed in this calculus are too stringent for the random variables (infima) considered here. Indeed, as noted in Privault and Wei [22], Remark 5.2, the differentiability conditions imposed above are not satisfied e.g. by the infimum inf Rx(t), 0≤t≤T corresponding to a finite time ruin probability, whose density may not be continuous. In Section 4 we propose to use a direct integration by parts on the Poisson space to compute the density of the infimum = inf f(t) S(t), t [0, T ] M[0,T ] { − ∈ } of the process (f(t) S(t)) − t∈[0,T ] between time 0 and some fixed time horizon T > 0. This provides an explicit probabilistic representation formula in the case of a compound Poisson process (which corresponds to the classical Crámer-Lundberg risk model) which is suitable for simulation purposes. In the case of a jump-diffusion process (with an independent Brownian component that models investment of the surplus into a risky asset), we obtain a formula involving the density of the Brownian bridge, which suggests how our method could be extended to such models after further research. In the compound Poisson risk model, the ruin probability ψ(x, T ) equals ψ(x, T )= P < x , x 0, M[0,T ] − ≥ and the density of at x< 0 equals M[0,T ] − ∂ψ (x, T ). − ∂x In Section 5, we show that our method is much more stable than classical finite-difference methods on several examples (for unit valued, exponential, and Pareto distributed claim amounts). 3 2 Malliavin calculus for density estimation We start by a summary of Malliavin’s calculus and integration by parts formulas applied to the existence and smoothness of probability densities of random variables. In particular, our goal is to determine more precisely the range of application of these techniques to the suprema of compensated jump processes. Existence of densities Here we state conditional versions of classical results on the existence of probability densities, see e.g. 3.1 of Nualart [16] or Corollary 5.2.3 of Bouleau § and Hirsch [3]. In the sequel we work on a probability space (Ω, , P). F Proposition 1 Let A such that P(A) > 0 and assume that F satisfies an integration by parts∈ formula F of the form E[Gf ′(F ) A]= E[Λ f(F ) A] f 1(R), (2.1) | F,G | ∈ Cb where G is a.s. positive on A and ΛF,G is an integrable weight depending on F and G, and independent of f 1(R). ∈ Cb Then: i) the law of F has a conditional density ϕF |A given A with respect to the Lebesgue measure. ii) if in addition G = 1 a.s. on A then this density is given by ϕ (y)= E[Λ 11 A], y R. (2.2) F |A F, {y≤F }| ∈ Proof. We have E[Gf ′(F ) A]= E[f(F )Λ A] f E[ Λ A], | F,G| ≤k k∞ | F,G|| ′ and taking f = 1B yields E[G1 (F )] m(B)E[ Λ ], B ≤ | F,G| where m(B) denotes the Lebesgue measure of any bounded Borel subset B of R, hence the law of F is absolutely continuous with respect to the Lebesgue measure since G> 0 a.s. If G = 1 a.s. on A, we have 4 F E[f(F ) A]= E f ′(y)dy A | "Z−∞ # 0 = E f ′(y + F ) A dy Z−∞ 0 = E ΛF,1f(y + F ) A dy Z−∞ ∞ = f(y)E 1{F ≥y}ΛF, 1 A dy. Z−∞ Formulas of the type (2.1) can be used to show that ∂ E[f(F ) A]= E[∂ F f ′(F ) A]= E[Λ f(F ) A], (2.3) ∂ξ ξ | ξ ξ ξ | Fξ,∂ξFξ ξ | where Fξ is a random variable depending on a real parameter ξ, and can be applied to sensitivity analysis in finance, see e.g. Fournié et al. [7] and in insurance, Privault and Wei [22]. Continuity of densities Proposition 1 ensures the existence of the density ϕF |A but not its smoothness. The next proposition provides a more precise statement. Proposition 2 Under the hypothesis of Corollary 2 with G = 1 on A, if in p addition Λ 1 L (A) for some p> 1, the conditional probability density ϕ F, ∈ F |A is continuous on R. Proof. Use the bound 1 1/q ϕ (y) ϕ (z) Λ p (E[1 (F )]) , y,z R, (2.4) | F |A − F |A | ≤ P(A)k F kL (A) [z,y] ∈ that follows from (2.2), with 1/p +1/q = 1. p The integrability of ΛF in L (A) for p > 1 can essentially be obtained under strong (second order) differentiability conditions in the Malliavin sense, cf.

Load more