Multiquadratic methods in the numerical. treatment of the integro-differential equations of collective non-ruin; the finite time case

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1 Multiquadratic methods in the numerical treatment of the integro-differential equations of collective non-ruin; the finite time case Athena Makroglou Division of Mathematics and Statistics, School of Computer Science and Mathematics, University of Portsmouth, Mercantile House, Portsmouth PO7 2EG, EngWd alpha2. iso.port.ac. uk Abstract An important problem of collective non-ruin is the estimation of the probabilities R(z,t) and R(z) of the finite and ultimate non-ruin respectively, where t is time and z is the initial reserve. The governing equations are first order Volterra integro-differential equations (VIDEs). Computational results for the finite time case are presented. These are obtained by using multiquadric approximation methods applied directly to the VIDE. Comparisons are made with results obtained by indirect approaches. 1 Introduction The problem of estimating the probability of non-ruin given an initial reserve is very important in modelling a risk business such as an insurance company. In the case offinitetime non-ruin, we are concerned with the estimation of the probability R(z,t) = P[Z(a)>Q,Q<8<t\Z(Q)=z], (1.1) where, using the notation of Peters and Mangel (1996)), Z(s) denotes the risk reserve at time s and z = Z(0) is the initial risk reserve.

2 496 Boundary Element Technology The simple model of the risk business of an insurance company considered, assumes that: Risk reserve = initial reserve + total premiums - total claims. In the absence of claims, it is assumed that the reserve satisfies the deterministic equation ^ = & + TZ«, (1.2) where /% denotes the rate (constant) at which the premiums come in, and 7 is the interest rate that applies to risk reserve until a claim is made. Claim sizes at time t denoted by Xt are assumed to have distribution function B(x) P[Xt < x] and are assumed to arrive according to a Poisson process Nt with parameter A. This means that Then St = ]Ci=i ^, the accumulated claims process is a compound Poisson process. Models derived under more complicated assumptions can be found for example in Seal (1974), Bade (1983). Using the above assumptions, an integro-differential equation model is derived for the probability R(z,t), see for examples Arfwedson (1950), Knessl and Peters (1994, 1996) and Grandell (1991) for more details. This problem has been solved by a number of theoretical and numerical approaches involving use of Laplace transforms (cf. Seal (1974), Knessl and Peters ( 1994, 1996), Makroglou (1998)). Solving the VIDEs numerically is very important since it allows treatment of the problem under more complicated assumptions for B(x) and /3(Z(t)) when analytical solutions are difficult to obtain. In this paper we are concerned with the computational treatment of the problem following the approach of solving directly the VIDE in R(z, t) numerically using multiquadric (MQ) methods. Solving numerically the VIDEs directly is also very important, since it is a more efficient and simpler way of approach than the indirect one based on the use of Laplace transforms. The organization of the paper is as follows: In section 2 the form of the VIDEs obeyed by R(z, t) is given together with an overview of existing methods of solution. In section 3, the method of solving directly the VIDE by using multiquadric (MQ) methods is described. In section 4, numerical results are presented. Conclusions and some comparisons with the results of Makroglou (1998) can be found in section 5.

3 Boundary Element Technology The integro-differential equations models Starting from equation (1.1) and using probabilistic arguments (cf. Knessl and Peters (1994)) the following equation is obtained for the probability of non-ruin R(z,t): with conditions R(z, t) = 1, R(z, 0) = l,z > 0. (2.2) We also have that R(z, t) = 0, z < 0. Most of the methods applied to the solution of (2.1)-(2.2) treat special cases of the problem concerning the choice of the model for the premiums ((3(Z(t))) and the distribution function B(x). The choices for the premiums are: and 0(Z(t)) =0 = constant (2.3) 0(Z(t)) = A> + 7^(*),A),7constants. (2.4) The choices for the distribution function B(x) and its density function b(x) include the Gamma distribution with v 1 (exponential) and v = 2, i.e., where {W-'ae-" jf _>n ' "W 'f*-0 (2.5) u otherwise (i/)= Jo is the Gamma function. Existing methods for the solution of equations (2.1)-(2.2) include Seal (1974, 1978)), Knessl and Peters (1994, 1996), and Makroglou (1998). The approach of Seal (1974, 1978) in brief, is to consider the Laplace transform of R(z,t) over z values, i.e. %*(s) = f e~'*r(z,t)dz and then obtain a partial differential equation for vt(s) which is solved analytically in terms of #(0, t) and w(s) = f e~'*b(y)dy. To find R(z, t), vt(s) is inverted to obtain a formula which involves the inverse Laplace transforms of three more functions which are obtained by numerical methods. Numerical integration is also used for the evaluation of an integral occuring in the formula

4 498 Boundary Element Technology for R(z,t). The approach of Knessl and Peters (1994) can be summarized as follows: They consider the Laplace transform Q(z,s) of the function R(z,t) over time, i.e. they take e~**r(z, t)dt. (2.6) For the special case that B(x) is the exponential distribution (B'(x) = b(x) ae'"*, p, = I/a, the mean) this leads to a linear integro-differential equation in Q(z, s) which they solve analytically by conversion to a second order ordinary differential equation. Analytical inversion of Q(z,s) is then applied to obtain R(z,t). Both cases (2.3) and (2.4) for the premiums are considered. Exact and asymptotic solutions are derived for case (2.3) and exact ones for the case (2.4). Asymptotic solutions for the case (2.4) were given in Knessl and Peters (1996). Some of the formulae are complicated and involve series and integrals. Some numerical results which are obtained by numerical evaluation of the final formulae for R(z,t) using MATHEMATICA can be found in Knessl and Peters (1996). Some of the results of Seal (1974) and Knessl and Peters (1994) were verified in Harper (1995, 1996) where the numerical inversion of the analytical expression for vt(s) and Q(z,s) respectively, was done using ACM routine 619 (Piessens and Huysmans (1984)). All the above approaches depend on the ability to be able to solve analytically the equation for the Laplace transforms of the unknown function R(z,t). For more general choices of B(x) and /3(Z(t)) though this is not always possible and even for the simple choices of B(x) and 0(Z(t)), considered the form of the resulting formulae is often complicated. So in Makroglou (1998), a fully numerical method was implemented using polynomial collocation methods to solve the integro-differential equation in Q(z, s) and then numerical inversion to obtain values of R(z,t). The analytical (exact) solution provided in Knessl and Peters (1994) was used for testing purposes and computation of the true errors. In this paper a method for the direct numerical solution of equations (2.1)-(2.2) is presented which does not use Laplace transforms. In particular, multiquadric approximation methods are used (see for example Powell (1987, 1992) for an introduction to the subject) extending the line of approach of Bonzani (1997) as applied to PDEs. The description of the method is given in next section.

5 Boundary Element Technology Description of the multiquadric method for solving (2.1)-(2.2) We are going to use multiquadric approximations (cf. Kansa (1990a,b), Makroglou (1994)) for the numerical solution of (2.1)-(2.2) following the approach of Bonzani (1997) who applied sine and Lagrange type approximations to second order semilinear partial differential equations in one space dimension. For fixed t, the following approximation (see also Kansa (1990a,b), to the unknown function R(z, t) of equation (2.1) is used: n+l (t)fi,(z), (3,1) where and where 0 = z\ <... < Zn < Zn+i = 2, * > 0, and c is the multiquadric parameter. The boundary condition g(t) = R(z, t) was found using the true solution. This boundary condition was used to express dn+i(t) in (3.1) in terms of a\ ( ),..., a,n(t). In practice, a z value has to be chosen to represent the oo, so that the given boundary condition lim^oo R(z^ t) = I can be used instead. The coefficients a,j(t) are then to be estimated so that (3.1)-(3.3) satisfies the equation (2.1)-(2.2), giving afirstorder system of ordinary differential equations in aj (t) of the form where 5, F are n x n matrices with elements s-'fa + S-'M*), (3.4) Skj = Pj(zk), 771=1 kj = l,2,...,n. (3.5)

6 500 Boundary Element Technology and b(t) is given by where k + XhY ' -^ m=l & = l,2,...,n (3.6) (3.7) To obtain equations (3.5) we also used a quadrature rule with weights Wkm and stepsize h for the evaluation of the integrals J** Pj(zk - y)b(y)dy and To solve the ODE system (3.4) we also need starting values a(0). These are found using the second of the conditions (2.2) for B, that is R(z, 0) = l,z > 0 and (3.1). We thus obtain the equations (3.8) After we solve the ODE system (3.4)-(3.8) for the values of t we need, the function R(z, t) is evaluated from (3.1)-(3.3). For other methods applied to hyperbolic partial differential equations of higher order which include error estimates (but no numerical results) we refer for example to Yanik and Fairweather (1988), Pani, Thomee and Wahlbin (1992), Fairweather (1994). 4 Numerical results Numerical results are presented for the solution R(z, t),z 1.0,0 < t < 10 of (2.1)-(2.2) using with a = 1.0, A) = 2.0, A = 7 = 1.0. The true solution is given (Knessl and Peters (1994)) as, (4.1)

7 Boundary Element Technology 501 (4.2) The results of method COL-LAP in Table 1 (Makroglou (1998)) were obtained using polynomial collocation methods with h = 0.1 and collocation parameters 0o = 0,0i = 1, which makes the method identical to that resulting from the use of the trapezoidal method, in combination with the ACM routine 619 (Piessens (1984)) with tolerance equal to 1.D - 4 for the inversions of the Laplace transforms. The results of the MQ method in Table 1 were obtained using the direct multiquadric method presented in this paper. Table 4.1: Results for R(l,t) t exact R(l,t) COL - LAP true error 0.74D D D D - 4 MQ,c = true error 0.25D J D-2

8 502 Boundary Element Technology The results for the direct MQ method were obtained using Matlab 5.0 with the ODE23 rourine for the solution of the system of differential equations for obtaining the a?(f),j = 1,...,n. The MQ approximations used n = 10. The integrals were evaluated using the trapezoidal quadrature rule with mesh points Z{, i 1,..., k and stepsize h 0.1. Prom Table 1 we may see that the results obtained by the MQ method and the collocation method of Makroglou (1998) are in good agreement with the exact solution of Knessl (1994) with these obtained by the collocation method considerably smaller. The accuracy of the MQ method might improve if the integrals were evaluated to convergence. That way, one could easily change the value of n and get results for different MQ approximations easily. Also, using different values of the c parameter for different t values or certain methods of estimation (cf. Golberg, Chen and Karur (1996)) might benefit the MQ method. Tests with different ODE solvers is also worth considering. 5 Conclusions The estimation of the probability offinitetime non - ruin R(z, t) was obtained by solving directly the governing integro-differential equation by using multiquadric approximations. The direct treatment of the problem described here and also the fully numerical indirect approach of Makroglou (1998), allow the easy extension for other types of distribution function B(x) and premiums function 0(z) whenfindingthe exact solutions for R or the corresponding Laplace transforms of R is not easy or possible, and it is hoped to be useful for the actuarial society. Global multiquadric approximations were used with respect to the "space" variable z. Next line of approach to follow would be the use of multiquadric approximations with compact support (cf. Wendland (1995, 1998)) or to extend other methods which are in use for the numerical solution of first order hyperbolic partial differential equations (see for example Morton and Mayers (1994, chapter 4) for an introduction). Keywords: Partial integro-differential equations,firstorder, numerical solution, multiquadrics, actuarial risk management. References [1] Arfwedson, G., Some problems in the collective theory of risk, Skand. Aktuar. Tidskr, 33 (1950), 1-38.

9 Boundary Element Technology 503 [2] Bonzani, I., Solution of nonlinear evolution problems by parallelized collocation-interpolation methods, Computers Math. Applic., 34 (1997), [3] Bade, J.P., The ruin problem for mixed Poisson risk processes, Scand. Actuarial J., (1983), [4] Fairweather, G., Spline collocation methods for a class of hyperbolic partial integro-differential equations, SIAM J. Numer. Anal., 31 (1994), [5] Golberg, M.A., Chen, C.S., Karur, S.R., Improved multiquadric approximation for partial differential equations, Engng Anal, with Boundary Elements, 1996, [6] Grandell, J., Aspects of risk theory, Springer-Verlag, [7] Harper, W., Numerical treatment of the problem of collective ruin, Final year project, School of Computer Science and Mathematics, Div. of Mathematics and Statistics, University of Portsmouth, UK, [8] Harper, W., On the estimation of the survival probability for the problem offinitetime collective non-ruin, M.Sc. thesis, School of Computer Science and Mathematics, Div. of Mathematics and Statistics, University of Portsmouth, UK, [9] Kansa, E.J., Multiquadrics - A scattered data approximation scheme with applications to computational fluid dynamics-i, Surface approximations and partial derivatives estimates, Computers Math. Applic., 19 (1990a), [10] Kansa, E.J., Multiquadrics - A scattered data approximation scheme with applications to computational fluid dynamics-ii, Solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers Math. Applic., 19 (1990b), [11] Knessl, C. and Peters, C.S., Exact and asymptotic solutions for the time dependent problem of collective ruin I, SIAM J. Appl. Math., 54 (1994), [12] Knessl, C. and Peters, C.S., Exact and asymptotic solutions for the time-dependent problem of collective ruin II, SIAM. J. Appl. Math., 56 (1996), [13] Makroglou, A., Radial basis functions in the numerical solution of nonlinear Volterra integral equations, J. Applied Science and Computations, 1 (1994),

10 504 Boundary Element Technology [14] Makroglou, A., Computer treatment of the problem of collective nonruin; the finite time case, contributed talk: 18th International Congress on Computational and Applied Mathematics (ICCAM '98), July 27 - August 1, 1998, Katholieke Universiteit Leuven, Belgium, and Conference HERCMA '98, September, 1998, AUEB, Athens, Greece. [15] Morton, K.W. and Mayers, D.F., Numerical solution of partial differential equations, Cambridge University Press, [16] Pani, A.K., Thomee, V. and Wahlbin, L.B., Numerical methods for hyperbolic and parabolic integro-differential equations, J. Integral Equations, 4 (1992), [17] Peters, C.S. and Mangel, M., New methods for the problem of collective ruin, SIAM J. Appl. Math., 50 (1990), [18] Piessens, R., Alg. 619, Automatic numerical inversion of the Laplace transform, ACM Trans. Math. Soft., 10 (1984), [19] Powell, M.J.D., Radial basis functions for multivariable interpolation: a review, pp in: Algorithms for Approximation, Mason, J.C. and Cox, M.G. (Eds), Clarendon Press, [20] Powell, M.J.D., The theory of radial basis functions in 1990, pp in: Advances in Numerical Analysis II: Wavelets, Subdivision, and Radial Basis Functions, W. Light (Ed.), Oxford University Press, [21] Seal, H.L., The numerical calculation of 7(w;,t), the probability of non-ruin in an interval (0,t), Scand. Actuarial J., (1974), [22] Seal, H.L., Survival probabilities. The goal of risk theory, John Wiley and Sons, [23] Wendland, H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in Computational Mathematics, 4 (1995), [24] Wendland, H., Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), [25] Yanik, E.G. and Fair weather, G., Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Analysis, Theory, Methods and Applications, 12 (1988), Acknowledgement. The author of this article would like to thank Professor C.S. Chen of the University of Nevada, for inviting her to the Conference BETECH 99 and for providing her with additional references on the theory and applications of multiquadric approximations.