Necessary and sufficient conditions of existence and uniqueness of solutions to integral equations of actuarial mathematics

Abstract

519.21Necessary and sufficient conditions of existence and uniqueness of solutions of integral equations are established for ruin probability as a function of the initial capital of an insurance company. Several sufficient conditions and correctness conditions of the problem of finding ruin probability are also established. A general method of successive approximations for solution of this problem is substantiated. Keywords: actuarial mathematics, insurance mathematics, risk process, ruin probability, integral equation, existence and uniqueness of a solution, necessary and sufficient conditions, method of successive approximations, correctness.We call a risk process a random process x t ³ 0 describing the stochastic evolution of the capital of an insurance company and denote by x 0 = u the initial capital of the company [1]. We denote by Y( ) u the ruin probability for the insurance company as a function of u ³ 0 and by j x ( ) ( ) u u P t t

Stochastic successive approximation method for assessing the insolvency risk of an insurance company

“…As is obvious, this also takes place for a collection k = 2 3 , ,K. Now, passing to the limit with respect to N ® ¥ in inequality (18), introducing the limit under the summation sign, and taking into account inequality (20), we obtain the statement of the theorem for estimates (14). The theorem is proved.…”

“…(2) is a Volterra equation of a special form and, hence, its solution and successive approximations to the solution possess specific properties. The method of successive approximations for solving integral equations of actuarial mathematics with a view to finding the bankruptcy probability was investigated in [18][19][20] for an infinite time interval and in [21,22] for a finite time interval. In particular, as is shown in [19], operator (3) provides the first degree of uniform contraction and possesses monotonicity.…”

“…where the right side in inequality (20) tends to zero with probability one as N ® ¥ for each k by virtue of the uniform law of large numbers on the class of convex sets [17, Appendix 1]. As is obvious, this also takes place for a collection k = 2 3 , ,K. Now, passing to the limit with respect to N ® ¥ in inequality (18), introducing the limit under the summation sign, and taking into account inequality (20), we obtain the statement of the theorem for estimates (14).…”

“…However, other initial approximations can also be used in the method of successive approximations, for example, well-known analytical approximations for a ruin probability function. Moreover, the method of successive approximations can also be applied to more general risk processes [20][21][22]. In this article, the uniform convergence with probability one is proved for the stochastic method of successive approximations in which integrals are approximated by the Monte-Carlo method during each iteration.…”

Mathematical Model of Banking Operation

“…In [31], the bankruptcy probability in a complex Poisson model was estimated by the Monte Carlo method. The papers [32][33][34] consider a risk process with arrival of claims in an insurance company, described by the general process of restoration and deterministic nonlinear monotonically increasing intensity of arrival of preiums. An algorithm is presented to construct successive approximations for nonruin probability.…”