High-order Asymptotic Approximations Research Articles

Accumulated claims and collective risk in insurance: Higher order asymptotic approximations

Abstract A stochastic process modelling the reserves of an insurance company is introduced. After a slight modification, the model can be reinterpreted for the analysis of the amount of claims against an insurance company as well. The flexibility this model provides may prove useful when approaching real problems in the insurance industry. With some simplifying assumptions, it becomes possible to compute the probability distribution of the claims at each time epoch. The epression for the exact distribution becomes complicated after a long period of time has elapsed. Hence the need for asymptotic expansions arises. Conditions are found under which a higher order asymptotic expression is valid. The Berry-Esseen approach is used in the proof of this result. There are problems with establishing an expansion for densities. These are briefly discussed. A result is provided for approximating the tails of the distribution F(x; t). The result constrains the size of |x|, as it must be compared to t. Finally, an equ...

Intrinsic modes in a nonseparable ocean waveguide

Waveguide structures encountered in underwater acoustics usually lead to nonseparable boundary value problems, which do not admit definition of conventional normal modes. However, it may nevertheless be possible to construct source-free modelike wavefunctions for such configurations, which are intrinsically determined by the waveguide geometry, are mutually decoupled, and therefore propagate independently of one another when used in superposition. We call such wavefunctions intrinsic modes. For the special case of a wedge-shaped ocean with penetrable bottom, we pursue an approach similar to our previous analysis [J. M. Arnold and L. B. Felsen, J. Acoust. Soc. Am. 73, 1105–1119 (1983)] to construct the intrinsic modes explicitly. Leading order asymptotic evaluation of an intrinsic mode for small bottom slope reveals its similarity to the conventional adiabatic mode, but with a uniform transition, through cutoff, between the bound and leaky mode regions. This construction proceeds without reference to any source of the acoustic field, thereby differing from our previous treatment wherein the existence of local modes was inferred from an independently calculated asymptotic form of a line source field. Moreover, by local normal mode expansion of an intrinsic mode, with the expansion coefficients derived from the next higher order asymptotic approximation, we demonstrate explicitly that the intrinsic mode can account for coupling between adiabatic modes, with coupling coefficients that are shown elsewhere to be consistent with those from conventional coupled mode theory.