Dynamic insurance models with investment: constrained singular problems for integrodifferential equationsстатья
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Дата последнего поиска статьи во внешних источниках: 6 декабря 2018 г.
Аннотация:Previous and new results are used to compare two mathematical insurance models with
identical insurance company strategies in a financial market, namely, when the entire current surplus
or its constant fraction is invested in risky assets (stocks), while the rest of the surplus is invested in a
risk-free asset (bank account). Model I is the classical Cramér–Lundberg risk model with an exponential
claim size distribution. Model II is a modification of the classical risk model (risk process with
stochastic premiums) with exponential distributions of claim and premium sizes. For the survival
probability of an insurance company over infinite time (as a function of its initial surplus), there arise
singular problems for second-order linear integrodifferential equations (IDEs) defined on a semiinfinite
interval and having nonintegrable singularities at zero: model I leads to a singular constrained
initial value problem for an IDE with a Volterra integral operator, while II model leads to a more complicated
nonlocal constrained problem for an IDE with a non-Volterra integral operator. A brief overview
of previous results for these two problems depending on several positive parameters is given, and
new results are presented. Additional results are concerned with the formulation, analysis, and numerical
study of “degenerate” problems for both models, i.e., problems in which some of the IDE parameters
vanish; moreover, passages to the limit with respect to the parameters through which we proceed
from the original problems to the degenerate ones are singular for small and/or large argument values.
Such problems are of mathematical and practical interest in themselves. Along with insurance models
without investment, they describe the case of surplus completely invested in risk-free assets, as well as
some noninsurance models of surplus dynamics, for example, charity-type models.