  The core of the research is to provide the explicit expression for the
expected net present values (NPVs) of double barrier strategies for regular
diffusions on the real line without solving differential equations. Under the
so-called bail-out setting, the value of the expected NPVs of an insurance
company varies according to the choice of a pair of policies, which consist of
dividend payments paid out and capital injections received. In the case of the
double barrier strategy, the expected NPVs are expressible with the help of
certain types of functions allowing explicit expression in some cases, which is
called the bivariate $q$-scale function in the article. This is accomplished by
making use of a perturbation technique in \cite{czarna2014dividend}, which
could lead to the linear equation system. In addition, a condition ensuring the
existence of an optimal (upper) barrier level is presented. In the end,
examples fitting the condition for selecting the optimal barrier are given.
