  We introduce a family of risk networks composed from a) several subsidiary
branches $U_i(t), i=1,...,I$ necessary for coping with different types of
risks, which must all be kept above $0$, and b) a central branch (CB) which
bails out the subsidiaries whenever necessary. Ruin occurs when the central
branch is ruined. We find out that with one subsidiary ($I=1$), the finite time
ruin probability of the central branch may be explicitly written out in terms
of the finite time ruin probability of the subsidiary, provided that the CB in
the absence of subsidiary bailouts is a deterministic drift. To study other
problems, like for example the optimization of dividends to the CB with one
subsidiary over a barrier, it is convenient to restrict to the case of
phase-type claims to the subsidiary, and study the Markovian phase process at
the moments when the CB process reaches new minima. The resulting structure is
quite close to that of the phase of a PH/G/1 queue at the moments when it
reaches new minima, and this yields in principle numeric approaches to several
problems, based on the iterative calculation of the Laplace transforms of the
upwards and downwards ladder times (also called excursions, and busy periods).
In this work we propose a different approach of further providing matrix
exponential approximations for the distributions of the ladder times. The
advantage of this approach is that once a SNMAP approximation is obtained, many
similar problems may be solved just by applying the recently developed scale
matrix methodology. A numeric experiment for the CB barrier optimization
problem is provided.
