  In this paper we consider a company whose assets and liabilities evolve
according to a correlated bivariate geometric Brownian motion, such as in
Gerber and Shiu (2003). We determine what dividend strategy maximises the
expected present value of dividends until ruin in two cases: (i) when
shareholders won't cover surplus shortfalls and a solvency constraint (as in
Paulsen, 2003) is consequently imposed, and (ii) when shareholders are always
to fund any capital deficiency with capital (asset) injections. In the latter
case, ruin will never occur and the objective is to maximise the difference
between dividends and capital injections.
  Developing and using appropriate verification lemmas, we show that the
optimal dividend strategy is, in both cases, of barrier type. Both value
functions are derived in closed form. Furthermore, the barrier is defined on
the ratio of assets to liabilities, which mimics some of the dividend
strategies that can be observed in practice by insurance companies. Existence
and uniqueness of the optimal strategies are shown. Results are illustrated.
