Before revision:  The surplus dynamics obeys a linearly controlled drifted Brownian motion that is reflected at zero , dividends give rise to time-dependent instantaneous marginal profits, whereas capital injections are subject to time-dependent instantaneous marginal costs.
After revision:  The surplus dynamics obeys a linearly controlled drifted Brownian motion that is reflected at the origin , dividends give rise to time-dependent instantaneous marginal profits, whereas capital injections are subject to time-dependent instantaneous marginal costs.