Dividends When Capital Is Slow: de Finetti’s Problem with a Sticky Boundary

Dividend models give a firm at zero surplus two fates: it is ruined, or capital arrives instantly and reflects it back into business. Neither is what happens when capital must be raised, which takes time, and while it is being raised the firm can neither pay nor operate. This paper models that interval exactly. The surplus is a Brownian motion with drift that is sticky at zero: while frozen there it is pushed off at the speed $\theta$ at which capital arrives, each unit of capital costs $\phi$, and each unit of time frozen costs $\kappa$. The capital raised is the local time of the surplus at zero, and the time frozen is that local time divided by $2\theta$ in the unit case. The equation has a weak but no strong solution, so the control problem is posed on weak solutions.

The optimal policy is a dividend barrier fixed by one explicit equation, the sticky boundary condition $\theta g_b'(0) – q g_b(0) = \phi\theta + \kappa$ applied to the smooth-fit function. Rescue is worth paying for exactly when $\theta(\bar\phi – \phi) > \kappa$, where $\bar\phi = \rho^{(\rho-1)/(\rho+1)}$ is a price ceiling for capital that depends on the model only through the ratio $\rho$ of the two characteristic roots. Below the critical speed $\theta^{\ast} = \kappa/(\bar\phi – \phi)$ the firm should be liquidated at zero and keep de Finetti’s buffer; above it the buffer falls continuously, leaving de Finetti’s level with a kink, and converges to the instant-injection barrier as $b_{LZ} + c_1/\theta$. The value lost at zero to slow capital is $\phi c_1/\theta$ to first order.

At the house parameters the critical speed is $\theta^{\ast} = 0.028906$, the ceiling is $\bar\phi = 8.1189$, the kink has slope $-10.800$ and $c_1 = 1.7351$. A lattice chain that is solved exactly and optimises its own barrier, never touching the closed forms, reproduces the optimal barrier, the value function and the critical speed, with errors of first order in the lattice spacing.