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Optimal Dividend Strategy in a Jump-Diffusion Model with a Linear Barrier Constraint

For a financial or insurance entity, the problem of finding theoptimal dividend distribution strategy and optimal firm value function is a widely discussedtopic. In the present paper, it is assumed that the firm faces two types of liquidity risks:a Brownian risk and a Poisson risk. The firm can control the time and amount of dividendspaid out to shareholders. By sufficiently taking into account the safety of the company,bankruptcy is said to take place at time $t$ if the cash reserve of the firm runs belowthe linear barrier b+kt (not zero), see 1. We deal with the problem of maximizingthe expected total discounted dividends paid out until bankruptcy. The optimal dividendreturn (or, firm value) function is identified as the classical solution of the associatedHamilton-Jacobi-Bellman (HJB) equation where a second-order differential-integro equationis involved. By solving the corresponding HJB equation, the analytical solution of theoptimal firm value function is obtained, the optimal dividend strategy is also characterized,which is of linear barrier type: at time t the firm keeps cash inside when the cashreserves level is less than a critical linear barrier and pays cash in excess ofthis linear barrier as dividends.