We revisit the dividend payment problem in the dual model of Avanzi et al. ([3], [2], and [4]).Using the fluctuation theory of spectrally positive Lévy processes, we give a short exposition in which we show the optimality of barrier strategies for all such Lévy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [4] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.

Abstract. We introduce a stochastic version of the classical Perron's method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub-and super-) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a martingale along the solutions of the stochastic differential equation. The unique viscosity solution is actually equal to the expected payoff. This amounts to a verification result (Itô's Lemma) for non-smooth viscosity solutions of the linear parabolic equation.

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We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power‐law and exponential‐decay order book models explicitly and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.

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We analyze the optimal dividend payment problem in the dual model under constant transaction costs. We show, for a general spectrally positive Lévy process, an optimal strategy is given by a (c 1 , c 2 )-policy that brings the surplus process down to c 1 whenever it reaches or exceeds c 2 for some 0 ≤ c 1 < c 2 . The value function is succinctly expressed in terms of the scale function. A series of numerical examples are provided to confirm the analytical results and to demonstrate the convergence to the no-transaction cost case, which was recently solved by Bayraktar et al. [8].

We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled diffusion evolving in a multi-dimensional Euclidean space. In this game, the controller affects both the drift and diffusion terms of the state process, and the diffusion term can be degenerate. Under appropriate conditions, we show that the game has a value and the value function is the unique viscosity solution to an obstacle problem for a Hamilton-Jacobi-Bellman equation.Key Words: Controller-stopper games, weak dynamic programming principle, viscosity solutions, robust optimal stopping. over all choices of τ . At the same time, however, the controller plays against her by maximizing (1.1) over all choices of α.Ever since the game of control and stopping was introduced by Maitra & Sudderth [25], it has been known to be closely related to some common problems in mathematical finance, such as pricing American contingent claims (see e.g. [17,21,22]) and minimizing the probability of lifetime ruin (see [5]). The game itself, however, has not been studied to a great extent except certain particular cases. Karatzas and Sudderth [20] study a zero-sum controller-and-stopper game in which the state process X α is a one-dimensional diffusion along a given interval on R. Under appropriate conditions they prove that this game has a value and describe fairly explicitly a saddle point of Date: January 6, 2013. We would like to thank Mihai Sîrbu for his thoughtful suggestions. We also would like to thank the two anonymous referees whose suggestions helped us improve our paper.