We study the scale function of the spectrally negative phase-type Lévy process. Its scale function admits an analytical expression and so do a number of its fluctuation identities. Motivated by the fact that the class of phase-type distributions is dense in the class of all positive-valued distributions, we propose a new approach to approximating the scale function and the associated fluctuation identities for a general spectrally negative Lévy process. Numerical examples are provided to illustrate the effectiveness of the approximation method.

We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions.2000 Mathematics Subject Classification. 60G40, 60J60, 93E20.

ABSTRACT. Sustaining efficiency and stability by properly controlling the equity to asset ratio is one of the most important and difficult challenges in bank management. Due to unexpected and abrupt decline of asset values, a bank must closely monitor its net worth as well as market conditions, and one of its important concerns is when to raise more capital so as not to violate capital adequacy requirements. In this paper, we model the tradeoff between avoiding costs of delay and premature capital raising, and solve the corresponding optimal stopping problem. In order to model defaults in a bank's loan/credit business portfolios, we represent its net worth by Lévy processes, and solve explicitly for the double exponential jump diffusion process and for a general spectrally negative Lévy process.

We study two practical optimization problems in relation to venture capital investments and/or Research and Development (R&D) investments. In the first problem, given the amount of the initial investment and the cash flow structure at the initial public offering (IPO), the venture capitalist wants to maximize overall discounted cash flows after subtracting subsequent investments, which keep the invested company solvent. We describe this problem as a mixture of singular stochastic control and optimal stopping problems. The singular control corresponds to finding an optimal subsequent investment policy so that the value of the investee company stays solvent. The optimal stopping corresponds to finding an optimal timing of making the company public. The second problem is concerned with optimal dividend policy. Rather than selling the company at an IPO, the investor may want to harvest technological achievements in the form of dividend when it is appropriate. The optimal control policy in this problem is a mixture of singular and impulse controls. . We thank Savas Dayanik for valuable comments and are most grateful to Anthea Au Yeung, Ikumi Koseki, Tōru Masuda, and Akihiko Yasuda for insight and good memories through helpful and enjoyable conversations.

We consider stochastic impulse control problems where the process is driven by a general onedimensional diffusions. Impulse control problems are widely used to financial engineering/decisionmaking problems such as dividend payout problem, portfolio optimization with transaction costs, and inventory control. We shall show a new mathematical characterization of the value function as a linear function in certain transformed space. Our approach can (1) relieve us from the burden of guessing and proving the optimal strategy, (2) present a simple method to find the value function and the corresponding control policies, and (3) handle systematically a broader class of reward and cost functions than the conventional methods of quasi variational inequalities, especially because the existence of the finite value function can be shown in much a simpler way. This paper proposes a general solution method of stochastic impulse control problems for one dimensional diffusion processes. Stochastic impulse control problems have attracted a growing interest of many researchers for the last two decades. Under a typical setting, the controller faces some underlying process and reward/cost structure. There exist continuous and instantaneous components of reward/cost functions. By exercising impulse controls, the controller moves the underlying process from one point to another. At the same time, the controller receives rewards associated with the instantaneous shifts of the process. Then the controller's objective is to maximize the total discounted expected net income.