This dissertation is concerned with stochastic optimization problems, especially stochastic impulse control and optimal stopping. This consists of three parts. In the first part, a new characterization of the value function of stochastic impulse control problem for one-dimensional diffusion is presented as a linear function in some transformed space. Based on this characterization, a new solution method is described in details. These results not only clarify mathematical principle that covers a general set of problems but also contain practical value in the literature of financial engineering because this new method enables one to solve a broad set of problems, facilitating proofs of the finiteness of the value function and its optimality.; In the second part, optimal stopping problems are considered in the setting of asset management contract between the investors and the asset manager. The manager invests the fund entrusted by the investors in some credit risky portfolio subject to downward jumps. The manager's optimal time to terminate the contract is presented explicitly under the given reward/cost structure. At the same time, economic value of limited protection for the investors is calculated.; In the third part, a combined problem of stochastic impulse control and optimal stopping is considered in the setting of venture capital investments, given the initial investment and reward schedule at the initial public offering (IPO) market. The impulse control part is related to the venture capitalist's subsequent investments, if necessary, to avoid bankruptcy of the invested start-up company, and the optimal stopping part addresses the problem of choosing the best timing of selling, at the IPO market, the venture capitalist's interest in the company. The quasi closed-form solution is presented and its optimality is proved.; It should be noted that in the last two parts, the underlying process includes stochastic jumps that make the problems more difficult to solve.
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