矩陣指數跳躍擴散的最佳停止問題

Abstract

在本論文之中， 我們在矩陣指數型跳躍擴散隨機過程以及一般報酬函數之下，考慮最佳停止問題。任給一個報酬函數，遵循平均問題的方法， (請參照 Alili and Kyprianou [1], Kyprianou and Surya [16], Novikov and Shiryaev [23], and Surya [28] )， 對於所對應之平均問題的解，我們提供了明確的公式。 透過此明確的公式， 對於美式最佳停止問題，我們得到最佳的執行邊界和最佳的報酬。 此外，在矩陣指數型跳躍擴散隨機過程之下，我們也考慮永久美式複合選擇權定價問題。遵循 Gapeev and Rodosthenous [12]，對於跳躍擴散的隨機過程而言，原先兩步的最佳停止問題可分解成，單步最佳停止問題的序列。在雙重指數型跳躍擴散模型之下，對於永久美式複合選擇權而言，我們得到了明確的定價公式。 藉由我們的方法，我們也涵蓋了Gapeev and Rodosthenous [12] 的結果。

In this dissertation, we consider the optimal stopping problems for a general class of reward functions under the matrix-exponential jump-diffusion processes. Given the reward function in this class, following the averaging problem approach(see, for example, Alili and Kyprianou [1], Kyprianou and Surya [16], Novikov and Shiryaev [22], and Surya [27] ), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal boundary and the value function for the American optimal stopping problems. Also, we consider the pricing problems of perpetual American compound options under the matrix-exponential jump-diffusion processes. Following Gapeev and Rodosthenous [12], the initial two-step optimal stopping problems are decomposed into sequences of one-step problems for the underlying jump-diffusion process. In the double-exponential jump-diffusion model, we obtain the explicit pricing formula for the perpetual American compound option pricing problems. By our approach, we also recover results obtained in Gapeev and Rodosthenous [12]