There are three chapters of manuscripts in this dissertation and all of them are talking about a speciﬁc theme: stochastic control, but with completely diﬀerent perspectives.
In the ﬁrst manuscript, we solve the optimal barrier strategy for dividend distribution in a complicated Lévy system. In this system, the capital of the company ﬂuctuates with one-sided jumping Lévy noise and the macroeconomic indicator, interest rate, is al-lowed to go negative and move with possibly inﬁnite jumping frictions in a short time. Both of them represent the current status of ﬁnancial insurance markets. The main techniques used, Itô excursion theory and ﬂuctuation identities of Lévy process, are purely probabilistic.
In the second manuscript, we take a look at a classical Hamilton-Jacobian-Bellman equation related to dividend distribution. It needs some technical analysis to ﬁnd the analytical solution but here we develop a novel numerical method: projected semismooth Newton with shooting-like method, which solves the approximate solution with desired superlinear convergence rate. It suggests a new way to solve numerically the constrained free-boundary variational inequality.
In the last manuscript, we come up with a new economic model on eﬃciently utilizing the ﬂexibility of renewable energy in its market. There are several pioneering considerations in this model. On the one hand, the electricity price and reservoir storage both ﬂuctuate based on sophisticated stochastic models. On the other hand, we could sell the excessive hydropower to market while we could buy the electricity from market in case of low reservoir level. We propose a “trigger price” strategy to manage the ﬂexibility of hydropower so as to optimize the net proﬁt. In this manuscript, we ﬁnd one explicit strategy in a classical setting and prove the existence and uniqueness of some intricate strategy in a complicated setting via viscosity solution technique.