Impulse、singular Stochastic Optimal Control And Application

This thesis applies stochastic analysis, stochastic differential equations, martingale to study optimal stochastic control. I apply these results in the areas of of Finance, Insurance, Risk control, Income Distribution. This thesis is structured as follows:In Chapter1, we give brief introduction to the background and current situation of stochastic optimal control problem, as well as, it summarizes main research results in this thesis.In Chapter2, we study some complex analysis problems associated with differen-tial equation and obtains a series of conclusions, these conclusions will be the key point in the studying of variational equations.In Chapter3, we study a class of steady problem related to the Richard model, and make a further extension to cost functions in the model. Due to the extension, the cost function in the Richard model becomes suitable for both symmetric and nonsymmetric conditions, so that it can meet more general conditions of the model. By using methods such as stochastic analysis, advanced probability theory, the stochastic process, and modern martingale theory, the optimal analytical solution was found for the extended model, and the optimal control and control interval were derived. Because the cost function is nonsymmetric, the extended model successfully resolves problems related to stochastic control and also provides wider prospects for the application of this model.In Chapter4, we examine an optimal impulse control problem of stochastic sys-tem, whose state follows a Brownian motion. Here we want to maximum the objective function. The main feature of our model is that the controlled state process includes an impulse control governed by a Poisson process. In other words, the set of possible intervention times are discrete, random and determined by the signal process. Here we not only present a theorem giving a sufficient condition on the existence of an optimal control and its corresponding objective function, but also provide an explicit solution obtained under some simplified conditions.In Chapter5, we study availability of process deeply. It shows the connection between availability of process and sequential process. We also discuss uniformly inte-grable with process.In Chapter6, it is the specific application of stochastic optimal control theory in financial management. By analyzing the existing paid services of the financial manage-ment system and the main problems, we suggest that we should deepen revenue and expenditure management system, put forward the recommendations of implementation of the comprehensive income reporting, improve the management of bills standardized, establish automatic mechanisms for implementation of funding collection, establish the new performance appraisal system about economic value added. About paid services revenue collection, establishment of supervisory game model provide a theoretical basis for automated collection system, For income distribution, we help for the formulation of practical work in the income distribution policy.In Chapter7, we mainly expound KMV model of the basic ideas. It combined with the listed company to the actual KMV model to be amended. The article introduced into company assets value to our growth of the credit risk on the study, which analysis KMV model in our capital markets of the validity and evaluate the effectiveness of the model simply.