MC Methods For A Class Of Stochastic Convex Programming And Its Applications To Finance

This dissertation studies Monte Carlo (MC) methods for a class of stochastic convexprogramming and their applications to some special financial optimization problems, andresearches mainly the convergence of the optimal solution of the sequence sample averageapproximation (SAA) problems. The results obtained in this dissertation are summarizedas follows:1. Chapter 2 studied Monte Carlo methods for solving three special classes of the stochas-tic smooth convex programming. It is proved that the optimal solution of the sequenceSAA unconstrained problems is convergent to one of the true problems since the gra-dient of the SAA problems converges to the gradient of the true problems; for both theproblem with the expected value function as an inequality constraint and the problemwith deterministic constraints, the cluster point of the optimal solutions of the se-quence SAA problems is proved be the solution to the true problems with probabiltyone (w.p.1). Moreover, the optimality conditions of each problem are given.2. Chapter 3 studied Monte Carlo methods to four special classes of the stochastic non-smooth convex programming. We proved that the optimal solutions of the sequenceSAA unconstrained problem converge to one of the true problem w.p.1 via the ex-change between the expectation and the directional derivative. For the stochasticnonsmooth convex programming with a expecstion function as an inequality con-straint, under the assumption of the unique optimal solution, the cluster point of theoptimal solutions of the sequence SAA problems is almost surely the optimal solutionto the true problem. A special stochastic nonsmooth convex programming from thefinancial optimization problems is proposed, its optimality condition and the confi-dent interval of its optimal value are presented. The Monte Carlo penalty functionmethod is used to solve the stochastic nonsmooth convex programming with a ex-pected value affine function as an equality constraint, the cluster point of the optimalsulotions of the sequence SAA problems is almost surely the optimal solution to thetrue problem. Moreover, the optimality conditions of each problem are given. 3. In Chapter 4, the results obtained in the chapter 2 and chapter 3 are used to solve fourspecial financial optimization problems, which include the log-portfolio problem withthe sub-semivariance as a risk measure, the log-portfolio problem with the CVaR asa risk measure, the optimal reinsurance of the single-insured with the CVaR as a riskmeasure, the optimal reinsurance of the multi-insured with the CVaR as a risk mea-sure. Some numerical experiments are presented. The exact optimal solutions to thethird problem is obtained, the approximate nonsmooth problem of the fourth problemis solved via a certain smooth method and the upper bound of the approximate erroris given.