Performance Optimization Of Stochastic Process Based On Sensitivity Analysis Approach

Stochastic optimal problem is one important part of control theory.Relative modeling and applications are getting more and more attention in the academia and industry because of the universality.The thesis mainly introduces two optimal approaches of stochastic optimal control-dynamic programming and sensitivity approach,and then proposed that due to the unique resolving thought the sensitivity approach is more applicable and efficient when solving complex problems.This work studies three problems in portfolio selection: non-expected utility optimization with distorted probabilities,constrained linear quadratic optimal control problem,and optimization of stock trading with additional information by Limit Order Book,by utilizing the sensitivity analysis based approach.The sensitivity analysis based approach emphasizes the relationship between the performance variation and the control variable’s change on the back-stepping method.By analyzing the relationship of the work obtains the most important formula-Performance Difference Formula(PDF).In the optimization of portfolio selection,in order to balance returns and risks,the expected utility theory(EUT)is often used to model investors’ thoughts.The traditional expected utility theory is introduced into distorted probabilities,which becomes non-expected utility theory with distorted probability(distorted utility,for short),to describe the intuition of investors: tending to exaggerate the impact of rare events and reduce the impact of common events.Because distorted utility is nonlinear and lose the time-consistency property,iterative expectations are invalid.We demonstrate that after changing the underlying probability measure to the distortion function with investors’ subjective information,distorted utility is no more non-linear in probabilities.We call this property Mono-linearity.Based on the sensitivity analysis and performance potential,the search optimization algorithm is derived.Thereby,the optimal strategy for the optimization of distorted probability non-expected utility is obtained.This work provides a new direction for the distorted utility,expands the applications,and gives more choices for many portfolio optimization problems,which could only be solved by mean-variance model in the past.Meanwhile,it also reflects the huge advantages of stochastic optimization methods based on sensitivity analysis in nonlinear optimization.Because linear quadratic control optimal model has the graceful structure and the characteristic of closed loop feedback control system,it is widely used in the industry.Experts have carried out extensive research into Stochastic Linear Quadratic problem.However,regulators have many constraints for the financial products(shares,bonds,etc.),such as no shorting,boundary constraints for capital positions.The work adds the conditional constraints to the tradition SLQ model.Because the constraints destroy the graceful structure of LQ model,the analytical form of optimal strategy can not directly obtained.The thesis applies the sensitivity analysis based approach to solve the SLQ problem in the finite horizon.A general solution and performance difference formula of SLQ problem are developed by introducing three auxiliary problems and using the sensitivity analysis based approach.Then we prove that the method is applicable in the infinite horizon and develop the optimal strategies in the finite and infinite horizons.This work makes SLQ problems fit the reality of portfolio selection well and shows the advantages of sensitivity approach in the stochastic linear quadratic control optimal problems with constraints.In recent years,the impact of financial market micro-structure on trading behavior and asset prices has become an increasingly important hot topic in the academic and industrial circles.From the perspective of the trading microstructure,electronic trading has already occupied a dominant position in the world’s financial markets.Statistics show that in 2010 58% of US stocks were traded by algorithms.A number of new issues have emerged,such as,how to allocate assets optimally by using market microstructure when investors conduct high-frequency trading.Specifically,using external information as an observation,a direct comparison method is used to derive the optimal strategy.The conclusion is further extended to more general optimizing problems with observations.Poisson equation,Dynkin formula and performance difference formula in the finite time domain are derived based on the sensitivity analysis.Then optimal conditions are obtained through HJB equation.The work provides a new direction for optimization problems with external information and reflects the advantages of system optimization method based on sensitivity analysis in multi-variable optimization problems.