A Lagrange Method For Stochastic Conditional Extremum And Application Of Hidden Markov Model

Stock forecasting is a classic problem in financial markets.However,due to the instability and seasonality of stock trend,the prediction of stock price is influenced by a lot of uncertainty.The price of a stock depends on many factors that may be hidden from investors.The hidden Markov model can model the hidden state transition according to the ordered observation data,it is a model with double random process.Therefore,the hidden Markov model is suitable for stock price prediction.This paper studies the application of the hidden Markov model to stock prediction.Learning of hidden Markov model can be viewed as a constrained optimization problem.For example,Baum Welch algorithm is often used to find the model parameters.However,in practical problems,we will encounter the situation that the objective function and constraint function can not be accurately obtained,so how to solve the problem of conditional extremum using observations of objective function and constraint function with errors is particularly important.In this paper,a stochastic Lagrange multiplier method for equality constrained optimization is investigated,and almost sure convergence of the method is obtained by applying stochastic approximation.Aiming at solving the problem of equality constrained optimization with noise by stochastic approximation and taking advantages of the hidden Markov model in stock forecasting,the main results of this paper are summarized as follows:(1)Stochastic conditional extremum problem is considered,where the objective function,constraint function and their gradients are unknown,and their observations are corrupted by errors.A stochastic approximation algorithm is given by using Lagrange multiplier method.Under the condition that the Lagrangian is convex in the primal variable,it is proved that the algorithm converges almost surely to a saddle point of the Lagrangian.The performance of the algorithm is verified by a simulation.(2)This paper demonstrates the theory of hidden Markov model and applies it to American stock market.The demonstration part includes the definition of the model,the three basic problems and classic algorithms to solve the related problems.In the application part,the hidden Markov model is applied to the US stock market.Firstly,the stock market state predicted by the Gaussian hidden Markov model is obtained,and then the purchase operation of the next day after the hidden state result has been obtained is simulated.The simulation results show that when the state of the stock market is rising,buying can get earnings,that is to say,the model can provide a good reference for stock enthusiasts whether to buy or sell a stock.