Stein Method In Choosing Importance Weight

Importance sampling is a widely used statistical method.Its core idea is artificially modifying the probability distribution of sampling so that if a part of sample has a significant contribution to the estimation results,it will appears frequently.Compared with other sampling methods,importance sampling method has the advantage of reducing variance.However,the importance sampling method also has its own defects,such as it is difficult to calculate the analytic solution of the importance distribution,and the path dependence problem occurs when solving problems with multiple time nodes occasionally.To solve such problems we introduce Stein method,this method was put forward in the 1970s.It creatively constructed Stein discrepancy to define the similarity of two distributions,which can help us measure sample quality.The main purpose of this paper is to improve some defects of importance sampling method with the advantages of Stein method.Firstly,we makes some improvements on these two methods respectively,we propose a new variable to estimate the sample size required in importance sampling for sample distribution with large variance so that our method has a wider application space,then we prove the theorem about constructing Stein operator with Ito diffusion process so that we are able to enhance the theoretical basis of Stein method and obtain a form of Stein operator.Secondly,we combine the traditional Stein method with kernel function to give the characterization of the kernel Stein discrepancy,and construct the Stein importance sampling method.The importance distribution of this method is given by the optimization process to minimize the kernel Stein discrepancy between the sample distribution and the target distribution.After the construction of Stein importance sampling method,we theoretically prove the feasibility of Stein importance sampling method,by comparing the convergence rate of this new algorithm with that of the traditional importance sampling method,we prove that the new algorithm is superior to the traditional importance sampling method.Then we give the KKT condition for calculating the importance distribution,which is the basis for us to calculate the importance distribution by numerical calculation.Finally,through the comparative analysis of option pricing simulation examples,we prove that the Stein importance sampling method can further reduce the estimation variance.Furthermore,we explain how this method can solve the possible path dependence problem of traditional importance sampling method.