Handling The Approximation Sequences Of Hermite Expansions Of Transition Densities Of Diffusions Via Stein Variational Gradient Descent

In the parametric estimation methods of diffusion models,expanding the transition density function by Hermite polynomial(proposed by Ait-Sahalia(2002)),and then obtaining an explicit closed-form expession by approximating the expansion,thus forming the approximate likelihood function,which can deal with the case of the unknown transition density function,and the ap-proximation sequence with closed-form is very accurate.However,as an approximate likelihood method,it is impossible to avoid the unnomalized problem of the approximation sequence of transi-tion density function(i.e.,over the diffusion process's domain,the integration of the approximation sequence is not equal to 1).In this paper,the state-of-the-art Stein variational gradient descen-t(SVGD)is used to handle the approximation sequence for overcoming the unnomalized barrier,thereby we can realize sampling from the nomalized approximation sequence of transition density function and further simulating the path of the diffusion process and also solving the nomalized approximate maximum likelihood estimation via generative adversarial pattern(GA-NA-MLE),which is a novel approach.In addition,we give relevant convergence results and several subsequent applications of the nomalized approximation sequence of transition density function,including:1.financial derivatives pricing.2.parametric models specification testing.3.comparing discrepancy between parametric diffusion models.