**IMA Complex Systems Seminar**

**1:30, Thursday, February 5,
2004 **

On the existence and non-sample estimation of stationary densities of semimartingale reflecting Brownian motions

Wanyang Dai

Department of Mathematics,

Nanjing University,

Nanjing 210093, China

We will present algorithms to estimate the stationary densities of semimaringale reflecting Brownian motions (SRBMs) which have been demonstrated as diffusion approximation models in performance predictions and evaluations for numerous complex queueing networks. Our estimation algorithms are considered as non-sample nonparametric inference method since they do not depend on any observed sample data. When the state space for the SRBMs is a d-dimensional hypercube, we develop an approach to establish the weak convergence in certain sense for a slightly refined finite element estimator from our previous implemented algorithm (currently, complete convergence analysis is an open issue for all of existing estimators in this area). The modification is slight since it only has minor influence to the original algorithm in determining the entry values of some associated coefficient matrix by re-selecting basis functions and properly organizing them, however, the condition number for the matrix is reduced and hence the numerical stability is improved. When the state space is the d-dimensional positive orthant, we design an adaptive finite element estimator in certain sense to compute the stationary distribution and extend the developed weak convergence analysis approach to form a constructive method to establish the existence of a stationary distribution for an SRBM living in the orthant. Combining our finding with some existing achievement, the uniqueness of the stationary distribution for the SRBM is also established. All of the discussions are under a completely-S condition which is as general as the one to guarantee the existence (in distribution) of an SRBM and under a general Borel integrable condition for the stationary density of the SRBM.