It is well known that the nonlinear filter has important applications in military, engineering and commercial industries. In this paper, we propose efficient and accurate numerical algorithms for the realization of the Yau-Yau method for solving nonlinear filtering problems by using finite difference schemes. The Yau-Yau method reduces the nonlinear filtering problem to the initial-value problem of Kolmogorov equations. We first solve this problem by the implicit Euler method, which is stable in most cases, but costly. Then, we propose a quasi-implicit Euler method which is feasible for acceleration by fast Fourier transformations. Furthermore, we propose a superposition technique which enables us to deal with the nonlinear filtering problem in an off-time process and thus, save a large amount of computational cost. Next, we prove that the numerical solutions of Kolmogorov equations by our schemes are always nonnegative in each iteration. Consequently, our iterative process preserves the probability density functions. In addition, we prove convergence of our schemes under some mild conditions. Numerical results show that the proposed algorithms are efficient and promising.
This paper is concerned with a mean-reversion trading rule. In
contrast to most market models treated in the literature, the underlying market
is solely determined by a two-state Markov chain. The major advantage of
such Markov chain model is its striking simplicity and yet its capability of
capturing various market movements. The purpose of this paper is to study
an optimal trading rule under such a model. The objective of the problem
under consideration is to nd a sequence stopping (buying and selling) times
so as to maximize an expected return. Under some suitable conditions, explicit
solutions to the associated HJ equations (variational inequalities) are obtained.
The optimal stopping times are given in terms of a set of threshold levels. A
verication theorem is provided to justify their optimality. Finally, a numerical
example is provided to illustrate the results.
In this paper, a brief introduction of the nonlinear filtering problems and a review of the quasi-implicit Euler method are presented. The major contribution of this paper is that we propose a nonnegativity-preserving algorithm of Yau-Yau method for solving high-dimensional nonlinear filtering problems by applying quasi-implicit Euler method with discrete sine transform. Furthermore, our algorithms are directly applicable on the compact difference schemes, so that the number of spatial points can be substantially reduced and retain the same accuracy. Numerical results indicate that the proposed algorithm is capable of solving up to six-dimensional nonlinear filtering problems efficiently and accurately.
Chen S, Min J, Teng J, et al. Inventory and shelf-space optimization for fresh produce with expiration date under freshness-and-stock-dependent demand rate[J]. Journal of the Operational Research Society, 2016, 67(6): 884-896.
Abdelhak Mezghiche · Mustapha Moulai · Lotfi Tadj. Model Predictive Control of a Forecasting Production System with Deteriorating Items. 2015.
Xiaoyan Xu · Yanan Ji · Yiwen Bian · Yanhong Sun. Service outsourcing under co-opetition and information asymmetry. 2016.
Minghui Xu · Xiaode Zuo. An optimal dynamic advertising model with inverse inventory sensitive demand effect for deteriorating items. 2016.
This paper studies the inventory management problem of dual channels operated by one vendor. Demands of dual
channels are inventory-level-dependent.We propose a multi-period stochastic dynamic programming model which
shows that under mild conditions, the myopic inventory policy is optimal for the infinite horizon problem. To
investigate the importance of capturing demand dependency on inventory levels, we consider a heuristic where the
vendor ignores demand dependency on inventory levels, and compare the optimal inventory levels with those
recommended by the heuristic. Through numerical examples, we show that the vendor may order less for dual
channels than those recommended by the heuristic, and the difference between the inventory levels in the two cases
can be so large that the demand dependency on inventory levels cannot be neglected. In the end, we numerically
examine the impact of different ways to treat unmet demand and obtain some managerial insights.