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This paper discusses how to solve filtering problem for a class of continuous nonlinear time-varying systems via Duncan-Mortensen-Zakai (DMZ) equation. In this paper, the original DMZ equation is changed into Kolmogorov forward equation (KFE) by exponential transformations in each time interval, and then under some assumptions, the KFE can be transformed into time-varying Schr\"odinger equation which can be solved explicitly. The novelty of this paper lies in how to transform the KFE into Schr\"odinger equation. As a direct application, the results of \cite{yau 2004} are extended for time-varying Yau systems.

This paper focuses on pricing and vertical cooperative advertising decisions in a two-tier supply chain. Using a Stackelberg game model where the manufacturer acts as the game leader and the retailer acts as the game follower, we obtain closed-form equilibrium solution and explicitly show how pricing and advertising decisions are made. When market demand decreases exponentially with respect to the retail price and increases with respect to national and local advertising expenditures in an additive way, the manufacturer benefits from providing percentage reimbursement for the retailer’s local advertising expenditure when demand price elasticity is large enough. Whether the manufacturer benefits from cooperative advertising is also closely related to supply chain member’s relative advertising efficiency. In the decision for adopting coop advertising strategy, it is critical for the manufacturer to identify how market demand depends on national and local advertisements. The findings from this research can enhance our understanding of cooperative advertising decisions in a two-tier supply chain with price-dependent demand.

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.