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This work concerns the two-step Maruyama schemes for nonlinear stochastic differential delay equations (SDDEs). We first examine the strong convergence rates of the split two-step Maruyama scheme and linear two-step Maruyama scheme (including Adams-Bashforth and Adams-Moulton schemes) for nonlinear SDDEs with highly nonlinear delay variables, then we investigate the exponential mean square stability and exponential decay rates of the two classes of two-step Maruyama schemes. These results are important for three reasons: first, the convergence rates are established under the non-global Lipschitz condition; second, these stability results show that these two-step Maruyama schemes can not only reproduce the exponential mean square stability, but also preserve the bound of Lyapunov exponent for sufficient small stepsize; third, they are also suitable for the corresponding two-step Maruyama methods of stochastic ordinary differential equations (SODEs).

The adoption of new technologies often represents a crucial component of firms' investment decisions. This paper studies a dynamic duopoly model in which two firms compete in adoption of current technology with a further new technology anticipated. Here it is assumed that the operating costs are not zero which has more explanatory power of the real world. There exist three kinds of equilibria that may occur in adoption of current technology, which mainly depends on the level of operating costs and the first-move advantage. It shows that the faster technological substitution or innovation encourages the leader to invest earlier while induces the follower to invest later. Furthermore,like the investment costs, with the increase of operating costs the follower tends to invest later while the leader tends to invest earlier,the investment thresholds are more sensitive to the change of operating costs than that of investment costs. w 1 IntroductionTechnological innovation has become one of the key sources of firms' competition advantages in the market. The adoption of new technologies often represents a crucial component of firms' investment strategies. Technology adoption is a set of practices and factors related to organization selecting,deploying and sustaining the use of a technology.A firm that invests today often faces the risk that a much better technology will become available tomorrow, which provides an incentive to delay the investment. Especially, when a new technology is at least irreversible,the option of value of delay may be significant,so a firm considering investment in this technology must wrestle with a difficult tradeoff of whether to adopt now or later. For example,in the dynamic information technology (IT) industry,many corporations are mainly interested in early adoption of a current technology which may generate experience and knowledge for access to subsequent adoption of a new technology. Received : 2005-06-23. MR Subject Classification : 90A09,90All, 62L15,60G40,60J60.The investment in technology adoption is usually irreversible and faces an uncertainty setting. The firm that owns technological innovation can often have the monopoly position in market within some time, so it is very fierce for firms to compete in adoption of new technologies. When a firm has the opportunity to adopt a new uncertain technology,this opportunity can be viewed as a real option to the firm. By analogy with a financial call option,it is optimal to delay exercising the option in the hope of gaining a higher payoff in future. * Applying this insight, the real options approach improves the traditional netpresent-value(NPV) method by introducing the flexibility on decisionmaking (See,e. g. , McDonald and Siegel (1986) and [3-]). An excellent and elaborate overview of the real options approach is found in [2-] and Trigeorgis (1996). A collection of the recent work in this area is provided by Grenadier E4~.The first paper important to our work is the contribution of Fudenberg and Tirole (1985) in which they have studie...