Dynamic differential games have been widely applied to the timing of product and device innovations. Uncertainty is also inherent in the process of technological innovation: R expenditures will be engaged in an unforeseeable environment and possibly lead to innovations after a random time interval. Reinganum [Reinganum, 1982] enumerates such uncertainties and risks: feasibility, delays in the process, imitation by rivals. Uncertainties generally affect the fundamentals of the standard differential game problem: discounted profit functional, differential state equations of the system, initial states. Two ways of resolution may be taken [Dockner, 2000]: firstly, stochastic differential games with Wiener process and secondly differential games with deterministic stages between random jumps (Poisson driven probabilities) of the modes. The player will then maximize the expected flows of his discounted profits subject to the stochastic state constraints of the system. In this context, the state evolution is described by a stochastic differential equation SDE (the Ito equation or the Kolmogorov forward equation KFE). According to the Dasguspta and Stiglitz’s model [Dasguspta, 1980], R efforts exert direct and induced influences (through accumulated knowledge) about the chances of success of innovations. The incentive to innovate and the R competition can be supplemented by a competition around a patent. This presentation is focused on such essential economic and managerial problems (R investments by firms, innovation process , and patent protection) with uncertainties using stochastic differential games [Friedman, 2004], [Yeung, 2006], [Kythe, 2003], modeling with It SDEs [Allen, 2007] and queueing models [Gross, 1998]. The computations are carried out using the software Mathematica 5.1 and other specialized packages [Wolfram, 2003], [Kythe, 2003].
