Noise suppresses explosive solutions of differential systems: A new general polynomial growth condition

Recently, Liu and Shen (2012) have discussed the problem of suppression explosive solutions by noise for nonlinear deterministic differential system with coefficients satisfying a general one-sided polynomial growth condition. Liu and Shen introduced the Brownian noise feedback |x(t)|β∑x(t)dB(t)|x(t)|β∑x(t)dB(t) to the nonlinear differential system, and they showed that the corresponding stochastic system has a unique global solution. However, Liu and Shen's condition is still very restrictive and many more general nonlinear differential systems fail. This paper aims to introduce a new more general polynomial growth condition which can be applied to more nonlinear differential systems. Under the new polynomial growth condition, we show that the considered stochastic system has a unique global solution although the corresponding deterministic system may explode in a finite time. Moreover, we show that the flexible stochastic feedback g(x(t))g(x(t)) can guarantee the solution not only be limited in the sense of moment but also grows at most polynomially. Finally, we provide two examples to illustrate superiority of the new general polynomial growth condition.