Comparison between solutions of SDEs and ODEs (Q1113197)

The author considers stochastic differential equations driven by continuous semimartingales, and in particular their behavior as time goes to \(\infty\). In the case where the solution tends to \(\infty\) (sufficient conditions for this to happen are given), the asymptotic rate at which it diverges is found in terms of a solution of a randomized ordinary differential equation. Previous results along these lines were obtained when the driving terms were Brownian motion and Lebesgue measure, and the techniques of proof were related to the fact that the solutions were strong Markov processes; thus, this paper illustrates that the behavior is essentially a continuous semimartingale phenomenon, and not a Markov one.