Strong completeness for a class of stochastic differential equations with irregular coefficients. (English) Zbl 1307.60074

Chen, Xin; Li, Xue-Mei

Electron. J. Probab. 19, Paper No. 91, 34 p. (2014).

Summary: We prove the strong completeness for a class of non-degenerate SDEs, whose coefficients are not necessarily uniformly elliptic nor locally Lipschitz continuous nor bounded. Moreover, for each \(p>0\) there is a positive number \(T(p)\) such that for all \(t<T(p)\), the solution flow \(F_t(\cdot)\) belongs to the Sobolev space \(W_{\mathrm{loc}}^{1,p}\). The main tool for this is the approximation of the associated derivative flow equations. As an application a differential formula is also obtained.

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

strong completeness; stochastic differential equation; derivative flow equation; approximation; differential formula