Équation différentielle stochastique (EDS) sur \(R^ N\) et sur \(R^ N\cup \{\infty \}=S_ N\). (Stochastic differential equation (SDE) on \(R^ N\) and on \(R^ n\cup \{\infty \}=S_ N)\). (French) Zbl 0686.60046

Schwartz, Laurent

Ann. Inst. Henri Poincaré, Probab. Stat. 25, No. 3, 259-263 (1989).

The semi-martingale solution \(X_ t\) of the vector stochastic differential equation \[ (*)\quad dX_ t=H(t,\omega,X_ t)dZ_ t,\quad X_ 0=x, \] driven by a given continuous semi-martingale \(Z_ t\), with the function \[ H: {\mathbb{R}}^+\times \Omega \times {\mathbb{R}}^ N\to {\mathbb{R}}^ m \] satisfying a uniform Lipschitz condition in the last variable, has been studied by P. A. Meyer [Séminaire de Probabilités XV, Univ. Strasbourg 1979/80, Lect. Notes Math. 850, 103- 117 (1981; Zbl 0461.60076)], using probabilistic methods. The author here presents a purely geometric argument for one of the key properties of the solution by showing, namely, that the flow of the equation goes to infinity uniformly as the initial point x tends to infinity.

Reviewer: M.M.Rao

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
58J65	Diffusion processes and stochastic analysis on manifolds

Keywords:

stochastic flow; martingale driving force; stochastic differential equation; semi-martingale

Citations:

Zbl 0461.60076