Some remarks on stochastic differential equations in the plane with local Lipschitz coefficients (Q1083765)

The author considers the stochastic differential equation \[ dX_ z=A(X_ z)dW_ z+B(X_ z)dz \] for a vector-valued process \((X_ z;z\in {\mathbb{R}}_+^ 2)\), \((W_ z;z\in {\mathbb{R}}_+^ 2)\) being a Wiener process. She gives bounds for the moments of \(\sup_{z}| X_ z|\) in terms of the initial condition \((\alpha_ z;z\in {\mathbb{R}}_+^ 2)\) and shows that X depends regularly on \(\alpha\). Moreover she proves existence and uniqueness of a strong solution X up to a stopping line under local Lipschitz conditions on A and B. Finally she gives a sufficient condition in order that no explosions occur. Related results, also for other integrators than the Wiener process, are proved by \textit{M. L. Puri} and the reviewer in Stochastics 17, 19-41 (1986; Zbl 0592.60045).