On global in time solutions of stochastic algebraic-differerential equations with forward mean derivatives

The paper is devoted to the investigation of the completeness property of the flows generated by the stochastic algebraic-differential equations given in terms of forward Nelson's mean derivatives. This property means that all solutions of those equations exist for all $t\in[0,\infty)$. It is very important for the description of qualitative behavior of the solutions. This problem is new since previously it was investigated for equations given in terms of symmetric mean derivatives. The case of forward mean derivatives requires different methods of investigation and the cases of forward and symmetric mean derivatives have different important applications. We find conditions under which all solutions of stochastic algebraic-differential equations given in terms of forward Nelson's mean derivatives, exist for all $t\in[0,\infty)$. Some obtained conditions are necessary and sufficient.