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Based on stochastic curvilinear integrals in the Cairoli-Walsh sense and inthe It\^{o}-Udri\c{s}te sense, we develop an original theory regarding themultitime stochastic differential systems. The first group of the originalresults refer to the complete integrable stochastic differential systems, thepath independent stochastic curvilinear integral, the It\^{o}-Udri\c{s}testochastic calculus rules, examples of path independent processes, andvolumetric processes. The second group of original results include themultitime It\^{o}-Udri\c{s}te product formula, first stochastic integrals andadjoint multitime stochastic Pfaff systems. Thirdly, we formulate and we provea multitime maximum principle for optimal control problems based on stochasticcurvilinear integral actions subject to multitime It\^{o}-Udri\c{s}te processconstraints. Our theory requires the Lagrangian and the Hamiltonian asstochastic 1-forms.

Multitime stochastic maximum principle on curvilinear integral actions