University of Guilan Journal of Mathematical Modeling 2345-394X 12 3 2024 09 01 Lions's partial derivatives with respect to probability measures for general mean-field stochastic control problem 517 532 7804 10.22124/jmm.2024.27136.2390 EN Fatiha Korichi Laboratory of Mathematical Analysis, Probability and Optimizations, Department of Mathematics, University of Biskra, PO Box 145, Biskra 7000, Algeria Mokhtar Hafayed Laboratory of Mathematical Analysis, Probability and Optimizations, Department of Mathematics, University of Biskra, PO Box 145, Biskra 7000, Algeria Journal Article 2024 04 02 In this paper, a necessary stochastic maximum principle for stochastic model governed by mean-field nonlinear controlled It$\rm{\ddot{o}}$-stochastic differential equations is proved. The coefficients of our model are nonlinear and depend explicitly on the control variable, the state process as well as of its probability distribution. The control region is assumed to be bounded and convex. Our main result is derived by applying the Lions's partial-derivatives with respect to random measures in Wasserstein space. The associated It$\rm{\ddot{o}}$-formula and convex-variation approach are applied to establish the optimal control. Stochastic mean-field models stochastic control Lions's partial-derivatives with respect to measures maximum principle probability measure https://jmm.guilan.ac.ir/article_7804_f1e54aa063e314582a94b019f31c9ccd.pdf