A stochastic maximum principle for a stochastic differential game of a mean-field type
Necessary conditions for the existence of Nash equilibria in a certain N-agent stochastic differential game of a mean-field type are constructed for a state process of the form \[ X^{(u)}(\omega,t)=x_0 + \int^t_0 b(\omega. s, X^{(u)}(\omega,s), \int_ {\Omega} \Phi_b ( \omega^{\prime}.s,X^{(u)}(\omega, s), u( \omega^{\prime},s))P(d \omega^{\prime}), u( \omega, s)) ds \] \[ + \int^t_0 \sigma(\omega. s, X^{(u)}(\omega,s), \int_ {\Omega} \Phi_{\sigma} ( \omega^{\prime},s,X^{(u)}(\omega , s), u( \omega^{\prime},s))P(d \omega^{\prime}), u( \omega, s)) dB( \omega, s) \] and performance functionals \[ J_i (u)= \int_{\Omega} \Big[\int^T_0 f_{i}(\omega. s, X^{(u)}(\omega,s), \int_ {\Omega} \Phi_{f_i} ( \omega^{\prime}.s,X^{(u)}(\omega , s), u( \omega^{\prime},s))P(d \omega^{\prime}), u( \omega, s)) ds \] \[ + g_i ( \omega,X^{(u)}(\omega , T), \int_{\Omega} \Phi_{g_i} ( \omega^{\prime},s,X^{(u)}(\omega^{\prime} , T) P(d \omega^{\prime})\Big] P(d \omega), \;i=1, 2, \dots ,N \] for each \(u= (u_1, u_2, \dots , u_N)\) comprising of each agent's employed admissible controlprocess. A stochastic maximum principle (see e.g., [\textit{J. Li}, Automatica 48, No. 2, 366--373 (2012; Zbl 1260.93176)]) using second-order adjoint processes provides necessary conditions for each \(i=1, 2, \dots ,N\) is considered for the problem in its reduced one-dimentional form. An existence and uniqueness result for a class of backward stochastic equations related to adjoint processes is presented.
mean-field type backward stochastic equations
scientific article; zbMATH DE number 6136945
scientific article