A stochastic maximum principle for a stochastic differential game of a mean-field type
type: Item
instance of: scholarly article
title: A stochastic maximum principle for a stochastic differential game of a mean-field type
zbMATH Open document ID: 1261.49005
author: John Joseph Absalom Hosking
published in: Applied Mathematics and Optimization
publication date: 2013-02-18T00:00:00Z
full work available at URL: A stochastic maximum principle for a stochastic differential game of a mean-field type by John J. A. Hosking.pdf
review text: 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.
Mathematics Subject Classification ID: 49K45
Mathematics Subject Classification ID: 49N70
Mathematics Subject Classification ID: 91A15
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 6136945
zbMATH Keywords: stochastic maximum principle
zbMATH Keywords: stochastic differential games
zbMATH Keywords: Nash equilibria
zbMATH Keywords: mean-field type stochasic equations
zbMATH Keywords: mean-field type backward stochastic equations
zbMATH Keywords: adjoint processes
MaRDI profile type: Publication
OpenAlex ID: W2087989961
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DOI: 10.1007/S00245-012-9177-X
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description: scientific article; zbMATH DE number 6136945