Maximum Principle for Non-Zero Sum Stochastic Differential Game with Discrete and Distributed Delays
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Maximum Principle for Non-Zero Sum Stochastic Differential Game with Discrete and Distributed Delays∗ ZHANG Qixia
DOI: 10.1007/s11424-020-9068-1 Received: 1 March 2019 / Revised: 18 October 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract This technical note is concerned with the maximum principle for a non-zero sum stochastic differential game with discrete and distributed delays. Not only the state variable, but also control variables of players involve discrete and distributed delays. By virtue of the duality method and the generalized anticipated backward stochastic differential equations, the author establishes a necessary maximum principle and a sufficient verification theorem. To explain theoretical results, the author applies them to a dynamic advertising game problem. Keywords Distributed delay, generalized anticipated backward stochastic differential equations, maximum principle, Nash equilibrium point, non-zero sum stochastic differential game.
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Introduction
Differential game involves multiple players decision making in the context of dynamical systems. The study of differential games could be traced to the pioneering work of Isaacs[1] , and his work inspired much further researches and interests in this area. Among them, Nash’s theory[2] is now recognized as one of the outstanding results, and Nash equilibrium became a fundamental and important notion. Due to wide applications in economics and finance, the stochastic differential game theory attracted many researchers and was developed rapidly (see, for example, [3–6]). In the classical case, we use the stochastic differential equations (SDEs) to describe stochastic differential game systems. But there are also many phenomena whose behavior depend not only on the present but on the past. Such models can be simulated by stochastic differential delayed equations (SDDEs) and have wide range of applications (see [7–11] and the references therein). The infinite dimensional problem and the absence of Itˆ o’s formula to deal with the ZHANG Qixia School of Mathematical Sciences, University of Jinan, Jinan 250022, China. Email: [email protected]. ∗ This research was supported by the National Natural Science Foundation of China under Grant No. 11701214 and Shandong Provincial Natural Science Foundation, China under Grant No. ZR2019MA045. This paper was recommended for publication by Editor YOU Keyou.
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ZHANG QIXIA
delay part of SDDEs make the stochastic control problems with delay more complex. Recently, with the establishment of the theory of anticipated backward stochastic differential equations (ABSDEs, [12]), Chen and Wu[13] developed a necessary maximum principle and a sufficient verification theorem for stochastic systems involving delays in both the state variable and the control variable. Yu[14] , and Chen and Yu[15] obtained the Pontryagin’s maximum principle for delayed stochastic optimal control problems with random coefficients involving both continuous and impulse controls and the delayed non-zero sum stochastic differential ga
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