Linear quadratic nonzero‐sum stochastic differential game of a partially observed Markov jump linear systems

Abstract

In this study, a partially observed linear quadratic (LQ) non-cooperative stochastic differential game of Markov jump linear system (MJLS) is considered. First, a new proof of filtering theory is given, using the separation principle and filtering technology, partial observations can be transformed into complete observation. Second, nonzero-sum stochastic differential game problem for MJLS are studied using the dynamic programming principle and completion square method.Then, the sufficient and necessary conditions for the Nash equilibrium are obtained. Finally, a numerical simulation is performed.

Infinite‐time partially observed nonzero‐sum bilinear affine‐quadratic stochastic differential game and its application to competitive advertising

“…Recently, the authors have tackled the partially observed stochastic differential game on Markov jump linear system [1] and the bilinear competitive advertising problem on finite-time horizon [2,3]. The state equations (1) as follows describe the dynamics of advertising expenditure u i (t) and the market share x i (t) of player i and i = 1, 2 as there are two firms in the competition. This formulation originated from the bilinear model [4,5] and the extensions to the differential game [6,7], also the case of additive Gaussian white noise [8][9][10].…”

“…With the preset vector B [1] i 13) is identical to Equation (11), which is a time-invariant partially observed system with ( 14)-( 15), the Nash equilibrium trajectory after filtering X [1] and the solution set { û [1] i } can be derived as Corollary 13 in this paper. Then, replacing X [𝑗−1] by X [1] in Equation (12) or (13) as 𝑗 = 2, it can similarly obtain the renewed X [2] by ( 13)-( 15) with B [2] i…”

“…Lemma 6. As B [1] i △ = B i + N i X [0] is bounded with given B i , N i , and X [0] , the solution of ( 11) is estimated as follows if matrix A is Hurwitz:…”

“…(26) The estimate shows that the zero-input response decays to zero exponentially fast, while for bounded strategies u [1] i , the zero-state responses are bounded [58]. This bounded-input-bounded-state property means N i X [𝑗−1] and B [𝑗] i in ( 13) are bounded also with the bound X [𝑗−1] as 𝑗 > 2 accordingly.…”

“…Due to the conjunction of huge expenditure and latent inefficiencies, competitive advertising has always been occupying a front‐and‐center place in marketing, among which the bilinear system is one of the most important. Recently, the authors have tackled the partially observed stochastic differential game on Markov jump linear system [1] and the bilinear competitive advertising problem on finite‐time horizon [2, 3]. The state equations () as follows describe the dynamics of advertising expenditure $$$ {\boldsymbol{u}}_i(t) $$$ and the market share $$$ {\boldsymbol{x}}_i(t) $$$ of player $$$ i $$$ and $$$ i=1,2 $$$ as there are two firms in the competition.…”