A Partially Observed Nonzero‐Sum Stochastic Differential Game with Delays and its Application to Finance

Abstract

In this paper, we deal with a new kind of partially observed nonzero-sum differential game governed by stochastic differential delay equations. One of the special features is that the controlled system and the utility functionals involve both delays in the state variable and the control variables under different observation equations for each player. We obtain a maximum principle and a verification theorem for the game problem by virtue of Girsanov's theorem and the convex variational method. In addition, based on the theoretical results and Malliavin derivative techniques, we solve a production and consumption choice game problem.

Stochastic maximum principle for partially observed risk‐sensitive optimal control problems of mean‐field forward‐backward stochastic differential equations

“…This highlights the importance of studying partially observed optimal control problems and makes them widely used. Therefore, optimal control problem for partially observed risk‐neutral case has become a hot topic for decades, see References 49‐54 and references therein. Besides, forward‐backward SDEs have applications in various fields, including economics, mathematical finance, science, and engineering; see Carmona, Delarue, 55 Ma, Yong 56 .…”

“…(a) We mention that our article can be reviewed as an extension of References 49‐54, whereas they studied only the partially observed risk‐neutral problem.…”

Forward‐backward doubly stochastic differential equations with random jumps and related games

“…Yu [29] researched the linear‐quadratic nonzero‐sum differential game problem of the forward‐backward stochastic system. Yang et al [30] studied a new kind of partially observed nonzero‐sum differential game governed by stochastic differential delay equations. In this paper, inspired by the above results, we give the explicit form of a Nash equilibrium point for our backward doubly stochastic LQ game problem.…”

Infinite horizon optimal control for mean‐field stochastic delay systems driven by Teugels martingales under partial information

“…For instance, the population growth model 21 and the production and consumption choice game problem. 22 Shen et al 23 considered the optimal control of mean-field jump-diffusion systems with pointwise and average delays, where the adjoint equation consists of three-coupled jump-diffusion MF-BSDEs. Compared with the work of Shen et al, 23 Meng and Shen 24 discussed a more general case that the control variable is also required to pointwise delay both in the controlled system and the cost functional, and using a different method for the system, the adjoint equation is an MF-ABSDE with Poisson jump.…”