Nonlocality, nonlinearity, and time inconsistency in stochastic differential games. (English) Zbl 1536.91042
The article examines control in \(m\)-player nonzero-sum stochastic differential games (SDG) with time inconsistent (TIC) preferences on a time interval \([0,T]\). Each player wants to minimize his/her own cost functional resulting to the Nash equilibrium (NE) point. Time inconsistency means that NE point received at intervals \([s,T]\) and \([t,T]\) may be different on their common part. For such systems, time consistent NE is introduced as follows. Consider some player and assume that controls chosen by all other players are fixed. Then a desired control of this player over a time interval \([0,T]\) must be optimal over each small interval \([t,t+\epsilon]\) if it is chosen over \([t+\epsilon,T]\), \(0 \leq t <T\). This condition must be met for all players. In the paper, the equilibrium Hamilton-Jacobi-Bellman system is obtained that characterizes the time consistent NE of SDG. A general multidimensional Feynman-Kac formula is derived.
Considered SDG form a class of nonlocal fully nonlinear parabolic systems. The paper proves the well-posedness, i.e., existence, uniqueness and stability for the solutions to such systems. The results are first obtained for the linear cases and then extended to the quasilinear and fully nonlinear cases under suitable conditions. Two financial examples of TIC SDG are provided.
Considered SDG form a class of nonlocal fully nonlinear parabolic systems. The paper proves the well-posedness, i.e., existence, uniqueness and stability for the solutions to such systems. The results are first obtained for the linear cases and then extended to the quasilinear and fully nonlinear cases under suitable conditions. Two financial examples of TIC SDG are provided.
Reviewer: Alex V. Kolnogorov (Novgorod)
MSC:
| 91A15 | Stochastic games, stochastic differential games |
Keywords:
existence and uniqueness; Feynman-Kac formula; mathematics of behavioral economics; nonlocal nonlinear parabolic systems; stochastic differential games; time inconsistencyReferences:
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