Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLinear-Quadratic Stochastic Differential Games on Directed Chain Networks
116
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study linear-quadratic stochastic differential games on directed chains inspired by the directed chain stochastic differential equations introduced by Detering, Fouque, and Ichiba. We solve explicitly for Nash equilibria with a finite number of players and we study more general finite-player games with a mixture of both directed chain interaction and mean field interaction. We investigate and compare the corresponding games in the limit when the number of players tends to infinity. The limit is characterized by Catalan functions and the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain, with or without the presence of mean field interaction.
rate research
Read More
The study of linear-quadratic stochastic differential games on directed networks was initiated in Feng, Fouque & Ichiba cite{fengFouqueIchiba2020linearquadratic}. In that work, the game on a directed chain with finite or infinite players was defined as well as the game on a deterministic directed tree, and their Nash equilibria were computed. The current work continues the analysis by first developing a random directed chain structure by assuming the interaction between every two neighbors is random. We solve explicitly for an open-loop Nash equilibrium for the system and we find that the dynamics under equilibrium is an infinite-dimensional Gaussian process described by a Catalan Markov chain introduced in cite{fengFouqueIchiba2020linearquadratic}. The discussion about stochastic differential games is extended to a random two-sided directed chain and a random directed tree structure.
The paper studies the open-loop saddle point and the open-loop lower and upper values, as well as their relationship for two-person zero-sum stochastic linear-quadratic (LQ, for short) differential games with deterministic coefficients. It derives a necessary condition for the finiteness of the open-loop lower and upper values and a sufficient condition for the existence of an open-loop saddle point. It turns out that under the sufficient condition, a strongly regular solution to the associated Riccati equation uniquely exists, in terms of which a closed-loop representation is further established for the open-loop saddle point. Examples are presented to show that the finiteness of the open-loop lower and upper values does not ensure the existence of an open-loop saddle point in general. But for the classical deterministic LQ game, these two issues are equivalent and both imply the solvability of the Riccati equation, for which an explicit representation of the solution is obtained.
In this paper, we study the solvability of anticipated backward stochastic differential equations (BSDEs, for short) with quadratic growth for one-dimensional case and multi-dimensional case. In these BSDEs, the generator, which is of quadratic growth in Z, involves not only the present information of solution (Y, Z) but also its future one. The existence and uniqueness of such BSDEs, under different conditions, are derived for several terminal situations, including small terminal value, bounded terminal value and unbounded terminal value.
In this Note, assuming that the generator is uniform Lipschitz in the unknown variables, we relate the solution of a one dimensional backward stochastic differential equation with the value process of a stochastic differential game. Under a domination condition, a filtration-consistent evaluations is also related to a stochastic differential game. This relation comes out of a min-max representation for uniform Lipschitz functions as affine functions. The extension to reflected backward stochastic differential equations is also included.
The paper is concerned with adapted solution of a multi-dimensional BSDE with a diagonally quadratic generator, the quadratic part of whose $i$th component only depends on the $i$th row of the second unknown variable. Local and global solutions are given. In our proofs, it is natural and crucial to apply both John-Nirenberg and reverse Holder inequalities for BMO martingales.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
