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SIAM J. Control Optim,2014,52(1):451–492
2014年02月06日
The purpose of this paper is to study two-person zero-sum stochastic differential games, in which one player is a major one and the other player is a group of $N$ minor agents which are collectively playing, are statistically identical, and have the same cost functional. The game is studied in a weak formulation; this means in particular that we can study it as a game of the type “feedback control against feedback control." The payoff/cost functional is defined through a controlled backward stochastic differential equation, for which the driving coefficient is assumed to satisfy strict concavity-convexity with respect to the control parameters. This ensures the existence of saddle point feedback controls for the game with $N$ minor agents. We study the limit behavior of these saddle point controls and of the associated Hamiltonian, and we characterize the limit of the saddle point controls as the unique saddle point control of the limit mean-field stochastic differential game.
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Abstract and Applied Analysis ,2014,2014():Article ID
2014年03月23日
We get a new type of controlled backward stochastic differential equations (BSDEs), namely, the BSDEs, coupled with value function. We prove the existence and the uniqueness theorem as well as a comparison theorem for such BSDEs coupled with value function by using the approximation method. We get the related dynamic programming principle (DPP) with the help of the stochastic backward semigroup which was introduced by Peng in 1997. By making use of a new, more direct approach, we prove that our nonlocal Hamilton-Jacobi-Bellman (HJB) equation has a unique viscosity solution in the space of continuous functions of at most polynomial growth. These results generalize the corresponding conclusions given by Buckdahn et al. (2009) in the case without control.
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SIAM J. Control Optim.,2014,52(3):1622–1662
2014年05月13日
In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions to the associated generalized Hamilton--Jacobi--Bellman (HJB) equations. For this we generalize the notion of stochastic backward semigroup introduced by Peng Topics on Stochastic Analysis, Science Press, Beijing, 1997, pp. 85--138. We emphasize that when $\sigma$ depends on the second component of the solution $(Y, Z)$ of the BSDE it makes the stochastic control much more complicated and has as a consequence that the associated HJB equation is combined with an algebraic equation. We prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove a new local existence and uniqueness result for fully coupled FBSDEs when the Lipschitz constant of $\sigma$ with respect to $z$ is sufficiently small. We also establish a generalized comparison theorem for such fully coupled FBSDEs.
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【期刊论文】Lp estimates for fully coupled FBSDEs with jumps
Stochastic Processes and their Applications,2014,124(4):1582-1611
2014年04月01日
In this paper we study some useful estimates, in particular, Lp estimates, for fully coupled forward–backward stochastic differential equations (FBSDEs) with jumps. These estimates are proved at one hand for fully coupled FBSDEs with jumps under the monotonicity assumption for arbitrary time intervals and on the other hand for such equations on small time intervals. Moreover, the well-posedness of this kind of equation is studied and regularity results are obtained.
Fully coupled FBSDEs with jumps Lp estimates
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Journal of Mathematical Analysis and Applications,2014,413(1):47-68
2014年05月01日
Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections. On the other hand, we will prove that this type of reflected mean-field BSDEs can also be obtained as the limit equation of the mean-field BSDEs by penalization method. Finally, we give the probabilistic interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean-field BSDEs.
Backward stochastic differential equation Mean-field approach Mean-field BSDE Reflected BSDE Penalization method Viscosity solution
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