Some topics in stochastic control
Tyrone E. Duncan
Discrete & Continuous Dynamical Systems - B 2010, 14(4): 1361-1373 doi: 10.3934/dcdsb.2010.14.1361
Some stochastic optimal control problems in a Hilbert space are formulated and solved. The controlled equations are abstract equations in a HIlbert space that can model stochastic partial differential equations and stochastic delay equations. Both linear and semilinear equations are considered where the cylindrical Brownian motion can occur as distributed, boundary, or at discrete points in the domain. For the linear equations, the cost is an ergodic, quadratic functional of the state and the control. An optimal linear feedback control is given explicitly. For the semilinear equations, the cost is an ergodic functional. Some results for the null controllability of a stochastic parabolic equation are given. A control problem for a finite dimension linear stochastic system with an arbitrary fractional Brownian motion and a quadratic cost functional is formulated and explicitly solved.
keywords: Stochastic optimal control stochastic partial differential equations fractional Brownian motion. ergodic control
Some partially observed multi-agent linear exponential quadratic stochastic differential games
Tyrone E. Duncan
Evolution Equations & Control Theory 2018, 7(4): 587-597 doi: 10.3934/eect.2018028

Some multi-agent stochastic differential games described by a stochastic linear system driven by a Brownian motion and having an exponential quadratic payoff for the agents are formulated and solved. The agents have either complete observations or partial observations of the system state. The agents act independently of one another and the explicit optimal feedback control strategies form a Nash equilibrium. In the partially observed problem the observations are the same for all agents which occurs in broadcast situations. The optimal control strategies and optimal payoffs are given explicitly. The method of solution for both problems does not require solving either Hamilton-Jacobi-Isaacs equations or backward stochastic differential equations.

keywords: Stochastic differential games multi-agent games partially observed games risk sensitive games exponential quadratic payoffs

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