Almost sure stability of controlled degenerate diffusions

A converse Lyapunov theorem for almost sure stabilizability

We prove a converse Lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions: given a stochastic system a.s. stochastic open loop stabilizable at the origin, we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium. This result provides, with the direct Lyapunov theorems proved in a companion paper, a complete Lyapunov-like characterization of the a.s. stabilizability.

A Relative Value Iteration Algorithm for Non-degenerate Controlled Diffusions

The ergodic control problem for a non-degenerate controlled diffusion controlled through its drift is considered under a uniform stability condition that ensures the well-posedness of the associated Hamilton-Jacobi-Bellman (HJB) equation. A nonlinear parabolic evolution equation is then proposed as a continuous time continuous state space analog of White's `relative value iteration' algorithm for solving the ergodic dynamic programming equation for the finite state finite action case. Its convergence to the solution of the HJB equation is established using the theory of monotone dynamical syst…

Existence of an Optimal Control for a coupled FBSDE with a non degenerate diffusion coefficient

We a controlled system driven by a coupled forward-backward stochastic differential equation (FBSDE) with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation (BSDE), at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the ini…

带退化扩散的停止器与奇异控制器博弈

Stopper vs. singular-controller games with degenerate diffusions

针对退化扩散下奇异控制器与停止器的零和随机博弈，提出基于参数调节扩散扰动的概率方法，证明博弈值存在，构造出收敛到原博弈最优停时的近似最优停时。

Ergodicity Conditions For Controlled Stochastic Non-Linear Systems Under Information Constraints

Consider a stochastic nonlinear system controlled over a possibly noisy communication channel. An important problem is to characterize the largest class of channels for which there exist coding and control policies so that the closed-loop system is stochastically stable. In this paper, we consider the stability notion of (asymptotic) ergodicity. We prove lower bounds on the channel capacity necessary to achieve the stability criterion. Under mild technical assumptions, we obtain that the necessary channel capacity is lower bounded by the log-determinant of the linearization, double-averaged ov…

A variational characterization of the risk-sensitive average reward for controlled diffusions on $\mathbb{R}^d$

We address the variational formulation of the risk-sensitive reward problem for non-degenerate diffusions on $\mathbb{R}^d$ controlled through the drift. We establish a variational formula on the whole space and also show that the risk-sensitive value equals the generalized principal eigenvalue of the semilinear operator. This can be viewed as a controlled version of the variational formulas for principal eigenvalues of diffusion operators arising in large deviations. We also revisit the average risk-sensitive minimization problem and by employing a gradient estimate developed in this paper, w…