Ergodic BSDEs under weak dissipative assumptions and application to ergodic control

Abstract: In this talk, we first introduce the notion of ergodic BSDE which arises naturally in the study of ergodic control. The ergodic BSDE is a class of infinite-horizon BSDEs:
Y_{t}^{x}=Y_{T}^{x}+∫_{t}^{T}[ψ(X^{x}_{σ},Z^{x}_{σ})-λ]dσ-∫_{t}^{T}Z_{σ}^{x}dB_{σ}, P-<K1.1/>, ∀0≤t≤T<∞,
<K1.1 ilk="TEXTOBJECT" > <screen-nom>hbox</screen-nom> <LaTeX>\hbox{a.s.}</LaTeX></K1.1> where X^{x} is a diffusion process. We underline that the unknowns in the above equation is the triple (Y,Z,λ), where Y,Z are adapted processes and λ is a real number. We review the existence and uniqueness result for ergodic BSDE under strict dissipative assumptions.
Then we study ergodic BSDEs under weak dissipative assumptions. On the one hand, we show the existence of solution to the ergodic BSDE by use of coupling estimates for perturbed forward stochastic differential equations. On the other hand, we show the uniqueness of solution to the associated Hamilton-Jacobi-Bellman equation by use of the recurrence for perturbed forward stochastic differential equations.
Finally, applications are given to the optimal ergodic control of stochastic differential equations to illustrate our results. We give also the connections with ergodic PDEs.