Ergodic control bellman equation with Neumann boundary conditions

摘要

Let 0 be an open bounded smooth domain of R{sup}n, and let Γ= partial deriv O be its boundary. We denote by n the normal vector at the boundary F, oriented towards the outside of O. Let us consider the canonical process Ω=C{sup}0([0, ∞]; O{top}-) y(t, ω)≡ω(t), if ω∈Ω, F{sup}t=σ(y(s), 0≤s≤t). For any x∈O{top}-, there exists a probability measure P{sup}x on Ω, a standard Wiener process in R{sup}n, w(t), such that w(t) is an F{sup}t martingale, an increasing adapted process ξ(t) such that y(t)=x+w(t)-∫(1{sub}Γ(y(s))n(y(s)))dξ(s) (ξ from 0 to t) 1{sub}O(y(t))dξ(t)=0, (arbitrary)t, a.s.. Such a measure is uniquely defined. The process ξ(t) and w(t) also uniquely defined.