Stability of an invariant measure of stochastic differential equation with respect to bounded pertubations of its coefficients is investigated. The results as well as some earlier author's results on Liapunov type stability of the invariant measure are applied to a system describing molecular rotation.

In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time. More precisely, we are going to show that if (Wn) is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion W and if $\left({\tau }^n\right)$ is a sequence of stopping times that converges to a stopping time τ, then the solution of the BSDE driven by Wn with random terminal time ${\tau }^n$ converges to the solution...

In this paper we give sufficient conditions under which a nonlinear stochastic differential system without unforced dynamics is globally asymptotically stabilizable in probability via time-varying smooth feedback laws. The technique developed to design explicitly the time-varying stabilizers is based on the stochastic Lyapunov technique combined with the strategy used to construct bounded smooth stabilizing feedback laws for passive nonlinear stochastic differential systems. The interest of this...

In this paper we give sufficient conditions under which a nonlinear stochastic differential system without unforced dynamics is globally asymptotically stabilizable in probability via time-varying smooth feedback laws. The technique developed to design explicitly the time-varying stabilizers is based on the stochastic Lyapunov technique combined with the strategy used to construct bounded smooth stabilizing feedback laws for passive nonlinear stochastic differential systems. The interest of this...

The present paper addresses the problem of the stabilization (in the sense of exponential stability in mean square) of partially linear composite stochastic systems by means of a stochastic observer. We propose sufficient conditions for the existence of a linear feedback law depending on an estimation given by a stochastic Luenberger observer which stabilizes the system at its equilibrium state. The novelty in our approach is that all the state variables but the output can be corrupted by noises...

Let $L$ be a parabolic second order differential operator on the domain $\overline{\Pi }=\left[0,T\right]\times ℝ.$ Given a function $\widehat{u}:ℝ\to R$ and $\widehat{x}\>0$ such that the support of $\widehat{u}$ is contained in $(-\infty ,-\widehat{x}]$, we let $\widehat{y}:\overline{\Pi }\to ℝ$ be the solution to the equation:$$L\widehat{y}=0,\widehat{y}{|}_{\left\{0\right\}\times ℝ}=\widehat{u}.$$Given positive bounds $0\<x_0\<x_1,$ we seek a function $u$ with support in $\left[x_0,x_1\right]$ such that the corresponding solution $y$ satisfies:$$y(t,0)=\widehat{y}(t,0)\forall t\in \left[0,T\right].$$We prove in this article that, under some regularity conditions on the coefficients of $L,$ continuous solutions are unique and dense in the sense that $\widehat{y}{|}_{[0,T]\times \left\{0\right\}}$ can be $C^0$-approximated, but an exact solution does not...

Let L be a parabolic second order differential operator on the domain $\overline{\Pi }=\left[0,T\right]\times ℝ.$ Given a function $\widehat{u}:ℝ\to R$ and $\widehat{x}>0$ such that the support of û is contained in $(-\infty ,-\widehat{x}]$, we let $\widehat{y}:\overline{\Pi }\to ℝ$ be the solution to the equation: $$L\widehat{y}=0,\widehat{y}{|}_{\left\{0\right\}\times ℝ}=\widehat{u}.$$ Given positive bounds $0<x_0<x_1,$ we seek a function u with support in $\left[x_0,x_1\right]$ such that the corresponding solution y satisfies: $$y(t,0)=\widehat{y}(t,0)\forall t\in \left[0,T\right].$$ We prove in this article that, under some regularity conditions on the coefficients of L, continuous solutions are unique and dense in the sense that $\widehat{y}{|}_{[0,T]\times \left\{0\right\}}$ can be C0-approximated, but an exact solution...