On an extension of Lyapunov criterion of stability for quasi-linear systems via integral inequalities method. (English) Zbl 1074.60069

Du, Nguen Huu

Teor. Jmovirn. Mat. Stat. 70, 26-35 (2004) and Theory Probab. Math. Stat. 70, 29-40 (2005).

Let \(X_t(x)\) and \(Y_t(x)\) be solutions to stochastic differential equations with the initial conditions \(x\) and \(y\), respectively. Let \(M\) be a class of positive continuous functions on \([0,\infty)\). The trivial solution \(X\equiv 0\) is said to be more stable than \(Y\equiv 0\) in the comparing class \(M\) if for all \(q\in M\) the convergence \[ \lim_{y\to 0}\Pr\{| Y_t(y)| <q_t\;\forall t\geq 0\}=1 \] implies that \[ \lim_{x\to 0}\Pr\{| X_t(x)| <q_t\;\forall t\geq 0\}=1. \] The author describes conditions under which a quasi linear equation \(dX_t=(A_t X_t+f(t,X_t))dt+(B_t X_t+g(t,X_t))dW_t\) is more stable than the equation \(dY_t=a(t,Y_t)dt+\sigma(t,Y_t)dW_t\).

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05	Ordinary differential equations and systems with randomness

Keywords:

stochastic differential equation; growth rate; stochastic dynamical system