On the asymptotic degeneration of systems of linear inhomogeneous stochastic differential equations. (Ukrainian, English) Zbl 1199.60213

Il’chenko, Oleksander

Teor. Jmovirn. Mat. Stat. 76, 38-44 (2007); translation in Theory Probab. Math. Stat. 76, 41-48 (2008).

The author deals with the following system of differential equations \[ dx(t)=(A_0x(t)+f_0(t))dt+\sum_{k=1}^{m}(A_kx(t)+f_k(t))dW_k(t), \] where \(A_k\) are \(n\times{n}\) matrices; \(f_k(t)=(f_{k1}(t),\dots,f_{kn}(t)), t\geq0\), are vector functions; \(W_k(t),t\geq0\), are independent one-dimensional Wiener processes; \(x(t)=(x_1(t),\dots,x_n(t)),t\geq0\), is a solution to the equations.
Let \(H_s^t\) be a stochastic semigroup of nondegenerate operators such that \(dH_s^t=A_0H_0^tdt+\sum_{k=1}^mA_kH_s^tdW_k(t), H_s^s=I,s\leq t.\) It is assumed that the semigroup \(H_s^t\) is stable with probability one. A solution \(x(t),t\geq0,\) to the system of equations can be represented in the form \[ x(t)=H_0^tx+\int_0^t(H_u^t)\left(f_0(u)-\sum_{k=1}^mA_kf_k(u)\right)du+\sum_{k=1}^mH_0^t\int_0^t(H_0^u)^{-1}f_k(u)dW_k(u). \] The author obtains sufficient conditions for the convergence to zero, in probability as well as pathwise, of solutions \(x(t)\) of the indicated system of linear inhomogeneous stochastic differential equations.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

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

Keywords:

inhomogeneous stochastic differential equation; stochastic semigroup; nondegenerate operator; convergence in probability