Invariant sets for systems of stochastic differential equations without aftereffect. (English. Ukrainian original) Zbl 0990.60060

Kulinich, G. L.; Kushnirenko, S. V.

Theory Probab. Math. Stat. 63, 123-130 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 112-118 (2000).

The authors study the invariant sets for the system of stochastic differential equations \[ d\xi(t)=a(t,\xi(t))dt+\sum\limits_{k=1}^{n}b_{k}(t,\xi(t))dw_{k}(t)+ \int_{R^{n}}c(t,\xi(t),u)\nu(dt,du),\quad \xi(t_0)=x_0, \] where \(a(t,x)=(a_{i}(t,x)\), \(i=1,\ldots,n)\), \(b_{k}(t,x)=(b_{ik}(t,x),i=1,\ldots,n)\), \(c(t,x,u)=(c_{i}(t,x,u)\), \(i=1,\ldots,n)\) are real non-random functions; \(x\in R^{n}\), \(u\in R^{n}\); \(w_{k}(t)\) are independent one-dimensional Wiener processes; \(\nu([0,t),A)\) is a Poisson measure independent on \(w_{k}(t)\) with parameter \(t\Pi(A)\); \(A\) is a Borel set from \(R^{n}\); \(\Pi(R^{n})<\infty\). Let \(Q=[0,\infty)\times D\), where \(D\) is an open set in \(R^{n}\) such that \(\Pi\{u:x+c(t,x,u)\not\in D\}=0\) for all \((t,x)\in Q\). Let us denote \(\tau_{Q}(t_0,x_0)=\inf\{t\geq t_0:\xi(t)\not\in D\}\), \((t_0,x_0)\in Q\), and \(\Gamma_{Q}(G)=\{(t,x):G(t,x)=C\}\subset Q\), where sufficiently smooth function \(G(t,x)\) is defined in \(Q\) and there exist constants \(l>0, \delta>0\), such that in \(Q\) we have \[ \int_{R^{n}}|G(t,x+c(t,x,u))-G(t,x)|^{2+\delta}\Pi(du)\leq l. \] The set \(\Gamma_{Q}(G)\) is called invariant in the domain \(Q\) of the considered stochastic differential equation if for all \((t_0,x_0)\in \Gamma_{Q}(G)\) and for all \(t\geq t_0\) we have \([G(t,\xi(t))-G(t_0,x_0)]\varphi(t)=0\) a.s., where \(\varphi(t)=1\) as \(t_0\leq t<\tau_{Q}(t_0,x_0)\) and \(\varphi(t)=0\) as \(t\geq \tau_{Q}(t_0,x_0)\). The authors obtain necessary and sufficient conditions for invariance of the set \(\Gamma_{Q}(G)\). For some class of second order system the explicit form of invariant curves is presented.

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

invariant set; stochastic differential equations without aftereffect; necessary and sufficient conditions