The first integrals for systems of stochastic differential equations with jumps. (Ukrainian, English) Zbl 1199.60215

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

Teor. Jmovirn. Mat. Stat. 76, 84-91 (2007); translation in Theory Probab. Math. Stat. 76, 93-101 (2008).

The authors deal with the system of homogeneous stochastic differential equations \[ d\xi(t)=a(\xi(t))dt+\sum_{k=1}^nb_k(\xi(t))dw_k(t) +\int_{\Theta_1}c(\xi(t-),\theta)\tilde{\nu}(dt,d\theta) +\int_{\Theta_2}c(\xi(t-),\theta){\nu}(dt,d\theta),\,\xi(t_0)=x_0,\, t_0\geq0, \] \(x_0=(x_{10},x_{20},\dots,x_{n0})\), \(a(x)=(a_i(x),i=1,\dots,n)\), \(b_k(x)=(b_{ik}(x),i=1,\dots,n)\), \(c(x,\theta)=(c_i(x,\theta),i=1,\dots,n)\), \(\theta\in\Theta=\Theta_1\cup\Theta_2\), \(\Theta_1\cap \Theta_2=\emptyset\), are real valued nonrandom vector functions, \(w_k(t)\) are independent one-dimensional Wiener processes, \(\nu([0,t],A)\) is a Poisson measure, \(E\nu([0,t],A)=t\Pi(A)\), \(\tilde{\nu}(dt,d\theta)=\nu(dt,d\theta)-\Pi(d\theta)dt\), \(\Pi(\Theta_1)=\infty\), \(\int_{\Theta_1}|\theta|^2\Pi(d\theta)<\infty\), \(\Pi(\Theta_2)<\infty\).
A notion of the first integral for this system of homogeneous stochastic differential equations is introduced. The necessary and sufficient conditions are given that a function \(G(x)\) is a first integral of the system. A relationship between first integrals of the considered systems of homogeneous stochastic differential equations with jumps and those for some systems of ordinary differential equations is described. This extends some known results for Itô’s stochastic differential equations.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

stochastic differential equation; first integral; Itô’s stochastic differential equation