Existence and uniqueness theorem for the strong solution of Itô-Volterra stochastic differential equations with infinite delay (Q2761524)

The authors prove an existence and uniqueness theorem for the strong solution of the stochastic integro-differential equation NEWLINE\[NEWLINEdx(t)=\big[a_1(t,x^t)+\int_{t_0}^ta_2(t,s,x^s) ds+\int_{t_0}^ta_3(t,s,x^{s}) dw(s)NEWLINE\]NEWLINE NEWLINE\[NEWLINE+\int_{t_0}^{t}\int_{U}a_4(t,s,x^s,u) \widetilde\nu(du,ds)\big]dtNEWLINE\]NEWLINE NEWLINE\[NEWLINE+\big[b_1(t,x^t)+\int_{t_0}^{t}b_2(t,s,x^{s}) ds+\int_{t_0}^{t}b_3(t,s,x^{s}) dw(s)NEWLINE\]NEWLINE NEWLINE\[NEWLINE+\int_{t_0}^{t}\int_{U}b_4(t,s,x^{s},u) \widetilde\nu(du,ds)\big] dw(t)NEWLINE\]NEWLINE NEWLINE\[NEWLINE+\int_{U}\big[c_1(t,x^{t},u)+\int_{t_0}^{t}c_2(t,s,x^{s},u) ds+\int_{t_0}^{t}c_3(t,s,x^{s},u) dw(s)NEWLINE\]NEWLINE NEWLINE\[NEWLINE+\int_{t_0}^{t}\int_{U}c_4(t,s,x^{s},u,u_1) \widetilde\nu(du_1,ds)\big] \widetilde\nu(du,dt);NEWLINE\]NEWLINE \(x^{t_0}=\varphi^{t_0}\), where \(x(t)\in R^{n}\), \(x^{t}(s)=\begin{cases} x(t-s),&t_0\leq s\leq t,\\ \varphi(s-t),&s>t,\end{cases}\) \(w(t)\) is \(m\)-dimensional Wiener process; \(\widetilde\nu(du,dt)\) is a centered Poisson measure independent of \(w(t)\).