Neutral stochastic differential equations driven by Brownian motion and fractional Brownian motion in a Hilbert space (Q2834184)

The authors consider the neutral stochastic differential equation in a Hilbert space \(X\) NEWLINE\[NEWLINE\begin{multlined} d[x(t)-h(t,x(t-r(t)))]=\\ [Ax(t)+f(t,x(t-\rho(t)))]\,dt+g(t,x(t-\eta(t)))\,dW(t)+ \sigma(t)\,dB^{H}(t),\end{multlined}NEWLINE\]NEWLINE \(t\geq 0\), \(x(t)=\phi(t)\in C([-\tau,0],L^ 2(\Omega,X))\), \(t\in[-\tau,0]\), \(\tau>0\), where \(\phi\) is \({\mathcal F}_0\)-measurable; \(r,\rho,\eta:\;[0,\infty)\to[0,\tau]\) are continuous; \(f,h:\;[0,\infty)\times X\to X\); \(g:\;[0,\infty)\times X\to L^0_1 (Y_1,X)\), \(\sigma:\;[0,\infty)\to L_2^0(Y_2,X)\) are Borel measurable functions; the integral w.r.t. \(Q^{(1)}\)-Brownian motion \(\{W(t)\}_{t\geq0}\) on a real separable Hilbert space \(Y_1\) is an Itô integral; and the integral w.r.t. \(Q^{(2)}\)-fractional Brownian motion \(\{B^{H}\}_{t\geq0}\) on a real separable Hilbert space \(Y_2\) is a Wiener integral with Hurst parameter \(1/2<H<1\); \(A\) is the infinitesimal generator of an analytic semigroup \(\{S(t)\}_{t\geq0}\) on \(X\). The following existence and uniqueness theorem is proved: If \(A\) is the infinitesimal generator of an analytic semigroup of bounded linear operator \(\{S(t)\}_{t\in[0,T]}\) on \(X\) and \(0\in \rho(-A)\), where \(\rho(-A)\) is the resolvent set of \(-A\); the functions \(f\), \(g\) satisfy the Lipschitz and linear growth conditions; for the function \(h\) there exist positive constants \(C_{i}\), \(i=1,2\) and \(\alpha\in (1/2,1)\) such that \(h\) is \(D((-A)^{\alpha})\)-valued and satisfies, for all \(t\in[0,T]\) and \(x,y\in X\): NEWLINE\[NEWLINE\| (-A)^{\alpha}h(t,x)-(-A)^{\alpha}h(t,y)\|^2_{X}\leq C^2_1\| x-y\|^2_{X},NEWLINE\]NEWLINE NEWLINE\[NEWLINE\|(-A)^{\alpha}h(t,x)\|^2_{X}\leq C^2_2(1+\| x\|^2_{X}),NEWLINE\]NEWLINE \(C_1\|(-A)^{\alpha}\|<1\); the function \((-A)^{\alpha}h\) is continuous in the quadratic mean sense in time; the function \(\sigma\) satisfies \(\int_0^{T}\|\sigma(s)\|^2_{L_2^0}\,ds<\infty\) \(\forall T>0\), then for all \(T>0\) the considered neutral stochastic differential equation has a unique mild solution on \([-r,T]\) in the mean-square sense. Also the authors establish the sufficient conditions to ensure exponential stability for the solution.