d\Lambda(t) = dB(t), \ \ d\Lambda(t)
d\Lambda(t) = d\Lambda(t), \ \ d\Lambda(t) dB^\ast(t)=dB^\ast(t). \label{eq:Ito-sp}$$

The dynamics of the composite system is described by a unitary $U(t)$ solving the Schrödinger equation, or quantum stochastic differential equation (QSDE), $$dU(t) = \{ (S-I)d\Lambda(t) + L dB^\ast (t)- L^\ast S dB(t) - (\frac{1}{2}
L^\ast L +iH )dt \} U(t),  \label{eq:unitary}$$ with initial condition $U(0)=I$. Here, $H$ is a fixed self-adjoint operator representing the free Hamiltonian of the system, and $L$ and $S$ are system operators determining the coupling of the system to the field, with $S$ unitary. In this paper, for simplicity we assume that the parameters $S,L,H$ are bounded operators on the system Hilbert space $\mathfrak{H}_S$. However, we remark under some suitable additional conditions the results and equations obtained in this paper should also be extendable to some special classes of QSDEs with unbounded parameters, exploiting the results in [@Fagno90; @FW03].

A system operator $X$ at time $t$ is given in the Heisenberg picture by $X(t)=j_{t}( X) =U( t) ^{\ast } ( X\otimes I ) U( t) $ and it follows from the quantum Ito calculus that $$\begin{aligned}
dj_{t}( X) &=&j_{t}(S^{\ast }XS-X) d\Lambda ( t) +j_{t}(S^{\ast }[X,L]) dB(t) ^{\ast }  
\notag \\
&& 
