Quantum \(\Omega\)-semimartingales and stochastic evolutions (Q5953427)

Let \({\mathbf H}= \Gamma_+({\mathbf h})\) be Boson Fock space over \({\mathbf h}= L^2[0,\infty)\). \({\mathbf H}\) can be considered as the completion of the collection of exponential vector \(\varepsilon:\text{lin}\{\varepsilon(u):u\in{\mathbf h}\}\) with respect to the inner product \(\langle\varepsilon(u), \varepsilon(v)\rangle_H:= \exp(\langle u,v\rangle_h)\). The vacuum vector is defined by \(\Omega:= \varepsilon(0)\). A bounded process \(F\) is \(F= (F(t): t> 0)\subset{\mathbf B}({\mathbf H})\), satisfying some conditions. The conditional expectation process \(E= (E_{t]}:\geq 0)\) is defined. A process \(F\) is \(\Omega\)-adapted if \(F(t)= E_{t]}F(t) E_{t]} \forall t\geq 0\).NEWLINENEWLINENEWLINEQuantum stochastic integrals for \(\Omega\)-adapted processes are considered. It is shown that the quantum stochastic integrals of bounded \(\Omega\)-adapted processes are bounded and \(\Omega\)-adapted. \(\Omega\)-semimartingales are defined and studied. A polynomial Itô formula for \(\Omega\)-semimartingales is obtained. Quantum stochastic differential equations with bounded \(\Omega\)-adapted coefficients that are time dependent and act on the whole Fock space, are defined and studied. Solutions of such equations may be used to dilate quantum dynamical semigroups. Multidimensional processes are considered as well. The paper gives new insight into previous papers on quantum stochastic analysis.