Communications on Stochastic Analysis Communications on Stochastic Analysis

. We study the continuity property of multiple Q-adapted quantum stochastic integrals with respect to noncommuting integrands given by the non-adapted multiple integral kernels in Fock scale. The noncommutative algebra of relatively (exponentially) bounded nonadapted quantum stochastic processes is studied in the kernel form as introduced in [10]. The di(cid:11)erential Q-adapted formula generalizing It^o product formula for adapted integrals is presented in both strong and weak sense as a particular case of the quantum stochastic nonadapted It^o formula.

[1]  P.J.C. Spreij.  Stochastic Integration , 2021, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[2]  V. Belavkin.  Continuous non-demolition observation quantum filtering and optimal estimation , 2005, quant-ph/0509205.

[3]  L. Streit,et al.  Generalized Functionals in Gaussian Spaces: The Characterization Theorem Revisited☆ , 1996, math/0303054.

[4]  V. Belavkin.  Quantum stochastic calculus and quantum nonlinear filtering , 1992, math/0512362.

[5]  V. Belavkin.  Chaotic states and stochastic integration in quantum systems , 1992, math/0512265.

[6]  V. Belavkin.  A quantum nonadapted Ito formula and stochastic analysis in Fock scale , 1991, math-ph/0512076.

[7]  V. Belavkin.  A new form and a ⋆-algebraic structure of quantum stochastic integrals in Fock space , 1988 .

[8]  David Nualart,et al.  Stochastic calculus with anticipating integrands , 1988 .

[9]  K.R Parthasarathy,et al.  Stochastic Integral Representation of Bounded Quantum Martingales in Fock Space , 1986 .

[10]  Robin L. Hudson,et al.  Quantum Ito's formula and stochastic evolutions , 1984 .

[11]  A. Skorokhod.  On a generalization of the stochastic integral , 1976 .

[12]  Christopher Bergevin,et al.  Brownian Motion , 2006, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[13]  J. Maassen,et al.  The stochastic calculus of Bose noise , 1992 .

[14]  V. Belavkin.  A quantum stochastic calculus in fock space of input and output nondemolition processes , 1990 .

[15]  J. Lindsay.  On set convolutions and integral-sum kernel operators , 1990 .

[16]  Viacheslav P. Belavkin,et al.  Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes , 1989 .

[17]  P. Malliavin.  Stochastic calculus of variation and hypoelliptic operators , 1978 .

[18]  Alain Guichardet,et al.  Symmetric Hilbert spaces and related topics , 1972 .