$L^p$-solutions of QSDE driven by Fermion fields with nonlocal conditions under non-Lipschitz coefficients

Abstract

The main purpose of this paper is to obtain the existence and uniqueness of $L^p$-solution to quantum stochastic differential equation driven by Fermion fields with nonlocal conditions in the case of non-Lipschitz coefficients for $p>2$. The key to our technique is to make use of the Burkholder-Gundy inequality given by Pisier and Xu and Minkowski-type inequality to iterate instead of the fixed point theorem commonly used for nonlocal problems. Moreover, we also obtain the self-adjointness of the $L^p$-solution which is important in the study of optimal control problems.