Noncommutative $L^p$-integrators and stochastic differential equations in $L^p$

Source abstract

We propose a new framework for noncommutative stochastic integration in $L^p$, inspired by Bichteler's notion of $L^p$-integrators, that massively generalizes our previous theory of $L^2$-valued stochastic integration against $L^2$-decomposable processes. Central to our framework is a noncommutative analog of a predictable integrand we call a predictable linear process. We develop basic properties of stochastic integrals against "noncommutative $L^p$-integrators" and establish a useful criterion for a noncommutative stochastic process to be a noncommutative $L^p$-integrator. This criterion applies, in particular, to a rich class of processes we call measured decomposable processes, which includes free Brownian motion and, more generally, the $q$-Brownian motions. We also show that, when specialized appropriately, our framework recovers the classical notion of an $L^p$-integrator up to the fact that the noncommutative theory sees only the modification class of a classical stochastic process. As an application, we use our theory to study noncommutative stochastic differential equations (SDEs) in $L^p$. Under natural local-Lipschitz and boundedness assumptions, we establish the existence and uniqueness of maximal solutions and show that a maximal solution with finite lifetime must blow up in $L^p$ norm. As corollaries, we obtain new existence and uniqueness results for free SDEs and SDEs driven by $q$-Brownian motion. Notably, in the free case, our results apply to coefficients arising through the continuous functional calculus from merely locally Lipschitz scalar functions, as opposed to locally operator-Lipschitz functions.