A deterministic criterion for approximate controllability of stochastic differential equations with jumps

This paper investigates the approximate controllability and approximate null controllability of a class of linear stochastic systems driven by Gaussian random measures. The analysis focuses on controlled systems featuring both deterministic and stochastic components, where the control acts on the drift and jump terms. We establish the equivalence between approximate controllability and approximate null controllability by introducing an invariant subspace $V$, defined by the system’s parameters. The controllability of the system is shown to hinge on whether $V$ reduces to the trivial space $\left\{0\right\}$. These findings provide a unified framework for understanding the controllability properties of stochastic systems with jump and diffusion dynamics.