A note on the controllability of jump diffusions with linear coefficients

We investigate the approximate controllability property for a class of linear stochastic equations driven by independent Brownian motion and Poisson random measure. The paper generalizes recent results of Buckdahn et al. (2006, A characterization of approximately controllable linear stochastic differential equations. Stochastic Partial Differential Equations and Applications (G. Da Prato & L. Tubaro eds). Series of Lecture Notes in Pure and Applied Mathematics, vol. 245. London: Chapman & Hall, pp. 253–260) and Goreac (2008, A Kalman-type condition for stochastic approximate controllability. C. R. Math. Acad. Sci. Paris, 346, 183–188; 2009, Approximate controllability for linear stochastic differential equations in infinite dimensions. Appl. Math. Optim., 60, 105–132). An equivalent conditional invariance criterion is given. In general, this (explicit) criterion involves an L2-space, but, for particular cases, an iterative finite scheme is provided.