On analytic characteristic functions and processes governed by SDEs

We study the analyticity of a characteristic function of a process defined by means of SDEs. Namely, starting with the simple case of a scalar Ito SDE we show that the corresponding characteristic function is entire. The proof is based on Grönwall's inequality technique and the classic analyticity criterion in terms of moments. Further, we extend this criterion and derive a handy sufficient condition of analyticity in the multidimensional case, which is used to prove the corresponding general result. We assume that the drift vector obeys the linear growth condition and the diffusion matrix is time-only-dependent but possibly degenerate. The approach used in this article can be extended to more general types of SDEs.