Algebraic groups as difference Galois groups of linear differential equations

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field C(x) with derivation [Formula presented] and endomorphism f(x)↦f(x+1). Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over C(x).