Some remarks on local time–space calculus

Let X   be a reversible semi-martingale satisfying certain conditions and let F:R×R+⟶RF:R×R+⟶R be a locally bounded measurable function such that it admits locally bounded Radon–Nikodym derivatives ∂F/∂t∂F/∂t and ∂F/∂x∂F/∂x, then the following change-of-variable formula is valid: F(Xt,t)=F(X0,0)+∫0t∂F∂t(Xs-,s)ds+∫0t∂F∂x(Xs-,s)dXs+∑0