We consider a cadlag stochastic process $(X_r)_r$ with non-decreasing sample paths and $X_0=0.$ Let $Y_r:=X_r-r.$ We suppose that $(Y_r)_r,(Y^2_r-r)_r,$ and $(Y_r^3-3rY_r-r)_r$ are martingales relative to the canonical filtration of $(\mathcal{F}_r)_r$ of $(X_r)_r.$
Let $f:\mathbb{R} \to \mathbb{R}$ be a function of class $C^{\infty}$ with bounded derivatives.
Find, for $r \leq u,$ the expression of $E[f(X_u)|\mathcal{F}_r]$ in term of the derivatives of $f.$
So how could we find an expression for $E[f(X_u)|\mathcal{F}_r]$?