I am wondering if a non-decreasing càdlàg function $x$ can be recovered via "clever" integration when integrating over a general functional.
We know that for all absolutely continuous $x\in D((0,\infty))_\nearrow:=\{x: (0,\infty)\to\mathbb{R}\mid x \text{ càdlàg and non-decreasing}\}$ there exists a function $$k_x: (0,\infty)\to \mathbb{R};\ s\mapsto x^\prime(s)$$ such that $$x(t)=x(0)+\int_0^t k_x(s)\mathrm{d}s\ \ \forall\ t\in(0,\infty).$$ I am interested in a more general setting where $k$ can also take the whole path of $x$ as an argument. More specifically, I have two questions, where the second statement is stronger than the first.
- For a given $x$, does there exists a functional $$k_x: (0,\infty)\times D((0,\infty))_\nearrow \to \mathbb{R}$$ such that $$x(t)=x(0)+\int_0^t k_x(s,x)\mathrm{d}s\ \ \forall\ t\in(0,\infty)?$$
- Does there exists a universal functional $$k: (0,\infty)\times D((0,\infty))_\nearrow \to \mathbb{R}$$ such that $$x(t)=x(0)+\int_0^t k(s,x)\mathrm{d}s\ \ \forall\ t\in(0,\infty) \text{ and }x\in D((0,\infty))_\nearrow?$$
I guess that already the answer to 1. is negative, but coming up with a counterexample seems quite hard to me, because the function $k_x(\cdot,\cdot)$ can also take $x$ as an argument.
A further question would be for which types of functions $x$ 1. is true, i.e. if there is a nice regularity condition on $x$, apart from absolute continuity, such that 1. holds?