Define Lebesgue-Stieltjes integral for integrators of unbounded first variation

Question

In chapter 1.5 of Shreve & Karatzas's book, after proving that for continuous, square-integrable martingales, variations greater than 0 vanish and lower variations explode, it proceeds to argue that for this reason, they are not differentiable and we cannot define integrals of the form $\int_0^t Y_{s}(\omega)dX_{s}(\omega)$ pathwise. Question: why can't we define Lebesgue-Stieltjes integrals for integrators of unbounded first variation?

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Answer 1

I think I have found the answer. I will follow Exercise 2.21 in chapter 3.2 of Yor & Revuz.

Let $X$ be a real-valued function on $[0,1]$. Let $\Delta = \{t_{1}\ldots t_{n}, 0\leq t_{1} \leq \ldots \leq t_{n}\leq n\}$ be a partition. For each $h \in C([0,1],\mathbb{R})$, define

$S_{\Delta}(h) = \sum_{t_{i}\in \Delta}h(t_{i})(X_{t_{i+1}}-X_{t{i}})$.

It is obvious that the map $h\rightarrow S_{\Delta}(h)$ is a bounded linear functional on $C([0,1],\mathbb{R})$, with norm $||S_{\Delta}||:=\sum_{t_{i}\in \Delta}|X_{t_{i+1}}-X_{t{i}}|$.

The Riemann-style integral is defined only if $\{S_{\Delta_{n}}(h)\}$ converges as $|\Delta_{n}|\rightarrow0$, for every $h \in C([0,1],\mathbb{R})$. By the Principle of Uniform Boundedness (also called the Banach-Steinhaus theorem), this implies that $\sup_{n} ||S_{\Delta_{n}}|| < \infty$. But $\sup_{n} ||S_{\Delta_{n}}||$ is exactly the first variation of $X$, provided that it exists. Hence, $X$ must be of bounded variation in order that the integral exists.