Bounding a Riemann Stieltjes integral.

Question

I have little experience with Riemann Stieltjes integrals.

Any good reading material on it would be much appreciated (specifically a large summary of the material).

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Suppose $|k|_t$ is the total variation of $k:\mathbb{R}\to \mathbb{R}$. Where $k$ is non-negative and monotonically increasing. Consider the Riemann-Stieltjes integral (of a continuous $f:\mathbb{R}\to\mathbb{R}$ )

$$ \int_0^t f(s) d|k|_s $$

Is there any hope in bounding this by something like :

$$ \int_0^t f(s) d|k|_s\leq C |k|_t\int_0^tf(s)ds ~~~~?$$

Thanks in advance!

I would like to stress that $|k|_t$ is continuous and differentiable a.e.

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$\textbf{Edit :}$ Motivation : (you can kind of ignore the probability stuff just imagine everything as deterministic if you wish)

The reason for studying this is I have an integral

$$ \frac{\partial}{\partial t}\mathbb{E}\int_0^t |X_s| d|k|_s. $$

Here $X_s$ is a continuous random variable solution to the Skorohod reflection problem on a convex domain $D$. $k_t$ is the random ($X_t$ dependent) process which keeps $X_s$ in the domain. $|k|_t$ its total variation (also random but a.s finite).

Now I have bounds on $\mathbb{E}(X_t)$ and can bound $|k|_t$ by $|X_t|$ ! So I was hoping to write :

\begin{align*} \frac{\partial}{\partial t}\mathbb{E}\int_0^t |X_s| d|k|_s\leq \frac{\partial}{\partial t} C \mathbb{E} |k|_t \int_0^t |X_s| ds \end{align*}

Then use Cauchy Schwartz and my bounds previously mentioned.

Answer 1

This is not true in general. For a counterexample, take $f(s) = s$, $k(s) = 0$ for $s < 0$ and $k(s) = s$ for $s \geqslant 0$. In this case, $|k|_s = s$ for $s \geqslant 0$ and

$$\frac{t^2}{2}=\int_0^t f(s) \, d|k|_s \leqslant C |k|_t \int_0^t f(s) \, ds = C \frac{t^3}{2},$$

implies $C \geqslant 1/t$ for all $t \in (0,1)$.