On a generalisation of the Banach indicatrix theorem

We prove that for any regulated function $f:\left[a,b\right]\rightarrow\mathbb{R}$ and $c\geq 0$ the infimum of the total variations of functions approximating $f$ with accuracy $c/2$ is equal $\int_{\mathbb{R}} n_{c}^{y} \mathrm{d} y,$ where $n_{c}^{y}$ is the number of times that $f$ crosses the interval $[y,y+c].

On the Generalisation of the Hahn-Jordan Decomposition for Real C\`adl\`ag Functions

For a real c\`{a}dl\`{a}g function f and a positive constant c we find another c\`{a}dl\`{a}g function, which has the smallest total variation pos- sible among all functions uniformly approximating f with accuracy c/2. The solution is expressed with the truncated variation, upward truncated variation and downward truncated variation introduced in [L1] and [L2]. They are always finite even if the total variation of f is infinite, and they may be viewed as the generalisation of the Hahn-Jordan decomposition for real c\`{a}dl\`{a}g functions. We also present partial results for more general functions.Comment: arXiv admin note: substantial text overlap with arXiv:1106.319

A new theorem on the existence of the Riemann-Stieltjes integral and an improved version of the Lo\'eve-Young inequality

Using the notion of the truncated variation we obtain a new theorem on the existence and estimation of the Riemann-Stieltjes integral. As a special case of this theorem we obtain an improved version of the Lo\'{e}ve-Young inequality for the Riemann-Stieltjes integrals driven by irregular signals. Using this result we strenghten some results of Terry Lyons on the existence of solutions of integral equations driven by moderately irregular signals.Comment: More comprehensive paper, with similar results is "Integration of rough paths - the truncated variation approach, arXiv:1409.3757

BDG inequalities and their applications for model-free continuous price paths with instant enforcement

Shafer and Vovk introduce in their book \cite{ShaferVovk:2018} the notion of \emph{instant enforcement} and \emph{instantly blockable} properties. However, they do not associate these notions with any outer measure, unlike what Vovk did in the case of sets of ''typical'' price paths. In this paper we introduce an outer measure on the space [0, +\ns) \times \Omega which assigns zero value exactly to those sets (properties) of pairs of time $t$ and an elementary event $\omega$ which are instantly blockable. Next, for a slightly modified measure, we prove It\^o's isometry and BDG inequalities, and then use them to define an It\^o-type integral. Additionally, we prove few properties for the quadratic variation of model-free, continuous martingales, which hold with instant enforcement

On truncated variation, upward truncated variation and downward truncated variation for diffusions

The truncated variation, $TV^c$, is a fairly new concept introduced in [5]. Roughly speaking, given a c\`adl\`ag function $f$, its truncated variation is "the total variation which does not pay attention to small changes of $f$, below some threshold $c>0$". The very basic consequence of such approach is that contrary to the total variation, $TV^c$ is always finite. This is appealing to the stochastic analysis where so-far large classes of processes, like semimartingales or diffusions, could not be studied with the total variation. Recently in [6], another characterization of $TV^c$ was found. Namely $TV^c$ is the smallest total variation of a function which approximates $f$ uniformly with accuracy $c/2$. Due to these properties we envisage that $TV^c$ might be a useful concept to the theory of processes. For this reason we determine some properties of $TV^c$ for some well-known processes. In course of our research we discover intimate connections with already known concepts of the stochastic processes theory. Firstly, for semimartingales we proved that $TV^c$ is of order $c^{-1}$ and the normalized truncated variation converges almost surely to the quadratic variation of the semimartingale as $c\searrow0$. Secondly, we studied the rate of this convergence. As this task was much more demanding we narrowed to the class of diffusions (with some mild additional assumptions). We obtained the weak convergence to a so-called Ocone martingale. These results can be viewed as some kind of large numbers theorem and the corresponding central limit theorem. All the results above were obtained in a functional setting, viz. we worked with processes describing the growth of the truncated variation in time. Moreover, in the same respect we also treated two closely related quantities - the so-called upward truncated variation and downward truncated variation.Comment: Added Remark 6 and Remark 15. Some exposition improvement and fixed constant