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Rafał M. Łochowski

Publications and source records attributed to Rafał M. Łochowski.

At least 19 recordsLinked to original sources
arXiv2026

Majorizing-measure bounds in the realized square-function metric

The predictable square function measures the size of a martingale. Applied to parameter differences, it also defines a random pseudometric. We explain how to perform a majorizing-measure argument directly in this realized geometry, without conditioning the terminal field to be Gaussian and without first replacing the metric by a random multiple of a deterministic one. For a finite family of predictable Gaussian sums in a $(2,D)$-smooth Banach space, and a fixed probability measure $μ$ on the parameter set, we prove \[ \norm{\max_j\osc_T f_j}_{L^p} \le CD\norm{\G_μ(d)+\sqrt p\,Δ_d}_{L^p},\qquad p\ge1, \] where $Δ_d$ is the diameter of the parameter space in the terminal square-function metric and $\G_μ(d)$ is its ball-mass integral. The proof combines a localized exponential inequality, pathwise averaging over random balls, and a stopping-time argument. Atomic measures recover logarithmically weighted maximal inequalities; Haar measure connects the result with homogeneous entropy and continuity estimates. We also give extensions, examples, and a precise account of the restrictions on choosing the averaging measure.

math.PR↗
arXiv2025

Concentration of the truncated variation of fractional Brownian motions of any Hurst index, their $1/H$-variations and local times

We obtain bounds for probabilities of deviations of the truncated variation functional of fractional Brownian motions (fBm) of any Hurst index $H \in (0,1)$ from their expected values. Obtained bounds are optimal for large values of deviations up to multiplicative constants depending on the parameter $H$ only. As an application, we give tight bounds for tails of $1/H$-variations of fBm along Lebesgue partitions and establish the a.s. weak convergence (in $L^1$) of normalized numbers of strip crossings by the trajectories of fBm to their local times for any Hurst parameter $H \in (0,1)$.

math.PR↗
arXiv2025

Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)

We show that the horizontal component of the Peano curve has quadratic variation equal the limit of quadratic variations along the Lebesgue partitions for grids of the form $3^{-n}p\mathbb{Z}+3^{-n}r$, $n=1,2,\ldots$, where $p$ is a rational number, while $r$ is irrational number, but the value of such quadratic variation depends on $p$. This also yields that the horizontal component of the Peano curve is an example of a deterministic function possessing local time (density of the occupation measure) with respect to the Lebesgue measure, whose local time can be expressed as the limit of normalized numbers of interval crossings by this function but the normalization is not a smooth function of the width of the intervals. These two features distinct the horizontal component of the Peano curve from the trajectories of the Wiener process, which is widely used in financial models.

math.CA↗
arXiv2024

Local times of deterministic paths with finite variation

In this note, we define the numbers of level crossings by a c{à}dl{à}g (RCLL) real function $x: [0,+\infty) \rightarrow R$ and, in analogy to the work of Bertoin and Yor [BY14] we prove that for $x$ with locally finite total variation these numbers are densities of relevant occupation measures associated with $x$. Next, depending on the regularity of $x$ and $f: R \rightarrow R$, we derive change of variable formulas, which may be seen as analogous of the Itô or Tanaka-Meyer formulas. Some of these formulas are present in [BY14] but we also present some generalizations.

math.CA↗
arXiv2023

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 $[0, +\ns) \times Ω$ which assigns zero value exactly to those sets (properties) of pairs of time $t$ and an elementary event $ω$ which are instantly blockable. Next, for a slightly modified measure, we prove Itô's isometry and BDG inequalities, and then use them to define an Itô-type integral. Additionally, we prove few properties for the quadratic variation of model-free, continuous martingales, which hold with instant enforcement.

q-fin.GN↗
arXiv2022

On SDEs with Lipschitz coefficients, driven by continuous, model-free martingales

We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard's iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingales. We work with a new outer measure which assigns zero value exactly to those properties which are instantly blockable.

q-fin.MF↗
arXiv2021

Local times and Tanaka--Meyer formulae for càdlàg paths

Three concepts of local times for deterministic c{à}dl{à}g paths are developed and the corresponding pathwise Tanaka--Meyer formulae are provided. For semimartingales, it is shown that their sample paths a.s. satisfy all three pathwise definitions of local times and that all coincide with the classical semimartingale local time. In particular, this demonstrates that each definition constitutes a legit pathwise counterpart of probabilistic local times. The last pathwise construction presented in the paper expresses local times in terms of normalized numbers of interval crossings and does not depend on the choice of the sequence of grids. This is a new result also for c{à}dl{à}g semimartingales, which may be related to previous results of Nicole El~Karoui and Marc Lemieux.

math.PR↗
arXiv2021

One-dimensional game-theoretic differential equations

We provide a very brief introduction to typical paths and the corresponding Itô type integration. Relying on this robust Itô integration, we prove an existence and uniqueness result for one-dimensional differential equations driven by typical paths with non-Lipschitz continuous coefficients in the spirit of Yamada--Watanabe as well as an approximation result in the spirit of Doss--Sussmann.

math.PR↗
arXiv2020

On tails of symmetric and totally asymmetric $α$-stable distributions

We estimate up to universal constants tails of symmetric and totally asymmetric 1-dimensional $α$-stable distributions in terms of functions of the parameters of these distributions. In particular, for values of $α$ close to $2$ we specify where exactly the tail changes from being Gaussian and starts to behave like in the Pareto distribution

math.PR↗
arXiv2020

Optimal uniform approximation of Lévy processes on Banach spaces with finite variation processes

For a general càdlàg Lévy process on a separable Banach space $V$ we estimate values of $\inf_{Y\in{\cal A}_X} \mathbb{E}\left\{ ψ\left( \Vert X - Y \Vert_\infty\right) + \mathrm{TV}(Y[0,T]) \right\}$, where ${\cal A}_X$ is the family of processes on $V$ adapted to the natural filtration of $X$, $ψ$ has polynomial growth and TV$(Y[0,T])$ denotes the total variation of the process $Y$ on the interval $[0,T]$. Next, we apply obtained estimates in three specific cases: a Brownian motion with drift on $\mathbb{R}$, a standard Brownian motion on $\mathbb{R}^d$ and a symmetric $α$-stable process ($α\in(1,2)$) on $\mathbb{R}$.

math.PR↗
arXiv2019

Quadratic variation of càdlàg semimartingales as a.s. limit of the normalized truncated variations

For a real càdlàg path $x$ we define sequence of semi-explicit quantities, which do not depend on any partitions and such that whenever $x$ is a path of a càdlàg semimartingale then these quantities tend a.s. to the continuous part of the quadratic variation of the semimartingale. Next, we derive several consequences of this result and propose a new approach to define Föllmer's pathwise integral.

math.PR↗
arXiv2018

On the quadratic variation of the model-free price paths with jumps

We prove that the model-free typical (in the sense of Vovk) càdlàg price paths with mildly restricted downward jumps possess quadratic variation which does not depend on the specific sequence of partitions as long as these partitions are obtained from stopping times such that the oscillations of a path on the consecutive (half-open on the right) intervals of these partitions tend (in a specified sense) to 0. Finally, we also define quasi-explicit, partition independent quantities which tend to this quadratic variation.

q-fin.MF↗
arXiv2017

A superhedging approach to stochastic integration

Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation of "typical price paths" in the space of non-negative càdlàg paths and implies the existence of quadratic variation in the sense of Föllmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a model-free Itô integration is developed.

q-fin.MF↗
arXiv2016

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 $[y,y+c].$

math.CA↗
arXiv2015

A new theorem on the existence of the Riemann-Stieltjes integral and an improved version of the Loéve-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 Loéve-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.

math.CA↗
arXiv2014

Integration of rough paths - the truncated variation approach

Using truncated variation techniques we obtain an improved version of the Loeve-Young inequality for the Riemann-Stieltjes integrals driven by rough paths. This allowed us to strenghten some result on the existence of solutions of integral equations driven by moderately irregular signals. We introduce also a new Banach space, containing as a proper subspace the paths with finite $p$-variation, and develop, in a systematic way, several parallel results for the paths from this space, obtained so far for the paths with finite $p$-variation. We start the paper with a general theorem on the existence of the Riemann-Stieltjes integral.

math.FA↗