14 citations · 68 across the 25 of their papers we have counts for
29 papers
Majorizing-measure bounds in the realized square-function metric
Witold M. Bednorz, Rafał M. Łochowski
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-m…
Soohak: A Mathematician-Curated Benchmark for Evaluating Research-level Math Capabilities of LLMs
Guijin Son, Seungone Kim, Catherine Arnett +73
Following the recent achievement of gold-medal performance on the IMO by frontier LLMs, the community is searching for the next meaningful and challenging target for measuring LLM…
Concentration of the truncated variation of fractional Brownian motions of any Hurst index, their -variations and local times
Witold M. Bednorz, Rafał M. Łochowski
We obtain bounds for probabilities of deviations of the truncated variation functional of fractional Brownian motions (fBm) of any Hurst index from their expected val…
The It{ô}-Föllmer formula -- nonstandard cases
W. M. Bednorz, R. M. Łochowski, P. L. Zondi +2
The purpose of this note is to prove the It{ô}-Föllmer formula for the càdlàg paths possessing quadratic variation in a possibly ``weakest'' sense along some sequence of partitions…
Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)
Phumlani L. Zondi, Darlington Hove, Rafał M. Łochowski +1
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}…
Local times of deterministic paths with finite variation
Darlington Hove, Farai J. Mhlanga, Rafał M. Łochowski +1
In this note, we define the numbers of level crossings by a c{à}dl{à}g (RCLL) real function and, in analogy to the work of Bertoin and Yor [BY14] we…
Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings
Witold Bednorz, Purba Das, Rafał Łochowski
We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a càdlàg path and an occupation measure associated with this path. Using…
Level crossings of fractional Brownian motion
Purba Das, Rafał Łochowski, Toyomu Matsuda +1
Since the classical work of Lévy, it is known that the local time of Brownian motion can be characterized through the limit of level crossings. While subsequent extensions of this…
Moments and tails of hitting times of Bessel processes and convolutions of elementary mixtures of exponential distributions
W. M. Bednorz, R. M. Łochowski
We present explicit estimates of right and left tails and exact (up to universal, multiplicative constants) estimates of tails and moments of hitting times of Bessel processes. The…
One-dimensional game-theoretic differential equations
Rafał M. Łochowski, Nicolas Perkowski, David J. Prömel
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 resul…
On tails of exit times of multidimensional Lévy processes
Rafał Marcin Łochowski
Using a very simple argument based on the indepenence of increments and the fact that in a finite dimensional space there are not too many directions, we derive a theorem s…
On SDEs with Lipschitz coefficients, driven by continuous, model-free martingales
Lesiba Ch. Galane, Rafał M. Łochowski, Farai J. Mhlanga
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 ite…
On tails of symmetric and totally asymmetric -stable distributions
Witold M. Bednorz, Rafał M. Łochowski, Rafał Martynek
We estimate up to universal constants tails of symmetric and totally asymmetric 1-dimensional -stable distributions in terms of functions of the parameters of these distribution…
Quadratic variation of càdlàg semimartingales as a.s. limit of the normalized truncated variations
Rafał M. Łochowski
For a real càdlàg path we define sequence of semi-explicit quantities, which do not depend on any partitions and such that whenever is a path of a càdlàg semimartingale the…
Moments approach to the concentration properties of truncated variation
Witold Bednorz, Rafal Lochowski
In this paper we show an alternative approach to the concentration of truncated variation for stochastic processes on a real line. Our method is based on the moments control and ca…
Integration with respect to model-free price paths with jumps
Rafał M. Łochowski
For every adapted, càglàd process (strategy) and typical càdlàg price paths whose jumps satisfy some mild growth condition we define integral as a limit of simple in…
Asymptotics of the truncated variation of model-free price paths and semimartingales with jumps
Rafał M. Łochowski
We prove that typical (in the model-free finance setting) price paths with jumps may be uniformly approximated with accuracy by paths whose total variation is of order …
On a generalisation of the Banach indicatrix theorem
Rafał M. Łochowski
We prove that for any regulated function and the infimum of the total variations of functions approximating with accuracy $c…
Integration of rough paths - the truncated variation approach
Rafał M. Łochowski
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 stren…
A new theorem on the existence of the Riemann-Stieltjes integral and an improved version of the Loéve-Young inequality
Rafał M. Łochowski
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 a…
The Play Operator, the Truncated Variation and the Generalisation of the Jordan Decomposition
R. Ghomrasni, R. Łochowski
The play operator minimalizes the total variation on intervals , of functions approximating uniformly given regulated function with given accuracy and starting from a…
Integrability and concentration of the truncated variation for the sample paths of fractional Brownian motions, diffusions and Lévy processes
Witold Marek Bednorz, RafaŁ Marcin Łochowski
For a real càdlàg function defined on a compact interval, its truncated variation at the level is the infimum of total variations of functions uniformly approximating …
Pathwise stochastic integration with finite variation processes uniformly approximating càdlàg processes
Rafał M. Łochowski
For any real-valued stochastic process with cŕdlŕg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the…
On the Generalisation of the Hahn-Jordan Decomposition for Real Càdlàg Functions
Rafał M. Łochowski
For a real càdlàg function f and a positive constant c we find another càdlàg function, which has the smallest total variation pos- sible among all functions uniformly approximatin…