Rafał M. Łochowski

The play operator, the truncated variation and the generalisation of the Jordan decomposition

The play operator minimalizes the total variation on intervals, [0,T],T > 0, of functions approximating uniformly given regulated function with given accuracy and starting from a given point. In this article, we link the play operator with the so-called truncated variation functionals, introduced recently by the second-named author, and provide a semi-explicit expression for the play operator in terms of these functionals. Generalisation for time-dependent boundaries is also considered. This gives the best possible lower bounds for the total variation of the outputs of the play operator and its Jordan-like decomposition. Copyright

On a generalisation of the Hahn-Jordan decomposition for real càdlàg functions

For a real càdlàg function f and a positive constant c we find another càdlàg function, which has the smallest total variation possible 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 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 càdlàg functions. We also present partial results for more general functions.

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

The truncated variation, TVc, is a fairly new concept introduced in Łochowski (2008) [5]. Roughly speaking, given a cadlag 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, TVc 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 Łochowski (2011) [6], another characterization of TVc has been found. Namely TVc is the smallest possible total variation of a function which approximates f uniformly with accuracy c/2. Due to these properties we envisage that TVc might be a useful concept both in the theory and applications of stochastic processes.

On the double Laplace transform of the truncated variation of a Brownian motion with drift

For any real-valued stochastic process X with c\`adl\`ag paths we define non-empty family of processes, which have finite total variation, have jumps of the same order as the process X and uniformly approximate its paths: This allows to decompose any real-valued stochastic process with c\`adl\`ag paths and infinite total variation into a sum of uniformly close, finite variation process and an adapted process, with arbitrary small amplitude but infinite total variation. Another application of the defined class is the definition of the stochastic integral with respect to the process X as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term.For a real càdlàg function f and positive constant c we find another càdlàg function, which has the smallest total variation possible among the 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 [3] and [4]. They are analogs of Hahn-Jordan decomposition of a càdlàg function with finite total variation but are always finite even if the total variation is infinite. We apply obtained results to general stochastic processes with càdlàg trajectories and in the special case of Brownian motion with drift we apply them to obtain full characterisation of its truncated variation by calculating its Laplace transform. We also calculate covariance of upward and downward truncated variations of Brownian motion with drift.

On a generalisation of the Banach Indicatrix Theorem

We prove that for any regulated function f : [a, b]→ R and c ≥ 0, the infimum of the total variations of functions approximating f with accuracy c/2 is equal ∫ R n y cdy, where n y c is the number of times that f crosses the interval [y, y + c].

A new inequality for the Riemann-Stieltjes integrals driven by irregular signals in Banach spaces

We prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces, and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any p≥1

On Truncated Variation of Brownian Motion with Drift

p\ge1

Moment and Tail Estimates for Multidimensional Chaos Generated by Positive Random Variables with Logarithmically Concave Tails

, we introduce the space of regulated signals f:[a,b]→W

A superhedging approach to stochastic integration

f:[a,b]\rightarrow W

Asymptotics of the truncated variation of model-free price paths and semimartingales with jumps

(a<b