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Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift-their mean value and their applications

100%
Łochowski R.
Banach Center Publications
|
2010
|
tom 90
|
nr 1
67-78
EN
In [L 2008] for c > 0 we defined the truncated variation, $TV_{μ}^{c}$, of a Brownian motion with drift, $W_t = B_t + μt$, t ≥ 0, where $(B_t)$ is a standard Brownian motion. In this article we define two related quantities: the upward truncated variation
$UTV^{c}_{μ}[a,b] = sup_{n} sup_{a≤t₁ and, analogously, the downward truncated variation
$DTV^{c}_{μ}[a,b] = sup_{n} sup_{a≤t₁ We prove that the exponential moments of the above quantities are finite (in contrast to the regular variation, corresponding to c = 0, which is infinite almost surely). We present estimates of the expected value of $UTV_{μ}^{c}$ up to universal constants. As an application we give some estimates of the maximal possible gain from trading a financial asset in the presence of flat commission (proportional to the value of the transaction) when the dynamics of the prices of the asset follows a geometric Brownian motion process. In the presented estimates the upward truncated variation appears naturally.
2
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Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails

100%
Łochowski R.
Banach Center Publications
|
2006
|
tom 72
|
nr 1
161-176
EN
Two kinds of estimates are presented for tails and moments of random multidimensional chaoses $S = ∑ a_{i₁,..., i_{d}} X_{i₁}^{(1)} ⋯ X_{i_{d}}^{(d)}$ generated by symmetric random variables $X_{i₁}^{(1)},...,X_{i_{d}}^{(d)}$ with logarithmically concave tails. The estimates of the first kind are generalizations of bounds obtained by Arcones and Giné for Gaussian chaoses. They are exact up to constants depending only on the order d. Unfortunately, suprema of empirical processes are involved. The second kind estimates are based on comparison between moments of S and moments of some related Rademacher chaoses. The estimates for pth moment are exact up to a factor $(max(1,ln p))^{d²}$.
3
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On a generalisation of the Hahn-Jordan decomposition for real càdlàg functions

100%
Łochowski R.
Colloquium Mathematicum
|
2013
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tom 132
|
nr 1
121-138
EN
For a real càdlàg function f and a positive constant c we find another càdlàg function which has the smallest total variation among all functions uniformly approximating f with accuracy c/2. The solution is expressed in terms of truncated variation, upward truncated variation and downward truncated variation introduced in earlier work of the author. They are always finite even if the total variation of f is infinite, and they may be viewed as a generalisation of the Hahn-Jordan decomposition for real càdlàg functions. We also present partial results for more general functions.
4
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Full cooperation applied to environmental improvements

51%
Jeanblanc M., Łochowski R., Szatzschneider W.
Banach Center Publications
|
2015
|
tom 104
|
nr 1
121-131
EN
We analyse the case of certificates of environmental improvements and full cooperation of two identical agents. We model pollution levels as geometric Brownian motions with quadratic costs of improvements. Our main result is the construction of the optimal improvements strategy in the case of separate actions, collusive actions and fusion. In certain range of the model parameters, the fusion solution generates lower pollution levels than separate and collusive actions.
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