We introduce the concept of truncated variation of Brownian motion with drift, which differs from regular variation by neglecting small jumps (smaller than some c > 0). We estimate the expected value of the truncated variation. The behaviour resembling phase transition as c varies is revealed. Truncated variation appears in the formula for an upper bound for return from any trading based on a single asset with flat commission.
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...
Two kinds of estimates are presented for tails and moments of random multidimensional chaoses generated by symmetric random variables 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...
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.
Page 1