We study the tails of the distribution of the maximum of a stationary Gaussian process on a bounded interval of the real line. Under regularity conditions including the existence of the spectral moment of order 8, we give an additional term for this asymptotics. This widens the application of an expansion given originally by Piterbarg [CITE] for a sufficiently small interval.

Let N ≥ 2 be a given integer. Suppose that $df={\left(dfₙ\right)}_{n\ge 0}$ is a martingale difference sequence with values in $\ell {₁}^N$ and let ${\left(\epsilon ₙ\right)}_{n\ge 0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $\mathbb{P}(su{p}_{n\ge 0}\left|\right|{\sum }_{k=0}^n{\epsilon }_{k}d{f}_k{\left|\right|}_{\ell {₁}^N}\ge 1)\le (lnN+ln\left(3lnN\right))/(1-{\left(2lnN\right)}^{-1})su{p}_{n\ge 0}\left|\right|{\sum }_{k=0}^{n}d{f}_k{\left|\right|}_{\ell {₁}^N}$. It is shown that this result is asymptotically sharp in the sense that the least constant $C_N$ in the above estimate satisfies $li{m}_{N\to \infty }{C}_N/lnN=1$. The novelty in the proof is the explicit verification of the ζ-convexity of the space $\ell {₁}^N$.

Upper estimates are presented for the universal constant in the Katz-Petrov and Osipov inequalities which do not exceed 3.1905.

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.

Consider the following inhomogeneous fragmentation model: suppose an initial particle with mass x₀ ∈ (0,1) undergoes splitting into b > 1 fragments of random sizes with some size-dependent probability p(x₀). With probability 1-p(x₀), this particle is left unchanged forever. Iterate the splitting procedure on each sub-fragment if any, independently. Two cases are considered: the stable and unstable case with $p\left(x₀\right)=x{₀}^a$ and $p\left(x₀\right)=1-x{₀}^a$ respectively, for some a > 0. In the first (resp. second) case, since smaller...

Discrete autoregressive process of the first order is considered. The process is observed at unequally spaced time instants. Both least squares estimate and maximum likelihood estimate of the autocorrelation coefficient are analyzed. We show some dangers related with the estimates when the true value of the autocorrelation coefficient is small. Monte-Carlo method is used to illustrate the problems.

This paper proposes a Lie group analytical approach to tackle the problem of pricing derivative securities. By exploiting the infinitesimal symmetries of the Boundary Value Problem (BVP) satisfied by the price of a derivative security, our method provides an effective algorithm for obtaining its explicit solution.