The problem of risk measures in a discrete-time market model with transaction costs is studied. Strategy effectiveness and shortfall risk are introduced. This gives a generalization of quantile hedging presented in [4].

This paper presents a new model of asymmetric bifurcating autoregressive process with random coefficients. We couple this model with a Galton−Watson tree to take into account possibly missing observations. We propose least-squares estimators for the various parameters of the model and prove their consistency, with a convergence rate, and asymptotic normality. We use both the bifurcating Markov chain and martingale approaches and derive new results in both these frameworks.

We determine the optimal constants $C_{p,q}$ in the moment inequalities ${\left|\right|g\left|\right|}_p\le C_{p,q}{\left|\right|f\left|\right|}_q$, 1 ≤ p< q< ∞, where f = (fₙ), g = (gₙ) are two martingales, adapted to the same filtration, satisfying |dgₙ| ≤ |dfₙ|, n = 0,1,2,..., with probability 1. Furthermore, we establish related sharp estimates ||g||₁ ≤ supₙΦ(|fₙ|) + L(Φ), where Φ is an increasing convex function satisfying certain growth conditions and L(Φ) depends only on Φ.

Let f be a conditionally symmetric martingale and let S(f) denote its square function. (i) For p,q > 0, we determine the best constants $C_{p,q}$ such that $su{p}_n{\left(\right|fₙ|}^p{)/(1+Sₙ²\left(f\right))}^q\le C_{p,q}$. Furthermore, the inequality extends to the case of Hilbert space valued f. (ii) For N = 1,2,... and q > 0, we determine the best constants $C_{N,q}^{\text{'}}$ such that $su{p}_n\left(f{ₙ}^{2N-1}\right){(1+Sₙ²\left(f\right))}^q\le C_{N,q}^{\text{'}}$. These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if the conditional...

Let ${\left(h_k\right)}_{k\ge 0}$ be the Haar system on [0,1]. We show that for any vectors $a_k$ from a separable Hilbert space and any ${\epsilon }_k\in [-1,1]$, k = 0,1,2,..., we have the sharp inequality $\left|\right|{\sum }_{k=0}^n{\epsilon }_k{a}_k{h}_k{\left|\right|}_{W\left(\right[0,1\left]\right)}\le 2\left|\right|{\sum }_{k=0}^n{a}_k{h}_k{\left|\right|}_{L^{\infty }\left([0,1]\right)}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${\left|\right|Y\left|\right|}_{W\left(\Omega \right)}{\le 2\left|\right|X\left|\right|}_{L^{\infty }\left(\Omega \right)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.