Dimension Independent Atomic Decomposition for Dyadic Martingale $$ \pmb {{\mathbb {H}}}^{1}$$

A Martingale Decomposition Theorem

On an atomic decomposition in Banach spaces

An atomic decomposition is considered in Banach space.  A method for constructing an atomic decomposition of Banach  space, starting with atomic decomposition of  subspaces  is presented. Some relations between them are established. The proposed method is used in the  study  of the  frame  properties of systems of eigenfunctions and associated functions of discontinuous differential operators.

Martingale Families and Dimension in P

We introduce a new measure notion on small complexity classes (called F -measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0’s and 1’s. As applications to F -measure, we answer a question raised...

Stochastic Volatility and Epsilon-Martingale Decomposition

We address the problems of pricing and hedging derivative securities in an environment of uncertain and changing market volatility. We show that when volatility is stochastic but fast mean reverting Black-Scholes pricing theory can be corrected. The correction accounts for the effect of stochastic volatility and the associated market price of risk. For European derivatives it is given by explic...

Outlier Detection with Streaming Dyadic Decomposition

In this work we introduce a new algorithm for detecting outliers on streaming data in R. The basic idea is to compute a dyadic decomposition into cubes in R of the streaming data. Dyadic decomposition can be obtained by recursively bisecting the cube the data lies in. Dyadic decomposition obtained under streaming setting is understood as streaming dyadic decomposition. If we view the streaming ...