Martingales

This chapter summarizes the essentials of sequential conditioning and martingale theory. After a review with examples of the basic properties of martingales and semi‐martingales, including the Doob decomposition, the upcrossing inequality and martingale convergence are studied and also the role of the conditional variances in establishing convergence. The important martingale inequalities of Kolmogorov, Doob, Burkholder, and Azuma are proved.

Martingale convergence theorem for a conditional intuitionistic fuzzy mean value

The aim of this contribution is to show a representation of a conditional intuitionistic fuzzy mean value of intuitionistic fuzzy observables by a conditional mean value of random variables. We formulate a martingale convergence theorem for a conditional intuitionistic fuzzy mean value, too.

Martingale Convergence Theorem for the Conditional Intuitionistic Fuzzy Probability

For the first time, the concept of conditional probability on intuitionistic fuzzy sets was introduced by K. Lendelová. She defined the conditional intuitionistic fuzzy probability using a separating intuitionistic fuzzy probability. Later in 2009, V. Valenčáková generalized this result and defined the conditional probability for the MV-algebra of inuitionistic fuzzy sets using the state and probability on this MV-algebra. She also proved the properties of conditional intuitionistic fuzzy probability on this MV-algebra. B. Riečan formulated the notion of conditional probability for intuitionistic fuzzy sets using an intuitionistic fuzzy state. We use this definition in our paper. Since the convergence theorems play an important role in classical theory of probability and statistics, we study the martingale convergence theorem for the conditional intuitionistic fuzzy probability. The aim of this contribution is to formulate a version of the martingale convergence theorem for a conditional intuitionistic fuzzy probability induced by an intuitionistic fuzzy state m. We work in the family of intuitionistic fuzzy sets introduced by K. T. Atanassov as an extension of fuzzy sets introduced by L. Zadeh. We proved the properties of the conditional intuitionistic fuzzy probability.

Conditional intuitionistic fuzzy probability and martingale convergence theorem using IF-probability

The aim of this paper is to formulate the conditional intuitionistic fuzzy probability and a version of martingale convergence theorem with respect an intuitionistic fuzzy probability. Since the intuitionistic fuzzy probability can be decomposed to two intuitionistic fuzzy states, we can use the results holding for intuitionistic fuzzy states.

Unifying the Dynkin and Lebesgue–Stieltjes formulae

We establish a local martingaleMassociate withf(X,Y) under some restrictions onf, whereYis a process of bounded variation (on compact intervals) and eitherXis a jump diffusion (a special case being a Lévy process) orXis some general (càdlàg metric-space valued) Markov process. In the latter case,fis restricted to the formf(x,y)=∑k=1Kξk(x)ηk(y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes anL2-martingale. Convergence of the product of this Martingale with some deterministic function ( of time ) to 0 both inL2and almost sure is also considered and sufficient conditions for functions for which this happens are identified.