Fuzzy Martingales Research Articles

Properties of fuzzy Ito integrals with respect to fuzzy Brownian motion

Integral of a stochastic process with respect to Brownian motion is called Ito integral. Here the stochastic process and Brownian motion are random as well as fuzzy. Hence the Ito integral is fuzzy Ito integral. This paper deals with the properties of fuzzy Ito integral for simple adapted process with respect to fuzzy Brownian motion. The quadratic variance and covariance of FII are discussed. The concept of fuzzy simple adapted process, fuzzy martingale, fuzzy functions are used to derive the properties of fuzzy Ito integrals.

Some moment inequalities for fuzzy martingales and their applications

Abstract Martingales are a class of stochastic processes which has had profound influence on the development of probability theory and stochastic processes. Some recent developments are related to mathematical finance. In the real world, some information about these phenomena might be imprecise and represented in the form of vague quantities. In these situations, we need to generalize classical methods to vague environment. Thus, fuzzy martingales have been extended as a vague perception of real-valued martingales. In this paper, some moment inequalities are presented for fuzzy martingales. Several convergence theorems are established based on these inequalities. As an application of convergence theorems, a weak law of large numbers for fuzzy martingales is stated. Furthermore, a few examples are devoted to clarify the main results.

Positive parts of convergent fuzzy set-valued martingales

For any given filtration (Σi), we give a description of the space of D∞-convergent set-valued fuzzy martingales with respect to (Σi). We show that this space of martingales can be endowed with a canonical metric in such a way that it is a complete metric space. We also show that this space of martingales is a join-semilattice. In particular, we derive a formula to calculate the join of any two such martingales. This enables us to also calculate positive parts of such martingales. Our order theoretic approach yields, as a bonus, a necessary and sufficient condition for a regular fuzzy set-valued martingale to be D∞-convergent. A description, in terms of constant sequences, is also given of the D∞-convergent fuzzy set-valued reversed martingales.

Regularity and stopping theorem for fuzzy martingales with continuous parameters

The notion of graph convergence for fuzzy-valued stochastic processes is introduced. Regularity for fuzzy (super) martingales with continuous parameters in the sense of gr · KM -convergence is discussed. Finally, we establish Doob's stopping theorem for continuous parameter (super, sub) martingales whose cut sets may be unbounded.

Doob's stopping theorem for fuzzy (super, sub) martingales with discrete time

Fuzzy random variables are introduced as random variables whose values are not reals but fuzzy sets. The concepts of fuzzy conditional expectation and fuzzy (super, sub) martingales are given. In this paper we study Doob's stopping theorem on fuzzy (super, sub) martingales with discrete parameters.

On the convergence of fuzzy martingales

The purpose of this paper is to prove the convergence theorem for fuzzy martingales without assuming that their values are Lipschitz or continuous fuzzy numbers on R n . This approach allows us to deduce many results on convergence of fuzzy numbers and fuzzy random variables from the relevant results on real numbers and real-valued random variables that appear as their support functions. Based on the concept of fuzzy downward martingale and applications of convergence theorem of fuzzy martingales, the strong law of large numbers for fuzzy random variables is proved, here the convergence is in the uniform metric but not the separable metric on fuzzy number space.

Decomposition theorems for fuzzy supermartingales and submartingales

Doob decomposition and Riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales. The concepts of two types of Riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale (resp. supermartingale) has Doob decomposition (resp. Riesz decomposition) are discussed in detail.

Convergence theorems for fuzzy random variables and fuzzy martingales

In this paper we discuss the properties of fuzzy random variables and fuzzy conditional expectation. Some extended results of dominated convergence theorems for fuzzy random variables are proved. We define the concept of a right-closed fuzzy martingale and give the necessary and sufficient conditions of right closability for fuzzy martingales. The convergence theorem for fuzzy martingales is also extended.

Fuzzy martingales - a simple form of fuzzy processes∗

Fuzziness is discussed in the context of fuzzy random variable and a corresponding view of fuzzy martingale and some their properties are given. Fuzzy random variables are introduced as random variables whose values are not reals but fuzzy sets. In this paper we study fuzzy valued random processes in discrete time and with values in a set of fuzzy sets of separable Banach space.

Random fuzzy sets and fuzzy martingales

The present paper is motivated by the definitions and properties developed by Puri and Ralescu (1986). We study the random fuzzy sets and their relationship to fuzzy set valued measures in a separable Banach space. Using the conditional expectations of random fuzzy sets, we introduce the concept of fuzzy martingales. Some properties and convergence theorems of fuzzy martingales are investigated.

Fuzzy Random Variables, Expectation, and Martingales

Banach space valued fuzzy random variables on a complete σ-finite measure space (Ω, F , μ) are studied. Their fuzzy valued integral is defined and its properties are examined. Some connections between fuzzy integral and conditional expectation are given. The definition and some properties of fuzzy martingales are investigated.

Convergence theorem for fuzzy martingales

We study fuzzy set-valued measures in a Banach space and their relationships to fuzzy random variables. Our main result is a convergence theorem for fuzzy martingales. Our tool is a Radon-Nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures.