On existence and uniqueness of solutions to uncertain backward stochastic differential equations

Abstract

On existence and uniqueness of solutions to uncertain backward stochastic differential equations FEI Wei-yin Abstract. This paper is concerned with a class of uncertain backward stochastic differential equations (UBSDEs) driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in modelling hybrid systems, where the phenomena are simultaneously subjected to two kinds of uncertainties: randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coefficients are proved. §1 Introduction Randomness is a basic type of objective uncertainty, and probability theory is a branch of mathematics for studying the behavior of random phenomena. The study of probability theory was started by Pascal and Fermat in 1654, and an axiomatic foundation of probability theory given by Kolmogorov in 1933. The concept of fuzzy set was initiated by Zadeh [37] via membership function in 1965. In order to measure a fuzzy event, Zadeh [38] introduced the theory of possibility. Moreover, fuzzy random variables are mathematical descriptions for fuzzy stochastic phenomena (i.e., a mixture of fuzziness and randomness) and can be defined in several ways on the basis of probability theory and fuzzy mathematics. The concept of fuzzy random variables are introduced by Kwakernaak [21,22] and Puri and Ralescu [33]. Furthermore, the theory of fuzzy-valued (or set-valued) random systems was also investigated by many researchers, such as Fei [6,7,9,12,[14][15][16], Li and Guan [23], Malinowski et al. [28,29] and references therein.However, some information and knowledge are usually represented by human language like "about 100km", "roughly 80kg", "low speed", "middle age", and "big size". A lot of surveys MR Subject Classification: 60H10, 94D05.

Almost Sure Exponential Stability of Uncertain Stochastic Hopfield Neural Networks Based on Subadditive Measures

“…Consequently, we can deduce from Refs. [28,29] that for k ≥ 0, Equation ( 8) possesses a unique global solution z(k, z 0 ), assuming g(0) = h(0) = 0 for the sake of stability in this paper. As a result, Equation ( 8) possesses an equilibrium solution z(k, 0) = 0.…”

“…Liu [27] first introduced chance theory to investigate a hybrid system with both uncertainty about belief degree and randomness. To investigate the uncertain stochastic differential systems, Fei [29] extended a filtered chance space (Γ × Ω, L ⊗ F , (L k ⊗ F k ) k∈[0,T] , M × P ) on which some concepts, theorems, are presented as follows.…”

“…When faced with a system that exhibits both uncertainty and randomness simultaneously, the noise should be modeled using the Wiener-Liu process, and the system evolution can be described through a hybrid differential equation, leading to the development of uncertain stochastic hybrid neural network systems [26]. In 2013, Liu [27] first introduced chance theory to investigate such uncertain stochastic systems based on subadditive measures, and subsequent works by Fei et al [28,29] have further explored the use of the Wiener-Liu process and the Itô-Liu formula in uncertain stochastic differential equations. Researchers have made progress in studying various forms of the stability of stochastic neural networks based on additive measures, but the analysis of indeterminate neural networks, including both random and uncertain factors, requires chance theory's subadditive measures.…”

Uncertain Stochastic Hybrid Age-Dependent Population Equation Based on Subadditive Measure: Existence, Uniqueness and Exponential Stability

“…With demand of practical issues need, recently investigators have examined the effects of uncertain random elements on the differential equation system. In 2014, similar to the analysis method of the fuzzy random process, Fei [21] first considered uncertain stochastic differential equations (USDEs), proved the Itô-Liu formula and applied it to control systems with Markovian switching. In the same year, Fei [22] proved the existence and uniqueness theorems of backward USDEs.…”

Uncertain Stochastic Optimal Control with Jump and Its Application in a Portfolio Game

“…Uncertain stochastic analysis is a branch of pure mathematics that studies the integral and differential of uncertain stochastic processes. Fei [13] considered a class of uncertain backward stochastic differential equations driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Fei [14] first described a class of uncertain stochastic control systems with Markov switching, and derived an Itô-Liu formula for Markov-modulated processes.…”