Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

Abstract

Abstract:We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L 2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A; M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari's inequality.

Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

“…Such results were established in [38]. Further considerations for fuzzy stochastic equations in the metric space (F ( 2, ), 2, ) setup and under weaker assumptions than Lipschitz type can be found in [39].…”

“…The aim of this paper is twofold. Firsly, we present a survey of extensions of some of approaches for deterministic fuzzy differential equations to fuzzy stochastic equations driven by semimartingales presented mainly in [35][36][37] and developed further in [38,39]. Secondly, we present an analysis of the notion of fuzzy Itô's stochastic integral understood as fuzzy random variable.…”

Bilateral set-valued stochastic integral equations

“…The following properties of stochastic trajectory integrals (see e.g. [29,30]) are useful in studying setvalued stochastic integral equations.…”

“…Such an observation with equations only with the left-hand side was explicitly emphasized in [31]. The equation above combines two different types of equations previously investigated in the literature, i.e., widening equations [29] and [30] and narrowing equations [31], so in this case the solutions can change the type of monotonicity of diameter over time.…”

On Bipartite Fuzzy Stochastic Differential Equations

“…However, these models are not enough when some stochastic noises in terms of Brownian motions appear. In such the situations fuzzy stochastic differential equations are more appropriate to be applied (Malinowski, 2012c;Malinowski, 2013a;Malinowski, 2013b;Malinowski, 2015d;Malinowski, 2015e;Malinowski, 2015a;Malinowski, 2015b;Malinowski and Agarwal, 2015).…”