Set-valued stochastic differential equation in M-type 2 Banach space

Abstract

Abstract. Let (Ω, F, {F t }, P ) be a complete probability space with filtration {F t }, (X , H, µ) an abstract Wiener space of M-type 2, and {B t : t ≥ 0} an X -valued Brownian motion such that the distribution of the random function t −1/2 B t : Ω → X is µ for any t > 0. We consider the strong solutions to a set-valued stochastic differential equation with a set-valued drift and a single valued diffusion driven by dB t . Under some suitable conditions, the existence and uniqueness of strong solutions are obtained.

Existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non-Lipschitzian condition

“…randomness and fuzziness, are incorporated. Furthermore, in the systems disturbed by the so-called white noise, stochastic set differential equations (see [18][19][20]) and FSDEs could be some of the best tools in modelling of phenomena with uncertainties. There are only few papers on the latter topic, i.e.…”

On uniqueness and existence of solutions to stochastic set-valued differential equations with fractional Brownian motions

“…Fei, 2005aFei, , 2005bFei, , 2007aFei, , 2007bFei, , 2013Fei & Wu, 2004;Fei et al, 2003;Hiai & Umegaki, 1977;Malinowski, 2010Malinowski, , 2012aMalinowski, , 2012bMalinowski, , 2012cMalinowski, , 2013aMalinowski, , 2013bMalinowski, , 2013cPapageorgiou, 1986). Recently, the set-valued random differential equations are investigated in , Fei and Liang (2013), , Malinowski and Michta (2010), Michta (1995Michta ( , 1997Michta ( , 2011 and Mitoma et al (2010). Also, Li et al (2010) studied the strong solution of set-valued stochastic differential equations of Itô type.…”

Properties of Solutions to Stochastic Set Differential Equations under Non-Lipschitzian Coefficients

“…Since then the subject has attracted the interest of many mathematicians and further contributions are made from both the theoretical and applied viewpoints (see, e.g., [17][18][19][20][21][22][23][24][25][26]). In [27][28][29][30][31], the set-valued random differential equations are studied. The strong solution of Itô type set-valued stochastic differential equation is analyzed in [32].…”