Fuzzy stochastic differential equations of decreasing fuzziness: Non-Lipschitz coefficients

Article type: Research Article

Authors: Malinowski, Marek T.^*

Affiliations: Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24, 31-155 Kraków, Poland

Correspondence: [*] Corresponding author. Marek T. Malinowski, Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24, 31-155 Kraków, Poland. Tel.: +48 12 6282994; E-mail: malinowskimarek@poczta.fm.

Abstract: We study fuzzy stochastic differential equations driven by multidimensional Brownian motion with solutions of decreasing fuzziness. The drift and diffusion coefficients are random. Under a non-Lipschitz condition, the existence and pathwise uniqueness of solutions to such the equations are proven. The solutions are considered to be fuzzy stochastic processes. The main result is obtained with a help of a sequence of approximate solutions that converge to a desired unique local solution with trajectories having decreasing fuzziness. A parallel assertion for solutions to fuzzy stochastic differential equations of increasing fuzziness is stated as well. We indicate that our considerations of fuzzy stochastic differential equations of decreasing fuzziness can be applied to examine non-Lipschitz set-valued stochastic differential equations with solutions being set-valued stochastic processes.

Keywords: Fuzzy stochastic differential equation, fuzzy stochastic integral equation, fuzzy stochastic process, set-valued stochastic differential equation, set-valued stochastic integral equation, set-valued stochastic process, uncertainty

DOI: 10.3233/IFS-152102

Journal: Journal of Intelligent & Fuzzy Systems, vol. 31, no. 1, pp. 13-25, 2016