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Extension of a stochastic Gronwall lemma

Makasu Cloud

Bulletin of the Polish Academy of Sciences. Mathematics

2020

Vol. 68, no. 1

97--104

A stochastic Gronwall lemma is proved in Scheutzow (2013) in the case when the exponent p lies in the interval 0 < p <1. In this paper, we extend the lemma to the entire interval 0 < p < ∞. We construct simple examples to illustrate the present result.

Ulam-Hyers stability of a parabolic partial differential equation

Marian Daniela, Ciplea Sorina Anamaria, Lungu Nicolaie

Demonstratio Mathematica

2019

Vol. 52, nr 1

475--481

The goal of this paper is to give an Ulam-Hyers stability result for a parabolic partial differential equation. Here we present two types of Ulam stability: Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability. Some examples are given, one of them being the Black-Scholes equation.

The abstract Gronwall lemma for some nonlinear operators

Zima M.

Demonstratio Mathematica

1998

Vol. 31, nr 2

325-332

The aim of this paper is to prove a generalization of abstract Gronwall lemma for a nonlinear operator. The applications to the differential-functional equations are given.