Soft Computing https://doi.org/10.1007/s00500-018-3054-8 METHODOLOGIES AND APPLICATION A stock model with jumps for Itô–Liu financial markets Frank Ranganai Matenda 1 · Eriyoti Chikodza 2 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract Over the years, various types of differential equations have been employed to describe a myriad of processes driven by the respective forms of indeterminacy. This paper presents and examines uncertain stochastic differential equations and their important characteristics. An uncertain stochastic differential equation is a differential equation driven by both a Brownian motion and a canonical Liu process. Moreover, an uncertain stochastic differential equation with jumps is a differential equation driven by a Brownian motion, a canonical Liu process and an uncertain random renewal process. Based on an uncertain stochastic differential equation with jumps, this study suggests a stock model with jumps for Itô–Liu financial markets. Generalised stock models for Itô–Liu financial markets are introduced as well. Keywords Indeterminacy · Uncertain stochastic differential equation · Uncertain stochastic differential equation with jumps · Uncertain stochastic finance · Stock model with jumps · Itô–Liu financial markets 1 Introduction The concept of indeterminacy is critical in the field of finance. In practice, financial decisions are made under the state of indeterminacy. Indeterminacy is defined as phenomena whose outcomes cannot be distinctly determined in advance (Peng 2013). Matenda et al. (2015) propounded that indeter- minacy is the state of being unpredictable in advance as far as events’ outcomes are concerned. Randomness and uncertainty are two basic forms of inde- terminacy. Kolmogorov (1933) proposed probability theory, and Liu (2007) introduced uncertainty theory in order to model randomness and uncertainty, respectively. Random- ness refers to any phenomenon which can be quantified by a probability measure (Liu 2012b). In other words, random- ness is defined as a characteristic of anything that can be described by probabilistic laws. On the other hand, uncer- Communicated by V. Loia. B Frank Ranganai Matenda frmatenda@gmail.com Eriyoti Chikodza eriyoti.chikodza677@gmail.com 1 Department of Banking and Finance, Great Zimbabwe University, P. O. Box 1235, Masvingo, Zimbabwe 2 Department of Mathematics and Computer Science, Great Zimbabwe University, P. O. Box 1235, Masvingo, Zimbabwe tainty refers to any phenomenon which can be quantified by an uncertain measure (Liu 2012b). Alternatively, uncertainty is defined as an attribute of anything that can be described by belief degrees. The theory of probability is used when the sample size is large enough to estimate the probability distribution from available frequency. Probability theory is an axiomatic branch of pure mathematics that models the behaviour of dynamic random phenomena. The application of probability theory in finance theory gave birth to stochastic finance the- ory. Conversely, if the sample size is too small or non-existent to estimate the probability distribution, uncertainty theory is adopted. Domain experts are invited to evaluate their belief degrees of each event occurring. Human beings usually over- estimate unlikely events, and as a result, belief degrees may have larger variance than the real frequency. Belief degrees are not subjective probability. Adopting probability theory in this case leads to counter-intuitive results. The theory of uncertainty is a branch of axiomatic pure mathematics that models human uncertainty. Uncertainty theory is the foun- dation of uncertain finance theory. In order to model the dynamics of random phenomena which vary with time, stochastic processes were introduced in probability theory. More generally, a stochastic process is a sequence of random variables indexed by time or space. A Brownian motion, pioneered by Robert Brown in 1827, is one of the prominent examples of stochastic processes. Einstein 123