Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itô’s integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms

Yet Another Shade of Deduction. On measuring deductive flexibility and how it may relate to other cognitive abilities

The article describes the construction process of Deductive Flexibility Test considered a difficult deductive reasoning measure – and the research on correlations between fluency in difficult deductive reasoning and other cognitive abilities. The main goal in the research was to examine the relations between Deductive Flexibility Test scores and results of Raven’s Advanced Progressive Matrices – fluid intelligence test. Additionally, the measures of the need for cognitive closure and epistemological understanding were included in the study. The results of the study revealed that Deductive Flexibility Test is a reliable instrument and thus can be used for research purposes. We found low or even no statistically significant correlations between the chosen variables. The directions of further research are discussed

Weak solutions of stochastic differential inclusions and their compactness

In this paper, we consider weak solutions to stochastic inclusions driven by a semimartingale and a martingale problem formulated for such inclusions. Using this we analyze compactness of the set of solutions. The paper extends some earlier results known for stochastic differential inclusions driven by a diffusion process

Continuity properties of solutions of multivalued equations with white noise perturbation

In the paper, we consider a set-valued stochastic equation with stochastic perturbation in a Banach space. We prove first the existence theorem and then study continuity properties of solutions

On risk reserve under distribution constraints

The purpose of this work is a study of the following insurance reserve model: $R(t) = η + ∫_{0}^{t} p(s,R(s))ds + ∫_{0}^{t} σ(s,R(s))dW_{s} - Z(t)$, t ∈ [0,T], P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0. Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: $inf_{0≤t≤T} P{R(t) ≥ c} ≥ γ$ is considered

On weak solutions of random differential inclusions

In the paper we study the existence of solutions of the random differential inclusion x˙t∈G(t,xt)     P.1,t∈[0,T]-a.e.x0=dμ, where G is a given set-valued mapping value in the space Kn of all nonempty, compact and convex subsets of the space ℝn, and μ is some probability measure on the Borel σ-algebra in ℝn. Under certain restrictions imposed on F and μ, we obtain weak solutions of problem (I), where the initial condition requires that the solution of (I) has a given distribution at time t=0