In this paper we consider a space fractional reaction-advection-diffusion equation which is actually a semi-linear Cauchy problem with a spatial fractional derivative operator of order α, 0 < α < 2. We establish and prove assertions concerning the existence and uniqueness of solution within certain Colombeau space. The proofs are given in the case when the left Liouville fractional derivative is involved, but the results are also valid in the case of the right Liouville fractional derivative as well as for the Riesz fractional derivative. The solutions are obtained by using the theory of generalized uniformly continuous semigroups of operators. We have also proved that the non-regularized and corresponding regularized operators, as well as solutions of non-regularized and corresponding regularized equations are L 2 −associated.MSC 2010 : 26A33, 35R11, 46F30
Application of stochastic control theory to the optimal portfolio selection problem, in the case when portfolio consists of two assets with different level of risk is illustrated. Choosing power functions and natural logarithmic for the utility function, and using a converse of Hamilton-Jacobi-Bellman (HJB) theorem, the formula for optimal portfolio is derived.
We consider a nonlinear stochastic heat equation with Riesz space-fractional derivative and variable thermal conductivity, on infinite domain. First we approximate the original problem by regularizing the Riesz space-fractional derivative. Then we prove that the approximate problem has almost surely a unique solution within a Colombeau generalized stochastic process space. In our solving procedure we use the theory of Colombeau generalized uniformly continuous semigroups of operators. At the end, we study the relation of the original and the approximate problem and prove that, under certain conditions, the derivative operators appearing in these two problems are associated. Even more, we prove that under some additional conditions, solutions of the original and the approximate problem are almost certainly associated as well (assuming that the first one almost surely exists).
Uloga fazi skupa u modeliranju neodreĎenostiTradicionalno, ekonomski modeli su zasnovani na klasičnoj matematici utemeljenoj na aristotelovskoj dvoelementnoj logici. U aristotelovskom konceptu, određeni element ili pripada ili ne pripada nekom skupu, treće mogućnosti nema. Međutim, ovakav egzaktan pristup nepodesan je za modeliranje pojava koje su prožete neodređenošću. Poznat je Wang-ov paradoks: Ako je x mali broj, onda je to i x+1. Ako je x+1 mali broj, onda je to i (x+1)+1. Tako dolazimo do zaključka da je i pet biliona mali broj, Ivan Pavkov, saradnik u nastavi, Visoka poslovna škola strukovnih studija, Novi Sad. Mr Miloš Japundžić, asistent, Visoka poslovna škola strukovnih studija, Novi Sad.
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