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

Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

Theory of photospheric emission from relativistic outflows

In this paper we reexamine the optical depth of ultrarelativistic spherically symmetric outflows and reevaluate the photospheric radius for each model during both the acceleration and coasting phases. It is shown that for both the wind and the shell models there are two asymptotic solutions for the optical depth during the coasting phase of the outflow. In particular we show that quite counterintuitively a geometrically thin shell may appear as a thick wind for photons propagating inside it. For this reason we introduce notions of photon thick and photon thin outflows, which appear more general and better physically motivated with respect to winds and shells. Photosphere of relativistic outflow is a dynamic surface. We study its geometry and find that the photosphere of photon thin outflow has always a convex shape, while in the photon thick one it is initially convex (there is always a photon thin layer in any outflow) and then it becomes concave asymptotically approaching the photosphere of an infinitely long wind. We find that both instantaneous and time integrated observed spectra are very close to the thermal one for photon thick outflows, in line with existing studies. It is our main finding that the photospheric emission from the photon thin outflow produces non thermal time integrated spectra, which may be described by the Band function well known in the GRB literature. We find that energetic GRBs should produce photon thin outflows with photospheric emission lasting less than one second for the total energy $E_0\leq10^{54}$ erg and baryonic loading parameter $B\leq10^{-2}$. It means that only time integrated spectra may be observed from such GRBs.Comment: Revision of the previous version, new effect is discussed. Conclusions remain unchange

Finite Difference Methods For Linear Fuzzy Time Fractional Diffusion And Advection-Diffusion Equation

Fractional differential equations have attracted considerable attention in the last decade or so. This is evident from the number of publications on such equations in various scientific and engineering fields. Crisp quantities in fractional differential equations which are deemed imprecise and uncertain can be replaced by fuzzy quantities to reflect imprecision and uncertainty. The fractional partial differential equation can then be expressed by fuzzy fractional partial differential equations which can give a better description for certain phenomena involving uncertainties. The analytical solution of fuzzy fractional partial differential equations is often not possible. Therefore, there is great interest in obtaining solutions via numerical methods. The finite difference method is one of the more frequently used numerical methods for solving the fractional partial differential equations due to their simplicity and universal applicability. In this thesis, the focus is the development, analysis and application of finite difference schemes of second order of accuracy and compact finite difference methods of fourth order of accuracy to solve fuzzy time fractional diffusion equation and fuzzy time fractional advection-diffusion equation. Two different fuzzy computational techniques (single and double parametric form of fuzzy number) are investigated. The Caputo formula is used to approximate the fuzzy time fractional derivative. The consistency, stability, and convergence of the finite difference methods are investigated. Numerical experiments are carried out and the results indicate the effectiveness and feasibility of the schemes that have been developed

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515