We consider nonlinear integro-differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C 1; ̨ regularity for general fully nonlinear integro-differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen ...

in this paper, we present a comparative study between the modified variational iteration method (mvim) and a hybrid of fourier transform and variational iteration method (ftvim). the study outlines the efficiencyand convergence of the two methods. the analysis is illustrated by investigating four singular partial differential equations with variable coefficients. the solution of singular partia...

We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump L evy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is o...

Abstract This research apparatuses an approximate spectral method for the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel (TFPIDE). The main idea of this approach is to set up new Hilbert space that satisfies initial and boundary conditions. collocation applied obtain precise numerical approximation using basis functions based on shifted first-kind ...

In this paper we consider a risk model with two classes of insurance risks in the presence of multiple thresholds. We assume that the two claim counting processes are, respectively, Poisson and Sparre Andersen with generalized Erlang(2) claim inter-arrival times. We derive an integro-differential system for the Gerber-Shiu functions for surplus-dependent premium rates and a piecewise integro-di...

In this paper, we prove the existence and uniqueness of a nonlinear perturbed stochastic fractional integro-differential equation of Volterra-Itô type involving nonlocal initial condition by using the theory of admissibility of integral operator and Banach fixed-point principle. Also the stability and boundedness of the second moments of the stochastic solution are studied. In addition, an appl...