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 ...

We consider the problem of valuation of certain Asian options in the geometric jump-diffusion models with continuously dividend-paying assets. With the sources of diffusion risks and two primitive tradeable assets, the market in this model is, in general, incomplete, and so, there are more than one equivalent martingale measures and no-arbitrage prices. For this jump-diffusion model, we adopt t...

This article considers the dividend optimization problem for an insurer with a jumpdiffusion risk process in the presence of fixed and proportional transaction costs. Due to the presence of a fixed transaction cost, the mathematical problem becomes an impulse stochastic control problem. Using a stochastic impulse control approach, we transform the stochastic control problem into a quasi-variati...

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...

this paper has been devoted to apply the reconstruction of variational iteration method (rvim) to handle the systems of integro-differential equations. rvim has been induced with laplace transform from the variational iteration method (vim) which was developed from the inokuti method. actually, rvim overcome to shortcoming of vim method to determine the lagrange multiplier. so that, rvim method...

in this paper, we propose the chebyshev wavelet approximation for the numerical solution of a class of integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields. we show that the chebyshev approximation transform an integral equation to an explicit system of linear algebraic equations. illustrative examples are included t...

In this paper we consider the Cauchy boundary value problem for the integro-differential equation utt −m ( ∫