We derive and analyze monotone difference-quadrature schemes for Bellman equations of controlled Lévy (jump-diffusion) processes. These equations are fully non-linear, degenerate parabolic integro-PDEs interpreted in the sense of viscosity solutions. We propose new “direct” discretizations of the non-local part of the equation that give rise to monotone schemes capable of handling singular Lévy...

We derive and analyze monotone difference-quadrature schemes for Bellman equations of controlled Lévy (jump-diffusion) processes. These equations are fully non-linear, degenerate parabolic integro-PDEs interpreted in the sense of viscosity solutions. We propose new “direct” discretizations of the non-local part of the equation that give rise to monotone schemes capable of handling singular Lévy...

In this article, we consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. In the first part, we deal with two-dimensional integralalgebraic equations. Next, we analyze Volterra integral equations of the first kind with a degenerate matrix-kernel on the diagonal. Finally, the third part of the work is devoted to the analysis of degenerate integro...

In this paper, we present a Taylor-series expansion method for a class of Fredholm singular integro-differential equation with Cauchy kernel. This method uses the truncated Taylor-series polynomial of the unknown function and transforms the integro-differential equation into an nth order linear ordinary differential equa.tion with variable coefficients: ~y Galerkin method we use the orthogonal ...

We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to provide explicit estimates for the continuous dependence on the coefficients and the “Lévy measure” in the Bellman/Isaacs integro-PDEs arising in stochastic control/differential games. M...

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...

Two numerical algorithms based on variational iteration and decomposition methods are developed to solve a linear partial integro-differential equation with a weakly singular kernel arising from viscoelasticity. In addition, analytic solution is re-derived by using the variational iteration method and decomposition method.

For quasilinear integro-diierential equations of the form u t ?aA(u) = f, where a is a scalar singular integral kernel that behaves like t ? , 1 2 < 1 and A is a second order quasilinear elliptic operator in divergence form, solutions are found for which A(u) is integrable over space and time.

A simple method for solving Prandtl's integro-differential equation is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. Compared with known investigations, its ...