We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of order $\lambda \in (0,1)$. The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some mild assumptions on the noise, we prove that the solution has moments of all orders. In addition, we provide its connection to the solution of some Skorokhod reflection problem. As an illustration of our results and motivation for applications, we also suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes. We complete the study by providing a numerical scheme for the solution.

In this paper, a new numerical method for solving the decoupled forward-backward stochastic differential equations (FBSDEs) is proposed based on some specially derived reference equations. We rigorously analyze errors of the proposed method under general situations. Then we present error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method; in particular, we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE. Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.

A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovskyequation (ut + cux + α uux + α1u2ux + βuxxx)x = γuwhich is also known as the extended rotation-modified Korteweg–de Vries(KdV) equation. This equation is used for the description of internal oceanic wavesaffected by Earth’ rotation. Particular versions of this equation with zero some ofcoefficients, α, α1, β, orγ are also known in numerous applications. The proposed numericalscheme is a further development of the well-known finite-difference scheme earlier usedfor the solution of the KdV equation. The scheme is of the second order accuracy both ontemporal and spatial variables. The stability analysis of the scheme is presented forinfinitesimal perturbations. The conditions for the calculations with the appropriateaccuracy have been found. Examples of calculations with the periodic boundary conditionsare presented to illustrate the robustness of the proposed scheme.

The multi-mode approximation is presented to compute the interior wave function of Schrödinger equation. This idea is necessary to handle the multi-barrier and high dimensional resonant tunneling problems where multiple eigenvalues are considered. The accuracy and efficiency of this algorithm is demonstrated via several numerical examples.

In this article, we investigate numerical schemes for solvinga three component Cahn-Hilliard model. The space discretization isperformed by usinga Galerkin formulation and the finite element method.Concerning the time discretization,the main difficulty is to write a scheme ensuring,at the discrete level, the decrease of the free energyand thus the stability of the method.We study three different schemes and proveexistence and convergence theorems. Theoretical results areillustrated by various numerical examples showing that the new semi-implicitdiscretization that we propose seems to be a good compromise between robustnessand accuracy.

We study in this paper some numerical schemes for hyperbolic systemswith unilateral constraint. In particular, we deal with the scalar case, the isentropicgas dynamics system and the full-gas dynamics system.We prove the convergence of the scheme to an entropy solutionof the isentropicgas dynamics with unilateral constraint on the density and mass loss.We also study the non-trivial steady states of the system.