Nikolaos Halidias

Semi-discrete approximations for stochastic differential equations and applications

In this paper, we propose a new point of view in numerical approximation of stochastic differential equations. By using Ito–Taylor expansions, we expand only a part of the stochastic differential equation. Thus, in each step, we have again a stochastic differential equation which we solve explicitly or by using another method or a finer mesh. We call our approach as a semi-discrete approximation. We give two applications of this approach. Using the semi-discrete approach, we can produce numerical schemes which preserves monotonicity so in our first application, we prove that the semi-discrete Euler scheme converge in the mean square sense even when the drift coefficient is only continuous, using monotonicity arguments. In our second application, we study the square root process which appears in financial mathematics. We observe that a semi-discrete scheme behaves well producing non-negative values.

A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient

The existence of a mean-square continuous strong solution is established for vector-valued Ito stochastic differential equations with a discontinuous drift coefficient, which is an increasing function, and with a Lipschitz continuous diffusion coefficient. A scalar stochastic differential equation with the Heaviside function as its drift coefficient is considered as an example. Upper and lower solutions are used in the proof.

A novel approach to construct numerical methods for stochastic differential equations

In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.

On the Numerical Solution of Some Non-Linear Stochastic Differential Equations Using the Semi-Discrete Method

Abstract We are interested in the numerical solution of stochastic differential equations with non-negative solutions. Our goal is to construct explicit numerical schemes that preserve positivity, even for super-linear stochastic differential equations. It is well known that the usual Euler scheme diverges on super-linear problems and the tamed Euler method does not preserve positivity. In that direction, we use the semi-discrete method that the first author has proposed in two previous papers. We propose a new numerical scheme for a class of stochastic differential equations which are super-linear with non-negative solution. The Heston 3/2-model appearing in financial mathematics belongs to this class of stochastic differential equations. For this model we prove, through numerical experiments, the “optimal” order of strong convergence at least 1/2 of the semi-discrete method.

Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions

Abstract In this paper we consider a quasilinear second-order differential inclusion with a convex-valued multivalued term and nonlinear, multivalued boundary conditions. Using the Leray–Schauder fixed-point theorem and techniques from multivalued analysis and from nonlinear analysis, we prove the existence of a solution. Our formulation of the problem is general and includes as special cases the Dirichlet, the Neumann, the periodic problems, as well as certain Sturm–Liouville-type problems.

Multiple Solutions for Quasilinear Elliptic Neumann Problems in Orlicz-Sobolev Spaces

We investigate the existence of multiple solutions to quasilinear elliptic problems containing Laplace like operators (-Laplacians). We are interested in Neumann boundary value problems and our main tool is Brézis-Nirenbergs local linking theorem.

The Method of Upper and Lower Solutions of Stochastic Differential Equations and Applications

Abstract The purpose of this article is to consider a stochastic integral equation driven by semimartingale with discontinuous and increasing drift part. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered topological space. Finally we present some applications in finance.

A new numerical scheme for the CIR process

Abstract In this paper we generalize an explicit numerical scheme for the CIR process that we have proposed before. The advantage of the new proposed scheme is that preserves positivity and is well posed for a (little bit) broader set of parameters among the positivity preserving schemes. The order of convergence is at least logarithmic in general and for a smaller set of parameters is at least 1/4.

Approximating Explicitly the Mean-Reverting CEV Process

We are interested in the numerical solution of mean-reverting CEV processes that appear in financial mathematics models and are described as nonnegative solutions of certain stochastic differential equations with sublinear diffusion coefficients of the form where . Our goal is to construct explicit numerical schemes that preserve positivity. We prove convergence of the proposed SD scheme with rate depending on the parameter . Furthermore, we verify our findings through numerical experiments and compare with other positivity preserving schemes. Finally, we show how to treat the two-dimensional stochastic volatility model with instantaneous variance process given by the above mean-reverting CEV process.

Elliptic problems with discontinuities

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals.