Ioannis S. Stamatiou

Ioannis S. Stamatiou

We apply the semi-discrete method, c.f. \emph{N. Halidias and I.S. Stamatiou (2016), On the numerical solution of some non-linear stochastic differential equations using the semi-discrete method, Computational Methods in Applied Mathematics, 16(1)}, to a class of non-colliding particle systems. The proposed numerical scheme preserves the non-colliding property and strongly converges to the exact solution.

Ioannis S. Stamatiou

In this note we study the asymptotic mean-square stability for two-step schemes applied to a scalar stochastic differential equation (sde) and applied to systems of sdes. We derive necessary and sufficient conditions for the asymptotic MS-stability of the methods in terms of the parameters of the schemes. The stochastic Backward Differentiation Formula (BDF2) scheme is asymptotically mean-square stable for any step-size whereas the two-step Adams-Bashforth (AB2) and Adams-Moulton (AM2) methods are unconditionally stable. The improved versions of the schemes do not perform better w.r.t their stability behavior in the scalar case, as expected, but the situation is different in more dimensions. Numerical experiments confirm theoretical results.

Ioannis S. Stamatiou

We use the semi-discrete method, originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics, 89(6), to reproduce qualitative properties of a class of nonlinear stochastic differential equations with nonnegative, non-globally Lipschitz coefficients and a unique equilibrium solution. The proposed fixed-time step method preserves the positivity of solutions and reproduces the almost sure asymptotic stability behavior of the equilibrium with no time-step restrictions.

Ioannis S. Stamatiou

We are interested in the numerical approximation of non-linear stochastic differential equations (SDEs) with solution in a certain domain. Our goal is to construct explicit numerical schemes that preserve that structure. We generalize the semi-discrete method \emph{Halidias N. and Stamatiou I.S. (2016), On the numerical solution of some non-linear stochastic differential equations using the Semi-Discrete method, Computational Methods in Applied Mathematics,16(1)} and propose a numerical scheme, for which we prove a strong convergence result, to a class of SDEs that appears in population dynamics and ion channel dynamics within cardiac and neuronal cells. We furthermore extend our scheme to a multidimensional case.

Nikolaos Halidias, Ioannis S. Stamatiou

In this paper 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. In this class of stochastic differential equations belongs the Heston $3/2$-model that appears in financial mathematics, for which we prove %theoretically and through numerical experiments the "optimal" order of strong convergence at least $1/2$ of the Semi-Discrete method.