My research interest (google scholar) over the last 10 years has primarily focused on Stochastic Analysis (see works 1 – 8, 10, 12, 14 and 17 below), Financial Mathematics (see corresponding works below), Markov Chains (see works 13 and 20 below) as well as topics in Fourier and Laplace transforms (work 9) and Linear Algebra (work 16) that are directly related to Differential Equations and Markov Chains.
Semi-discrete approximations for stochastic differential equations and applications
Int. J. Computer Mathematics, Vol. 89, 6, 2012
In which we propose a new numerical scheme for the CIR stochastic differential equation that appears in finance (see also Heston volatility model). The difficulty is to construct a numerical scheme such that the approximate solution is also positive, like the true solution. However, other properties are also desirable (see arxiv). Subsequently, the above work was generalized,
A novel approach to construct numerical methods for stochastic differential equations
Numerical Algorithms, Springer
so that we are able to construct numerical schemes for some super-linear stochastic differential equations as well.
Construction of positivity preserving numerical schemes for multidimensional stochastic differential equations
Discrete and Continuous Dynamical Systems, Series B (maple code)
A generalization of [2] to multiple dimensions.
On the numerical solution of some nonlinear stochastic differential equations using the semi discrete method
Computational Methods in Applied Mathematics
In collaboration with PhD candidate Ioannis Stamatiou.
A new numerical scheme for the CIR process
Monte Carlo Methods and Applications, preprint (maple code)
Continuing work [1], we generalize the numerical scheme proposed in [1] so that it is well-defined for a larger set of parameters. The convergence order is logarithmic, similar to the convergence order of the simple Euler method, while for a limited set of parameters, the order is at least 1/4.
Constructing positivity preserving numerical schemes for the two factor CIR model
Monte Carlo Methods and Applications
We work on a system of stochastic differential equations that appears in finance, the so-called two factor CIR model. For this system, we propose two positivity-preserving numerical schemes using the basic idea from work [3] above.
An explicit and positivity preserving numerical scheme for the mean reverting CEV model
Japan Journal of Industrial and Applied Mathematics, Springer, arXiv
We propose an explicit scheme for the mean-reverting CEV model.
Approximating explicitly the mean-reverting CEV process
Journal of Probability and Statistics
An effort for the mean-reverting CEV model with PhD candidate I. Stamatiou.
An elementary approach to the option pricing problem
Asian Research Journal of Mathematics, 2016
We study option pricing in discrete time using basic mathematical tools.
On the construction of boundary preserving numerical schemes
Monte Carlo Methods and Applications, Vol. 22, issue 4, 2016, arXiv
We describe some thoughts for further generalization of the semi-discrete method.
A generalization of Laplace and Fourier transforms
Asian Journal of Mathematics and Computer Research24(1), pp. 32-41, 2018, arXiv (see also this paper)
Particular attention is needed when applying transforms to solve differential equations and others. Given that we do not know if the unknown quantity has the required properties, we apply the transform to calculate a **potential** solution. The proof of whether this quantity is indeed a solution to the problem will come through **verification** and not from the application of the transform!
Convergence rates of the Semi-Discrete method for stochastic differential equations
On the absorption probabilities and mean time to absorption for discrete Markov chains
Monte Carlo Methods Appl. 2021; 27(2): 105–115, also at researchgate.
A note on the asymptotic stability of the Semi-Discrete method for stochastic differential equations