In this paper, we present an intelligent combination of partition of unity (PU) and finite element (FE) methods for valuing American option pricing problems governed by the Black-Scholes (BS) model. The model is based on a partial differential equation (PDE) from which one can deduce the Black-Scholes formula, which gives a theoretical estimated value of options using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to expiration and expected volatility. Although the finite element method (FEM) seems to be an alternative tool for pricing options with a few applications reported in the literature, this combination called the Partition of Unity Finite Element Method (PUFEM) appears to offer many of the desired properties. The main advantage of the proposed approach is its ability to locally refine the solution by adapting an incorporated specific class of enrichment in the finite element space instead of generating a new fine mesh for the problem under study. Numerical computations are carried out to show a huge reduction in the number of degrees of freedom required to achieve a fixed accuracy which confirms that the PUFE method used is very efficient and gives better accuracy than the conventional FE method.

Mathematical epidemiology is one of the most important research areas, it has contributed to understanding the behavior and the impact also the prediction of infectious disease. One of the fundamental methods intended to see the behavior of the pandemic is the susceptible-infectious-recovered epidemic model. However, the deterministic approach of this model has some limitations in mathematical modeling, for that we propose to add a stochastic variation in SIR equations. In this paper we present a stochastic differential equation with jump-diffusion formula for COVID-19, then we estimate the parameters of our stochastic susceptible-infected-recovered model. Finally, we compare our result with real covid19 spread in Morocco.