Our aim in this paper is to study the interaction between surface and subsurface flows. The model considered is a system coupling Navier-Stokes and Darcy equations. We make use of a discontinuous Galerkin finite element method for the discretisation of this problem. Then we develop a posteriori error analysis for the resulting discrete problem. Numerical experimentations confirm our analytical results.
In this work we propose an approximate numerical method for pricing of options for the constant elasticity of variance (CEV) diffusion model. We prove firstly the existence and uniqueness of the solution in weighted Sobolev space, and then we propose the finite element method and finite difference method to solve the considered problem. Therefore, we compare the obtained results by the two approaches, with those given by the Monte Carlo method in Broadie-Kaya [6], using two simulation techniques : the exact method and the Euler discretization. A comparative numerical study is done using some values of the coefficient of elasticity.
We consider a problem modeling a porous medium with a random perturbation. This model occurs in many applications such as biology, medical sciences, oil exploitation, and chemical engineering. Many authors focused their study mostly on the deterministic case. The more classical one was due to Biot in the 50s, where he suggested to ignore everything that happens at the microscopic level, to apply the principles of the continuum mechanics at the macroscopic level. Here we consider a stochastic problem, that is, a problem with a random perturbation. First we prove a result on the existence and uniqueness of the solution, by making use of the weak formulation. Furthermore, we use a numerical scheme based on finite differences to present numerical results.
The aim of this work is to evaluate a European option with a stochastic volatility. For, we have a system of two stochastic differential equations (SDEs), where the first one describes the price of the underlying while the second one modelises the stochastic volatility. First we set the inconvenience of Black and Scholes model [1] and its limits, then we propose a model with a stochastic volatility. For this purpose, we use Garman partial differential equation (GPDE) to evaluate the option price where solution is approached by a finite difference method.
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