We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle...
We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle...
We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group.
The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme,
we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like where is the mesh step. Such...
This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of
with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary.
Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained.
These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous...
We discuss a parallel implementation of the domain
decomposition method based on the macro-hybrid formulation
of a second order elliptic
equation and on an approximation by the mortar element method.
The discretization leads to an algebraic saddle- point problem.
An iterative method with a block- diagonal
preconditioner is used for solving the saddle- point problem.
A parallel implementation of the method is emphasized.
Finally the results of numerical experiments are presented.
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