Band 19, Heft 2

Saddle point problems with singular or indefinite pivot blockmatrix arise in various applications. In some problems it may be necessary to regularize the corresponding matrix is some way. By straightforward derivations it is shown that use of certain preconditioners on block diagonal and block triangular form can be very efficient, both for the original matrix and for its regularized forms. As the value of regularization parameter increases, the eigenvalues cluster at unity or at just two points on the real line.

We consider European double barrier basket call options on two underlyings with an upper and a lower knock-out barrier featuring a finite number of cash settlements at prespecified values of the underlyings between the strike and the upper barrier. The bilaterally constrained cash settlements are considered as controls that have to be chosen such that the Delta of the option is as close as possible to a predefined constant profit/loss. This leads to a control constrained optimal control problem for the two-dimensional Black–Scholes equation with Dirichlet boundary control and finite time control. Based on the variational formulation of the problem in an appropriate Sobolev space setting, we prove the existence of a unique solution and state the first order necessary optimality conditions. A semi-discretization in space by conforming P1 finite elements with respect to a simplicial triangulation of the computational domain gives rise to a semi-discrete control constrained optimal control problem for a linear system of first order ordinary differential equations. A further discretization in time by the backward Euler scheme results in a fully discrete optimization problem that is solved numerically by the projected gradient method with Armijo line search. Numerical examples for some selected test cases illustrate the benefits of hedging with European double barrier basket options in case of optimally controlled cash settlements.

In this paper we use the discontinuous Galerkin finite element method for the space-semidiscretization of a nonlinear nonstationary convection–diffusion problem defined on a nonpolygonal two-dimensional domain. Using Zlámal's concept of the ideal curved elements, we define a finite element space . We prove the ‘ideal’ versions of the inverse and the multiplicative trace inequalities known for standard straight triangulations. Further, we define a projection on the finite element space and study its approximation properties. The obtained results allow us to derive an H 1 -optimal error estimate for the discontinuous Galerkin method employing the ideal curved elements.