Parabolic variational inequalities: The Lagrange multiplier approach

Parabolic variational inequalities are discussed and existence and uniqueness of strong as well as weak solutions are established. Our approach is based on a Lagrange multiplier treatment. Existence is obtained as the unique asymptotic limit of solutions to a family of appropriately regularized nonlinear parabolic equations. Two regularization techniques are presented resulting in feasible and unfeasible approximations respectively. Monotonicity results of the regularized solutions and convergence rate estimate are established. The results are applied to the Black-Scholes model for American options. The case of the bilateral constraints is also treated. Numerical results for the Black-Scholes model are presented and prove the practical efficiency of our results. (C) 2005 Elsevier SAS. All rights reserved.