An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering
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In this work we study an interior penalty method for a finite-dimensional large-scale linear complementarity problem (LCP) arising often from the discretization of stochastic optimal problems in financial engineering. In this approach, we approximate the LCP by a nonlinear algebraic equation containing a penalty term linked to the logarithmic barrier function for constrained optimization problems. We show that the penalty equation has a solution and establish a convergence theory for the approximate solutions. A smooth Newton method is proposed for solving the penalty equation and properties of the Jacobian matrix in the Newton method have been investigated. Numerical experimental results using three non-trivial test examples are presented to demonstrate the rates of convergence, efficiency and usefulness of the method for solving practical problems.
KeywordsVariational inequality Stochastic optimal control American option pricing HJB equation Linear complementarity problem Interior penalty method
Kai Zhang wishes to thank the supports from the Philosophy and Social Science Program of Guangdong Province (Grant No. GD13YYJ01) and the MOE Project of Key Research institute of Humanities and Social Sciences at Universities (Grant No. 14JJD790041). Project 11001178 partially supported by National Natural Science Foundation of China.
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