| Purdue Computational Finance Program | |
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On the Sharp Rate of Numerical Methods for Degenerate ODEs
April 3, 2003 |
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Abstract:
In mathematical finance (and many other areas) many problems often come down to numerically solving a PDE. There are typically two types of approaches: the deterministic approach such as finite difference methods (FDM, in short), and the stochastic approach such as Monte-Carlo methods. For low dimensional PDEs, the former approach is much more efficient. However, when the (ellicptic or parabolic) PDE is degenerate, there are few results on the rate of convergence of FDM, mainly due to the lack of smoothness of the solution. We note that a degenerate PDE rises when the underlying asset market is stable, or if the risk of bankruptcy is taken into consideration. In fact, even if the original PDE is nondegerate, it may lead to some degenerate equations along the approximations. In this talk, we study the simplest degenerate differential equations, namely the degenerate ODEs. We shall propose a numerical method for it, in the spirit of FDM. By using some probabilistic approach, we obtain the rate of convergence of our method and prove that such a rate is sharp. |
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