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Abstract
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We give a new proof of the fact that the value function of the finite time horizon American put option for a jump diffusion, when the jumps are from a compound Poisson process, is the classical solution of a quasi-variational inequality and it is C^1 across the optimal stopping boundary. Our proof only uses the classical theory of parabolic partial differential equations of Friedman and does not use the theory of vicosity solutions, since our proof relies on constructing a sequence of functions, each of which is a value function of an optimal stopping time for a diffusion. The sequence is constructed by iterating a functional operator that maps a certain class of convex functions to smooth functions satisfying variational inequalities (or to value functions of optimal stopping problems involving only a diffusion). The approximating sequence converges to the value function exponentially fast, therefore it constitutes a good approximation scheme, since the optimal stopping problems for diffusions can be readily solved. |
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