Linear hyperfinite LÃ©vy integrals
This article shows that the nonstandard approach to stochastic integration with respect to (C^2 functions of) LÃ©vy processes is consistent with the classical theory of pathwise stochastic integration with respect to (C^2 functions of) jump-diffusions with finite-variation jump part. It is proven that internal stochastic integrals with respect to hyperfinite LÃ©vy processes possess right standard parts, and that these standard parts coincide with the classical pathwise stochastic integrals, provided the integrator's jump part is of finite variation. If the integrator's LÃ©vy measure is bounded from below, one can obtain a similar result for stochastic integrals with respect to C^2 functions of LÃ©vy processes. As a by-product, this yields a short, direct nonstandard proof of the generalized ItÃ´ formula for stochastic differentials of smooth functions of LÃ©vy processes.
|Date of creation:||15 Aug 2011|
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