A change of variable formula with Itô correction term
نویسندگان
چکیده
We consider the solution u(x, t) to a stochastic heat equation. For fixed x, the process F (t) = u(x, t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions g(x, t), a stochastic integral ∫ g(F (t), t) dF (t) exists as a limit of discrete, midpoint style Riemann sums, where the limit is taken in distribution in the Skorohod space of cadlag functions. Moreover, we show that this integral satisfies a change of variables formulas with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of F . AMS subject classifications: Primary 60H05; secondary 60G15, 60G18, 60H15
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تاریخ انتشار 2008