Abbreviated Journal Title
Stochastic integration; quartic variation; quadratic variation; stochastic partial differential equations; long-range dependence; iterated Brownian motion; fractional Brownian motion; self-similar; processes; FRACTIONAL BROWNIAN-MOTION; STOCHASTIC INTEGRALS; LIMIT-THEOREMS; HURST; INDEX; Statistics & Probability
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 Ito sense. We show that for sufficiently differentiable functions g(x, t), a stochastic integral integral g(F(t), t)d F(t) exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary Ito integral with respect to a Brownian motion that is independent of F.
Annals of Probability
Burdzy, Krzysztof and Swanson, Jason, "A Change Of Variable Formula With Ito Correction Term" (2010). Faculty Bibliography 2010s. 7023.