A CHANGE OF VARIABLE FORMULA WITH ITO CORRECTION TERM

• 成果类型: Article
• 署名作者: Burdzy, Krzysztof; Swanson, Jason
• 署名单位: University of Washington; University of Washington Seattle; State University System of Florida; University of Central Florida
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP523
• 发表日期: 2010
• 页码: 1817-1869
• 关键词: fractional brownian-motion; STOCHASTIC INTEGRALS; LIMIT-THEOREMS; hurst index

摘要

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.