弱可微函数的逐段伊藤公式
A pathwise Ito formula for weakly differentiable functions
AI总结：
该研究推广了Föllmer的逐段伊藤公式，构造了依赖路径的索伯列夫空间，证明了变量替换公式，在布朗运动情形下验证了函数的空间归属，并得到多维Föllmer–Protter公式的逐段版本。
AI中文摘要：
我们将Föllmer的逐段伊藤公式推广到沿一列分割具有有限二次变差的连续路径的弱可微函数。对每条这样的路径ω，我们引入依赖于路径的索伯列夫空间W_{ω,π}^{2}，该空间通过路径ω的离散加权占据测度生成的半范数对弱黑塞矩阵进行光滑逼近而定义。对F∈W_{ω,π}^{2}，我们构造逐段积分∫∇F(ω)d^{π}ω和共变差[∇F(ω),ω]_{π}，并证明变量替换公式F(ω(t))−F(ω(0))=∫₀ᵗ∇F(ω(s))d^{π}ω(s)+½[∇F(ω),ω]_{π}(t)。对于布朗运动，我们证明W^{2+,p}(ℝᵈ)∩W^{2,1}(ℝᵈ)中的函数几乎必然属于对应的依赖于路径的空间，仅在起点的极例外集之外。若F还属于W_{loc}^{1,2}(ℝᵈ)，则逐段积分与随机伊藤积分一致，得到多维Föllmer–Protter公式的逐段版本。
英文摘要：
We prove a version of Föllmer's pathwise Itô formula for weakly differentiable functions of continuous paths with finite quadratic variation along a sequence of partitions. For each such path $ω$, we introduce a path-dependent Sobolev space $W_{ω,π}^{2}$ defined through smooth approximation of the weak Hessian in a seminorm generated by discrete weighted occupation measures of the path $ω$. For $F\in W_{ω,π}^{2}$, we construct the pathwise integral $\int \nabla F(ω)\,d^πω$ and the covariation $[\nabla F(ω),ω]_π$, and prove the change-of-variable formula $$ F(ω(t))-F(ω(0)) = \int_{0}^{t}\nabla F(ω(s))\,d^πω(s) + \frac12[\nabla F(ω),ω]_π(t). $$ Our result does not require any assumption on the existence of local time for the path; the Ito term appears as a quadratic covariation. For Brownian motion, we show that functions in $W^{2+,p}(\mathbb{R}^{d})\cap W^{2,1}(\mathbb{R}^{d})$ belong almost surely to the corresponding path-dependent space, outside a polar exceptional set of starting points. If, in addition, $F\in W_{\mathrm{loc}}^{1,2}(\mathbb{R}^{d})$, the pathwise integral agrees with the stochastic Itô integral, yielding a pathwise version of the multidimensional Föllmer--Protter formula.
