Regularities of local times of two-parameter martingales
Let \(X\) be the space of all continuous \(\omega :[0,1]^{2}\rightarrow \)\textbf{R}, vanishing on the axes, \(H\) be the set of all absolutely continuous \(h\in X\), with finite \(\| h\|_{H}^{2}=\int h_{s t}dsdt\) and \(\mu\) be the 2-parameter Wiener measure on \(X\). Let \(L\) be the Ornstein Uhlenbeck operator, \(D_{r}^{p}\) be the Sobolev space with the norm \(\| F\|_{p,2 r}=\) \(\| (1-L)^{r}F\|_{p}\), \(w_{z}(\omega )=\omega (z)\) for \(z\in [0,1]^{2}\) and \(M\) be an \(L^{2}\)-bounded martingale with respect to the filtration of \((w_{z})\), vanishing on the axes, represented (Wong-Zakai) as \[ M_{z}=\int_{[0,z]}\varphi (\xi )dw(\xi )+\iint_{[0,z]\times [0,z]}\psi (\xi ,\eta )dw(\xi )dw(\eta ), \] with adapted \(\varphi ,\psi\). \(M_{z}\) can also be represented as \(\int_{[0,z]}q_{1}(z,\eta )dw_{\eta }=\int_{[0,z]}q_{2}(z,\xi )dw_{\xi }\) and let \(M_{z}^{\sim }\) denote \(\iint_{[0,z]\times [0,z]}q_{1}(\xi \vee \eta ,\eta )q_{2}(\xi \vee \eta ,\xi )\,dw_{\xi }\,dw_{\eta}\). If, for \(\omega \in X\), \(\tau_{1}(\cdot ,\omega )\), \(\tau_{2}(\cdot ,\omega )\) denote the measures on \([0,1]^{2}\) defined by \(M^{1}= M^{\sim }\), \(M^{2}= M\), respectively, then \(L_{i}(z,a,\omega )\), \(z\in [0,1]^{2}\), \(a\in {\mathbb R}\), \(i=1,2\), are known to exist such that \(\int F(a)L_{i}(z,a,\omega )da=\int_{[0,z]}F(M_{\xi }^{i})(\omega )\tau_{i}(d\xi ,\omega )\). The brackets \(\langle M\rangle\), \(\langle M_{\cdot t}\rangle\) can be represented as \(\langle M\rangle _{z}=\int_{[0,z]}g(u,v)\,dudv\) and \(\langle M_{\cdot t}\rangle _{s}=\int_{[0,s]}f(v,t)dv\). If \(L^{1}(s,t,a)\) is the \(M_{\cdot t}\)-local time, let \[ L^{1,(n)}(u,v,a)=L^{1}(u,v,a)-L^{1}(1/n,v,a), \] \[ L_{2,n}(z,a)=\int_{[(1/n,1/n),z]}g(u,v)f(u,v)^{-1}L^{1,(n)}(du,v,a)dv. \] The authors prove the following theorems: 3.1. If \(\varphi (\xi ),\psi (\xi ,\eta )\in \bigcap_{1< p<\infty }D_{1}^{p}\) and \(\int_{_{_{[0,1]}2}}\| \varphi (\xi )\|_{p,1}^{p}d\xi +\int \int_{[0,1]^2\times [0,1]^2}\| \psi (\xi ,\eta )\|_{p,1}^{p}d\xi d\eta <\infty\) for all \(p> 1\), then \(L_{1}(s,t,a)\in D_{r}^{p}\) for all \(p> 1\), \(r<1/2\), \(s,t\in [0,1]\), \(a\in\mathbb R\). 3.2. If, for every \(p> 1\), \(\int_{[0,1]^2}\| \varphi (\xi )\|_{p}^{p}d\xi +\iint_{[0,1]^2 \times [0,1]^2}\| \psi (\xi ,\eta )\|_{p}^{p}d\xi d\eta <\infty\), then for every \(p> 1\), \(\delta > 0\), \(N> 0\), \(| a| ,| a'| \leq N\), \[ \| L_{1}(s,t,a)-L_{1}(s',t',a')\|_{p}\leq C(p,N,\delta )(| s-s'| ^{(1/2)-\delta }+| t-t'| ^{(1/2)-\delta}+| a-a'| ^{1/2}) \] for some constant \(C(p,N,\delta )\). 3.3. Suppose the hypothesis of 3.1 holds and that \(\varphi\) and \(\psi\) are continuous, that \(g(\cdot ,v),f(\cdot ,v)\) are strictly positive semimartingales, with integrable total variations of the finite variation components, with \(\langle g(\cdot v),g(\cdot v)\rangle _{\cdot }=\zeta_{v}(u)du\), \(\langle f(\cdot v),f(\cdot v)\rangle _{\cdot }=\beta_{v}(u)du\), \(\langle g(\cdot v),f(\cdot v)\rangle _{\cdot }=\gamma_{v}(u)du,\) \(\zeta\),\(\beta\) and \(\gamma\) jointly continuous and adapted. Then, if \[ E(\chi_{[(1/n,1/n),z]}| \zeta f^{- 2}| ^{p})<\infty, \quad E(\chi_{[(1/n,1/n),z]}| \beta g^{2}f^{-4}| ^{p}<\infty \] for all \(p> 4\), the authors prove that \(L_{2,n}\in D_{r}^{p}\) for all \(p> 1\) and \(r<1/3\). 4.1. In the hypothesis of 3.3, for \(L=L_{1}+ L_{2}\) and \(L_{(n)}=L(1/n,\cdot ,\cdot )-L(\cdot ,1/n,\cdot )+ L(1/n,1/n,\cdot )\), \(p> 1\), \(r<1/3\), \(\alpha <1/3-r\), it follows that \[ \| L_{(n)}(s,t,a)-L(s',t',a')\|_{p,r}\leq C(| s-s'| ^{3\alpha / 2}+| t- t'| ^{3\alpha / 2}+| a-a'| ^{3\alpha / 2}) \] for \(s,s',t,t'\geq 1/n\) and \(0<K_{1}\leq | a| ,| a'| \leq K_{2}\), where \(C\) depends on \(K_{1},K_{2},p,r,\alpha\). 4.2. There exists a \(B\subset X\) such that \(\inf_{V\supset B,V\text{open}} \inf\{\| u\|_{p,r}; u\geq 0,u| _{V}\geq 1\;\;\mu \text{-a.e.}\}= 0\) for all \(p> 1\) and \(r<1/3\) and \(\int_{[0,s]\times [0,t]}F(M_{\xi })(\omega )d(\langle M^{\sim }\rangle _{\xi }+\langle M\rangle _{\xi })=\int F(x)L(s,t,x,\omega )\,dx\) for all \(\omega \in \mathbb C B\), \(F\geq 0\).
Wong-Zakai representation of two parameter martingales
scientific article; zbMATH DE number 5010261