Applications of integration by parts formula for infinite-dimensional semimartingales
Let T be any element of \({\mathcal D}'(R^ d)\), b be a \(C^{\infty}\)-vector field on \(R^ d\) with bounded derivatives, \((W_ t)\) be the d- dimensional Wiener process, \(W_ 0=0\). It is proved that the \({\mathcal D}'(R^ d)\)-valued semimartingale \[ X_ t(\phi)=<T,\phi (\cdot +W_ t)\exp \{\int^{t}_{0}(b(\cdot +W_ t),dW_ s)- \frac{1}{2}\int^{t}_{0}(b,b)(\cdot +W_ s)ds\}>, \] \(t\geq 0\), \(\phi\in {\mathcal D}(R^ d)\), is a unique solution of the following stochastic differential equation in partial derivatives: \[ dX_ t(\phi)=[-D_ iX_ t(\phi)+(b_ iX_ t)(\phi)]dW^{i}_{t}+[\frac{1}{2}\Delta X_ t(\phi)-D_ i(b^ i{\mathbb{X}}_ t)(\phi)]dt,\quad X_ 0(\phi)=T(\phi). \] Analogously it is proved that the \({\mathcal D}'(R^ d)\)-valued semimartingale \[ X_ t(\phi)=<T,\phi (x_ t(\cdot))\exp \{\int^{t}_{0}((\sigma^{- 1}b)(x_ s(\cdot),dW_ s)-\frac{1}{2}\int^{t}_{0}(a^{-1}b,b)(x_ s(\cdot)ds\}>, \] \(t\geq 0\), \(\phi \in D(R^ d)\), is a unique solution of the equation: \[ dX_ t(\phi)=[\frac{1}{2}D_{ij}(a^{ij}X_ s)(\phi)-D_ i(b^ iX_ t)(\phi)]dt-[D^ j(\sigma_{ji}X_ t)(\phi)+(b^ i\sigma^{-1}_{ij}X_ t)(\phi)]dW^{i}_{t}, \] \(X_ 0(\phi)=T(\phi)\), where \(\{x_ t(x),t\geq 0\}\) is the solution of the equation: \(dx_ t(x)=\sigma (x_ t(x)dW_ t\), \(x_ 0(x)=x\in R^ d\), b,\(\sigma\) and \(\sigma^{-1}\) are \(C^{\infty}\) and have bounded derivatives of all orders; \(a=\sigma \sigma^*\).
scientific article; zbMATH DE number 3971900
scientific article