Filtering on manifolds
The main object of the article is the following system of Stratonovich stochastic differential equations \[ \begin{aligned} dx_ t & = b(t,x_ t,y_ t)dt + \sigma^ i(t,x_ t,y_ t)\circ dW^ i_ t + \overline{\sigma}^ j(t,x_ t) \circ dV^ j_ t,\\ dy_ t & = B(t,x_ t,y_ t)dt + dV_ t,\\ x_ 0 & = \xi,\quad y_ 0 \in \mathbb{R}^ d,\end{aligned} \] where \((W,V)\) is a \((d_ 1 + d_ 2)\)- dimensional Wiener process, \(b(t,\cdot,y)\), \(\sigma^ i(t,\cdot,y)\), \(\overline{\sigma}^ j(t,\cdot,y)\) are smooth vector fields, such that the solution \(x\) lies in the submanifold \(M\) of \(\mathbb{R}^{d_ 1}\), and \(\xi_ 0\) is a random vector in \(M\). The author considers the conditional distribution of \(x_ t\) under given \(\{y_ s; s\in [0;t]\}\). To describe this distribution the author considers a certain vector field on \(M\). He constructs a variant of the stochastic calculus in which the derivative is a derivative with respect to this field. The corresponding partial differential equations are considered. The main result is the following. For special \(b\), \(\sigma\), \(\overline{\sigma}\) the conditional distribution of \(x\) has a density with respect to the initial distribution \(\xi_ 0\) and this density satisfies the differential equation in the sense of the constructed stochastic calculus.
Stratonovich stochastic differential equations
scientific article; zbMATH DE number 627456