Triangular stochastic differential equations with boundary conditions
The author considers the general equation \[ {d \over dt} X_ t + f(X_ t)={d \over dt} W_ t; \quad h(X_ 0,X_ 1)=0, \] where \(t \in [0,1]\), \(\{W_ t:t \in [0,1]\}\) is a \(d\)-dimensional Brownian motion, \(\{X_ t\}\) takes values in \(\mathbb{R}^ d\) and (note) the boundary condition involves both \(X_ 0\) and \(X_ 1\). It is also assumed that for each \(i \in \{1,\dots,d\}\), \(f_ i (x_ 1, \dots,x_ d)\) depends only on the first \(i\) variables, with a similar condition being imposed upon the \(h_ i\). A necessary condition is provided for the solution \(X_ t\) to be a Markov field. Then two ``necessary and sufficient'' results are given: in the first of these, \(X_ t\) is shown to be a Markov field process if and only if the \(f_ i\) are linear in the last variable, while in the second \(X_ t\) is a Markov field process if and only if the boundary condition is of a specific form.
scientific article; zbMATH DE number 510104