An example of a non-Markovian stochastic two-point boundary value problem (Q1380396)

The authors define the conditional independence of two \(\sigma\)-fields with respect to a third \(\sigma\)-field and characterize that in terms of factorization property. As an application, they prove that the solution \((X^1(t), X^2(t))\) of the following two-dimensional stochastic differential equation of Stratonovich type \[ dX^1(t) =X^1(t) \circ dW^1(t) +X^2(t) \circ dW^2 (t),\;dX^2(t) =X^2(t) \circ dW^2(t), \quad t\in (0,1), \] with the boundary condition \(X^1(0)+ X^2(0) =1\), \(X^1(1) +X^2(1) =1\) is not a Markov field.