A second-order Stratonovich differential equation with boundary conditions
The second-order SDE of the form \[ \ddot X_t= f(X_t)+ \sigma \dot X_t \circ \dot W_t,\qquad 0\leq t\leq 1, \] with boundary conditions \(X_0=0\), \(X_1=1\) is studied where \(W_t\) is a standard Wiener process, \(\sigma >0\) is a constant, and \(f:R\to R\) is a continuous function. Because of the terminal condition on \(X\), the solution is not expected to be a process adapted with respect to the Wiener process \(W_t\). Using the techniques of Malliavin calculus and the method of change of probability, the authors tackle the problem of existence and uniqueness of solutions and they show that the solution is a 2-Markov field if and only if the drift \(f\) is an affine function.
scientific article; zbMATH DE number 1134833