Boundary value problems for stochastic differential equations (Q1176364)

The paper is concerned with stochastic differential equations of the form \[ dX_ t=f(X_ t)dt+BdW_ t, \] where \(0\leq t\leq 1\), \(W_ t\) is a standard \(K\)-dimensional Wiener process, \(X_ t\) takes values in \(R^ d\), and \(B\) is a matrix. The basic difference with respect to customary stochastic problems is that a two-points boundary condition of the form \(h(X_ 0,X_ 1)=0\) is imposed, involving both \(X_ 0\) and \(X_ 1\). The first purpose of the paper is to prove the existence and uniqueness of solutions to this boundary value problem. The solution method is based on a pathwise analysis, which extends to more general processes than \(W_ t\). Existence and uniqueness are mostly obtained under monotonicity assumptions. The second aim of the paper is the analysis of the Markov properties of solutions. For \(d=1\) the following dichotomy is established: the solution is a Markov field if and only if \(f''=0\). For \(d>1\), when \(f\) is linear the solution is a Markov field, but for nonlinear \(f\) examples are given of problems with Markov solutions and problems without the Markov property. The proofs are based on highly technical preliminary results on a Radon-Nikodym derivative and the application of an extended Girsanov theorem of Kusuoka.