STOCHASTIC INVARIANT IMBEDDING - APPLICATION TO STOCHASTIC DIFFERENTIAL-EQUATIONS WITH BOUNDARY-CONDITIONS

We study stochastic differential equations of the type: dx(t) = f(t,x(t))dt + Sigma(k=1)(d) sigma(k)(t,x(t)) omicron dw(t)(k), x is an element of R(d), Instead of the customary initial value problem, where the initial value x(0) is fixed, we impose an affine boundary condition: h(0)x(0) + h(1)xT(0) = upsilon(0), where h(0), h(1) are deterministic matrices and upsilon(0) is a fixed vector. Our main aim is to prove existence and uniqueness results for such anticipating stochastic differential equations.