A Conditional Independence Property for the Solution of a Linear Stochastic Differential Equation with Lateral Conditions

Abstract

Let L be an nth order linear differential operator with smooth coefficients and {W(t) : t ∈ [0, 1]} a standard Wiener process. We consider the stochastic differential equation

$$L[X] = \dot W$$

on [0, 1], with the lateral condition

$$\sum\limits_{j = 1}^m {{\alpha _{ij}}X({t_j})} = {c_i},1 \leqslant i \leqslant n$$

where 0 ≤ t ₁ < … < t _m ≤ 1 and α _ij , c _i ∈ ℝ. We prove that the solution to this system, considered as the vector Y(t) = (X ⁽ⁿ⁻¹⁾(t),…, X′(t), X(t)), is not a Markov field in general but satisfies a weaker conditional independence property.

Supported by a grant of the CIRIT No. BE 94–3–221

Partially supported by a CNR short-term fellowship