Functional Solutions of Stochastic Differential Equations

We present an integration condition ensuring that a stochastic differential equation d X t = μ ( t , X t ) d t + σ ( t , X t ) d B t , where μ and σ are sufficiently regular, has a solution of the form X t = Z ( t , B t ) . By generalizing the integration condition we obtain a class of stochastic differential equations that again have a functional solution, now of the form X t = Z ( t , Y t ) , with Y t an Ito process. These integration conditions, which seem to be new, provide an a priori test for the existence of functional solutions. Then path-independence holds for the trajectories of the process. By Green’s Theorem, it holds also when integrating along any piece-wise differentiable path in the plane. To determine Z at any point ( t , x ) , we may start at the initial condition and follow a path that is first horizontal and then vertical. Then the value of Z can be determined by successively solving two ordinary differential equations. Due to a Lipschitz condition, this value is unique. The differential equations relate to an earlier path-dependent approach by H. Doss, which enables the expression of a stochastic integral in terms of a differential process.