Markov property for systems driven by the Brownian sheet

Source abstract

We study stochastic systems driven by a Brownian sheet and formulate a Markov property adapted to their two-parameter structure. For time-space homogeneous Itô sheets, the future evolution from a point $(t,x)$ is not determined by the single value $Y(t,x)$ alone, but also by the boundary data along the horizontal and vertical edges issuing from $(t,x)$. We therefore show that the enlarged state consisting of the triple $(Y,D_1Y,D_2Y)$ satisfies a natural Markov property with respect to the past sigma-algebra generated by the two coordinate histories of the Brownian sheet. The result is illustrated by examples showing the strong martingale property for the Brownian sheet and for simple Brownian sheet systems. We also recall a Brownian sheet Girsanov theorem and use it to treat drifted sheets.