Differentia Ion Formulas for Stochastic Integrals in the Plane*

For a one-parameter process of the form X, = X0 + & (b, d W, + & & ds, where W is a Wiener process and I+ d W is a stochastic integral, a twice continuously differentiable function f(X,) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process. Let {W,, t E Rt) be a Wiener process with a two-dimensional parameter. :-3rstwhile, we have defined stochastic integrals I +d W and I $d Wd .& as well as mixed integrals s h dz d W and $g d W dt. Now, let X; be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(X.) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(X,) on increasing paths in Rz.