Differentiation formulas for stochastic integrals in the plane

Eugene Wong and Moshe Zakai

Stochastic Processes and their Applications, 1978, vol. 6, issue 3, 339-349

Abstract: For a one-parameter process of the form Xt=X0+[integral operator]t0[phi]sdWs+[integral operator]t0[psi]sds, where W is a Wiener process and [integral operator][phi]dW is a stochastic integral, a twice continuously differentiable function f(Xt) 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 {W2, z[set membership, variant]R2+} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals [integral operator] [phi]dW and [integral operator][psi]dWdW, as well as mixed integrals [integral operator]h dz dW and [integral operator]gdW dz. Now let Xz 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(Xz) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(Xz) on increasing paths in R2+.

Date: 1978
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