Semimartingale characterization of generalized derivatives

Abstract

Let W be a Wiener process and let be a measurable function of two variables. In this paper we show that the function f admits a generalized derivative in x, a weakgeneralized derivative in t and a second order weak generalized derivative in x if and only if thetransformed process is a semimartingale and the quadratic covariation of theintegrand of the martingale part of and W _t exists in a weak sense. Under these conditions a generalized Itô formula is proved

Keywords:

• Wiener process
• semimartingale
• quadratic covariation
• Itô's formula
• generalized derivative
• AMS 1991 Subject Classification: 60G44, 60H05

^*Research supported by ISF grant MXI 200

^*Research supported by ISF grant MXI 200

Notes

^*Research supported by ISF grant MXI 200