平方协变差和连续半鞅的推广It（o）公式[英文]

令X是连续半鞅,f是R上的局部可积函数.本文我们将证明,只要∫otf(Xs)ds存在,那么平方协变差存在且等于-∫Rf(a)daLta,Lat是X的局部时.因此对具有导数f的绝对连续函数F,有推广的It6公式F(Xt)=F(X0)+∫ot f(Xs)dXs+1/2[f(X),X]t.

Quadratic Covariation and Extended It&s Formula for Continuous Semimartingales

Let X be a continuous semimartingale, f be a locally integrable function on R. In this paper we show that the quadratic convariation [f(X),X]t exists and is equal to ∫ f(a)daLia, where Lia is the local time of X, whenever ∫tf(Xs)dXs exists. It fol R 0lows that for an absolutely continuous function F with derivative f, the extended It6F(Xt) = F(X0) + ∫t0 f(Xs)dXs + 1/2[f(X)sformulatakestheform,X]to