平方协变差和连续半鞅的推广It公式(英文)

Quadratic Covariation and Extended It's Formula for Continuous Semimartingales

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∫t0 f(Xs)ds存在 ,那么平方协变差存在且等于 - ∫Rf(a)daLat,Lta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It 公式F(Xt) =F(X0 ) + ∫t0 f(Xs)dXs+ 12 [f(X) ,X]t.

Let X be a continuous semimartingale, f be a locally integrable function on R. In this paper we show that the quadratic convariation \ t exists and is equal to -∫ Rf(a)d aL a t, where L a t is the local time of X, whenever ∫ t 0f(X s) d X s exists. It follows that for an absolutely continuous function F with derivative f, the extended It's formula takes the form F(X t)=F(X 0)+∫ t 0f(X s) d X s+12\ t.