A Useful Extension of Ito＇s Formula with Applications to Optimal Stopping

Given a continuous semimartingale M = （Mt）t≥〉0 and a d-dimensional continuous process of locally bounded variation V = （V^1,……, V^d）, the multidimensional Ito Formula states that f（Mt, Vt） - f（M0, V0） = ∫[0, t] Dx0f（Ms, Vs）dMs＋∑i=1^d∫[0, t] Dxi F（Ms, Vs）dVs^i＋1/2∫[0, t] Dx0^2 f（Ms, Vs）d 〈M〉s if f（x0,……,xd） is of C^2-type with respect to x0 and of C^1-type with respect to the other arguments This formula is very useful when solving various optimal stopping problems based on Brownian motion. However, in such application the function f typically fails to satisfy the stated conditions in that its first partial derivative with respect to x0 is only absolutely continuous. We prove that the formula remains true for such functions and demonstrate its use with two examples from Mathematical Finance.