A Useful Extension of Its Formula with Applications to Optimal Stopping

Given a continuous semimartingale M = (M t )t0 and a d-dimensional continuous process of locally bounded variation V = (V 1, . . . , V d), the multidimensional It Formula states that f(Mt,Vt)f(M0,V0)=[0,t]Dx0f(Ms,Vs)dMs*i=1d[0,t]Dxif(Ms,Vs)dVis*12 [0,t]D2x0f(Ms,Vs)dMs if f(x 0, . . . , x d ) 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 x 0 is only absolutely continuous. We prove that the formula remains true for such functions and demonstrate its use with two examples from Mathematical Finance.