Variation Of Local Time And New Extensions To It(?)'s Formula

The classical It(o|^)'s formula (1944) for twice differentiable functions has played a central role in stochastic analysis and almost all aspects of its applications and connection with analysis, PDEs, geometry, dynamical systems, finance and physics. But the restriction of It(o|^)'s formula to functions with twice differentiability often encounter difficulties in applications. Extensions to less smooth functions are useful in studying many problems such as partial differential equtions with some singularities and mathematics of finance. Generally speaking, for any absolutely continuous function whose derivative f' exists almost everywhere, and a continuous semi-martingale X_t, there exists A_t such thatand for the time dependent case, the corresponding formula isTo find A_t in both cases especially a pathwise formula becomes key to establish a useful extension to It(o|^)'s formula. In fact investigations already began in Tanaka [46] with a beautiful use of local times introduced in Levy [29]. The generalized It(o|^)'s formula in one-dimension for time independent convex functions was developed in Meyer [36] and for superharmonic functions in multidimensions in Brosamler [5] and for distance function in Kendall [26] and more recently for time dependent functions in Peskir [39] and Elworthy, Truman and Zhao [7]. Meyer [36] and Elwortby-Truman-Zhao [7] proved(Tanaka-Meyer(1976)) Let f : R → R be a convex function (or difference of two convex functions) and μ its second derivative measure defined as μ([a, b)) := (?)ˉf(b) - (?)ˉf(a), -∞