On the quadratic variation of semi-martingales

Let X be a semi-martingale. Techniques of [1] and El Karoui (in [3] and [4]) are used to study the convergence of 2∊ times the number of upcrossings of [x, x+∊] by X to its local time at x. If X is continuous and if there exists a bicontinuous version of its local time process, then off a single null set, the convergence is shown to be uniform in x (and time). If X is such that the sum of the absolute value of its jumps over any finite time interval is almost surely finite, then, off a single null set, the convergence holds at all but countably many x. A notion of generalized arc length, is introduced, in the spirit of the quadratic arc length of [1], and the last result above is used to show that is the almost sure arc length of X, a uniform limit recoverable from the geometry of the trajectories.Science, Faculty ofMathematics, Department ofGraduat