The Calculus of Random Motion

To calculate an accumulated quantity along a fixed curve, we usually divide time into small pieces and multiply a value on each piece by the curve's increment. If the curve is sufficiently regular, choosing left endpoints, right endpoints, or slightly different partition points does not change the final answer. Brownian motion removes this familiar guarantee: its paths are continuous, yet the absolute amounts of their small changes have no finite sum. Refining a partition does not by itself make the error disappear.

Another distinction has nothing to do with smoothness. Consider deciding how many units of an asset to hold before its next price change, as opposed to filling in the holding for that interval after observing the change. Both records can be written along the same path. Both may use only data that have appeared by a particular time. They nevertheless describe different experiments. The first rule faces an unknown change; the second can exploit the change that has just occurred. On a discrete time grid, before and after remain distinct. When the grid disappears and an event arrives at a single instant, this distinction needs a mathematical expression of its own.

Rebuilding integration therefore requires more than a better choice of partitions. We must specify when the rule being integrated has already been determined, which changes form an accumulation identifiable in advance, and which fluctuations still have no bias given current information. The same path equipped with different information structures may have different decompositions. Temporal order is mathematical data of the model here, not an unconditional claim about causation in the world.

Roughness is not merely an obstacle to exclude. The quantity left by squared increments under refinement suggests that an ordinary first-order approximation may miss a contribution that really persists. Repeated crossings near a position suggest that ordinary residence time may not reveal all the geometry. Calculation must follow these emerging structures, rather than substitute random paths into formulas for smooth curves. Compensation, stochastic integration, and new rules for changes of variables will have different jobs; construction and proof must establish how they fit together.

This route will also test equations in the reverse direction. Specifying local rules of change does not make the symbols on the page an existing motion. Changing a boundary, a noise, or the permitted information can change the answer. A reliable stochastic calculus should explain how to accumulate and transform quantities, and under what conditions those calculations actually describe a random path.

This volume begins with a distinction that is small but indispensable: what we already know while something is happening. Only after drawing that boundary can calculation along random motion respect both the roughness of paths and the order in which information unfolds.