Stochastic integration and its relationship to stochastic differential equations are a huge area of research particularly in the financial sector. There is a lot of interesting mathematics behind stochastic integrals, and one of the more interesting ones is that Riemann sums don’t work with integrating random variables — so called Ito calculus and Stratonovich calculus use the limit of a sum using the left end point and midpoint respectively, and give different answers. Numerical approximations are challenging, because the most useful random variable in these types of problems is brownian motion, or the Weiner process, where each realization is a continuous nowhere differentiable function. Relevant Wikipedia articles on Ito and Stratonovich are https://en.wikipedia.org/wiki/It%C3%B4_calculus<https://en.wikipedia.org/wiki/Itô_calculus> and https://en.wikipedia.org/wiki/Stratonovich_integral but these are very bare bones. Steve On Jul 11, 2018, at 2:12 AM, Dan Asimov <dasimov@earthlink.net<mailto:dasimov@earthlink.net>> wrote: I had almost never thought about stochastic integrals, and the idea of a probabilistic vector field is cool. ----- the speed at one moment is NOT independent of the speed at the next moment ----- I'm not sure if I'm interpreting this right. One way might be (let's say for curves {A(t)} in the complex plane with A(0) = 0, A'(0) = 1: Then something like Brownian motion defines a measure mu_t, for each t > 0, on all possible continuous paths p : [0, t] —> C so that certain probabilistic consistency equations are satisfied. This could lead to really interesting possibilities. —Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...