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Differential equations sometimes describe things that are uncertain. For example, differential equations in finance often describe uncertain phenomena because the return on many investments is unknown. Stochastic differential equations are differential equations involving stochastic (random) processes, which are used to account for uncertainty. Stochastic differential equations have found widespread use in all areas of pure and applied mathematics such as finance, geometry, physics, geometry and so on.
Unfortunately, most interesting stochastic processes have very poor analytic properties. In particular, most interesting stochastic processes lack differentiability. The fundamental problem of stochastic differential equations lies mostly in making sense of differential equations involving derivatives of non-differentiable stochastic processes.
The celebrated Itô calculus was developed in the 1950s by Kiyoshi Itô as one approach to making sense of stochastic differential equations but it comes with several limitations. In 1998 Terry Lyons introduced rough paths theory as an alternative approach. In this talk, I will introduce the basics of rough paths theory with a basic example of the Weierstrass function. I will conclude with several open problems that can be approached with calculus and basic real analysis. I suspect these problems should be imminently doable by anyone with a strong background on undergraduate analysis.