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Stochastic Differential Equations (SDEs) were originally devised by It\^o toprovide a pathwise construction of diffusion processes. A less exploredapproach to represent them is through Time Change Equations (TCEs) as put forthby Doeblin. TCEs are a generalization of Ordinary Differential Equations drivenby random functions. We present a simple example where TCEs have some advantageover SDEs.  We represent sticky L\'evy processes as the unique solution to a TCE drivenby a L\'evy process with no negative jumps. The solution is adapted to thetime-changed filtration of the L\'evy process driving the equation. This is incontrast to the SDE describing sticky Brownian motion, which is known to haveno adapted solutions as first proved by Chitashvili. A known consequence ofsuch non-adaptability for SDEs is that certain natural approximations to thesolution of the corresponding SDE do not converge in probability, even thoughthey do converge weakly. Instead, we provide strong approximation schemes forthe solution of our TCE (by adapting Euler's method for ODEs), whenever thedriving L\'evy process is strongly approximated.

The Sticky L\'evy Process as a solution to a Time Change Equation