  We revisit the classical problem of approximating a stochastic differential
equation by a discrete-time and discrete-space Markov chain. Our construction
iterates Caratheodory's theorem over time to match the moments of the
increments locally. This allows to construct a Markov chain with a sparse
transition matrix where the number of attainable states grows at most
polynomially as time increases. Moreover, the MC evolves on a tree whose nodes
lie on a "universal lattice" in the sense that an arbitrary number of different
SDEs can be approximated on the same tree. The construction is not tailored to
specific models, we discuss both the case of uni-variate and multi-variate case
SDEs, and provide an implementation and numerical experiments.
