Curved Schemes for SDEs on Manifolds
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Given a stochastic differential equation (SDE) in ℝ^n whose solution is constrained to lie in some manifold M ⊂ℝ^n, we propose a class of numerical schemes for the SDE whose iterates remain close to M to high order. Our schemes are geometrically invariant, and can be chosen to give perfect solutions for any SDE which is diffeomorphic to n-dimensional Brownian motion. Unlike projection-based methods, our schemes may be implemented without explicit knowledge of M. Our approach does not require simulating any iterated Itô interals beyond those needed to implement the Euler–Maryuama scheme. We prove that the schemes converge under a standard set of assumptions, and illustrate their practical advantages by considering a stochastic version of the Kepler problem.
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