Stochastic control problems in finance having complex controls inevitably give rise to low order accuracy, usually at most second order. Fourier methods are efficient at advancing the solution between control monitoring dates, but are not monotone. This gives rise to possible violations of arbitrage inequalities. We devise a preprocessing step for Fourier methods which involves projecting the Green's function onto the set of linear basis functions. The resulting algorithm is guaranteed to be monotone (to within a tolerance), infinity norm stable and satisfies an epsilon-discrete comparison principle. The algorithm has the same complexity per step as a standard Fourier method and has second order accuracy for smooth problems.
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