M/Ph&IdZddlZddlmZmZddlmZGddZ Gdde Z Gdd e Z Gd d e Z Gd d e Z Gdde ZGdde ZGddZGddZ edkrcdZdZgZdevr.e ZedeZereejejedjZejedddZejddZe ede Z!erRejeje!djZeje!ddZejd!ej"#e!dej$d"e!dzd#z z ej%Z&e'e&e d$d%d&'Z(e()d(d( Z*erejeje*jZeje*ddZejd)ejejej+e*jZejej+e*ddZejd*e dd+d'Z,e,)d(d( Z-erYejeje-jZeje-ddZejd,eddd&d-.Z.e.)Z/e.0ddej12d/0Z3e.0dej4dd1d2de.5dd1e'e.5dd1d-d13de.5dd(d-d(3Z6erYejeje6jZeje6ddZejd4eddd&d-5Z7e75dd6d-d(3Z8e'e8de'ej+e8ddZerYejeje8jZeje8ddZejd7e'e79e8dddfd-8e75ddd-d93Z:e'd:e;d9D](Z<e'e79e:e<d-8)eZ=e=>dd&d;dd$d-<\Z?Z@ZAerejeje?jZeje?ddZejd=ejejeAd>d?@eje?BddA@ejdBejCeddgej1j2ej1j2gZDeD>dCdD ZEe'eEdFdEFdEe'eEdFe'eEddEdEe'eEde'eEdjGe'eEdFdSdS)Fa6getting started with diffusions, continuous time stochastic processes Author: josef-pktd License: BSD References ---------- An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations Author(s): Desmond J. Higham Source: SIAM Review, Vol. 43, No. 3 (Sep., 2001), pp. 525-546 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/3649798 http://www.sitmo.com/ especially the formula collection Notes ----- OU process: use same trick for ARMA with constant (non-zero mean) and drift some of the processes have easy multivariate extensions *Open Issues* include xzero in returned sample or not? currently not *TODOS* * Milstein from Higham paper, for which processes does it apply * Maximum Likelihood estimation * more statistical properties (useful for tests) * helper functions for display and MonteCarlo summaries (also for testing/checking) * more processes for the menagerie (e.g. from empirical papers) * characteristic functions * transformations, non-linear e.g. log * special estimators, e.g. Ait Sahalia, empirical characteristic functions * fft examples * check naming of methods, "simulate", "sample", "simexact", ... ? stochastic volatility models: estimation unclear finance applications ? 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