sde_npsml_euler.m

SIMULATION AND ESTIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH MATLAB

function xhat_endpoint = SDE_NPSML_euler(bigtheta,OWNTIME,TIME,VRBL,XOBS,PROBLEM,NUMSIM,SDETYPE,NUMDEPVARS,SEED)

% Returns the Euler-Maruyama (read the warning below) xhat estimated values at n time-points t_1,t_2,...,t_n for 
% the stochastic model dX(t) =... defined by a system of (Ito) SDEs considered on the time-intervals [t_0,t_1],...,[t_(n-1),t_n] where, 
% for each time-interval, the initial condition is given by X(t_(k)) = XOBS(t_(k)),    k=1,...,n-1.
%
% Warning: To be used ONLY with the implementation of the Non-Parametric SML algorithm as described in [1] (see SDE_NPSML).
%          For different purposes use SDE_euler.m
%
% IN:
%     bigtheta; complete structural parameter vector
%      OWNTIME; vector containing the equispaced simulation times sorted in ascending order. 
%               It has starting simulation-time in first and ending simulation-time in last position. 
%               Thus OWNTIME(i) - OWNTIME(i-1) = h, where h is the fixed stepsize 
%               for the numerical intregration (i=2,3,...)
%         TIME; the array of unique observation times
%         VRBL; the array of unique label-variables 
%         XOBS; the matrix-shaped observed data
%       NUMSIM; the number of desired simulations for the SDE numerical integration 
%      PROBLEM; the user defined name of the current problem/experiment/example etc. (e.g. 'mySDE')
%      SDETYPE; the SDE definition: must be 'Ito'
%   NUMDEPVARS; the number of dependent variables, i.e. the SDE dimension
%         SEED; the seed for the generation of pseudo-random normal variates, i.e. the argument for randn('state',SEED);
%               type 'help randn' for details;
% OUT: xhat_endpoint; the Euler-Maruyama approximation at the intervals end-points
%
% References:
% [1] A.S. Hurn, K.A. Lindsay and V.L. Martin "On the effficacy of simulated maximum likelihood for estimating the parameters of 
%     stochastic differential equations", J. Time Series Analysis vol. 24, n