Inference for systems of stochastic differential equations from discretely sampled data: a numerical maximum likelihood approach

Maximum likelihood estimation of discretely observed diffusion processes is mostly hampered by the lack of a closed form solution of the transient density. It has recently been argued that a most generic remedy to this problem is the numerical solution of the pertinent Fokker–Planck (FP) or forward Kolmogorov equation. Here we expand existing work on univariate diffusions to higher dimensions. We find that in the bivariate and trivariate cases, a numerical solution of the FP equation via alternating direction finite difference schemes yields results surprisingly close to exact maximum likelihood in a number of test cases. After providing evidence for the efficiency of such a numerical approach, we illustrate its application for the estimation of a joint system of short-run and medium-run investor sentiment and asset price dynamics using German stock market data. Copyright Springer-Verlag Berlin Heidelberg 2013