Abstracts

Likelihood based parameter estimation for diffusion processes is an important topic in many areas of mathematical finance. For a general irreducible diffusion model it is common to approximate the transition density using either Monte Carlo based methods or by finite difference discretization of the Fokker-Planck equation. These methods require the evaluation of an approximate probability density between each observation, which quickly becomes very time consuming as the number of observations increase. Instead of approximating the transition density explicitly in the construction of the likelihood a simple strategy is to replace the exact, but unknown density by an approximate density with exact moments. This technique is known as the quasi-likelihood method and many previous studies have been on diffusion models where one can obtain analytical expression for the moments. Monte Carlo estimation is the standard method of choice when analytical moments are unavailable.
Instead of using Monte Carlo based methods we here suggest to estimate the moments for the quasi-likelihood from the approximate solution of the Kolmogorov-backward equation using finite differences. The Kolmogorov-backward equation is the adjoint to the Fokker-Planck equation. The immediate advantage of this is that we need only to solve one backward equation for any number of observations, which is a dramatic reduction in computational complexity. Another nice property of the backward equation is the well-behaved initial condition in terms of moments, which should be contrasted to the initial condition of the Fokker-Planck equation given by a Dirac measure.
The quasi-likelihood method together with approximate moments from the discrete backward equation is tested on common models e.g. CIR, GEN1 and the low computational complexity and high performance is demonstrated.