ML Estimation Of The Parameters Of SDE's By Numerical Solution Of The Fokker-Planck Equation

Maximum likelihood (ML) estimates of the parameters of SDEs are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed form expression for the transitional probability density function (PDF) of the process is not available. One popular way to obtain the transitional PDF is to solve the Fokker-Planck equation numerically. However the treatment of the delta function initial condition and zero-flux boundary conditions, both of which are necessary to implement these numerical schemes, is not straightforward. By reformulating the problem in terms of the transitional cumulative distribution function (CDF), it is shown that these conditions are handled easily. The transitional PDF is subsequently computed by numerical differentiation of the transitional CDF and used to construct a likelihood function in the usual way. Consider the general one-dimensional, timehomogeneous stochastic differential equation (SDE) dX = μ(X; θ) dt + √ g(X; θ) dW where X is a stochastic Markov process, μ(x; θ) and g(x;θ) are respectively the instantaneous drift and instantaneous diffusion of X , dW is the differential of the Wiener process and θ is a vector of parameters to be estimated. The aim of ML estimation is to minimise the negative log-likelihood function with respect to the parameter vector θ. ML estimation relies on the fact that the transitional density of X at time t is the solution of the Fokker-Planck equation ∂f ∂t = ∂ ∂x [ 1 2 ∂(g(x; θ)f) ∂x − μ(x; θ)f ] satisfying a delta function initial condition and gradient-like boundary conditions. This paper is concerned with an equivalent statement of this problem in terms of the transitional CDF, F (x, t) , which is defined in terms of the transitional PDF, f(x, t), by F (x, t) = ∫ x f(u, t) du . When expressed in terms of F (x, t), the FokkerPlanck equation takes the form ∂F ∂t = 1 2 ∂ ∂x ( g ∂F ∂x ) − μ ∂x with a step function initial condition and Dirichlet boundary conditions. Both the PDF and CDF approaches to the solution of the Fokker-Planck equation are implemented using a finite-difference method. The latter is easier to implement than the former, because the initial condition is more amenable to numerical work and the gradient-like boundary conditions associated with the Fokker-Planck equation are replaced by Dirichlet boundary conditions in the modification. The efficacy of the numerical solution is evaluated by means of two Monte Carlo exercises based on simulating the CIR equation dX = α(β −X)dt + σ √ X dW with α = 0.2, β = 0.08 and σ = 0.1, using Milstein’s scheme with 1000 time steps of size 0.001 between observations. The first experiment compares the PDF and CDF approaches in terms of the accuracy of loglikelihood computation, while the second simulation experiment involves the estimation of the parameters of the underlying CIR model. The most significant finding is that the CDF approach using the step function initial condition can be implemented successfully in practice. By contrast, there is no equivalent result for the PDFbased procedure, because it is always necessary to approximate the initial condition. The main empirical result to emerge from the simulation exercises in this paper is that, given equivalent starting information, the CDF approach is always superior to the PDF approach in terms of the accuracy of likelihood evaluation.