Transition Density and Simulated Likelihood Estimation for Time-Inhomogeneous Diffusions

Abstract

We propose a method to estimate the transition density of a nonlinear time-inhomogeneous diffusion. Expressing the transition density as a functional of a Brownian bridge allows us to estimate the density through Monte Carlo simulations with any level of precision. We show how these transition density estimates can be effectively used to estimate the parameters of the time-inhomogeneous diffusion and the conditional moments of the process. In this article, we prove that our method is asymptotically equivalent to the maximum likelihood estimator and more reliable than the closed-form approximation approach largely used in the literature.

Keywords:

• Conditional moments estimation
• Simulated maximum likelihood estimator
• Stochastic differential equations
• Time-inhomogeneous diffusions

Mathematics Subject Classification:

• 60H10
• 60H35
• 62F10

Acknowledgment

This research was supported by Fundação para a Ciência e a Tecnologia - FCT (FEDER/POCI2010).

Notes

⁽*⁾ N = S = 30; ⁽**⁾ N = S = 60; Her(3): Hermite Expan., m = 3.

⁽*⁾ N = S = 30; ⁽**⁾ N = S = 60; Her(3): Hermite Expan., m = 3.

RMSE100: Root mean squared error multiplied by 100.

SEML100: Std. Error of MLE multiplied by 100.