Maximum-likelihood estimation for diffusion processes via closed-form density expansions

This paper proposes a widely applicable method of approximate maximum-likelihood estimation for multivariate diffusion process from discretely sampled data. A closed-form asymptotic expansion for transition density is proposed and accompanied by an algorithm containing only basic and explicit calculations for delivering any arbitrary order of the expansion. The likelihood function is thus approximated explicitly and employed in statistical estimation. The performance of our method is demonstrated by Monte Carlo simulations from implementing several examples, which represent a wide range of commonly used diffusion models. The convergence related to the expansion and the estimation method are theoretically justified using the theory of Watanabe [Ann. Probab. 15 (1987) 1-39] and Yoshida [J. Japan Statist. Soc. 22 (1992) 139-159] on analysis of the generalized random variables under some standard sufficient conditions.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1118 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity

We consider the class of non-linear stochastic partial differential equations studied in \cite{conusdalang}. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point (t,x)\in[0,T]\times \Rd is differentiable in the Malliavin sense. For this, an extension of the integration theory in \cite{conusdalang} to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise integrals are proved. In the particular case of equations with additive noise, we establish the existence of density for the law of the solution at (t,x)\in]0,T]\times\Rd. The results apply to the stochastic wave equation in spatial dimension $d\ge 4$.Comment: 34 page

Modeling delay in genetic networks: From delay birth-death processes to delay stochastic differential equations

Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove that the distance between these two descriptions, as measured by expectations of functionals of the processes, converges to zero with increasing system size. Further, we prove that the delay birth-death process converges to the thermodynamic limit as system size tends to infinity. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the spatial and temporal distributions of transition pathways in metastable systems, oscillatory behavior in negative feedback circuits, and cross-correlations between nodes in a network. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay