We consider the problem of option pricing under stochastic volatility models, focusing on the two processes known as exponential Ornstein-Uhlenbeck and Stein-Stein. We show they admit the same limit dynamics in the regime of low fluctuations of the volatility process, under which we derive the expressions of the characteristic function and the first four cumulants for the risk neutral probabili...

We consider Ornstein-Uhlenbeck processes (OU-processes) related to hypoelliptic diffusion on finite-dimensional Lie groups: let L be a hypoelliptic, left-invariant “sum of the squares”-operator on a Lie group G with associated Markov process X, then we construct OU-type processes by adding horizontal gradient drifts of functions U . In the natural case U(x) = − log p(1, x), where p(1, x) is the...

We study a least squares estimator b θT for the Ornstein-Uhlenbeck process, dXt = θXtdt+σdB H t , driven by fractional Brownian motion B H with Hurst parameter H ≥ 1 2 . We prove the strong consistence of b θT (the almost surely convergence of b θT to the true parameter θ). We also obtain the rate of this convergence when 1/2 ≤ H < 3/4, applying a central limit theorem for multiple Wiener integ...

In spatial modeling the presence of measurement error, or “nugget”, can have a big impact on the sample behavior of the parameter estimates. This article investigates the nugget effect on maximum likelihood estimators for a onedimensional spatial model: Ornstein-Uhlenbeck plus additive white noise. Consistency and asymptotic distributions are obtained under infill asymptotics, in which a compac...

An Ornstein-Uhlenbeck diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that the neuron is subject to a sequence of inhibitory and excitatory post-synaptic potentials that occur with time-dependent rates. The resulting process is characterized by time-dependent drift. For this model, we construct the return process describing the membra...