We study rates of convergence in central limit theorems for the partial sum of squares of general Gaussian sequences, using tools from analysis on Wiener space. No assumption of stationarity, asymptotically or otherwise, is made. The main theoretical tool is the so-called Optimal Fourth Moment Theorem [19], which provides a sharp quantitative estimate of the total variation distance on Wiener chaos to the normal law. The only assumptions made on the sequence are the existence of an asymptotic variance, that a least-squares-type estimator for this variance parameter has a bias and a variance which can be controlled, and that the sequence's auto-correlation function, which may exhibit long memory, has a no-worse memory than that of fractional Brownian motion with Hurst parameter H < 3/4. Our main result is explicit, exhibiting the trade-off between bias, variance, and memory. We apply our result to study drift parameter estimation problems for subfractional Ornstein-Uhlenbeck and bifractional Ornstein-Uhlenbeck processes with fixed-time-step observations. These are processes which fail to be stationary or self-similar, but for which detailed calculations result in explicit formulas for the estimators' asymptotic normality.
We construct a long-memory non-Gaussian stochastic process by aggregation, as limit of the empirical mean of identically distributed copies of Ornstein-Uhlenbeck processes with Hermite noise and random coefficients. We also study the asymptotic behavior of the process with respect to its parameter.
In this paper, we focus on mean-field anticipated backward stochastic differential equations (MF-BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H > 1/2. First, the existence and uniqueness of this new type of BSDEs are established using two different approaches. Then, a comparison theorem for such BSDEs is obtained. Finally, as an application of this type of equations, a related stochastic optimal control problem is studied.
We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than 1/2. Stochastic optimal control problems driven by fractional Brownian motion cannot be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional white noise calculus combined with some special tools related to differentiation for functions of measures, we establish necessary and sufficient stochastic maximum principles. To illustrate our study, we consider two applications: we solve a problem of optimal consumption from a cash flow with delay and a linear-quadratic (LQ) problem with delay.
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