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We explore the limit of stochastic differential equations driven by some random processes satisfying singularly perturbed second order stochastic differential equations. The main tool we employ is the universal limit theorem in rough path…
This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that…
We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are important because they are relatively easy to simulate. One does not…
Stochastic approximation is a foundation for many algorithms found in machine learning and optimization. It is in general slow to converge: the mean square error vanishes as $O(n^{-1})$. A deterministic counterpart known as quasi-stochastic…
We study the problem of parameter estimation for the homogenization limit of multiscale systems involving fractional dynamics. In the case of stochastic multiscale systems driven by Brownian motion, it has been shown that in order for the…
In this paper we prove the strong averaging principle for a slow-fast system of rough differential equations. The slow and the fast component of the system are driven by a rather general random rough path and Brownian rough path,…
Surface roughness is an important quantity to many engineering and precision manufacturing disciplines. In this paper we investigate the problem of estimating the root-mean-square roughness of a sample by passive linear optical methods. By…
We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983)…
We propose a novel framework for analyzing convergence rates of stochastic optimization algorithms with adaptive step sizes. This framework is based on analyzing properties of an underlying generic stochastic process, in particular by…
Randomized algorithms, such as randomized sketching or stochastic optimization, are a promising approach to ease the computational burden in analyzing large datasets. However, randomized algorithms also produce non-deterministic outputs,…
We examine a mean-reverting Ornstein-Uhlenbeck process that perturbs an unknown Lipschitz-continuous drift and aim to estimate the drift's value at a predetermined time horizon by sampling the path of the process. Due to the time varying…
Regularly varying stochastic processes model extreme dependence between process values at different locations and/or time points. For such processes we propose a two-step parameter estimation of the extremogram, when some part of the domain…
We show how averages of exponential functions of path dependent quantities, such as those of Work Fluctuation Theorems, detect phase transitions in deterministic and stochastic systems. State space truncation -- the restriction of the…
The signature of a path, as a fundamental object in Rough path theory, serves as a generating function for non-commutative monomials on path space. It transforms the path into a grouplike element in the tensor algebra space, summarising the…
We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process (not necessarily a semi-martingale). No adaptedness of initial point or vector fields is assumed. Under a simple condition on the…
Local dependence random graph models are a class of block models for network data which allow for dependence among edges under a local dependence assumption defined around the block structure of the network. Since being introduced by…
Motivated by empirical evidence from the joint behavior of realized volatility time series, we propose to model the joint dynamics of log-volatilities using a multivariate fractional Ornstein-Uhlenbeck process. This model is a multivariate…
We study the implicit regularization of mini-batch stochastic gradient descent, when applied to the fundamental problem of least squares regression. We leverage a continuous-time stochastic differential equation having the same moments as…
We extend the theoretical results for any FOU(p) processes for the case in which the Hurst parameter is less than 1/2 and we show theoretically and by simulations that under some conditions on T and the sample size n it is possible to…
In this paper, in a multivariate setting we derive near optimal rates of convergence in the minimax sense for estimating partial derivatives of the mean function for functional data observed under a fixed synchronous design over H\"older…