Boris Rozovskii

Stochastic evolution equations

The theory of strong solutions of Ito equations in Banach spaces is expounded. The results of this theory are applied to the investigation of strongly parabolic Ito partial differential equations.

A novel approach to detection of intrusions in computer networks via adaptive sequential and batch-sequential change-point detection methods

Large-scale computer network attacks in their final stages can readily be identified by observing very abrupt changes in the network traffic. In the early stage of an attack, however, these changes are hard to detect and difficult to distinguish from usual traffic fluctuations. Rapid response, a minimal false-alarm rate, and the capability to detect a wide spectrum of attacks are the crucial features of intrusion detection systems. In this paper, we develop efficient adaptive sequential and batch-sequential methods for an early detection of attacks that lead to changes in network traffic, such as denial-of-service attacks, worm-based attacks, port-scanning, and man-in-the-middle attacks. These methods employ a statistical analysis of data from multiple layers of the network protocol to detect very subtle traffic changes. The algorithms are based on change-point detection theory and utilize a thresholding of test statistics to achieve a fixed rate of false alarms while allowing us to detect changes in statistical models as soon as possible. There are three attractive features of the proposed approach. First, the developed algorithms are self-learning, which enables them to adapt to various network loads and usage patterns. Secondly, they allow for the detection of attacks with a small average delay for a given false-alarm rate. Thirdly, they are computationally simple and thus can be implemented online. Theoretical frameworks for detection procedures are presented. We also give the results of the experimental study with the use of a network simulator testbed as well as real-life testing for TCP SYN flooding attacks

Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics

In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier-Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner. The separation principle effectively reduces a stochastic equation to its associated propagator, a system of deterministic equations for the coefficients of the Wiener Chaos expansion. Simple formulas for statistical moments of the stochastic solution are presented. These formulas only involve the solutions of the propagator. We demonstrate that for short time solutions the numerical methods based on the Wiener Chaos expansion are more efficient and accurate than those based on the Monte Carlo simulations.

Stochastic Navier--Stokes Equations for Turbulent Flows

This paper concerns the fluid dynamics modelled by the stochastic flow \left\{ \begin{array}{l} \boldsymbol{\dot{\eta}}\left( t,x\right) =\boldsymbol{u}\left( t,\boldsymbol{\eta} \left( t,x\right) \right) +\boldsymbol{\sigma}\left( t,\boldsymbol{\eta}\left( t,x\right) \right) \circ\dot{W}, \\ \\ \boldsymbol{\eta}(0,x)=x, \end{array} \right. where the turbulent term is driven by the white noise

Nonlinear Filtering Revisited: A Spectral Approach

\dot{W}

Global L2-solutions of stochastic Navier–Stokes equations

. The motivation for this setting is to understand the motion of fluid parcels in turbulent and randomly forced fluid flows. Stochastic Euler equations for the undetermined components

Stochastic partial differential equations : six perspectives

\boldsymbol{u}(t,x)

Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness, and Ergodicity

and

Estimates of turbulence parameters from Lagrangian data using a stochastic particle model

\boldsymbol{\sigma}(t,x)

Stochastic modelling in physical oceanography

of the spatial velocity field are derived from the first principles. The resulting equations include as particular cases the deterministic and randomly forced counterparts of these equations.In the second part of the paper, we prove the existence and uniqueness of a strong local solution to the stochastic Navier--Stokes equation in