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Papers by Sergey Lototsky
ACM Transactions on Management Information Systems
Service liability interconnections among networked IT and IoT-driven service organizations create... more Service liability interconnections among networked IT and IoT-driven service organizations create potential channels for cascading service disruptions due to modern cybercrimes such as DDoS, APT, and ransomware attacks. These attacks are known to inflict cascading catastrophic service disruptions worth billions of dollars across organizations and critical infrastructure around the globe. Cyber-insurance is a risk management mechanism that is gaining increasing industry popularity to cover client (organization) risks after a cyber-attack. However, there is a certain likelihood that the nature of a successful attack is of such magnitude that an organizational client’s insurance provider is not able to cover the multi-party aggregate losses incurred upon itself by its clients and their descendants in the supply chain, thereby needing to re-insure itself via other cyber-insurance firms. To this end, one question worth investigating in the first place is whether an ecosystem comprising a...
Stochastics and Partial Differential Equations: Analysis and Computations
For stochastic evolution equations with fractional derivatives, classical solutions exist when th... more For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise, there can be a generalized solution in suitable weighted chaos spaces. Presence of fractional derivatives in time leads to various modifications of the stochastic parabolicity condition. Interesting new effects appear when the order of the time derivative in the noise term is less than or equal to one-half.
Asymptotic Analysis
It has been known for a while that a nonlinear equation driven by singular noise must be interpre... more It has been known for a while that a nonlinear equation driven by singular noise must be interpreted in the re-normalized, or Wick, form. For the stochastic Burgers equation, Wick nonlinearity forces the solution to be a generalized process no matter how regular the random perturbation is, whence the curse. On the other hand, certain multiplicative random perturbations of the deterministic Burgers equation can only be interpreted in the Wick form, whence the cure. The analysis is based on the study of the coefficients of the chaos expansion of the solution at different stochastic scales.
Asymptotic Analysis
ABSTRACT A multichannel model is considered, with each channel represented by a linear second-ord... more ABSTRACT A multichannel model is considered, with each channel represented by a linear second-order stochastic equation with two unknown coefficients. The channels are interpreted as the Fourier coefficients of the solution of a stochastic hyperbolic equation with possibly unbounded damping. The maximum likelihood estimator of the coefficients is constructed using the information from a finite number of channels. Necessary and sufficient conditions are determined for the consistency of the estimator as the number of channels increases, while the observation time and noise intensity remain fixed.
Statistical Inference for Stochastic Processes
We study the statistical properties of stochastic evolution equations driven by space-only noise,... more We study the statistical properties of stochastic evolution equations driven by space-only noise, either additive or multiplicative. While forward problems, such as existence, uniqueness, and regularity of the solution, for such equations have been studied, little is known about inverse problems for these equations. We exploit the somewhat unusual structure of the observations coming from these equations that leads to an interesting interplay between classical and non-traditional statistical models. We derive several types of estimators for the drift and/or diffusion coefficients of these equations, and prove their relevant properties.
Communications on Stochastic Analysis
Both Wick-Itô-Skorokhod and Stratonovich interpretations of the Parabolic Anderson model (PAM) le... more Both Wick-Itô-Skorokhod and Stratonovich interpretations of the Parabolic Anderson model (PAM) lead to solutions that are real analytic as functions of the noise intensity ε, and, in the limit ε → 0, the difference between the two solutions is of order ε 2 and is non-random.
Stochastics and Partial Differential Equations: Analysis and Computations
Even though the heat equation with random potential is a well-studied object, the particular case... more Even though the heat equation with random potential is a well-studied object, the particular case of time-independent Gaussian white noise in one space dimension has yet to receive the attention it deserves. The paper investigates the stochastic heat equation with space-only Gaussian white noise on a bounded interval. The main result is that the space-time regularity of the solution is the same for additive noise and for multiplicative noise in the Wick-Itô-Skorokhod interpretation.
A parameter estimation problem is considered for a linear stochastic hyperbolic equation driven b... more A parameter estimation problem is considered for a linear stochastic hyperbolic equation driven by additive space-time Gaussian white noise. The damping/amplification operator is allowed to be unbounded. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.
Statistical Inference For Stochastic Processes, Feb 1, 2003
Siam Journal on Mathematical Analysis, 2009
ABSTRACT We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white ... more ABSTRACT We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Even the simplest equations driven by this noise often do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition is an efiective tool to study both stationary and evolution equations driven by space-only noise. The paper presents results about solvability of such equations in weighted Wiener chaos spaces and studies the long-time behavior of the solutions of evolution equations with space-only noise.
The Journal of Heart Valve Disease, Oct 1, 2003
A parameter estimation problem is considered for a stochastic evolution equation on a compact smo... more A parameter estimation problem is considered for a stochastic evolution equation on a compact smooth manifold. Unlike previous works on the subject, no commutativity is assumed between the operators in the equation. The estimate is based on finite di- mensional projections of the solution. Under certain non-degeneracy assumptions the estimate is proved to be consistent and asymptotically normal as the
Siam Journal on Control and Optimization, 1997
A recursive in time Wiener chaos representation of the optimal nonlinear filter is derived for a ... more A recursive in time Wiener chaos representation of the optimal nonlinear filter is derived for a continuous time diusion model with uncorrelated noises. The existing rep- resentations are either not recursive or require a prior computation of the unnormalized filtering density, which is time consuming. An algorithm is developed for computing a re- cursive approximation of the filter, and the
Methods and Applications of Analysis, 2000
A family of Banach spaces is introduced to control the interior smoothness and boundary behavior ... more A family of Banach spaces is introduced to control the interior smoothness and boundary behavior of functions in a general domain. Interpolation, embedding, and other properties of the spaces are studied. As an application, a certain degenerate second-order elliptic partial differential equation is considered.
The paper discusses various methods of solving the nonlinear filtering problems using expansions ... more The paper discusses various methods of solving the nonlinear filtering problems using expansions of the optimal filter in the chaos space of the observa- tion process. The elements of the expansion can be either multiple integrals or the Cameron-Martin basis. Two particular filtering algorithms are discussed for the time-homogeneous diusion filtering model with possible correlation between the state process and
Interdisciplinary Mathematical Sciences, 2007
Statistical Inference For Stochastic Processes, 2003
A parameter estimation problem is considered for a one-dimensional stochastic wave equation drive... more A parameter estimation problem is considered for a one-dimensional stochastic wave equation driven by additive space-time Gaussian white noise. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.
We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Ev... more We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Even the simplest equations driven by this noise often do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition is an effective tool to study both stationary and evolution equations driven by space-only noise. The paper presents results about solvability of such equations in weighted Wiener chaos spaces and studies the long-time behavior of the solutions of evolution equations with space-only noise.
ACM Transactions on Management Information Systems
Service liability interconnections among networked IT and IoT-driven service organizations create... more Service liability interconnections among networked IT and IoT-driven service organizations create potential channels for cascading service disruptions due to modern cybercrimes such as DDoS, APT, and ransomware attacks. These attacks are known to inflict cascading catastrophic service disruptions worth billions of dollars across organizations and critical infrastructure around the globe. Cyber-insurance is a risk management mechanism that is gaining increasing industry popularity to cover client (organization) risks after a cyber-attack. However, there is a certain likelihood that the nature of a successful attack is of such magnitude that an organizational client’s insurance provider is not able to cover the multi-party aggregate losses incurred upon itself by its clients and their descendants in the supply chain, thereby needing to re-insure itself via other cyber-insurance firms. To this end, one question worth investigating in the first place is whether an ecosystem comprising a...
Stochastics and Partial Differential Equations: Analysis and Computations
For stochastic evolution equations with fractional derivatives, classical solutions exist when th... more For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise, there can be a generalized solution in suitable weighted chaos spaces. Presence of fractional derivatives in time leads to various modifications of the stochastic parabolicity condition. Interesting new effects appear when the order of the time derivative in the noise term is less than or equal to one-half.
Asymptotic Analysis
It has been known for a while that a nonlinear equation driven by singular noise must be interpre... more It has been known for a while that a nonlinear equation driven by singular noise must be interpreted in the re-normalized, or Wick, form. For the stochastic Burgers equation, Wick nonlinearity forces the solution to be a generalized process no matter how regular the random perturbation is, whence the curse. On the other hand, certain multiplicative random perturbations of the deterministic Burgers equation can only be interpreted in the Wick form, whence the cure. The analysis is based on the study of the coefficients of the chaos expansion of the solution at different stochastic scales.
Asymptotic Analysis
ABSTRACT A multichannel model is considered, with each channel represented by a linear second-ord... more ABSTRACT A multichannel model is considered, with each channel represented by a linear second-order stochastic equation with two unknown coefficients. The channels are interpreted as the Fourier coefficients of the solution of a stochastic hyperbolic equation with possibly unbounded damping. The maximum likelihood estimator of the coefficients is constructed using the information from a finite number of channels. Necessary and sufficient conditions are determined for the consistency of the estimator as the number of channels increases, while the observation time and noise intensity remain fixed.
Statistical Inference for Stochastic Processes
We study the statistical properties of stochastic evolution equations driven by space-only noise,... more We study the statistical properties of stochastic evolution equations driven by space-only noise, either additive or multiplicative. While forward problems, such as existence, uniqueness, and regularity of the solution, for such equations have been studied, little is known about inverse problems for these equations. We exploit the somewhat unusual structure of the observations coming from these equations that leads to an interesting interplay between classical and non-traditional statistical models. We derive several types of estimators for the drift and/or diffusion coefficients of these equations, and prove their relevant properties.
Communications on Stochastic Analysis
Both Wick-Itô-Skorokhod and Stratonovich interpretations of the Parabolic Anderson model (PAM) le... more Both Wick-Itô-Skorokhod and Stratonovich interpretations of the Parabolic Anderson model (PAM) lead to solutions that are real analytic as functions of the noise intensity ε, and, in the limit ε → 0, the difference between the two solutions is of order ε 2 and is non-random.
Stochastics and Partial Differential Equations: Analysis and Computations
Even though the heat equation with random potential is a well-studied object, the particular case... more Even though the heat equation with random potential is a well-studied object, the particular case of time-independent Gaussian white noise in one space dimension has yet to receive the attention it deserves. The paper investigates the stochastic heat equation with space-only Gaussian white noise on a bounded interval. The main result is that the space-time regularity of the solution is the same for additive noise and for multiplicative noise in the Wick-Itô-Skorokhod interpretation.
A parameter estimation problem is considered for a linear stochastic hyperbolic equation driven b... more A parameter estimation problem is considered for a linear stochastic hyperbolic equation driven by additive space-time Gaussian white noise. The damping/amplification operator is allowed to be unbounded. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.
Statistical Inference For Stochastic Processes, Feb 1, 2003
Siam Journal on Mathematical Analysis, 2009
ABSTRACT We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white ... more ABSTRACT We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Even the simplest equations driven by this noise often do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition is an efiective tool to study both stationary and evolution equations driven by space-only noise. The paper presents results about solvability of such equations in weighted Wiener chaos spaces and studies the long-time behavior of the solutions of evolution equations with space-only noise.
The Journal of Heart Valve Disease, Oct 1, 2003
A parameter estimation problem is considered for a stochastic evolution equation on a compact smo... more A parameter estimation problem is considered for a stochastic evolution equation on a compact smooth manifold. Unlike previous works on the subject, no commutativity is assumed between the operators in the equation. The estimate is based on finite di- mensional projections of the solution. Under certain non-degeneracy assumptions the estimate is proved to be consistent and asymptotically normal as the
Siam Journal on Control and Optimization, 1997
A recursive in time Wiener chaos representation of the optimal nonlinear filter is derived for a ... more A recursive in time Wiener chaos representation of the optimal nonlinear filter is derived for a continuous time diusion model with uncorrelated noises. The existing rep- resentations are either not recursive or require a prior computation of the unnormalized filtering density, which is time consuming. An algorithm is developed for computing a re- cursive approximation of the filter, and the
Methods and Applications of Analysis, 2000
A family of Banach spaces is introduced to control the interior smoothness and boundary behavior ... more A family of Banach spaces is introduced to control the interior smoothness and boundary behavior of functions in a general domain. Interpolation, embedding, and other properties of the spaces are studied. As an application, a certain degenerate second-order elliptic partial differential equation is considered.
The paper discusses various methods of solving the nonlinear filtering problems using expansions ... more The paper discusses various methods of solving the nonlinear filtering problems using expansions of the optimal filter in the chaos space of the observa- tion process. The elements of the expansion can be either multiple integrals or the Cameron-Martin basis. Two particular filtering algorithms are discussed for the time-homogeneous diusion filtering model with possible correlation between the state process and
Interdisciplinary Mathematical Sciences, 2007
Statistical Inference For Stochastic Processes, 2003
A parameter estimation problem is considered for a one-dimensional stochastic wave equation drive... more A parameter estimation problem is considered for a one-dimensional stochastic wave equation driven by additive space-time Gaussian white noise. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed.
We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Ev... more We study bilinear stochastic parabolic and elliptic PDEs driven by purely spatial white noise. Even the simplest equations driven by this noise often do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition is an effective tool to study both stationary and evolution equations driven by space-only noise. The paper presents results about solvability of such equations in weighted Wiener chaos spaces and studies the long-time behavior of the solutions of evolution equations with space-only noise.