Sergey Lototsky - Academia.edu (original) (raw)

Uploads

Papers by Sergey Lototsky

Research paper thumbnail of Will Catastrophic Cyber-Risk Aggregation Thrive in the IoT Age? A Cautionary Economics Tale for (Re-)Insurers and Likes

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...

Research paper thumbnail of Classical and generalized solutions of fractional stochastic differential equations

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.

Research paper thumbnail of Wick product in the stochastic Burgers equation: A curse or a cure?

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.

Research paper thumbnail of Parameter estimation in hyperbolic multichannel models

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.

Research paper thumbnail of Statistical analysis of some evolution equations driven by space-only noise

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.

Research paper thumbnail of An Asymptotic Comparison of Two Time-Homogeneous PAM Models

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.

Research paper thumbnail of Time-homogeneous parabolic Wick–Anderson model in one space dimension: regularity of solution

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.

Research paper thumbnail of Parameter Estimation in Diagonalizable Stochastic Hyperbolic Equations

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.

Research paper thumbnail of Parameter Estimation for Stochastic Parabolic Equations: Asymptotic Properties of a Two-Dimensional Projection-Based Estimator

Statistical Inference For Stochastic Processes, Feb 1, 2003

Research paper thumbnail of Problems in Statistics of Stochastic Di erential Equations

Research paper thumbnail of Stochastic Partial Differential Equations Driven by Purely Spatial Noise

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.

Research paper thumbnail of Comment on derivation of sample size requirements for evaluating heart valves with constant risk events

The Journal of Heart Valve Disease, Oct 1, 2003

Research paper thumbnail of Parameter Estimation for Stochastic Evolution Equations with Noncommuting Operators

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

Research paper thumbnail of Nonlinear Filtering Revisited: a Spectral Approach, II

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

Research paper thumbnail of Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations

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.

Research paper thumbnail of Chaos Approach to Nonlinear Filtering

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

Research paper thumbnail of Stochastic Differential Equations: Theory and Applications

Interdisciplinary Mathematical Sciences, 2007

Research paper thumbnail of Parameter Estimation for Stochastic Parabolic Equations: Asymptotic Properties of a Two-Dimensional Projection Based Estimate

Statistical Inference For Stochastic Processes, 2003

Research paper thumbnail of Estimating Speed and Damping in the Stochastic Wave Equation

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.

Research paper thumbnail of Stochastic Differential Equations Driven by Purely Spatial Noise

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.

Research paper thumbnail of Will Catastrophic Cyber-Risk Aggregation Thrive in the IoT Age? A Cautionary Economics Tale for (Re-)Insurers and Likes

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...

Research paper thumbnail of Classical and generalized solutions of fractional stochastic differential equations

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.

Research paper thumbnail of Wick product in the stochastic Burgers equation: A curse or a cure?

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.

Research paper thumbnail of Parameter estimation in hyperbolic multichannel models

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.

Research paper thumbnail of Statistical analysis of some evolution equations driven by space-only noise

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.

Research paper thumbnail of An Asymptotic Comparison of Two Time-Homogeneous PAM Models

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.

Research paper thumbnail of Time-homogeneous parabolic Wick–Anderson model in one space dimension: regularity of solution

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.

Research paper thumbnail of Parameter Estimation in Diagonalizable Stochastic Hyperbolic Equations

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.

Research paper thumbnail of Parameter Estimation for Stochastic Parabolic Equations: Asymptotic Properties of a Two-Dimensional Projection-Based Estimator

Statistical Inference For Stochastic Processes, Feb 1, 2003

Research paper thumbnail of Problems in Statistics of Stochastic Di erential Equations

Research paper thumbnail of Stochastic Partial Differential Equations Driven by Purely Spatial Noise

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.

Research paper thumbnail of Comment on derivation of sample size requirements for evaluating heart valves with constant risk events

The Journal of Heart Valve Disease, Oct 1, 2003

Research paper thumbnail of Parameter Estimation for Stochastic Evolution Equations with Noncommuting Operators

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

Research paper thumbnail of Nonlinear Filtering Revisited: a Spectral Approach, II

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

Research paper thumbnail of Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations

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.

Research paper thumbnail of Chaos Approach to Nonlinear Filtering

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

Research paper thumbnail of Stochastic Differential Equations: Theory and Applications

Interdisciplinary Mathematical Sciences, 2007

Research paper thumbnail of Parameter Estimation for Stochastic Parabolic Equations: Asymptotic Properties of a Two-Dimensional Projection Based Estimate

Statistical Inference For Stochastic Processes, 2003

Research paper thumbnail of Estimating Speed and Damping in the Stochastic Wave Equation

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.

Research paper thumbnail of Stochastic Differential Equations Driven by Purely Spatial Noise

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.