Leszek Gawarecki

Stochastic Differential Equations in Infinite Dimensions

Preface.- Part I: Stochastic Differential Equations in Infinite Dimensions.- 1.Partial Differential Equations as Equations in Infinite.- 2.Stochastic Calculus.- 3.Stochastic Differential Equations.- 4.Solutions by Variational Method.- 5.Stochastic Differential Equations with Discontinuous Drift.- Part II: Stability, Boundedness, and Invariant Measures.- 6.Stability Theory for Strong and Mild Solutions.- 7.Ultimate Boundedness and Invariant Measure.- References.- Index.

The monotonicity inequality for linear stochastic partial differential equations

We prove the monotonicity inequality for differential operators A and L that occur as coefficients in linear stochastic partial differential equations associated with finite-dimensional Ito processes. We characterize the solutions of such equations. A probabilistic representation is obtained for solutions to a class of evolution equations associated with time dependent, possibly degenerate, second-order elliptic differential operators.

Stochastic Differential Equations

There exist different notions of a solution to a semilinear stochastic differential equation (SSDE). We define strong, weak (in the sense of duality), mild, and martingale solutions, and study the problem of existence and uniqueness. As in the deterministic case, for example in Pazy (Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York 1983), we first study solutions to a stochastic counterpart of the deterministic inhomogeneous Cauchy problem and we highlight the role played by the stochastic convolution. The SSDE’s we investigate are allowed to depend on the entire past of the solution which significantly broadens the field of applications. The existence result for mild solutions is first obtained for equations with Lipschitz coefficients. In the special case of equations depending only on the presence, we discuss the Markov property, dependence of the solution on the initial condition, including differentiability, and the Kolmogorov backward equation. We also study SSDE’s with continuous coefficients, and present an existence result for martingale solutions, but due to the failure of the Peano theorem, a compactness assumption is added for the associated semigroup, as in DaPrato and Zabczyk (Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge 1992). We also present an existence result for SSDE’s driven by a cylindrical Wiener process.

On the Zakai Equation of Filtering with Gaussian Noise

The purpose of this work is to present an analogue of the Zakai type equation in case the noise is a Gaussian process, including fractional Brownian motion (fBm). The problem is of interest in view of the fact that the signal sent through the internet is contaminated by the noise given by fBm ([6]). Recently, a similar filtering problem with fBm noise was considered in [1]. However, the authors considered a non—Markovian signal process. The assumption of the Markov property on the signal is realistic. This also leads to a recursive equation, which is easily tractable, and is widely studied ([11, 5]).

Proper Moving Average Representations and Outer Functions in Two Variables

Abstract In this work, we consider the problem of moving average representations for random fields. As in the Kolmogorov–Wiener case, such representations lead to interesting questions in harmonic analysis in the polydisc. In particular, we study outer functions with respect to half-space, semigroup and quarterplane and their interrelations.

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Several results concerning the existence and uniqueness of solutions of Ito SDEs in a real separable Hubert space have recently been reported. In this work we first obtain the existence and uniqueness of strong solutions to (not necessarily) Ito SDEs in a Hubert space under Lipschitz-type conditions on the coefficients. We assume usual continuity and linear growth conditions. For nonLipschitz coefficients an approximation technique of Gikhman and Skorokhod is then used to prove the existence of weak solutions taking values in a larger Hubert space H-\. This result depends on an assumption that H can be compactly embedded in H-\, such that the coefficients satisfy regularity conditions with respect to H-\. This assumption is not a limitation to our method as it is necessary even in the deterministic case. Additionally, we prove the existence of martingale solutions to infinite dimensional semilinear SDEs. In both cases, coefficients F(£, ·) and J9(£, ·) may depend on the entire past of X £ C([0, T], H) and not on the value of x(t) alone.

Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

Abstract We use the Yosida approximation to find an Itô formula for mild solutions { X x ⁢ ( t ) , t ≥ 0 } {\{X^{x}(t),t\geq 0\}} of SPDEs with Gaussian and non-Gaussian colored noise, with the non-Gaussian noise being defined through a compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in C 1 , 2 ⁢ ( [ 0 , T ] × H ) {C^{1,2}([0,T]\times H)} , as in the case considered for SDEs in [15]. Using this Itô formula, we prove exponential stability and exponential ultimate boundedness properties, in the mean square sense, for mild solutions. We also compare this Itô formula to an Itô formula for mild solutions introduced by Ichikawa in [12], and an Itô formula written in terms of the semigroup of the drift operator [5], which we extend to the non-Gaussian case.

Paired vehicle occupant analysis indicates age and crash severity moderate likelihood of higher severity injury in second row seated adults in frontal crashes

The majority of advances in occupant protection systems for motor vehicle occupants have focused on occupants seated in the front row of the vehicle. Recent studies suggest that these systems have resulted in lower injury risk for front row occupants as compared to those in the second row. However, these findings are not universal. In addition, some of these findings result from analyses that compare groups of front and second row occupants exposed to dissimilar crash conditions, raising questions regarding whether they might reflect differences in the crash rather than the front and second row restraint systems. The current study examines factors associated with injury risk for pairs of right front seat and second row occupants in frontal crashes in the United States using paired data analysis techniques. These data indicate that the occupant seated in the front row frequently experiences the more severe injury in the pair, however there were no significant differences in the rate of occurrence of these events and events where the more severe injury occurs in the second row occupant of the pair. A logistic regression indicated that the likelihood of the more severe injury occurring in the second row seated occupant of the pair increased as crash severity increased, consistent with data from anatomic test dummy (ATD) tests. It also indicated that the second row occupant was more likely to have the more severe injury in the pair if that occupant was the older occupant of the pair. These findings suggest that occupant protection systems which focus on providing protection specifically for injuries experienced by older occupants in the second row in higher severity crash conditions might provide the greatest benefit.

Injury severity data for front and second row passengers in frontal crashes

The data contained here were obtained from the National Highway Transportation Safety Administration׳s National Automotive Sampling System – Crashworthiness Data System (NASS-CDS) for the years 2008–2014. This publically available data set monitors motor vehicle crashes in the United States, using a stratified random sample frame, resulting in information on approximately 5000 crashes each year that can be utilized to create national estimates for crashes. The NASS-CDS data sets document vehicle, crash, and occupant factors. These data can be utilized to examine public health, law enforcement, roadway planning, and vehicle design issues. The data provided in this brief are a subset of crash events and occupants. The crashes provided are exclusively frontal crashes. Within these crashes, only restrained occupants who were seated in the right front seat position or the second row outboard seat positions were included. The front row and second row data sets were utilized to construct occupant pairs crashes where both a right front seat occupant and a second row occupant were available. Both unpaired and paired data sets are provided in this brief.

Ultimate Boundedness and Invariant Measure

We study ultimate boundedness in the m.s.s. of solutions to SDE’s, namely the following property, \(E\| X^{x}(t)\|^{2}_{H}\leq c\mathrm{e}^{-\beta t}\| x\|_{H}^{2}+M\), \(c,\; \beta >0\), M<∞.