Stochastic Delay Equations Research Articles

Abstract stochastic integrodifferential delay equations

We investigate a class of abstract stochastic integrodifferential delay equations dependent upon a family of probability measures in a separable Hilbert space. We establish the existence and uniqueness of a mild solution, along with various continuous dependence estimates and Markov (weak and strong) properties of this solution. This is followed by a convergence result concerning the strong solutions of the Yosida approximations of our equation, from which we deduce the weak convergence of the measures induced by these strong solutions to the measure induced by the mild solution of the primary problem under investigation. Next, we establish the pth moment and almost sure exponential stability of the mild solution. Finally, an analysis of two examples, namely a generalized stochastic heat equation and a Sobolev‐type evolution equation, is provided to illustrate the applicability of the general theory.

A Note on Limit Theorems for Multivariate Martingales

Multivariate versions of the law of large numbers and the central limit theorem for martingales are given in a generality that is often necessary when studying statistical inference for stochastic process models. To illustrate the usefulness of the results, we consider estimation for a multidimensional Gaussian diffusion, where results on consistency and asymptotic normality of the maximum likelihood estimator are obtained in cases that were not covered by previously published limit theorems. The results are also applied to martingales of a different nature, which are typical of the problems occurring in connection with statistical inference for stochastic delay equations.

Wong-Zakai approximations for stochastic differential equations

The aim of this paper is to give a wide introduction to approximation concepts in the theory of stochastic differential equations. The paper is principally concerned with Zong-Zakai approximations. Our aim is to fill a gap in the literature caused by the complete lack of monographs on such approximation methods for stochastic differential equations; this will be the objective of the author's forthcoming book. First, we briefly review the currently-known approximation results for finite- and infinite-dimensional equations. Then the author's results are preceded by the introduction of two new forms of correction terms in infinite dimensions appearing in the Wong-Zakai approximations. Finally, these results are divided into four parts: for stochastic delay equations, for semilinear and nonlinear stochastic equations in abstract spaces, and for the Navier-Stokes equations. We emphasize in this paper results rather than proofs. Some applications are indicated.

Absolute stability approach to stochastic stability of infinite-dimensional nonlinear systems

Recent applications of absolute stability methods to robustness analysis have led to a rebirth of interest in this topic. This paper is concerned with the robust stability of feedback distributed parameter systems against nonlinear and random disturbances. Our analysis is based on the Lyapunov direct method. To derive a stochastic Lyapunov function, we introduce the special stochastic infinite-dimensional counterparts of the Kalman-Yakubovich lemma. They are proved by use of dynamic programming methods in combination with the properties of Hilbert-space-valued Wiener processes. Examples exhibit extensions of the Popov and circle criteria to the feedback stochastic heat and delay equations, respectively.

Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift

We establish the existence of smooth densities for solutions of $\mathbf{R}^d$-valued stochastic hereditary differential systems of the form $dx(t) = H(t,x)dt + g(t,x(t - r))dW(t).$ In the above equation, $W$ is an $n$-dimensional Wiener process, $r$ is a positive time delay, $H$ is a nonanticipating functional defined on the space of paths in $\mathbf{R}^d$ and $g$ is an $n \times d$ matrix-valued function defined on $\lbrack 0,\infty) \times \mathbf{R}^d$, such that $gg^\ast$ has degeneracies of polynomial order on a hypersurface in $\mathbf{R}^d$. In the course of proving this result, we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent principal part.

Coupled map lattices as models of deterministic and stochastic differential delay equations

We discuss the probabilistic properties of a class of differential delay equations (DDE's) by first reducing the equations to coupled map lattices, and then considering the spectral properties of the associated transfer operators. The analysis is carried out for the deterministic case and a stochastic case perturbed by additive or multiplicative white noise. This scheme provides an explicit description of the evolution of phase space densities in DDE's, and yields an evolution equation that approximates the analog for delay equations of the generalized Liouville and Fokker-Planck equations. It is shown that in many cases of interest, for both stochastic and deterministic delay equations, the phase space densities reach a limit cycle in the asymptotic regime. This statistical cycling is observed numerically in continuous time systems with delay and discussed in light of our analytical description of the transfer operators.

An extension of the Wong-Zakai theorem for stochastic evolution equations in Hilbert spaces

In this paper we examine an approximation theorem of the Wong–Zakai type for stochastic evolution equations in a Hilbert space with the noise being the generalized derivative of the Wiener process with values in another Hilbert space. As a consequence of the approximation of the Wiener process we get in the limit equation the Ito correction term for the infinite dimensional case. The obtained result includes the case of stochastic delay equations. The uniqueness and existence of solutions are guaranteed by known theorems for the mild solutions

The Malliavin calculus and stochastic delay equations

We consider stochastic delay systems dx( t) = g( x( t − r)) dW( t) driven by multi-dimensional Brownian motion W. The diffusion coefficient g is smooth with a possible degeneracy at 0. For a large class of deterministic initial paths we show that the solution x( t) admits a smooth density with respect to Lebesgue measure. The proof is based on Malliavin calculus together with new probabilistic lower bounds on the solution x.

Solution and Control of a Bilinear stochastic Delay Equation

A bilinear stochastic delay equation with delay in the coefficients of the noise is considered. When the equation is rewritten in abstract form in Hubert spaces, the coefficients of the noise are unbounded operators. A direct solution of the abstract equation is presented, under a general assumption which allows unification of this class of delay equations with a class of parabolic equations. Finally, the linear quadratic regulator problem governed by this equation is studied by a direct solution of the associated Riccati equation.

Lyapunov exponents and stationary solutions for affine stochastic delay equations

A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.

The lyapunov spectrum and stable manifolds for stochastic linear delay equations

A class of linear stochastic retarded functional differential equations is considered. These equations have diffusion coefficients that do not look into the past. It is shown that the trajectories of such equations form a continuous linear cocycle on the underlying state space. At times greater than the delay the cocycle is almost surely compact. Consequently, using an infinite-dimensional Oseledec multiplicative ergodic theorem of Ruelle, the existence of a countable non-random Lyapunov spectrum is proved. In the hyperbolic case it is shown that the state space admits an almost sure saddle-point splitting which is cocycle-invariant and corresponds to an exponential dichotomy for the stochastic flow

Unstable invariant distributions for a class of stochastic delay equations

In our article [8] we examined asymptotic mean square stability for linear retarded f.d.e.'s which are perturbed by white noise. It is shown in [8] and [10] that if the deterministic linear retarded f.d.e. is asymptotically stable, then so is the perturbed stochastic f.d.e.

Discrete approximation of nonlinear filtering for stochastic delay equations

This paper deals with discrete–time approximation of nonlinear filtering problems in which both the state and observation processes are described by stochastic delay equations. The approach consists in approximating the original problem by Euler–type discrete–time model for which an optimal filter can be obtained by an explicit recursive procedure. The validity of the approach is substantiated by verifying the weak convergence of the approximating model to our original problem

Nonlinear flows of stochastic linear delay equations

The sample non-linearity of the classical Wiener integral as a function of continuous integrands is pointed out. As an application it is shown that the solution of a linear stochastic delay equation is an almost surely non-linear function of the initial trajectory segment.

Hyperbolic State Space Decomposition for a Linear Stochastic Delay Equation

We consider the equation $dX(t) = (\int_{ - r}^0 {X(s + t)d\mu (s))dt} + GdW(t)$ where $r > 0$, $\mu $ is a matrix-valued signed measure on $[ - r,0]$, W is an n-dimensional Wiener process and G is a fixed $n \times n$ matrix. Using the known decomposition $C = S \oplus U$ of $C = C[ - r,0]$ into a stable closed subspace S and a finite dimensional unstable subspace U for the analogous deterministic equation, we prove that the projection of the solution X onto S converges exponentially fast to a stationary Gaussian process. Also the speed of explosion of the projection onto U is exhibited.

Qualitative behaviour of stochastic delay equations with a bounded memory

A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists an invariant measure π on the state space C which is unique (up to a multiplicative constant) and the tail-field is trivial. If π happens to be a probability measure, then for every initial condition, the distribution of the process converges to it as t→∞. We will formulate a sufficient condition for the existence of an invariant probability measure (ipm) in icrnia of Lyapunov junctionals and give two examples, one Heing the stochastic-delay version of the famous logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.

Stochastic Differential Equations in Hilbert Space and their Application to Delay Systems

We consider infinite dimensional stochastic differential equations with state dependent martingale noise using semigroup approach. Several kinds of solutions to such equations are studied and the interrelationsbetween them investigated. We prove also some existence theorems for a general class of stochastic evolution equations. New results on equations with state dependent Gaussian white noise are obtained. The general theory is applied to stochastic delay equations with state dependent noise. A representation theorem as well as an existence and uniqueness theorem for such equations are proved.