Stochastic Differential Systems with Memory: Theory, Examples and Applications

The purpose of this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic time-lag model with Gaussian noise , and the classical ``heat-bath model of R. Kubo , modeling the motion of a ``large molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde\u27s). We then establish pathwise existence and uniqueness of solutions to these classes of sfde\u27s under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde\u27s is not covered by classical results of Protter, Metivier and Pellaumail and Doleans-Dade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from zero. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel $\sigma$-algebra of the state space. This chapter also contains a derivation of a formula for the weak infinitesimal generator of the semigroup for sufficiently regular functions, and for a large class of quasitame functions. In Chapter III, we study pathwise regularity of the trajectory random field in the time variable and in the initial path. Of note here is the non-existence of the stochastic flow for the singular sdde $dx(t)= x(t-r) dW(t)$ and a breakdown of linearity and local boundedness. This phenomenon is peculiar to stochastic delay equations. It leads naturally to a classification of sfde\u27s into regular and singular types. Necessary and sufficient conditions for regularity are not known. The rest of Chapter III is devoted to results on sufficient conditions for regularity of linear systems driven by white noise or semimartingales, and Sussman-Doss type nonlinear sfde\u27s. Building on the existence of a compacting stochastic flow, we develop a multiplicative ergodic theory for regular linear sfde\u27s driven by white noise, or general helix semimartingales (Chapter IV). In particular, we prove a Stable Manifold Theorem for such systems. In Chapter V, we seek asymptotic stability for various examples of one-dimensional linear sfde\u27s. Our approach is to obtain upper and lower estimates for the top Lyapunov exponent. Several topics are discussed in Chapter VI. These include the existence of smooth densities for solutions of sfde\u27s using the Malliavin calculus, an approximation technique for multidimensional diffusions using sdde\u27s with small delays, and affine sfde\u27s

Solving the Poisson disorder problem

The Poisson disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ’disorder ’ when the intensity of an observed Poisson process changes from one value to another. Partial answers to this question are known to date only in some special cases, and the main purpose of the present paper is to describe the structure of the solution in the general case. The method of proof consists of reducing the initial (optimal stopping) problem to a freeboundary differential-difference problem. The key point in the solution is reached by specifying when the principle of smooth fit breaks down and gets superseded by the principle of continuous fit. This can be done in probabilistic terms (by describing the sample path behaviour of the a posteriori probability process) and in analytic terms (via the existence of a singularity point of the free-boundary equation). 1

A weak stochastic integral in Banach space with application to a linear stochastic differential equation

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48089/1/245_2005_Article_BF01448381.pd