We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.

We prove continuity in generalized parabolic Morrey spaces of sublinear operators generated by the parabolic Calderón—Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. As a consequence, we obtain a global -regularity result for the Cauchy—Dirichlet problem for linear uniformly parabolic equations with vanishing mean oscillation (VMO) coefficients.

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

Motivated by a class of nonlinear nonlocal equations of interest for string theory, we introduce Sobolev spaces on arbitrary locally compact abelian groups and we examine some of their properties. Specifically, we focus on analogs of the Sobolev embedding and Rellich–Kondrachov compactness theorems. As an application, we prove the existence of continuous solutions to a generalized bosonic string equation posed on an arbitrary compact abelian group, and we also remark that our approach allows us to solve very general linear equations in a $p$-adic context.

We study the existence of solutions for a class of abstract impulsive differential equations. Our technical framework allows us to study partial differential equations with impulsive conditions involving partial derivatives and nonlinear expressions of the solution. Some applications to impulsive partial differential equations are presented.

We prove a new global stability estimate for the Gel’fand–Calderón inverse problem on a two-dimensional bounded domain. Specifically, the inverse boundary value problem for the equation ${- }\Delta \psi + v\hspace{0.167em} \psi = 0$ on $D$ is analysed, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The main feature of this estimate is that it shows that the smoother a potential is, the more stable its reconstruction is. Furthermore, the stability is proven to depend exponentially on the smoothness, in a sense to be made precise. The same techniques yield a similar estimate for the Calderón problem for electrical impedance tomography.

Our aim in this paper is to identify the limit behavior of the solutions of random degenerate equations of the form −div Aε(x′,∇Uε)+ρεω(x′)Uε=F with mixed boundary conditions on Ωε whenever ε→0, where Ωε is an N-dimensional thin domain with a small thickness h(ε), ρεω(x′)=ρω(x′/ε), where ρω is the realization of a random function ρ(ω) , and Aε(x′,ξ)=a(Tx′ /εω,ξ) , the map a(ω,ξ)being measurable in ω and satisfying degenerated structure conditions with weight ρ in ξ. As usual in dimension reduction problems, we focus on the rescaled equations and we prove that under the condition h(ε)/ε→0 , the sequence of solutions of them converges to a limit u0, where u0 is the solution of an (N−1) -dimensional limit problem with homogenized and auxiliary equations.

We define an analytic index and prove a topological index theorem for a non-compact manifold M0 with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the group K0(C*(M)), where C*(M) is a canonical C*-algebra associated to the canonical compactification M of M0. Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah–Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of the K-theory groups K0(C*(M))of the groupoid C*-algebra of the manifolds with corners M. We also prove that an elliptic operator P on M0 has an invertible perturbation P+R by a lower-order operator if and only if its analytic index vanishes.

Our focus in this work is to investigate an efficient state estimation scheme for a singularly perturbed stochastic hybrid system. As stochastic hybrid systems have been used recently in diverse areas, the importance of correct and efficient estimation of such systems cannot be overemphasized. The framework of nonlinear filtering provides a suitable ground for on-line estimation. With the help of intrinsic multiscale properties of a system, we obtain an efficient estimation scheme for a stochastic hybrid system.

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

We present qualitative and quantitative comparisons of various analytical and numerical approximation methods for calculating a position of the early exercise boundary of American put options paying zero dividends. We analyse the asymptotic behaviour of these methods close to expiration, and introduce a new numerical scheme for computing the early exercise boundary. Our local iterative numerical scheme is based on a solution to a nonlinear integral equation. We compare numerical results obtained by the new method to those of the projected successive over-relaxation method and the analytical approximation formula recently derived by Zhu [‘A new analytical approximation formula for the optimal exercise boundary of American put options’, Int. J. Theor. Appl. Finance9 (2006) 1141–1177].

This paper deals with two-species convolution diffusion-competition models of Lotka–Volterra type with delays which describe more accurate information than the Laplacian diffusion-competition models. We first investigate the existence of travelling wave solutions of a class of nonlocal convolution diffusion systems with weak quasimonotonicity or weak exponential quasimonotonicity by a cross-iteration technique and Schauder’s fixed point theorem. When the results are applied to the convolution diffusion-competition models with delays, we establish the existence of travelling wave solutions as well as asymptotic behaviour.

In this paper we establish a priori ${{h}^{1}}$ -estimates in a bounded domain for parabolic equations with vanishing $\text{LMO}$ coefficients.

In this paper we discuss the existence of solutions for a class of abstract degenerate neutral functional differential equations. Some applications to partial differential equations are considered.

In this paper, a theory is developed of generalized oscillatory integrals (OIs) whose phase functions and amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need for a general framework for partial differential operators with non-smooth coefficients and distribution dataffi The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wavefront sets.

A bounded continuous function is said to be S-asymptotically ω-periodic if . This paper is devoted to study the existence and qualitative properties of S-asymptotically ω-periodic mild solutions for some classes of abstract neutral functional differential equations with infinite delay. Furthermore, applications to partial differential equations are given.

Let W n be a simple Markov chain on the integers. Suppose that X n is a simple Markov chain on the integers whose transition probabilities coincide with those of W n off a finite set. We prove that there is an M > 0 such that the Markov chain W n and the joint distributions of the first hitting time and first hitting place of X n started at the origin for the sets {-M, M} and {-(M + 1), (M + 1)} algorithmically determine the transition probabilities of X n .

In this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

A field comprising uniformly porous soil overlying an impervious subsoil is drained through equally spaced tile drains placed on the boundary between the two layers of soil. When this field is subject to uniform irrigation, a free boundary forms in the porous region above the zone of saturation. We study the free boundary value problem which thus arises using the theory of variational inequalities. Existence and uniqueness results are established.