Stochastic morphological evolution equations

The inadequacy of locally defined set-valued differential equations to describe the evolution of shapes and morphological forms in biology, which are usually neither convex or nondecreasing, was recognised by J.-P. Aubin, who introduced morphological evolution equations, which are essentially nonlocally defined set-valued differential equations with the inclusion vector field also depending on the entire reachable set. This concept is extended here to the stochastic setting of set-valued Itô evolution equations in Hilbert spaces. Due to the nonanticipative nature of Itô calculus, the evolving reachable sets are nonanticipative nonempty closed random sets. The existence of solutions and their dependence on initial data are established. The latter requires the introduction of a time-oriented semi-metric in time-space variables. As a consequence the stochastic morphological evolution equations generate a deterministic nonautonomous dynamical system formulated as a two-parameter semigroup with the complication that the random subsets take values in different spaces at different time instances due to the nonanticipativity requirement. It is also shown how nucleation processes can be handled in this conceptual framework.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Logarithmic derivatives of invariant measure for stochastic differential equations in hilbert spaces

We consider a process X solution of a semilinear stochastic evolution equation in a Hilbert space. Assuming that X has an invariant measure ν, we investigate its regularity properties. Logarithmic derivatives of ν in certain directions, are shown to exist under appropriate conditions on the nonlinear term in the equation. A set of directions of differentiability for ν is explicitly described in terms of the coefficients of the equation. In some cases, logarithmic derivatives are represented as conditional expectations of random variables related to an appropriate stationary process. An application to a system of stochastic partial differential equations in one space variable is given     

Measure-valued Markov processes and stochastic flows on abstract spaces

We consider measure-valued processes with constant mass in Hilbert space. The stochastic flow which carries the mass satisfies a stochastic differential equation with coefficients depending on the mass distribution. This mass distribution can be considered as the conditional distribution of the solution of a certain SDE. In contrast to the filtration equation, in our case the random measure cannot diffuse: a single particle cannot break up or turn into clouds. The Markov structure of the measure-valued processes obtained is studied and a comparison with Fleming–Viot processes is presented.     

Regularly varying multivariate time series

Extreme values of a stationary, multivariate time series may exhibit dependence across coordinates and over time. The aim of this paper is to offer a new and potentially useful tool called tail process to describe and model such extremes. The key property is the following fact: existence of the tail process is equivalent to multivariate regular variation of finite cuts of the original process. Certain remarkable properties of the tail process are exploited to shed new light on known results on certain point processes of extremes. The theory is shown to be applicable with great ease to stationary solutions of stochastic autoregressive processes with random coefficient matrices, an interesting special case being a recently proposed factor GARCH model. In this class of models, the distribution of the tail process is calculated by a combination of analytical methods and a novel sampling algorithm.     

Approximation of stationary solutions of Gaussian driven stochastic differential equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.     

Hilbert space-valued forward-backward stochastic differential equations with Poisson jumps and applications

In this paper, we study a class of Hilbert space-valued forward-backward stochastic differential equations (FBSDEs) with bounded random terminal times; more precisely, the FBSDEs are driven by a cylindrical Brownian motion on a separable Hilbert space and a Poisson random measure. In the case where the coefficients are continuous but not Lipschitz continuous, we prove the existence and uniqueness of adapted solutions to such FBSDEs under assumptions of weak monotonicity and linear growth on the coefficients. Existence is shown by applying a finite-dimensional approximation technique and the weak convergence theory. We also use these results to solve some special types of optimal stochastic control problems.     

Well-Posedness of the Cauchy Problem for Stochastic Evolution Functional Equations

We prove the existence, uniqueness, and continuous dependence on the initial data of the solutions of the Cauchy problem for stochastic evolution functional equations with random coefficients in Hilbert spaces. We propose a method for constructing an approximating sequence for the solution of the Cauchy problem and obtain an estimate for the rate of convergence to the exact solution.     

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.     

Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces

The stochastic integrals of M- type 2 Banach valued random functions w.r.t. compensated Poisson random measures introduced in (Rüdiger, B., 2004, In: Stoch. Stoch. Rep., 76, 213–242.) are discussed for general random functions. These are used to solve stochastic integral equations driven by non Gaussian Lévy noise on such spaces. Existence and uniqueness of the path wise solutions are proven under local Lipshitz conditions for the drift and noise coefficients on M-type 2 as well as general separable Banach spaces. The continuous dependence of the solution on the initial data as well as on the drift and noise coefficients are shown. The Markov properties for the solutions are analyzed.     

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.     

Tail dependence between order statistics

In this work, we introduce the s,k-extremal coefficients for studying the tail dependence between the s-th lower and k-th upper order statistics of a normalized random vector. If its margins have tail dependence then so do their order statistics, with the strength of bivariate tail dependence decreasing as two order statistics become farther apart. Some general properties are derived for these dependence measures which can be expressed via copulas of random vectors. Its relations with other extremal dependence measures used in the literature are discussed, such as multivariate tail dependence coefficients, the coefficient η of tail dependence, coefficients based on tail dependence functions, the extremal coefficient ?, the multivariate extremal index and an extremal coefficient for min-stable distributions. Several examples are presented to illustrate the results, including multivariate exponential and multivariate Gumbel distributions widely used in applications.     

Linear stochastic differential equations and Wick products

Summary We establish the existence and uniqueness of the solution to a multidimensional linear Skorohod stochastic differential equation with deterministic diffusion matrix, using the notions of Wick product andStransform. If the diffusion matrix is constant and has real eigenvalues, the solution is a stochastic process with moments of all orders, provided that the initial condition is differentiable up to a suitable order. The case of a diffusion matrix in the first Wiener chaos is discussed in the last section.Supported by the Deutsche Forschungsgemeninschaft/Heisenberg ProgrammSupported by the DGICYT grant PB 90-0452     

Malliavin regularity of solutions to mixed stochastic differential equations

For a mixed stochastic differential equation driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of the solution are established. It is also proved that the solution possesses exponential moments.     

Linear forward-backward stochastic differential equations with random coefficients

Solvability of linear forward-backward stochastic differential equations (FBSDEs, for short) with random coefficients is studied. A decoupling reduction method is introduced via which a large class of linear FBSDEs with random or deterministic time-varying coefficients is proved to be solvable. On the other hand, by means of Four Step Scheme, a Riccati backward stochastic equation (BSDE, for short) for (m×n) matrix-valued processes is derived. Global solvability of such Riccati BSDEs is discussed for some special (but nontrivial) cases, which leads to the solvability of the corresponding linear FBSDEs. This work is supported in part by the NSFC, under grant 10131030, the Chinese Education Ministry Science Foundation under grant 2000024605, the Cheung Kong Scholars Programme, and Shanghai Commission of Science and Technology under grant 02DJ14063.     

Point processes in the plane

In this survey paper, two-parameter point processes are studied in connection with martingale theory and with respect to the partial-order induced by the Cartesian coordinates of the plane. Point processes are characterized by jump stopping times and by their two-parameter compensators. Properties of the doubly stochastic Poisson process, such as predictability, are discussed. A definition for the Palm measure of a two-parameter stationary point process is proposed.     

Quasi sure analysis and Stratonovich anticipative stochastic differential equations

Summary Consider a stochastic differential equation on ^d with smooth and bounded coefficients. We apply the techniques of the quasi-sure analysis to show that this equation can be solved pathwise out of a slim set. Furthermore, we can restrict the equation to the level sets of a nondegenerate and smooth random variable, and this provides a method to construct the solution to an anticipating stochastic differential equation with smooth and nondegenerate initial condition.     

A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation

In this paper a new version of the chain rule for calculating the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the corresponding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.     

Stochastic evolution equations and related measure processes

Basic results on stochastic differential equations in Hilbert and Banach space, linear stochastic evolution equations and some classes of nonlinear stochastic evolution equations are reviewed. The emphasis is on equations relevant to the study of spacetime stochastic processes. In particular the class of measure processes, the continuous analogs of spacetime population processes, is studied in detail.     

On rates of convergence and asymptotic normality in the multiknapsack problem

In Meanti et al. (1990) an almost sure asymptotic characterization has been derived for the optimal solution value as function of the knapsack capacities, when the profit and requirement coefficients of items to be selected from are random variables. In this paper we establish a rate of convergence for this process using results from the theory of empirical processes.