Martingale representation and the Malliavin calculus

Using stochastic flows and the Itô differentiation rule, the integrand in the representation of a martingale as a stochastic integral is identified. By iterating this representation result a homogeneous chaos type expansion is obtained. Using the stochastic integral representation, an integration by parts formula is obtained without using any calculus of variations in function space. If the inverse of the Malliavin matrix belongs to all spacesL ^p() it follows that a random variable has a smooth density.The research of R. J. Elliott was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-7964 and the Air Force Office of Scientific Research, United States Air Force, under Grant AFOSR-86-0332. The research of M. Kohlmann was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-7964.     

The optimal control of diffusions

Using a differentiation result of Blagovescenskii and Freidlin calculations of Bensoussan are simplified and the adjoint process identified in a stochastic control problem in which the control enters both the drift and diffusion coefficients. A martingale representation result of Elliott and Kohlmann is then used to obtain the integrand in a stochastic integral, and explicit forward and backward equations satisfied by the adjoint process are derived.This research was partially supported by NSERC under Grant A7964, the U.S. Air Force Office of Scientific Research under Contract AFOSR-86-0332, and the U.S. Army Research Office under Contract DAAL03-87-K-0102.     

Sufficient and Necessary Conditions for Stochastic Comparability of Jump Processes

Abstract This note is devoted to the study of the stochastic comparability of jump processes. On the basis of [2] and [3], it is proved that two jump processes are stochatically comparable if and only if their q-pairs are comparable. Meanwhile, the result concerning the uniqueness given in [6] is also improved upon. Research supported in part by DPFIHE(Grant No.96002704), NNSFC(Grant No.19771008), MCSEC, Ying-Tung Fok Educational Foundation and Youth Science Foundation of BNU     

On stochastic bang bang control

We consider a constrained stochastic control problem (stochastic bang-bang control) studied recently by many authors Ruzicka [1], Zwonkin-Krylov [2], and Ikeda-Watanabe [3], among others. Our techniques are quite different from theirs, and we obtain in addition some results complementing theirs which would appear to be of independent interest as well.Research supported in part under Grant No. 78-3550 AFOSR, Applied Math Division, USAF.     

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.A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.Research supported by National Science Foundation under Grant No. ECS-8005960.     

A function theoretic method forΔ 4 2 u+Q(x)u=0

Summary The approach used in this paper generalizes Colton's treatment [4] of certain second order elliptic equations in four independent variables to the fourth order case. This method is essentially a function theoretic one that is based on the earlier work of Tjong [11]. An integral operator is found that permits one to construct a complete family of solutions with respect to uniform convergence in compact sets of R⁴. Consequently, one is provided with a useful numerical procedure for solving the associated boundary value problems. This research was supported in party by the Air Force Office of Scientific Research through AF-AFOSR Grant no. 74-2592. Entrata in Redazione il 26 marzo 1973. Indiana University and University of Delaware. Indiana University and Wichita State University.     

Characterization of biorthogonal cosine wavelets

This paper is devoted to the study of characterization of two-overlapping dual window functions that give rise to biorthogonal Schauder bases, frames, and Riesz bases by modulation of the cosines. We show that in this case any frame is a Riesz basis and our characterization of Riesz bases may be considered as a generalization of the theorems established by Coifman, et al. [6] and by Jawerth and Sweldens [9]. First named author: Research supported by NSF Grant #DMS-95-05460 and ARO Contract #DAAH 04-95-10193. Second named author: Research supported by the Texas Higher Education Coordinating Board under Grant Numbers 999903-066, 999903-067, and 999903-109.     

Pricing problems with a continuum of customers as stochastic Stackelberg games

The pricing problem where a company sells a certain kind of product to a continuum of customers is considered. It is formulated as a stochastic Stackelberg game with nonnested information structure. The inducible region concept, recently developed for deterministic Stackelberg games, is extended to treat the stochastic pricing problem. Necessary and sufficient conditions for a pricing scheme to be optimal are derived, and the pricing problem is solved by first delineating its inducible region, and then solving a constrained optimal control problem.The research work reported here as supported in part by the National Science Foundation under Grant ECS-81-05984, Grant ECS-82-10673, and by the Air Force Office of Scientific Research under AFOSR Grant 80-0098.     

Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs

In this paper, we consider M-estimators of the regression parameter in a spatial multiple linear regression model. We establish consistency and asymptotic normality of the M-estimators when the data-sites are generated by a class of deterministic as well as a class of stochastic spatial sampling schemes. Under the deterministic sampling schemes, the data-sites are located on a regular grid but may have aninfill component. On the other hand, under the stochastic sampling schemes, locations of the data-sites are given by the realizations of a collection of independent random vectors and thus, are irregularly spaced. It is shown that scaling constants of different orders are needed for asymptotic normality under different spatial sampling schemes considered here. Further, in the stochastic case, the asymptotic covariance matrix is shown to depend on the spatial sampling density associated with the stochastic design. Results are established for M-estimators corresponding to certain non-smooth score functions including Huber’s ψ-function and the sign functions (corresponding to the sample quantiles). Research of Lahiri is partially supported by NSF grant no. DMS-0072571. Research of Mukherjee is partially supported by the Academic Research Grant R-155-000-003-112 from the National University of Singapore.     

Boundedness of Commutators of Marcinkiewicz Integral with Rough Variable Kernel

In this paper the authors give the boundedness of the commutator of Marcinkiewicz integral with rough variable kernel, which is an extension of the result in [5] . Research supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions

A superprocess with dependent spatial motion and interactive immigration is constructed as the pathwise unique solution of a stochastic integral equation carried by a stochastic flow and driven by Poisson processes of one-dimensional excursions. Supported by an NSERC Research Grant and a Max Planck Award.Supported by the NSFC (No. 10121101 and No. 10131040).Mathematics Subject Classification (2000): Primary 60J80; Secondary 60G57 60H20     

Randomized system trajectories

The notion of system trajectory of a time-varying input-output, dynamical system is reviewed. By introducing a probability measure on a class of such systems a stochastic system, the randomized system, is defined. The randomized system has a trajectory induced by the trajectories of the original systems. A theorem is proved giving fairly general conditions under which the randomized system trajectory is generated by a strongly continuous semigroup of bounded linear operators in a Banach space. An example is presented for a system represented by a quadratic integral operator.Research supported in part by National Science Foundation under Grant No. ECS-8005960.     

Approximation of infinite delay and Volterra type equations

Summary We consider a class of infinite delay equations of neutral type which includes Volterrra type integral und integrodifferential equations. Using abstract approximation results (cf. Trotter-Kato-type) for strongly continuous semigroups we develop an approximation scheme which is based on approximation of the system state by Laguerre (and Legendre) polynomials. Numerical examples demonstrate the feasibility of the scheme and show infinite order convergence for smooth data.Supported in part by the Air Force Office of Scientific Research under Contracts AFORS-84-0398 and AFORS-85-0303 and the National Aeronautics and Space Administration under NASA Grant NAG-1-517 and by NSF under Grant UINT-8521208Supported in part by the Air Force Office of Scientific Research under Contract AFORS-84-0398 and in part by the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under project No. S3206     

Consistent parameter estimation for partially observed diffusions with small noise

In this paper we provide a consistency result for the MLE for partially observed diffusion processes with small noise intensities. We prove that if the underlying deterministic system enjoys an identifiability property, then any MLE is close to the true parameter if the noise intensities are small enough. The proof uses large deviations limits obtained by PDE vanishing viscosity methods. A deterministic method of parameter estimation is formulated. We also specialize our results to a binary detection problem, and compare deterministic and stochastic notions of identifiability.This research was supported: by Systems Research Center, University of Maryland through NSF Grant CDR-85-00108 and AFOSR-URI Grant 87-0073; by Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, under ARO/MIT Grant DAAL-03-86-K-0171; by INRIA Sophia Antipolis, under ERO/INRIA Grant DAJA45-90-C-0008, and by the CNRS-GRAutomatique.     

On the chernoff-savage theorem for dependent sequences

Summary Given a sequence of ϕ-mixing random variables not necessarily stationary, a Chernoff-Savage theorem for two-sample linear rank statistics is proved using the Pyke-Shorack [5] approach based on weak convergence properties of empirical processes in an extended metric. This result is a generalization of Fears and Mehra [4] in that the stationarity is not required and that the condition imposed on the mixing numbers is substantially relaxed. A similar result is shown to hold for strong mixing sequences under slightly stronger conditions on the mixing numbers. Research partially supported by the National Research Council of Canada under Grant No. A-3954.     

Maximal domains of quasi-convexity and pseudo-convexity for quadratic functions

We study quasi-convex and pseudo-convex quadratic functions on solid convex sets. This generalizes Martos' results in [12] and [13] by using Koecher's results in [8].This research was supported by Hydro—Quebec; University of Montreal; Office of Naval Research, Contract N-00014-47-A0112-0011; National Science Foundation, Grant GP 25738.     

Continuous approximation schemes for stochastic programs

One of the main methods for solving stochastic programs is approximation by discretizing the probability distribution. However, discretization may lose differentiability of expectational functionals. The complexity of discrete approximation schemes also increases exponentially as the dimension of the random vector increases. On the other hand, stochastic methods can solve stochastic programs with larger dimensions but their convergence is in the sense of probability one. In this paper, we study the differentiability property of stochastic two-stage programs and discuss continuous approximation methods for stochastic programs. We present several ways to calculate and estimate this derivative. We then design several continuous approximation schemes and study their convergence behavior and implementation. The methods include several types of truncation approximation, lower dimensional approximation and limited basis approximation.His work is supported by Office of Naval Research Grant N0014-86-K-0628 and the National Science Foundation under Grant ECS-8815101 and DDM-9215921.His work is supported by the Australian Research Council.     

The ergodic theorem for Markov processes

Most of the material in Sections 4-5-6-8-11 has been published in [4]-[10]. We shall deal with the asymptotical behavior of the iterates of a Markov transition function. Our aim is to generalize the results about the ‘cyclic’ convergence of the iterates of a Markov matrix. Throughout the paper functional analytic methods are used and not probabilistic arguments. The report is self contained, modulo standart results from functional analysis, except for the decomposition into conservative and dissipative parts. Also we assume the existence of an invariant σ finite measure on the conservative part. This has been proved, under some restrictions, by several authors using probabilistic methods. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR) United States Air Force.     

Hammersley's interacting particle process and longest increasing subsequences

Summary In a famous paper [8] Hammersley investigated the lengthL _n of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly “soft” arguments that limn ^′1/2 EL _n =2. This is a known result, but previous proofs [14, 11] relied on hard analysis of combinatorial asymptotics. Research supported by NSF Grant MCS 92-24857 and the Miller Institute for Basic Research in Science Research supported by NSF Grant DMS92-04864     

Nonlinear transformations of the canonical gauss measure on Hilbert space and absolute continuity

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(,H,B) is absolutely continuous with respect to the abstract Wiener measure. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Itô integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL 03 92 G 0008.