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## A functional limit theorem for waves reflected by a random medium

### Applied Mathematics and Optimization (1994-11-01) 30: 307-334 , November 01, 1994

We introduce a class of distribution-valued stochastic processes that arise in the study of pulse reflection from random media and we analyze their asymptotic properties when they are scaled in a natural way.

## A finitely convergent “row-action” method for the convex feasibility problem

### Applied Mathematics and Optimization (1988-01-01) 17: 225-235 , January 01, 1988

We present a modification of the Cyclic Subgradient Projection (CSP) method by Censor and Lent, which solves the convex feasibility problem in a finite number of steps when a Slater type condition holds, while preserving its “row-action” properties. A linear rate of convergence for the CSP method is established assuming the same hypothesis.

## On Aronsson Equation and Deterministic Optimal Control

### Applied Mathematics and Optimization (2009-04-01) 59: 175-201 , April 01, 2009

When Hamiltonians are nonsmooth, we define viscosity solutions of the Aronsson equation and prove that value functions of the corresponding deterministic optimal control problems are solutions if they are bilateral viscosity solutions of the Hamilton-Jacobi-Bellman equation. We characterize such a property in several ways, in particular it follows that a value function which is an absolute minimizer is a bilateral viscosity solution of the HJB equation and these two properties are often equivalent. We also determine that bilateral solutions of HJB equations are unique among absolute minimizers with prescribed boundary conditions.

## Likelihood ratios for signals in additive white noise

### Applied Mathematics and Optimization (1976-12-01) 3: 341-356 , December 01, 1976

We present a formula for likelihood functionals for signals in which the corrupting noise is modelled as white noise rather than the usual Wiener process. The main difference is the appearance of an additional term corresponding to the conditional mean square error. By way of one application we consider the ‘order-disorder’ problem of Shiryayev.

## Integro-differential operators associated with diffusion processes with jumps

### Applied Mathematics and Optimization (1982-10-01) 9: 177-191 , October 01, 1982

We show existence and*W*_{loc}^{2,p}
⋂ W^{1,∞}-regularity results for the integro-differential equation, associated with a diffusion process with jumps on a bounded domain. The second order elliptic partial differential operator and the integral operator involved here are both maximum principle type operators, which enables us to make*W*^{1,∞} a priori estimates.

## A System of Poisson Equations for a Nonconstant Varadhan Functional on a Finite State Space

### Applied Mathematics and Optimization (2006-01-01) 53: 101-119 , January 01, 2006

Given a discrete-time Markov chain with finite state space and a stationary transition matrix, a system of "local" Poisson equations characterizing the (exponential) Varadhan's functional J(·) is given. The main results, which are derived for an arbitrary transition structure so that J(·) may be nonconstant, are as follows: (i) Any solution to the local Poisson equations immediately renders Varadhan's functional, and (ii) a solution of the system always exist. The proof of this latter result is constructive and suggests a method to solve the local Poisson equations.

## Stochastic variational formula for fundamental solutions of parabolic PDE

### Applied Mathematics and Optimization (1985-04-01) 13: 193-204 , April 01, 1985

Let*p(t, x, y)* be the fundamental solution of a linear, second order partial differential equation of parabolic type. The function*I* = −log*p* satisfies a nonlinear parabolic equation, which is the dynamic programming equation associated with a control problem of stochastic calculus of variations type. This gives a stochastic variational formula for*p.* The proof depends on a result of Molchanov about the asymptotic behavior of*p(t, x, y)* for small*t.*

## How to count and guess well: Discrete adaptive filters

### Applied Mathematics and Optimization (1994-07-01) 30: 51-78 , July 01, 1994

A discrete state and time Markov chain is observed through a finite state function which is subject to random perturbations. Such a situation is often called a Hidden Markov Model. A general filter is obtained which provides recursive updates of estimates of processes related to the Markov chain given the observations. In the unnormalized, or Zakai, form this provides particularly simple equations. Specializing this result provides recursive estimates and smoothers for the state of the process, for the number of jumps from one state to another, for the occupation time in any state and for a process related to the observations. These results allow a re-estimation of the parameters of the model, so that our procedures are adaptive or “self tuning” to the data. The main contributions of this paper are the introduction of an equivalent measure under which the observation values are independent and identically distributed, and the use of the idempotent property when the state space of the Markov chain is identified with canonical unit vectors in a Euclidean space.

## The Rate of Convergence of Finite-Difference Approximations for Bellman Equations with Lipschitz Coefficients

### Applied Mathematics and Optimization (2005-10-01) 52: 365-399 , October 01, 2005

We consider parabolic Bellman equations with Lipschitz coefficients.
Error bounds of order h^{1/2} for certain types of finite-difference schemes are obtained.

## On the boundary values of the solutions of elliptic equations

### Applied Mathematics and Optimization (1980-03-01) 6: 193-199 , March 01, 1980

In May 1978 Professor A. V. Balakrishnan invited me to report about boundary values of the solutions of elliptic equations in his seminar at the University of California, at UCLA. I thank him for this invitation. The present article is a summary of my report.