Papers

From this page you can access postscript or pdf files of the papers listed below.
Learning with Side Information: Part I
Pirkko Kuusela and Daniel Ocone. January, 2002.

Abstract: We explore a new learning setting in which each randomly generated sample gives rise to an additional deterministic sample, called side information, that is also classified by the oracle. Hence a learning algorithm utilizing side information chooses from a smaller and more accurate set of concepts, and is expected to operate more efficiently. In general, side information learning utilizes dependent data and the training space differs from the evaluation space, as the the output of the algorithm will only need to classify a typical observation. We analyze a simple problem of learning open intervals and compare exact learning rates with and without side information. Many times side information yields exponentially better learning rates, but in some cases this improvement may vanish.


A Leavable Bounded-Velocity Stochastic Control Problem
Ioannis Karatzas and Daniel Ocone. June, 2000.

Abstract: This paper studies bounded-velocity control of a Brownian motion when discretionary stopping, or `leaving', is allowed. The goal is to choose a control law and a stopping time in order to minimize the expected sum of a running and a termination cost, when both costs increase as a function of distance from the origin. There are two versions of this problem: the fully-observed case , in which the control multiplies a known gain, and the partially-observed case, in which the gain is random and unknown. Without the extra feature of stopping, the fully-observed problem originates with Benes (1974), who showed that the optimal control takes the `bang-bang' form of pushing with maximum velocity toward the origin. We show here that this same control is optimal in the case of discretionary stopping; in the case of power-law costs, we solve the variational equation for the value function, explicitly determine the optimal stopping policy, and derive the main qualitative features of the solution.

When no discretionary stopping is allowed, the partially-observed case has been solved by Benes, Karatzas and Rishel (1991) and Karatzas and Ocone (1993). The full solution when stopping is allowed remains open. In this paper, we obtain lower bounds on the optimal stopping region using stopping regions of related, fully-observed problems.


Entropy Inequalities and entropy dynamics in
nonlinear filtering of diffusion processes
Daniel Ocone. January, 1999.

Abstract: Suppose that the nonlinear filtering equations are solved using an incorrect initial condition. It is known that the relative entropy of the actual conditional distribution with respect to this incorrectly initialized filter is a positive supermartingale. In this paper, we study the filtering of diffusion signals. Using the Kushner-Stratonovich equations, we decompose the relative entropy supermartingale into decreasing and local martingale terms, and we derive an entropy bound on information and error measures of the difference between conditional distribution and incorrectly initialized filter.


Asymptotic Stability of Benes Filters
Daniel Ocone. November, 1998.

Abstract: The asymptotic behavior of the explicit solution to the Bene\v s filtering problem is studied. It is shown that there is a universal, data-dependent change of location that renders any Bene\v s filter asymptotic to a fixed normal distribution. Asymptotic stability of Bene\v s filters follows as a result; that is, the variational distance between any two, differently initialized solutions of the Kushner-Stratonovich equation converges to zero in the infinite time limit. It is also shown the relative entropy between differently initialized solutions converges to zero. More careful relative entropy bounds are used to derive exponential convergence of the variational distance between filters.


Exponential Stability of Discrete Time Nonlinear Filters for Bounded Observation Noise
Amarjit Budhiraja and Daniel Ocone. July, 1996.

Abstract: This paper proves exponential asymptotic stability of discrete-time filters for the estimation of solutions to stochastic difference equations, when the observation noise is bounded. No assumption is made on the ergodicity of the signal. The proof uses the Hilbert projective metric, introduced into filter stability analysis by Atar and Zeitouni. It is shown that when the signal noise is sufficiently regular, boundedness of the observation noise implies that the filter update operation is, on average, a strict contraction with respect to the Hilbert metric. Asymptotic stability then follows.


A Symmetry Characterization of Conditionally Independent Increment Martingales
Daniel Ocone. 1992.

Abstract: We show that a cadlag, local martingale has conditionally independent increments and symmetric jumps if and only if its law is invariant under integral transformations which preserve quadratic variation.