Piecewise Constant Processes Research Articles

Coherent Multiple-Antenna Block-Fading Channels at Finite Blocklength

In this paper, we consider a channel model that is often used to describe mobile wireless scenarios: multiple-antenna additive white Gaussian noise channels subject to random (fading) gains with full channel state information at the receiver. The dynamics of the fading process are approximated by a piecewise-constant process (frequency non-selective isotropic block fading). This paper addresses the finite blocklength fundamental limits of this channel model. Specifically, we give a formula for the channel dispersion—a quantity governing the delay required to achieve capacity. The multiplicative nature of the fading disturbance leads to a number of interesting technical difficulties that required us to enhance traditional methods for finding the channel dispersion. Alas, one difficulty remains: the converse (impossibility) part of our result holds under an extra constraint on the growth of the peak-power with blocklength. Our results demonstrate, for example, that while the capacities of $n_{t}\times n_{r}$ and $n_{r} \times n_{t}$ antenna configurations coincide (under fixed received power), the coding delay can be sensitive to this switch. For example, at the received SNR of 20 dB, the $16\times 100$ system achieves capacity with codes of length (delay) which is only 60% of the length required for the $100\times 16$ system. Another interesting implication is that for the MISO channel, the dispersion-optimal coding schemes require employing orthogonal designs such as Alamouti’s scheme—a surprising observation considering the fact that Alamouti’s scheme was designed for reducing demodulation errors, not improving coding rate. Finding these dispersion-optimal coding schemes naturally gives a criteria for producing orthogonal design-like inputs in dimensions where orthogonal designs do not exist.

Covariance Control Problems over Martingales with Fixed Terminal Distribution Arising from Game Theory

We study several aspects of covariance control problems over martingale processes in ${\mathbb{R}}^d$ with constraints on the terminal distribution, arising from the theory of repeated games with incomplete information. We show that these control problems are the limits of discrete-time stochastic optimization problems called problems of maximal variation of martingales, meaning that sequences of optimizers for problems of length $n$, seen as piecewise constant processes on the uniform partition of $[0,1]$, define relatively compact sequences having all their limit points in the set of optimizers of the control problem. Optimal solutions of this limit problem are then characterized using convex duality techniques, and the dual problem is shown to be an unconstrained stochastic control problem characterized by a second order nonlinear PDE of HJB type. We deduce from this dual relationship that solutions of the control problem are the images by the spatial gradient of the solution of the HJB equation of the solutions of the dual stochastic control problem using tools from optimal transport theory.

Noisy heteroclinic networks

We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic network. We show that under the logarithmic time rescaling the diffusion converges in distribution in a special topology to a piecewise constant process that jumps between saddle points along the heteroclinic orbits of the network. We also obtain precise asymptotics for the exit measure for a domain containing the starting point of the diffusion.

First jump approximation of a Lévy-driven SDE and an application to multivariate ECOGARCH processes

The first jump approximation of a pure jump Lévy process, which converges to the Lévy process in the Skorokhod topology in probability, is generalised to a multivariate setting and an infinite time horizon. It is shown that it can generally be used to obtain “first jump approximations” of Lévy-driven stochastic differential equations, by establishing that it has uniformly controlled variations. Applying this general result to multivariate exponential continuous time GARCH processes of order (1, 1), it is shown that there exists a sequence of piecewise constant processes determined by multivariate exponential GARCH(1, 1) processes in discrete time which converge in probability in the Skorokhod topology to the continuous time process.

A numerical scheme for BSDEs

In this paper we propose a numerical scheme for a class of backward stochastic differential equations (BSDEs) with possible path-dependent terminal values. We prove that our scheme converges in the strong $L^2$ sense and derive its rate of convergence. As an intermediate step we prove an $L^2$-type regularity of the solution to such BSDEs. Such a notion of regularity, which can be thought of as the modulus of continuity of the paths in an $L^2$ sense, is new. Some other features of our scheme include the following: (i) both components of the solution are approximated by step processes (i.e., piecewise constant processes); (ii) the regularity requirements on the coefficients are practically minimum; (iii) the dimension of the integrals involved in the approximation is independent of the partition size.

Mean value theorems for stochastic integrals

The distributions of stochastic integrals are approximated by the distributions of stochastic integrals of piece-wise constant processes. The rate of approximation in some negative Sobolev spaces is estimated. Generalizations are given for problems arising in control theory.

The detection and estimation of the change point in a disccrete-time stochastic system

A piecewise-constant process containing a single jump is observed under noise in the context of discrete time. The conditional density and maximum a posteriori (MAP) estimator of the jump time as well as the Bayes detector of the jump itself are determined using the powerful measure transformation approach. The Bayes detector provides a convenient sequential detection rule for practical on-line implementation. An asymptotic result for the distribution of the MAP estimator's estimation error and the corresponding convergence rate are derived. This result provides a reference measure of optimal performance for jump-time estimators in discrete-time stochastic systems that does not depend on the jump time's prior distribution

On ϵ-coupling and piecewise constant processes

We show that if a stochastic process on a general state space with piecewise constant paths having finitely many jumps in finite intervals admits ϵcoupling to a stationary process for each ϵ≷0 then tends in total variation to as . This result is applied in renewal theory to the total life process, to processes regenerative in the wide sense (regeneration as in Harris chains), and to the queue GI/GI/k with traffic intensity strictly between 0 and 1 but without assuming that the system ever empties

A filtering problem with counting observations: approximation with error bounds

We consider a pure jump Markov process (Xt Yt ) with discrete state space. We suppose that the state Xt is not observable and that the observation Yt is a counting process. We construct an approximation for the filter of Xt given (Ys s ≤ t), by means of a family of piecewise constant processes, depending on the value of Yt and on the time discretization parameter. Moreover we give an explicit error bound for the convergence of the scheme

Stochastic Integration and $L^p$-Theory of Semimartingales

If $X$ is a bounded left-continuous and piecewise constant process and if $Z$ is an arbitrary process, both adapted, then the stochastic integral $\int X dZ$ is defined as usual so as to conform with the sure case. In order to obtain a reasonable theory one needs to put a restriction on the integrator $Z$. A very modest one suffices; to wit, that $\int X_n dZ$ converge to zero in measure when the $X_n$ converge uniformly or decrease pointwise to zero. Daniell's method then furnishes a stochastic integration theory that yields the usual results, including Ito's formula, local time, martingale inequalities, and solutions to stochastic differential equations. Although a reasonable stochastic integrator $Z$ turns out to be a semimartingale, many of the arguments need no splitting and so save labor. The methods used yield algorithms for the pathwise computation of a large class of stochastic integrals and of solutions to stochastic differential equations.