Analysis and Optimization of Systems pp 533-543 | Cite as

# A Continuous—Time Distributed Version of Wald’s Sequential Hypothesis Testing Problem

## Abstract

*H*

_{0}and

*H*

_{1}, where

*μ*

_{ i }≠ 0 and

*σ*

_{ i }≠ 0,

*i*= 1, 2, non-random; here the noises {

*W*

_{ t }

^{1},

*t*≥ 0} and {

*W*

_{ t }

^{2},

*t*≥ 0} are independent Brownian motions.

Data is observed continuously and at each instant in time, each decision-maker can either declare one of the hypotheses to be true or continue collecting data. In either case, they base their individual decisions on the data collected by their own sensors up to that time; they do not communicate with each other and so do not share information. The decisions are selected to minimize a *joint* cost function with two components, the first one capturing the cost for *collecting* data, and the second assessing the cost for incorrect decisions. This is the simplest problem of its type, for the coupling between the two decision—makers occurs only through the cost structure. This problem was considered first in discrete—time by Teneketzis [6] who showed that the person-by-person optimal strategy was of threshold type for each sensor. Here a similar result is derived by *simple* and direct arguments based on well—known facts for the single detector problem. Moreover, *explicit* formulae are derived for this joint cost function when the detector policies are of threshold type, owing to the fact that at the decision times, the likelihood functionals assume *one* of two threshold values owing to the *continuity* of the paths of Brownian motion. This is in sharp contrast with the *overshoot* phenomena that leads in the discrete—time situation to the celebrated Wald approximations. These explicit formulae not only vividly display the cost interaction taking place between the two sensors but readily allow for a reduction of the original problem to a mathematical programming problem in four variables over a simple constraint set.

## Keywords

Incorrect Decision Mathematical Programming Problem Direct Argument Stochastic Dynamical System Electrical Engineer Department## Preview

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