Vol. 93, No. 1, 1981

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Sequential testing of several simple hypotheses for a diffusion process and the corresponding free boundary problem

Luis A. Caffarelli and Avner Friedman

Vol. 93 (1981), No. 1, 49–94

An n-dimensional stochastic process ξ(t) is observed. It is known that ξ(t) has the statistics of an n-dimensional Brownian motion with any one of possibly n + 1 drifts λ0,n (λi are given n-vectors). We observe the process at a running cost, per unit time, given by ci when the drift is λi, and after some (stopping) time τ make a decision which hypothesis to accept; the hypothesis Hj means accepting the drift λj; the drift changes in time in accordance with a Markov process with n + 1 states and a given transition probability matrix. The problem of finding the optimal stopping time and optimal final decision leads to a variational inequality for a degenerate elliptic operator. In this paper we study this variational inequality and the corresponding free boundary. We also consider, by purely probabilistic methods, the case where ξ(t) is k-dimensional, kn. The outline of the main results is given at the end of §2.

Mathematical Subject Classification 2000
Primary: 60J60
Secondary: 62L10
Received: 12 July 1979
Revised: 2 November 1979
Published: 1 March 1981
Luis A. Caffarelli
Department of Mathematics
University of Texas at Austin
1 University Station C1200
Austin TX 78712-0257
United States
Avner Friedman