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Bayesian problems of sequential discrimination between hypotheses for continuous random processes

In the monograph [1] the problem of sequential discrimination between two simple hypotheses about the mean value for the Wiener process was discussed. In the present paper we consider the same problem for diffusion type processes and for fractal Brownian motion. Let an F0-measurable random variable θ with two values such that Pπ[θ = 1] = π, Pπ[θ = 0] = 1−π, π ∈ [0, 1], be defined on a complete stochastic base space (Ω,F, (Ft)t≥0,Pπ). Suppose that there are locally equivalent measures P0 and P1 on the filtration (Ft)t≥0 that are singular on F∞ = ∨ t≥0 Ft and such that the original measure Pπ has the structure Pπ = πP1 + (1− π)P0. Suppose that a continuous process X = (Xt)t≥0 is observed, with distribution Pi[X ∈ A] = Pπ[X ∈ A|θ = i], i = 0,1, A ∈ FX = σ{Xt, t ≥ 0}, depending on θ. We shall assume that θ is independent of the processes with distributions Pi[X ∈ A], i = 0, 1, A ∈ FX , under the measure Pπ. Then the decomposition Pπ[X ∈ A] = πPπ[X ∈ A | θ = 1] + (1 − π)Pπ[X ∈ A | θ = 0], A ∈ FX , holds for the distribution ofX under Pπ, where π and (1−π) are the a priori probabilities of the simple statistical hypotheses H0: θ = 0 and H1: θ = 1. Our problem is to determine sequentially from observations of the process X = (Xt)t≥0 which of the hypotheses H0 or H1 holds. We introduce the notion of a decision rule δ = (τ, d), where τ = τ (ω) is a stopping time with respect to the filtration (FX t )t≥0, F X t = σ{Xs, s ∈ [0, t]}, FX t ⊆ Ft , t ≥ 0, and where d = d(ω) is an FX τ -measurable random variable with two values 0 and 1. The choice of the stopping time τ = τ (ω) determines the choice of a rule for terminating the observations of X, whereas the terminal solution function d = d(ω) (or the decision function) shows which of the hypotheses to adopt at time τ . We shall apply the following rule: if d(ω) = 1, then hypothesis H1 will be adopted, and if d(ω) = 0, then hypothesis H0 will be true. With each decision rule δ = (τ, d) we can naturally associate losses of two kinds: those connected with the cost of observations and those connectedwith the wrong choice of the final solution. The average losses of the first kind will be characterised by a quantity K(τ ), the function K(t), t ≥ 0 having the form