28-31 August 2018
Angers - France
Europe/Paris timezone

Some sequential problems for Brownian motion with random drift in statistics and finance.

28 Aug 2018, 09:00
Angers - France

Angers - France


Albert Shiryaev (Steklov Mathematical Institute)


We consider two models of observable process X = (X t ):
Model A: X t = μt + B t ,
Model B: X t = μt + ν(t − θ)^+ + B t ,
where B = (B t ) is a standard Brownian motion, μ and ν are unknown parameters,
and θ is a disorder time.
For Model A, we consider some sequential statistical problems with different
risk functions.
For Model B, we deal with sequential problems of the following type:
H 1 = sup EX τ
or H 2 = sup EE(X τ ),
τ ≤1
τ ≤1
where τ is a stopping time. We show that for such functionals H 1 and H 2 optimal
stopping times have the following form:
τ ∗ = inf{t ≤ 1: ψ(t) ≥ a ∗ (t)},
where ψ(t) is some statistic of observations and a ∗ (t) is a curvilinear boundary
satisfying the Fredholm integral equation of second order. These problems will
be applied to the real asset price models (Apple, Nasdaq).
The talk will gives a survey of the joint papers of authors with Četin,
Novikov, Zhitlukhin, and Muravlev.

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