10 February 2014

Stopping time property of hitting times

Suppose that Ft, t ≥ 0, is an increasing family of σ-fields of subsets of a set Ω,  such that ⋂ε > 0 Ft = Ft, for all t ≥ 0.

Let F be a σ-field on Ω such that FtF for all t ≥ 0.

Let P be a probability measure on (Ω, F). Assume that each Ft contains every subset N of Ω  included in some set ANF with P(AN)=0.

Let, for each t ≥ 0, Xt be a measurable function from (Ω, Ft)  into a Polish space S, where the Polish space is equipped with the Borel σ-field B, i.e., the smallest σ-field containing all its open sets.

Assume that, for all ω ∈ Ω, the function tXt(ω) is continuous or right-continuous with discontinuities of first kind only.

Let BB, and define TB := inf{t ≥ 0: Xt ∈ B}. Then TB is a measurable function from Ω into [0, ∞] such that {ω ∈ Ω: TB(ω) ≤ t} ∈ Ft, for all t.

Probabilities and Potential, by Claude Dellacherie and Paul-André Meyer.

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What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant