Balayage of Excessive Measures

Abstract

Recall [S, (12.1)] that an (F t ) stopping time, T, is a weak terminal time provided (i) for each t, \(t + T \circ {{\theta }_{t}} = T\) a.s. on t < T, and that T is exact if, in addition, (ii) \({{t}_{n}} + T \circ {{\theta }_{{{{t}_{n}}}}} \downarrow T\) a.s. whenever tn ↓↓ 0. If T is an exact weak terminal time, then according to [S, (55.20)] there exists an (F t ) stopping time S with S = T a.s. and such that (iii) t + S(θ t ω) = S(ω) for all t,ω with t < S(ω) and (iv) \(S(\omega ) = \downarrow \mathop{{\lim }}\limits_{{t \downarrow 0}} [t + S({{\theta }_{t}}\omega )]\) for every ω. It follows from (iii) and (iv) that t → t + S(θ t ω) is right continuous and increasing on [0, ∞[ for every ω. For simplicity we define a terminal time S to be an (Ft) stopping time that satisfies (iii), and we say that S is exact if, in addition, it satisfies (iv). In this language, [S, (55.20)] states that an exact weak terminal time is equal a.s. to an exact terminal. Note that if S is a terminal time, then (iii) implies that \(t \to t + S \circ {{\theta }_{t}}\) is increasing and one readily checks that \({{S}^{*}}: = \downarrow \mathop{{\lim }}\limits_{{t \downarrow 0}} (t + S \circ {{\theta }_{t}})\) is an exact terminal time called the exact regularization of S. Of course, if B ∈ εe then T B : = inf {t > 0: X t ∈ B} is an exact terminal time in the above sense, while D B : = inf {t ≥ 0: X t ∈ B} is a terminal time. Moreover T B is the exact regularization of D B .KeywordsStrong DualityPotential DensityTerminal TimeInvariant PartOuter MeasureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.